The Anomaly Structure of Regularized Supergravity

UCB-PTH-14/37 Nikhef-2014-044 October 2014

THE ANOMALY STRUCTURE OF REGULARIZED SUPERGRAVITY∗ Daniel Butter1 and Mary K. Gaillard2

arXiv:1410.6192v1 [hep-th] 22 Oct 2014

1 Nikhef

Theory Group, Science Park 105, 1098 XG Amsterdam, The Netherlands 2 Department of Physics and Theoretical Physics Group, Lawrence Berkeley Laboratory, University of California, Berkeley, California 94720 Abstract On-shell Pauli-Villars regularization of the one-loop divergences of supergravity theories is used to study the anomaly structure of supergravity and the cancellation of field theory anomalies under a U (1) gauge transformation and under the T-duality group of modular transformations in effective supergravity theories with three Kähler moduli T i obtained from orbifold compactification of the weakly coupled heterotic string. This procedure requires constraints on the chiral matter representations of the gauge group that are consistent with known results from orbifold compactifications. Pauli-Villars regulator fields allow for the cancellation of all quadratic and logarithmic divergences, as well as most linear divergences. If all linear divergences were canceled, the theory would be anomaly free, with noninvariance of the action arising only from Pauli-Villars masses. However there are linear divergences associated with nonrenormalizable gravitino/gaugino interactions that cannot be canceled by PV fields. The resulting chiral anomaly forms a supermultiplet with the corresponding conformal anomaly, provided the ultraviolet cut-off has the appropriate field dependence, in which case total derivative terms, such as Gauss-Bonnet, do not drop out from the effective action. The anomalies can be partially canceled by the four-dimensional version of the Green-Schwarz mechanism, but additional counterterms, and/or a more elaborate set of Pauli-Villars fields and couplings, are needed to cancel the full anomaly, including D-term contributions to the conformal anomaly that are nonlinear in the parameters of the anomalous transformations. ∗

This work was supported in part by the Director, Office of Science, Office of High Energy and Nuclear Physics, Division of High Energy Physics, of the U.S. Department of Energy under Contracts DE-AC02-05CH11231 and DE-AC03-76SF00098, in part by the National Science Foundation under grants PHY-0457315, PHY-1002399 and PHY-0098840, in part by the Australian Research Council (grant No. DP1096372), in part by a UWA Research Development Award, by the ERC Advanced Grant no. 246974, “Supersymmetry: a window to non-perturbative physics” and by the European Commission Marie Curie International Incoming Fellowship grant no. PIIF-GA-2012627976.

Disclaimer

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ii

1

Introduction

It has been shown [1]–[3] that on-shell Pauli-Villars (PV) regularization of one-loop quadratic and logarithmic ultraviolet divergences in general N = 1 supergravity [4, 5] is possible, subject to constraints on the matter representations of the gauge group that are consistent with the spectra found, for example in orbifold compactifications. Supergravity derived from weakly coupled string theory typically has classical symmetries that are broken at the quantum level by conformal and chiral anomalies which arise, respectively, from logarithmic and linear divergences in the light field loops. If they can be canceled by PV loops in the regulated theory, the remaining noninvariance under the classical symmetries arises from the noninvariance of the PV mass terms, provided all other PV couplings are invariant. In this paper we investigate the anomaly structure and anomaly cancellation in a class of ZN orbifolds with just three “diagonal” K¨ ahler moduli T I = T II (I = 1, 2, 3). The anomalous symmetries that we consider are the target space duality transformations, hereafter referred to as modular transformations, and a gauge transformation under an Abelian gauge group, hereafter referred to as U (1)X . These symmetries are perturbatively unbroken [6] in the underlying string theory and therefore must be canceled by some combination of loop contributions from heavy string and Kaluza-Klein modes (“string threshold corrections”) and of counterterms, including a four-dimensional version of the Green-Schwarz (GS) mechanism [7]. The anomaly canceling contributions to the Yang-Mills (YM) Lagrangian have been determined for large classes of orbifolds (including the ZN orbifolds considered here) by matching field theory and string loop calculations [8]–[13]. More recently, a string theory analysis [14] of a limited class of ZN heterotic orbifolds found cancellation of all anomalies through a universal GS mechanism. Here we approach the same problem from the point of view of the effective four dimensional supergravity theory [15]. In the following section we define our notation and display the ultraviolet divergent part of the low energy effective Lagrangian obtained from light particle loops [16]–[18] in the form of superfield operators that will be convenient for the subsequent analysis. In Section 3 we will use the results of [2] and [3], hereafter referred to as I and II, respectively, to construct invariant couplings of PV supermultiplets needed to cancel the light loop divergences. Mechanisms for anomaly cancellation and constraints on PV masses will be discussed in Section 4, and the explicit form of the anomalies will be displayed in Section 5. Our results are summarized in Section 6. Some calculational details are presented in a series of appendices. As discussed below, requiring the cancellation of quadratic and logarithmic UV divergences that were identified in [16]–[18] does

1

not uniquely fix the couplings of the PV sector. In Appendix A we derive additional constraints that assure the cancellation of almost all linear divergences. While the constraints from the requirement of cancellation of (on-shell) quadratic divergences automatically assures the cancellation of the chiral K¨ ahler anomaly arising from the fermion spin connection, there is a residual linear divergence associated with the affine connection of the gravitino. In addition there is an off-diagonal gravitino/gaugino connection that has no counterpart in the PV sector. The corresponding contributions to the chiral anomaly must have supersymmetric counterparts; these can be obtained by introducing a field-dependent UV cut-off Λ = µ0 eK/4 ,

(1.1)

where µ0 is a constant parameter that may be set to infinity at the end of the one-loop calculation, and K is the K¨ ahler potential of the light field theory. This has only the effect that total derivatives with nonvanishing coefficients of ln Λ do not drop out of the S-matrix elements of the regulated theory. Once these procedures have been implemented we recover the standard form of the anomaly coefficients of the Yang-Mills and curvature field strengths as well as agreement with string theory results. However the anomalous coefficients of operators that themselves depend on the modular weights of the light fields depend on details of the PV regularization procedure, and are not uniquely fixed by the cancellation of ultraviolet divergences. In Appendix B we calculate the chiral anomaly for a general supergravity theory. As has been recently emphasized [19], in supergravity the fermion connections and corresponding field strengths contain many more operators than the Yang-Mills [8]–[13] and space-time curvature [20]-[22] terms that have been studied previously in the context of anomaly cancellation. These additional operators include [21, 16, 17] the K¨ ahler U (1)K connection for all fermions, the reparameterization connection for chiral fermions, an axion coupling in the gaugino connection and a (matrix-valued) connection [17] linear in the Yang-Mills field strength in the gaugino-gravitino sector.1 In addition, besides the spin connection common to all fermions, the gravitino connection includes a term proportional to the affine connection, and the gauginos have an additional connection that involves the dilaton and its axionic superpartner. The anomaly is ill defined in an unregulated theory. The authors of Ref. [19] study supergravity theories in which a subgroup of the invariance group of the K¨ ahler metric (group of modular transformations) is gauged. They use consistency conditions analogous to those used to obtain the “consistent anomaly” [25]–[27] for Yang-Mills theories. They 1

In supergravity without a GS term to cancel the U (1)X anomaly, there is an additional connection [23] which must also be included [19, 24].

2

also note that some of the operators in their expression can be removed by counter terms [19]. Here we are working with a regulated theory in which the ambiguity is removed, except for those terms for which a linear divergence remains. The squared field strength Gµν G µν corresponding to the full fermion connection appears in the coefficient of ln Λ. It combines with other contributions from both light and PV loops to cancel the UV divergence, up to a total derivative. Part of the conformal anomaly, including the standard Yang-Mills term and part of the curvature term, is determined by the field-dependence of the PV masses in such a way that it combines with the part of the chiral anomaly arising from PV masses to form a superfield. The remainder of the conformal anomaly arises from the residual total derivative logarithmic divergence mentioned above, and combines with the part of the chiral anomaly associated with the residual linear divergence to form a superfield, provided the cut-off has the correct field-dependence. Specifically, the choice (1.1) assures a supersymmetric result for the full anomaly coefficient of the curvature field strength term. When combined with the PV contribution we recover an anomaly coefficient that is consistent with string loop calculations [20]. Similarly, the chiral anomaly associated with the off-diagonal gauginogravitino connection, which depends on the Yang-Mills field strength, forms a supermultiplet with a contribution to the conformal anomaly from a total derivative in the coefficient of ln Λ, when the field-dependence of the cut-off (1.1) is included. This contribution to the anomaly is canceled by a PV loop contribution for a particular choice of the relevant PV mass ratio. In the absence of linear divergences, the chiral anomaly arises solely from the noninvariance of PV masses. In Appendix B.2 we sketch how this may be demonstrated by comparing the direct chiral anomaly calculation with an indirect method which assumes that no linear divergences are present. We also show that under appropriate assumptions the “consistent anomaly” [25]–[27] is recovered. As described in [1]–[3], and reviewed in Section 3 below, all the quadratic and logarithmic UV divergences of supergravity can be regulated with PV fields in chiral supermultiplets and Abelian vector supermultiplets. The Abelian vector fields acquire U (1)X and modular invariant masses through the superhiggs mechanism and do not contribute to the anomaly, except for a contribution, mentioned above, that cancels the gravitino-gaugino mixed loop contribution. As a consequence, the part of the PV sector relevant to the bulk of the anomaly does not contain any connections associated with the nonrenormalizable couplings of the gravitino-gaugino sector, and the only field strength bilinears that appear in their contribution to the anomaly coefficient are those associated with the spin connection, the YM gauge connection, the U (1)K connection and the scalar reparameterization connections associated with the K¨ ahler metric for PV chiral superfields. As shown in Appendix C, the latter can be chosen such that PV fields with noninvariant masses 3

have very simple reparameterization connections. In the case of an anomalous Yang-Mills theory with constant PV masses, the anomaly is uniquely determined, as illustrated in Appendix B.2 by the standard result (B.58). However in PV regulated supergravity, some PV masses are necessarily field dependent, and there is considerable leeway in the choice of these masses, which can only be fixed by a detailed knowledge of string/Planck scale physics. As explained in Section 4.1, the QFT anomaly cannot be completely removed, even in the absence of linear divergences. Terms linear in gauge charges and modular weights are fixed by the requirement that quadratic divergences cancel; this uniquely fixes the PV contribution to the coefficient of rµνρσ r˜µνρσ as given in (C.43), but terms cubic in these parameters are not fixed by the requirement of cancellation of quadratic (or logarithmic) UV divergences. However the expansion of the parity odd part of the fermion determinant [see (B.1)], given explicitly in [17], has, in addition to a linear divergence proportional to TrA that generates an rµνρσ r˜µνρσ term in the anomaly, a linear divergence proportional to TrA3 , where A is the axial current in the fermion connection. This contains the U (1)K connection as well as the (symmetric) YM gauge connections and the (axial part of the) scalar reparameterization connections for chiral fermions.2 Requiring that this linear divergence vanish imposes cubic constraints, but as mentioned above, does not completely determine the anomaly. In Appendix C we consider simple parameterizations of the PV sector that are motivated by physical considerations, and are consistent with perturbative modular invariance of the K¨ ahler potential and cancellation of quadratic and logarithmic UV divergences, as well as cancellation of all linear divergences from chiral multiplet loops. Under these assumptions we calculate the bosonic part of the variation of the Lagrangian under a variation of the noninvariant PV masses. We then identify the corresponding anomaly superfields and compare them with operators that could potentially cancel anomalies in a generalized GS term; these include a generalization to K¨ ahler superspace of the (F-term) operator found in Ref. [22] for the case of pure supergravity, as well as new Dterm operators [15], that can be inferred [29] by first working in conformal supergravity and then gauge-fixing to K¨ ahler superspace supergravity which we use in this paper. In order to evaluate the anomaly coefficients for specific orbifold models, in Appendix D we construct simple examples of PV sectors that can be used to regularize the matter sector in such a way that those PV fields that contribute to the renormalization of the K¨ ahler potential have 2

As in any regularization procedure, the definition of γ5 is a priori ambiguous; as discussed in [1]–[3] the separation into “vector” and “axial” currents is dictated by the finiteness requirement and nonrenormalization of the θ-parameter [28].

4

modular and U (1)X invariant masses. Implementation of the GS anomaly cancellation mechanism for the F-term imposes constraints on the modular weights and gauge charges. Cancellation of UV divergences assures that some of these constraints are satisfied, but terms nonlinear in the charges under the anomalous symmetries are not completely determined by finiteness conditions and depend on the specifics of the PV spectrum and couplings. The simple procedure adopted in Appendix C and Appendix D is not sufficient to assure the factorization needed for implementation of anomaly cancellation by a universal GS term.3 In addition, some D-term anomaly operators are nonlinear in the parameters of the anomalous transformations and cannot obviously be canceled by a straightforward generalization of the standard four dimensional GS term; it is possible that the additional counterterms that may be needed could correspond to 4-d remnants of the conformal analogue of the 10-d GS [7] term that cancels the 10-d chiral anomaly. The construction of the GS term is discussed in Appendix E, and our notations and conventions are summarized in Appendix F.

2

One-Loop On-Shell Ultraviolet Divergences

In this paper we consider supergravity theories with classical K¨ ahler potential K, superpotential W and gauge kinetic function f given by ¯ = − ln(S + S) ¯ + G(T + T¯, Φ, Φ) ¯ = k + G, K(Z, Z) fab (Z) = δab S,

W (Z) = W (T, Φ),

S| = δab (x + iy),

(2.1)

which are the classical functions found in string compactifications with affine level one.4 S is the dilaton superfield in the chiral formulation used here, T i are the moduli chiral superfields, and Φa are superfields for gauge-charged matter. The ultraviolet divergent part of the one-loop corrected bosonic supergravity Lagrangian was calculated in [16]-[18]. As shown in II, the result for the logarithmically divergent part can be interpreted as the bosonic part of a superfield expression. After a Weyl redefinition to put the Einstein term in canonical form, one obtains Lef f = L (g, KR ) +

√ ln Λ2 (LD + LF ) + LQ , g 32π 2

3

(2.2)

The difficulty of obtaining factorization in a direct field theory approach was noted in [14]. The results can be generalized to higher affine level with fab = δab ka f, ka = constant, by making the substitutions 1 1 −1 a a , Aaµ → ka2 Aaµ , T a → ka 2 T a . Fµν → ka2 Fµν 4

5

where L(g, K) is the standard Lagrangian [4, 5] for N = 1 supergravity coupled to matter with space-time metric gµν , K¨ ahler potential K and superpotential W . If NG is the dimension of the gauge group, the renormalized K¨ ahler potential KR is given by5 i ln Λ2 h −K ij 2 a ¯ ˆ KR = K + e A A − 2 V + (N − 10)M − 4K − 16D , ij G a 32π 2 1 a i ¯ (T z) (Tb z¯)m Kim A = eK W = A¯† , Aij = Di Dj A, (2.3) Kba = ¯, x where V = Vˆ + D is the classical scalar potential with Vˆ

= e−K Ai A¯i − 3M 2 ,

Da = Ki (Ta z)i ,

Ai = Di A,

¯ M 2 = e−K AA,

D = (2x)−1 D a Da , (2.4)

where M 2 is the field-dependent squared gravitino mass, and Di is the scalar field reparameterization covariant derivative: Ai = ∂i A,

Aij = ∂i ∂j A − Γkij ∂k A,

...

(2.5)

im ¯ Scalar indices are lowered and raised with the K¨ ahler metric Kim ¯ and its inverse K . In the K¨ ahler U (1) superspace formulation [5] of supergravity, which we use throughout, a general “Fterm” Lagrangian takes the form Z 1 E LF = L(Φ) = (2.6) d4 θ Φ + h.c., 2 R

where Φ is a chiral superfield of K¨ ahler U (1) weight w(Φ) = 2, and a general “D-term” Lagrangian has the form Z Z 1 E ¯2 4 LD = L(φ) = d θEφ = − d4 θ (2.7) D − 8R φ + h.c., 16 R with φ a real superfield of K¨ ahler weight w(φ) = 0. These contributions to (2.2) are given by6 1 1 3 Φ = Φ1 + Φ2 + Ca ΦaY M + ΦX (N − 9NG − 79) + ΦW (41 + N − 3NG ) + NG Φ′g , 2 36 6 β αβγ a α ΦX = X Xβ , ΦW = W Wαβγ , ΦY M = Wa Wαa , i h 1 j j a , W + 2(T ) Γ Φ1 = − CaM ΦaY M + Γiα a j i α iα 2 1 α Φ2 = X Γα + 2(Ta )ii Wαa , Γα = Γiiα . (2.8) 3

We denote by z i = Z i the lowest (scalar) components of the light chiral superfields Z i . 6 The signs of the ΦaY M terms were inadvertently flipped in transcribing the results of I to II. In addition, the sign of Wab as defined in I is incorrect, as is the sign of (A12) (see Appendix E). Here we have normalized some of the operators differently from those in II: ΦYM |here = 4 ΦYM |II , ΦW |here = 6 ΦW |II , ΦX |here = −2 Φα |II . 5

6

for the F-terms with N the number of chiral supermultiplets Z i = S, T, Φ, and 1 φ = φ3 − 4φW − 4φWk + 8φˆ0 + φ′0 + NG φ′g + φχ (41 + N + 9NG ) , 2 1 αk l β˙ ˙ αk l −K/2 ¯ φ3 = R α R kβl Akl + h.c. , φ′0 = 3iWaα W βa Dαβ˙ (S − S) ˙ + R αe 2 1 ˙ αβ˙ ¯ , ¯ φχ = Gβα − 4RR φˆ0 = K αβ Kαβ˙ − 4RR, ˙ G 3 x2 α b a β˙ 1 α ˙¯ φWba = Wa Wα Wβ˙ Wb , φWk = Wa Dα SWβa˙ D β S, 4 8x

(2.9)

for the D-terms, where we used the on-shell relation Gαβ˙ = Kαβ˙ −

x Wα Wβ˙ 2

(2.10)

to simplify the expression for φχ given in (2.15) of II, and rewrote the “F-term” contribution Φ′0 of ¯ and that paper as the “D-term” contribution φ′0 . Here x is understood as the superfield 12 (S + S), † ¯ are auxiliary fields [5] of the supergravity supermultiplet. The chiral superfield G ˙ and R = R αβ

Wαa is the Yang-Mills superfield strength, with a a gauge index, and L(ΦY M ) gives the standard gauge charge renormalization. Ca and CaM are the quadratic Casimirs in the adjoint and matter representations, respectively, of the gauge subgroup Ga : Tr(Ta Tb )adj = δab Ca ,

Tr(Ta Tb )matter = δab CaM ,

(2.11)

with Ta a generator of Ga and Tb any generator. Wαβγ is the Weyl chiral superfield [5], and L(ΦW ) contains the Gauss-Bonnet (GB) term and the pseudoscalar operator that is bilinear in the Riemann tensor; its explicit expression is given by (E.4) (with H(Z) = 1). The other chiral superfields in (2.8) are constructed from superfields of the general form Tα = −

1 Dα˙ D α˙ − 8R Tˆα , 8

Tˆα = Ti Dα Z i ,

(2.12)

¯ is any (tensor-valued) zero-weight function of the chiral and anti-chiral superfields where Ti (Z, Z) Z i and Z¯¯ı , respectively. In particular, the chiral superfield Kα = Xα = −

1 Dα˙ D α˙ − 8R Dα K, 8

(2.13)

was introduced in [5]; the lowest component of its spinorial derivative − 12 D α Xα | is the kinetic term for matter fields in the classical Lagrangian. For the D-terms we further introduced the zero-weight 7

real superfields ˙

T αβ˙ = αβ

Tαβ˙

=

1 α i ¯ β˙ ¯ n D Z Dα Z j Dβ˙ Z¯ m D Z ¯ Tij m¯ ¯ n, 16 1 1 ¯ Dα Z i Dβ˙ Z¯ m Tim Tαα = D α Z i Dα Z j Tij + h.c. ¯, 4 2

(2.14)

Thus φ3 is constructed from the Riemann tensor i¯ n i i Rjk ¯ Γjk ¯ jk m ¯ = Dm m ¯ = K Rn

(2.15)

associated with the K¨ ahler metric. As discussed7 in II, the superfields Φ′g , φ′g are equivalent–up to terms that vanish on shell–to linear combinations of the the generic operators introduced above ˙¯ and D-terms that involve supergravity superfields: Gαβ˙ Ks¯s D α SD β S, . . .. As shown in Appendix C of II, these terms must be exactly canceled by PV Abelian gauge multiplets that couple to the dilaton. In the string-derived models considered here and in II, the masses of these PV fields are invariant under T-duality and the anomalous U (1), and therefore do not contribute to the field theory anomalies. The various terms in the bosonic part of (2.2) are given in component form in I and II, where total derivatives (such as the GB term) were dropped. These terms cannot be dropped if, after regularization, the constant cut-off is replaced by a field-dependent PV mass that plays the role of effective cut-off (or if the cut-off itself is field-dependent). All terms relevant to the anomaly structure and anomaly cancellation will be included in Sections 5. The quadratically divergent part LQ of (2.2) √ Λ2 1 α ˆ + M 2 (7 + 3NG − N ) (3 + N − N ) D X | + V LQ = − g G α 32π 2 2 2 α α a +NG D kα | + D Γα | + Da TrT + fermion terms (2.16) x can be interpreted as a further renormalization of the K¨ ahler potential KQ only after regularization [1, 2]; KQ is governed by the PV squared masses.

3

Invariant PV Regularization

In this section we construct gauge and modular invariant PV couplings to light fields that are needed to cancel the ultraviolet divergences from light field loops in the theory defined by (2.1). 7

In the expression (2.2) of II, L′g = L(Φ′g ) + L(φ′g ).

8

3.1

Chiral multiplet loops

To regulate the loops arising from dimension-four operators involving only chiral supermultiplets Z i , we need to introduce at least one set of PV chiral superfields Z˙ I with negative metric and the same gauge charges, K¨ ahler metric, and quadratic superpotential couplings as the light chiral multiplets. This assures cancellation of the ultraviolet divergences associated with Φ1 and Φ2 in (2.8), and D α Γα | in (2.16). Since the dilaton does not have a classical superpotential, and its K¨ ahler metric is easily reproduced, we concentrate in this subsection on the chiral supermultiplets p Z = T i , Φa and their PV counterparts Z˙ P = Z˙ I , Z˙ A : i ¯ X Q 1h P ˙Q Z˙ P ¯ ˙ ˙ ˙ K0 = Kp¯q Z Z + (∂p ∂q K − Kp Kq ) Z Z + h.c. , 2 P,Q=p,q

˙

Z W02 =

1 2

X

Z˙ P Z˙ Q Wpq ,

(3.1)

P,Q=p,q

where the subscript 2 on W02 signals the presence of a further contribution: we will denote by W1 the part of the PV superpotential that gives PV fields order Planck scale masses and that will be constructed explicitly in Section 4.2. From (3.1) it is straightforward to determine that [2] ˙ ˙ A¯IJ = A¯ij , AZ RIZmJ ¯ n ¯ + Kim ¯ Kj n ¯ + Kim ¯ Kj n ¯, IJ = Aij , ¯ n ¯ = Rimj Z˙ I J ˙ i j Z˙ = Rimj RIZmJ ¯ n ¯ R k ℓ + 4Rk mℓ¯ ¯ n + 2 (Kk m ¯ Kℓ¯ n + Kℓm ¯ Kk¯ n) , ¯ n ¯ R k ℓ Z˙ ¯IJ AIJ AZ˙

= Aij A¯ij ,

˙

¯IJ ¯ij ¯ ¯ n, RIZmJ ¯ n ¯ A + 2Am¯ ¯ n ¯ AZ˙ = Rimj

(3.2)

assuring [2] cancellation of the divergences arising from the first term in brackets in (2.3) and from φ3 in (2.9). The couplings in (3.1) are gauge invariant, provided Z˙ P transforms the same way as Z˙ p under gauge transformations. Now consider a chiral field redefinition that effects a K¨ ahler transformation Z p → Z ′p , W

K → K ′ = K(Z ′ , Z¯ ′ ) = K + F (Z) + F¯ (Z ′ ), dZ ′p = Mqp dZ q ,

→ W ′ = W (Z ′ ) = e−F W,

Mqp =

∂Z ′p . ∂Z q

(3.3)

The part of (3.1) involving the PV K¨ ahler metric KP Q¯ = Kp¯q is invariant under (3.3) provided Z˙ P → Z˙ ′P = Mqp Z˙ Q ,

(3.4)

but the other terms are not. Nevertheless the matrix elements (3.2) are covariant, depending only on the covariant scalar Riemann tensor and on the covariant holomorphic scalar derivatives of the 9

K¨ ahler covariant operator A = eK W,

¯

A′ = eF A,

¯

A′p = eF Ap ,

etc.

(3.5)

This suggests that (3.1) can be made invariant without modifying (3.2). For example, if we introduce uncharged fields Z˙ N and add to (3.1) expressions of the form   X X 2 1 ˙ ρN Kp Z˙ P Z˙ N + ρ′N Z˙ N + h.c. , ∆K Z = 2 P =p N   2 X X 1 ˙ ρN Wp Z˙ P Z˙ N − ρ′N Z˙ N W  , (3.6) ∆W2Z = 2 N

P =p

where ρ, ρ′ are constants, the one-loop corrections to KR and φ3 are unchanged: AP N = Rn¯ P N m ¯ = AN M = Rn ¯N M m ¯ = 0.

(3.7) ˙

˙

This trick was used in II to construct an explicitly invariant (covariant) expression for K Z (W2Z ) in no-scale models with the special properties: Gpq ∝ Gp Gq , Gp¯q Gp Gq¯ = 3, φp Wp = 3W . The general covariance of (3.2) suggests that this should be possible in general modular invariant models, e.g., including the twisted sector in orbifold compactification. In general we have under (3.3) Kp′ = Npq (Kq + Fq ) ,

k n ¯ Kp′ m n, ¯ = Np Nm ¯ Kk¯

N = M −1 ,

k ′ ], = Npn Nqm [Knm + Fmn − Nkl (Kl + Fl ) ∂n Mm Kpq

(3.8)

and Wp′ = e−F Npq (Wq − Fq W ) ,

′ Wpq = e−F Npk Nqm [Wkm − Fk Wm − Fm Wk − (Fkm − Fk Fm ) W n −Nnl (Wl − Fl ) ∂k Mm ].

(3.9)

Restoring invariance/covariance of these expressions in the general case would be cumbersome at best. For this reason, we restrict our attention to effective supergravity theories with three moduli T i , that have some promising features for a viable phenomenology (see for example [30]). These theories are classically invariant under the modular transformation: aT i − ib , ad − bc = 1, icT i + d a i a → Φ′a = e−qi F Φa = e−F Φa , F i = ln icT i + d ,

T i → T ′i = Φa

10

(3.10)

which effects the K¨ ahler transformation (3.3) with X F (Z) = F i (T i ).

(3.11)

i

The parameters qai are the modular weights of Φa . In the absence of a twisted sector potential (Wi = 0), the classical invariance is SL(2, R) : a, b, c, d, ∈ R. In the presence of a twisted sector potential, this is broken to SL(2, Z) : a, b, c, d, ∈ Z. We have i

a

i

a

Fi ≡ Fti = Ftii ,

j

Nja = Fja e2F Φa ,

Nbi = 0,

Nji = δji e2F ,

Nba = eF δba ,

a

Mja = −Fja e−F Φa ,

Mbi = 0,

Mji = δji e−2F ,

Mba = e−F δba ,

Fij = −δij Fi2 .

(3.12)

Writing the superpotential W as a sum of monomials: W =

X

W α,

W α = cα

α

Nα Y

Φa

n=1

P q Y [η(iT i )]2( n qi −1) ,

(3.13)

i

where η(iT ) is the Dedekind function: 1

i

η(iT ′i ) = e 2 F η(iT i ),

(3.14)

the PV K¨ ahler potential and superpotential can be made modular invariant and covariant, respectively, if we introduce three PV chiral superfields Z˙ N that transform under (3.10) as Z˙ ′N = Z˙ N − aF ˙ in Z˙ I ,

(3.15)

and modify (3.1) follows: ˙

KZ



˙

= K0Z − a˙ −1 2 +a˙ −1

X

A,N =a,n

X

N,M =n,m ˙ W2Z

=

Z˙ W02

−1

− 2a˙ +a˙

−2

(1 − qna ) Z˙ N Z˙ A Ka + 2

X

I,M =i,m

(1 − δim ) Ki Z˙ I Z˙ M 

[δnm (1 + qn ) − 1 − qnm ] Z˙ N Z˙ M + h.c. ,

α;A,N =a,n

X

X

α;N,M =n,m

(1 − qna ) Z˙ N Z˙ A Waα − 2a˙ −1

[δnm (1 +

qnα )

−1−

11

X

α;I,M =i,m

α qnm ] W α Z˙ N Z˙ M ,

(1 − δim ) WiαZ˙ I Z˙ M

qi =

X

qia φa Ka ,

qij =

a

qiα

=

X

X

qia qja φa Ka ,

a

qia φa ∂a ln W α ,

α qij =

a

X

qia qja φa ∂a ln W α ,

(3.16)

a

which reduces to the result found in II for the untwisted sector: φa → φia , qjia = δji . Note that α are constants, whereas q , q are not. Nevertheless we still get qiα , qij i ij AP N = Rn¯ P N m ¯ = AN M = Rn ¯N M m ¯ = 0.

(3.17) ¯

so the one-loop corrections to KR and φ3 are unchanged, provided K P Q does not change. We must ¯ also introduce a modular invariant K¨ ahler metric for Z˙ N ; leaving K P Q unchanged requires that it be of the form    X X ¯ X ¯ M Q ¯ ¯ ˙ , Z˙ N + K(Z˙ N ) = ξnm χnp Z˙ P  Z¯˙ + χ ¯m (3.18) ¯ q¯ Z n=N,m=M

p=P

q=Q

i ¯ ¯ where the metric ξnm ¯ (Z , Z ) is modular invariant and

′ ¯′ q n ¯ χ′n ˙ q . p (Z , Z ) = Np χq (Z, Z) + aF

(3.19)

As shown in Appendix D, for a general orbifold metric, this leads to unwanted contributions to n n ˙ p ln η(iT n ). Alternatively, LQ and ΦP1,2V unless ξnm ¯ = δnm and χp is holomorphic, e.g. χp = 2a∂ most of the unwanted terms can be canceled by including several copies of the above: Z˙ → Z˙ α , ˙ α = 1, . . . , 2nZ˙ + 1 with signatures ηαZ and parameters a˙ α such that X α

˙

ηαZ = −1,

X

˙

ηαZ a˙ 2α =

α

X

˙

ηαZ a˙ 4α = 0.

(3.20)

α

Only one of these with negative metric need have the superpotential and additional K¨ ahler potential couplings in (3.16) required to regulate the K¨ ahler potential corrections and φ3 . Then the only remaining unwanted contribution is from the curvature terms associated with the metric ξnm ¯ , which vanish if this metric is flat, or can be canceled by the introduction of additional gauge-singlet chiral PV multiplets with positive signature and just the metric ξnm ¯ . The correct procedure cannot be determined without further knowledge of the string loop corrections, and/or higher order terms in the classical twisted sector metric. We will consider only the case ξnm ¯ = δnm , with just a few n simple choices for χp .

12

3.2

Gauge couplings

To regulate light gauge field loops, we must introduce at least three Pauli-Villars chiral supermultiplets ϕa , ϕã , ϕâ , with positive signature, that transform according to the adjoint representation of the gauge group. Their contributions cancel the term proportional to Ca in (2.8). To cancel the divergences associated with the term proportional to Kaa in (2.3), we need at least one chiral supermultiplet YbP that transforms according to the representation of the gauge group conjugate to that of Z P , with positive signature, and superpotential coupling [2] W2φ = 2ϕa YbP (Ta Z)p ,

(3.21)

which is modular covariant if under (3.10) YbP and ϕa transforms as YbP′ = Npq YbQ ,

Then the K¨ ahler potentials b

K0Y =

X

P,Q=p,q

are gauge and modular invariant, and b

AYP aϕ = 2eK (Ta z)p ,

ϕ′a = e−F ϕa .

¯ ¯, K p¯q YbP Yb Q

K ϕ = eG

¯ A¯PYb aϕ = 2e−G (T a z¯)m Kpm ¯,

X a

2

X Pa

(3.22)

|ϕa |2

b AYP aϕ A¯PYb aϕ = 4Kaa .

(3.23)

(3.24)

However YbP loops contribute terms proportional to Φ1 , Φ2 and CaM ΦY M in (2.8), and D α Γα | in P b b (2.16). Therefore we must introduce a set YbPα of such fields with signatures ηαY such that α ηαY = 0, b with at least one of these, say YbP1 , η1Y = +1, participating in the coupling (3.21). In order to introduce PV masses that are gauge invariant under the nonanomalous gauge group, we have to introduce equal numbers of PV fields that transform like Z p and its conjugate representation. This requires that we also introduce additional charged PV fields UβA and UAβ that transform a and its conjugate, respectively, under the nonanomalous gauge according to the representation RA group factor Ga , and/or fields VβA that transform according to a (pseudo)real representation that is traceless and anomaly-free. Their gauge couplings must satisfy X a a = CM , (3.25) ηβA CA A∈U,V

where

a , TrR T a T b = δab CR 13

(3.26)

which may imply a constraint on the matter representations of the gauge group in the light spectrum, as discussed in I. This constraint can be trivially satisfied for all U (1)’s. It is satisfied in any supersymmetric extension of the Standard Model (SM). In addition to two Higgs doublets, its minimal supersymmetric extension, the MSSM, has 2Nf fundamental representations (reps) n of each factor Gn = SU (n), n = 2, 3, where Nf is the number of quark flavors. Their Casimirs can be mimicked by Nf real PV reps (n + n ¯). Further extensions necessarily involve real reps of the SM gauge group, so that the additional states can get SM gauge invariant masses. The condition (3.25) is also satisfied in the hidden sectors [31] that can accompany the SM-like ZN orbifolds found in [30]. These hidden sectors also come in even numbers of reps + antireps, except for one hidden SO(10) = 6, which can be mimicked by a sector that contain 3 16’s of SO(10) that contribute CM real PV rep with 6 10’s, and another hidden sector with 3 (5 + 10)’s and 6 ¯ 5’s of SU (5) with SU (5) = 9, that can be mimicked by 9 real PV reps (5 + ¯ 5). Since the underlying theory is finite CM when all degrees of freedom are included, one would expect (3.25) to have a solution for general superstring compactifications.

3.3

Nonrenormalizable couplings

In order to cancel the term proportional to Ap A¯p in (2.3) we need a PV superpotential coupling of the form [2] i Xh b eP Yb α , (3.27) a ˆα Wp ZeαP Yb0α + W Z W2 ∋ W2Y = α P α

eP , YbP transform like Z p , Z¯ p¯, respectively, under the gauge group, and (Gp¯q ≡ Kp¯q , Z q 6= S) where Z ¯ X e e¯ Q , Gp¯q ZeαP Z KαZ = α P,Q=p,q

b KαY

=

X

P,J=p,j b ηαY

=

e ηαZ ,

α

¯ Q¯ − a Gp¯q YbPα Yb ˆα ˆ = N

X

b ηαY

X α ¯ 0 Gp + h.c. + |Yb0α |2 1 + a ˆ2α Gp Gp , YbPα Yb

P =p

= 0,

Gp = Gp¯q Gq¯,

Gp¯q Gq¯k = δkp .

(3.28)

α

The superpotential (3.27) is covariant and the K¨ ahler potential (3.28) are invariant under modular transformations provided that under (3.3) e′Q = M q ZeP . Yb0′ = Yb0 , Z (3.29) YbP′α = Npq YbQα + aα Fq Yb0 , α p α

The same form of the K¨ ahler potential is in fact needed for the PV fields YbP that couple to ϕa in (3.21) in order to cancel [2, 3] the term proportional to D in (2.3). We therefore generalize these 14

to a subset of the Ybα ’s with the same signatures as a subset of ϕaα , with the full set satisfying P ϕ α ηα = 1, and take for the full superpotential W2 X b ˙ gα ϕa1+α YbPα (Ta Z)p + cα φSα φαS W, (3.30) W2 = W2Z + W2Y + 2 α

where

φS , φS

are gauge singlets needed to cancel dilaton loop contributions, with a K¨ ahler potential i Xh (3.31) KS = e−2k |φαS |2 + 2|φα0 |2 − e−k φ¯αS¯ φα0 + h.c. + e2k |φSα |2 , α

that involves an additional chiral superfield φ0 , analogous to Yb0 ; this field is also needed [see (3.33)] to regulate dilaton couplings. The superpotential (3.30) is modular covariant and the K¨ ahler r S potential (3.31) is modular invariant if the superfields φ = φ , φS , φ0 are modular invariant. Note that because F (Z) is holomorphic and gauge invariant it satisfies (T a Z)p Fp = 0,

(3.32)

so (3.21) remains modular covariant with the modification (3.28) of the K¨ ahler potential for YbI . To cancel the remaining divergences arising from nonrenormalizable couplings we introduce gauge singlets φγ , as well as U (1) vector supermultiplets Vγ = Vγ0 , Vγs , with signatures ηγ0 , ηγs , respectively, θγ0,s of the same signature that form massive vector supermultiplets with chiral multiplets Φ0,s γ = e and U (1)β charges qγ δγβ . We also need additional chiral multiplets ϕ ˜α , ϕˆα in the adjoint representation of the low energy gauge group; one of these sets, ϕ ˆα , participates in the matrix-valued gauge kinetic function that couples the PV superfield strengths Wγ0 , Wγs and the light gauge fields W a to one another in the Yang-Mills kinetic term: ! X fsaγ = 0, hSα φSα φα0 , f ab = δab S + α

0 fγβ

= δγβ ,

s fγβ

f0aγ =

= δγβ S,

X

eγβ ϕâβ .

(3.33)

β

These couplings are modular invariant provided θγ and ϕâα are modular invariant. The matrices dαβ , eαβ , are nonvanishing only when they couple fields of the same signature. Including the fields U, V , as well as additional chiral supermultiplets φγ needed to regulate gravity loops [1], the full K¨ ahler potential takes the form " # X 1 X 1 γ A 2 A 2 2 γ¯ 2 KP V = fφγ φ φγ + νγ (θγ + θ¯γ ) + fUγA |Uγ | + fU γ |UA | + fVγA |Vγ | A 2 2 γ A X X ˙ b e ˜¯αa + ˆ¯αa + ϕãα ϕ (3.34) KαZ + KαY + K Z + K S . + eG ϕaα ϕ¯αa + ek ϕâα ϕ α

α,a

15

As shown in I and II, the ultraviolet divergences from light loops are canceled if the PV coupling constants satisfy8 X α

X

X

b

ˆ2α = −2, ηαY a

S ηαφ c2α

b

ˆ4α = +2, ηαY a

α

X

= 5,

α

α

X

b

ˆ2α = −1, ηαY gα2 a

α

1 X ϕˆ 2 3 X ϕˆ β γ δ α ηα eαβ = −4 = ηγ eα eβ eγ eδ , 2 4 α,β αβγδ X S X ηαφ (hSα )2 = 2, ηγs = −NG , α

b

ηαY gα2 = 1,

X

S

ηαφ hSα cα = 1,

α

(3.35)

γ

and the PV signatures satisfy X

ηαϕ

a

=

α

X

a

ηαϕˆ =

α

˙ ηαZ

X γ

X

=

X

=

X

ηγs

γ

= −1,

+

X γ

a

a

ηγ0

X

a

b

ϕ η1+α = ηαY ,

ηαϕ˜ = 1,

α

S ηαφ

α

ηγθ

X

η1ϕ = +1, b

e

e

ηαZ = ηαY ,

ηαZ = 0,

α

= −12 − NG = NG′ ,

X P

A

ηαU = ηαUA ,

S

ηαφ = ηαφS = ηαφ0 ,

η C = −29 − N = N ′ ,

(3.36)

where C is any PV chiral superfield and N (NG ) is the number of chiral (gauge) superfields in the light sector. Cancellation of quadratic divergences requires that the pre-factors fC in (3.34) satisfy: X C

X

ηC ∂i ln fC = −10Ki ,

C

ηC ∂¯ı ln fC = −10K¯ı ,

(3.37)

and cancellation of logarithmic divergences requires X C

ηC ∂i ln fC ∂¯ ln fC = −4Ki K¯ + 2ki k¯.

˙ Z, e Yb , φS , φS,0 ; for example where in (3.37) φC is any PV chiral superfield except Z, fϕ = eK−k ,

fϕˆ = ek ,

fϕ˜ = 1.

(3.38)

(3.39)

The K¨ ahler potentials for ϕaα and ϕâα are determined by their couplings so as to cancel light field divergences, while the choice for ϕãα assures the K¨ ahler anomaly matching condition for the gauge 8

¯IJ There are extraneous factors of eK and W in the expression for A Zα ,Yα in (2.36) of I.

16

loop contribution to the term quadratic in the Yang-Mills field strength, which requires [12] that, averaged over the adjoint PV fields, hln f iϕ = K/3. The parameters νγ in (3.34) determine the squared PV masses of the PV vector supermultiplets and the corresponding eaten chiral supermultiplets θγ ; these masses play the role of effective (squared) cut-offs. The PV masses µα of the remaining PV chiral multiplets will be introduced through superpotential terms in section 4.2.

4

Anomaly Cancellation

In the context of orbifold compactification of the heterotic string the known mechanisms for cancellation of the effective field theory anomalies are the four-dimensional GS mechanism and string loop threshold corrections. The latter can be parametrized as moduli-dependent PV masses. In this section we outline the needed generalization of the GS mechanism, show that it restricts the form of PV mass terms, and construct an explicit PV superpotential that satisfies these restrictions.

4.1

Strategies for anomaly cancellation

The regularized theory would be invariant under the classical symmetries if it were possible to introduce PV mass terms that respect these symmetries and also cancel all the ultraviolet divergences. That this is not possible can easily be seen by looking at the quadratic divergences. For example, in (2.16) −K ¯p m ¯ D α Γα | = 2x−1 Da Dp (T a z)p − 2Rpm A A ¯ + Dµ z p D µ z¯m , ¯ e

Dp (T a z)p = TrT a + Γppq (T a z)q .

(4.1)

If U (1)X is anomalous, TrT X 6= 0, one cannot regulate the quadratic divergences without introduc′ 6= 0. ing at least one mass term for a pair of PV chiral superfields U, U ′ with U (1)X charges qX + qX Similarly, if the low energy theory possesses a classical invariance under a nonlinear symmetry that effects a K¨ ahler transformation (3.3), a mass term WU = µU U ′ with constant µ is covariant if KU U¯ KU ′ U¯ ′ = eK . In this case α U′ α −1 ′ D α ΓU + D Γ (4.2) α α = 2 D Xα | + 2x DX qX + qX .

The first term on the RHS of (4.2) cancels the U, U ′ counterpart of the term proportional to N D α Xα | in (2.16). Thus, as discussed in I, quadratic divergences associated with scalar curvature 17

cannot be regulated by PV fields with invariant masses. For the orbifold-derived supergravity models considered here, only a discrete group of modular transformations is expected to survive at the quantum level. As will be seen below it is always possible to construct PV masses that are invariant under this discrete group by including holomorphic functions of the moduli that have wellP i defined modular weights: w(T ′ ) = e i F ωi w(T ). The appearance of these factors in the effective one-loop Lagrangian would be interpreted as string-loop threshold effects. However it is known from string-loop calculations [8, 9] in orbifold compactifications that these effects do not fully cancel the anomaly. Indeed there are no threshold corrections to the Yang-Mills Lagrangian in Z3 and Z7 orbifolds, where the anomaly is completely canceled by the four dimensional analogue [10, 11] of the Green-Schwarz term. Moreover, the U (1)X anomalies can be canceled only by such a mechanism [32]. The GS mechanism is most easily displayed in the linear multiplet formalism [13] for the dilaton, where the GS term takes the form Z Z 1 4 ¯ − δX VX ≡ d4 θELVGS , LGS = d θEL bg(Z, Z) (4.3) 2 where b and δX are constants, VX is the U (1)X vector superfield, and g is a real superfield which, under a modular transformation (3.3), transforms as ¯ + F (Z) + F¯ (Z). ¯ g(Z ′ , Z¯ ′ ) = g(Z, Z)

(4.4)

The real superfield L satisfies the modified linearity conditions ¯ 2 − 8R)L = −Waα Wαa , (D

¯ = −Wαa˙ Waα˙ , (D 2 − 8R)L

(4.5)

where Wαa is the gauge superfield strength. Under (4.4) and a gauge transformation: ¯ , VX → VX′ = VX + Λ + Λ

(4.6)

where Λ is a chiral superfield, (4.3) transforms as, using (4.5),

LGS → L′GS = LGS + ∆LGS , Z Z δX δ X 4 4 E 2 ¯ − 8R L bF − ∆LGS = d θEL bF − Λ + h.c = − d θ Λ + h.c D 2 8R 2 Z δX α a 4 E W W bF − Λ + h.c, (4.7) = d θ 8R a α 2 18

which has the same form as the quantum anomaly from renormalizable couplings of particles charged under the gauge group Ga that also carry modular weights and/or U (1)X charge. The constants b, δX can be chosen to cancel the anomalous term for any one Ga . In orbifold compactifications of the heterotic string, the couplings satisfy constraints that allow universal U (1)X anomaly cancellation. Modular anomaly cancellation is also universal in orbifolds with no N = 2 twisted sector. Otherwise there are moduli-dependent threshold corrections that contribute to the anomaly cancellation. As discussed in I, these can be included in the PV regulator masses. The GS mechanism can be generalized to cancel anomalous coefficients of higher dimension operators by generalizing the modified linear condition: ¯ 2 − 8R)L = −Φ, (D

¯ = −Φ, ¯ (D 2 − 8R)L

(4.8)

where Φ is a chiral superfield with chiral K¨ ahler weight w(Φ) = 2, provided higher dimension operators are also included in the tree Lagrangian: Z Ltree (L) = − d4 θE [3 − 2Ls(L)] Z E ¯2 = d4 θ (D − 8R) [3 − 2Ls(L)] + h.c. 16R Z E = d4 θ s(L)Φ + h.c. + · · · , (4.9) 8R where the ellipsis represents the kinetic terms for the various components of L and the supergravity ¯ and the relation multiplet. The K¨ ahler potential (2.1) in this formalism is K = k(L) + G(Z, Z), k′ (L) + 2Ls′ (L) = 0

(4.10)

between the real functions k(L) and s(L) assures a canonical Einstein term. It follows from (4.8) that Φ is the chiral projection of a real field Ω with K¨ ahler weight wK (Ω) = 0 and Weyl weight wW (Ω) = wW (L) = −wW (E) = 2, and that it satisfies a generalized Bianchi identity: ¯ 2 − 24R Φ − D 2 − 24R ¯ Φ ¯ = total derivative. D

(4.11)

When Φ = Waα Wαa as in (4.5), the first term on the RHS of (4.9) reduces to the Yang-Mills kinetic term, and Ω = ΩY M , the Chern-Simons (CS) superfield. To obtain the Lagrangian for the chiral multiplet formulation we make a superfield duality transformation by writing [13] Z L = − d4 θE 3 − 2Ls(L) + (L + Ω) S + S¯ , (4.12) 19

where L complex linearity function

is unconstrained and S is chiral. Writing S as the chiral projection of an unconstrained ¯ 2 − 8R)Σ, the equations of motions for Σ and Σ† give the generalized field Σ: S = (D constraint (4.8). If instead we solve the equations of motion for L, we obtain [13] L as a of S + S¯ such that ¯ s(L) = (S + S)/2, (4.13)

and we recover the standard chiral superfield formulation [4] of supergravity, with a canonical Einstein term provided (4.10) holds, except that the standard Yang-Mills term is generalized to Z E LY M → d4 θ SΦ + h.c., (4.14) 8R provided that

δ (EΩ) = 0. δL When the GS term (4.3) is included, (4.13) is modified to s(L) = (S + S¯ − VGS )/2.

(4.15)

(4.16)

Since L is invariant under modular transformations (4.4) and gauge transformations (4.6), the chiral superfield S is not invariant: S → S ′ = S + bF (Z) − δX Λ/2.

(4.17)

Then from either (4.3) in the linear formulation or (4.14) in the chiral formulation, we get a variation in the Lagrangian Z E Φ (bF − δX Λ/2) + h.c.. (4.18) ∆L = d4 θ 8R

In Section 3 we specified the PV Lagrangian in terms of modular and U (1)X invariant K¨ ahler potential KP V and gauge kinetic functions (3.33). As a consequence the massive Abelian PV vector fields necessarily have modular invariant masses that do not contribute to the anomaly. We further imposed U (1)X invariance and modular covariance on the part W2 of the PV superpotential that cancels divergences from light chiral supermultiplet loops. The chiral and conformal anomalies of the low energy quantum field theory arise from linear and logarithmic divergences, respectively. If we assume that they are canceled when the theory is properly regulated, the only noninvariance in the regulated theory appears in the part W1 of the superpotential that gives a large supersymmetric mass matrix m to PV chiral supermultiplets. In this case the one loop action can be written as L1 = Linv + Lni ,

Lni =

i STr ln D 2 + H(m) ni + Tni (m), 2 20

(4.19)

where Linv is modular invariant, T is the helicity-odd fermion contribution, and Lni arises only from chiral supermultiplet loop contributions. As shown in I, under a transformation on the PV fields that leaves the tree Lagrangian, the PV K¨ ahler potential and the PV gauge couplings invariant, with W2 covariant: Φ′ = gΦ,

m′ (Φ) = m(Φ′ )

L′1 = Linv + Lni (m), ˜

m ˜ = g−1 m′ g,

(4.20)

because all the operators in the determinants are covariant except the matrix mni for the chiral multiplets whose PV masses arise from the noncovariant part of W1 . The residual quantum anomaly can be canceled by the GS term provided

Lni (m) =

Z

d4 θESTr [Ω ln(mm)] ¯ ,

P

m ˜ ni = e

i

Qi F i +QX ΛX

mni ,

(4.21)

with, for example STr (ΩYM Qi ) = −bΩYM ,

STr (ΩYM QX ) = δX ΩYM /2,

(4.22)

where ΩYM is the (matrix valued) Yang-Mills Chern-Simons form: ¯ 2 − 8R)ΩYM = Waα Wαa . (D

(4.23)

As mentioned in the introduction, the linear divergences are not completely canceled in the PV regulated theory. Cancellation of on-shell quadratic divergences imposed the two constraints given in (2.20) of I, which can be recast in the form X N + N ′ − NG − NG′ = 3 + 2α, N + N ′ − 3NG − 3NG′ = 7, α= ηC ln fC /K. (4.24) C

The first of these assures the cancellation of the linear divergence associated with the spin connection; as a consequence the associated anomaly arises only from PV masses, and, in the absence of threshold corrections, one gets a contribution [see (C.44)] ! X X 16π 2 1 χ N ′ − NG′ − 2α + 2 qnp ImF n √ ∆Lspin = − r · r˜ g 48 n p ! X X 1 p = (4.25) r · r˜ N − NG − 3 − 2 qn ImF n . 48 n p 21

However the linear divergence associated with the affine connection in gravitino loops is not canceled, and we get an additional contribution to the chiral anomaly: 16π 2 24 χ √ ∆Laffine = r · r˜ImF n . g 48 The conformal anomaly counterpart of (4.25), namely X X 16π 2 N − NG − 3 − 2 qnp √ ∆Lcspin = g n p

(4.26) !

LGB ReF n ,

(4.27)

1 rµνρσ r µνρσ − 4rµν r µν + r 2 , (4.28) 48 automatically combines with (4.25) to form the supersymmetric combination contained in the operator ΦW in (2.8). This is because Pauli-Villars regularization manifestly preserves supersymmetry, in contrast to a straight momentum cut-off procedure, which does not. The second constraint in (4.24) determines the PV mass-independent coefficient of the GaussBonnet term in the regularized one-loop effective Lagrangian: LGB =

16π 2 c √ LGB = LGB (N + N ′ − 3NG − 3NG′ + 41) ln Λ = 48LGB ln Λ. g

(4.29)

Since LGB is a total derivative, this drops out of the effective action if Λ is constant. If instead, Λ = µ0 eαΛ K , there is a finite, anomalous contribution to the effective action with 16π 2 √ ∆LcGB = 96αΛ LGB ReF. g

(4.30)

Supersymmetry requires that we take αΛ = 41 , in which case the residual contribution to the anomaly in K¨ ahler superspace is given by Z 4 1 res (4.31) d4 θEF n (T )ΩGB + h.c., ΩGB = −8ΩW − ΩX − 3φχ , ∆LGB = 2 8π 3

which is of the required form, (4.15), with ΩW the Chern-Simons form for the curvature superfield strength, and ΩX the Chern-Simons superfield for the chiral superfield ΦX introduced in (2.8): ¯ 2 − 8R)ΩW = W αβγ Wαβγ = ΦW , (D

¯ 2 − 8R)ΩX = ΦX , (D

(4.32)

and φχ is defined in (2.9). Including the PV contribution, the total anomaly involving the curvature strength becomes !Z X 1 1 X p ∆LGB = 2 N − NG + 21 − 2 qn d4 θEF n (T )ΩGB + h.c., (4.33) 8π 24 n p 22

The off-diagonal gravitino-gaugino connection leads to a contribution to the chiral anomaly (see Appendix B.3) 16π 2 χ √ ∆Loff g

a , = −2ImF D µ xFaρν Dρ F˜µν

(4.34)

which combines with the conformal anomaly associated with total divergences from mixed gaugegravity loop contributions: 16π 2 a a a − xFµν Dρ Faρν − 2Da D ν Fνµ + ···, √ ∆Lcmixed = −8αΛ ReF D µ xFaρν Dρ Fµν g

(4.35)

to give an overall contribution

LMx

1 = 2 8π

Z

d4 θEF ΩMx + h.c.,

(4.36)

where ΩMx = −

i 1 h α β˙ i h ¯ α W ˙ − 1 D 2 (S + S)W ¯ α Wα + h.c. + · · · , D ,D (S + S)W β 4 16

(4.37)

and the ellipses in (4.35) and (4.37) represent terms with derivatives of the dilaton superfield9 S (as well as fermionic terms). As discribed in Appendix B.3, this contribution to the anomaly is cancelled by mixed PV gauge-chiral matter loops for a specific choice of the V0 -θ0 /ϕa -ϕâ squared PV mass ratio. We will adopt this choice. In addition there are total derivatives with logarithmically divergent coefficients that generate a D-term conformal anomaly with no chiral counterpart. Specifially there is a contribution from chiral loops and mixed chiral-gravity loops 16π 2 µ ¯ ¯i ¯ ν i Dµ z¯m A Ae−K + Dν D ν z i − Dν Dµ z¯m D z + h.c. . √ ∆LcχG = −8αΛ ReF Kim ¯D g

(4.38)

Given that it is not trivial to identify all the total derivatives that were dropped in [16] and [17], we cannot guarantee that there are no other such D-term anomalies, so for present purposes we will subsume these into an operator that we will call Ω′D . In order to preserve the correct form of the anomaly, we require that PV chiral supermultiplet mass terms have well-defined modular weights. Those supermultiplets that regulate dilaton loops, and/or 9

Determination of the explicit expression for these terms requires restoring all the total derivatives dropped in [17]; we have done this only for the term appearing explicitly in (4.35).

23

that contribute to (3.33), must have invariant masses. Operators that do not satisfy (4.15) include those that contribute to the renormalization of the K¨ ahler potential. These get contributions from all fields with couplings in W2 . We therefore require that these PV fields also have modular and U (1)X invariant masses; a subset of these also regulate φ3 in (2.9). The requirement that these PV fields have invariant masses (covariant PV superpotential terms) has implications for soft supersymmetry breaking scalar masses. Other operators Ωn that satisfy (4.15) include those, like ΩY M , ΩW and ΩX , whose chiral projections Φn are bilinear in generalized chiral superfield strengths. In the present formalism ΩX , defined in (4.32) and (2.8), is generalized to the operator: ¯ 2 − 8R)ΩX m = X α X m , (D m α

3 ¯2 Xαm = (D − 8R)Dα ln M2 + Xα , 8

(4.39)

¯ VX ) = MM is the squared mass of a pair of PV fields. ΩX and where the real superfield M2 (Z, Z, ΩX m will be explicitly constructed in Appendix E.2 following the construction of ΩY M in [22]. In addition to the above “F-term” operators, which have no bosonic terms in their lowest components, there are “D-term” operators: −2 m ¯ ΩD = M2 (D 2 − 8R)M R + h.c.,

1 αβ˙ m −ΩG = Gµm Gm µ = Gm Gαβ˙ , 2

¯ m , (4.40) ΩR = R m R

where 1 ¯ 2 − 8R)M2 = − 1 M−2 D ¯ 2 M2 + R, Rm = − M−2 (D 8 8

= Gm αβ˙

1 M[Dα , Dβ˙ ]M−1 + Gαβ˙ , (4.41) 2

are generalizations of the supergravity auxiliary fields R, Gαβ˙ . Although the operators (4.40) satisfy the constraint (4.15), their chiral projections cannot be entirely included in Φ, i.e., the right hand side of the generalized linearity condition (4.8), because they contain terms that are not invariant under the anomalous transformations (or do not transform into a linear multiplet). Therefore full anomaly cancellation appears to require that we add additional counterterms. To determine which additional terms of the form (4.14) are actually present in the linearity condition (4.8) requires evaluating, in superstring derived supergravity, the higher dimension terms from the underlying string theory. However some insight might be gained by considering the zero-slope limit that gives a supergravity theory in 10 dimensions. The components of the linear multiplet defined in (4.5) include a three-form hλµν that is a linear combination of the curl of a two-form bµν and the YM . Similarly, the three-form H Yang-Mills CS form ωλµν LM N of 10-d supergravity includes the curl of YM a two-form BM N and the Yang-Mills CS form ωLM N . However in order to cancel all the anomalies 24

of 10-d supergravity, the 3-form H must be modified to include a term proportional to the Lorentz CS form as well. This must also then be the case in the 4-d effective theory, and supersymmetry then implies that in (4.8) and Φ = cYM ΦYM + cGB ΦGB + · · · ,

Ω = cYM ΩYM + cGB ΩGB + · · · .

(4.42)

Then (4.9) contains a term proportional to the GB term which must also have a 10-d counterpart. When compactified to four dimensions the 10-d Riemann tensor includes, in addition to the 4-d Riemann tensor, derivatives of the 10-d dilaton Φ and the breathing mode(s) (e.g., σ = ln det gmn , m, n = 5 . . . 9). However, in the class of models considered here, with only three diagonal moduli, 1

i

i gmn = δmn eσ = δmn Re(ti s 3 ),

(4.43)

we find for the Chern-Simons 3-forms YM YM (ωµνρ )10d = (ωµνρ )4d ,

Lor Lor (ωµνρ )10d = (ωµνρ )4d ,

(4.44)

and there are no nonvanishing elements of ωM N R involving scalar derivatives. The 10-d expression for the Green-Schwarz counter term is [7] Z Z 2 (10) Lc = B ∧ X8 − ω YM − ω Lor ∧ X7 +α 3 Z 1 YM Lor = H+ −ω −α ω ∧ X7 , (4.45) 3

where α is an arbitrary parameter, the 8-form curl X8 of the 7-form X7 , X8 = dX7 , is constructed from the Yang-Mills and curvature field strengths, and the 3-form H is H = dB + ω YM − ω Lor .

(4.46)

In compactifications of the heterotic string the 10-d field strengths hRmn i = hFmn i are nonvanishing in the 4-d vacuum, but

ω YM − ω Lor

= 0,

(4.47)

(4.48)

and no new couplings appear to be generated by direct truncation of (4.45) to four dimensions. However this expression, constructed to cancel the 10-d chiral anomaly, should have a 10-d conformal anomaly counterpart, and it is possible that its 4-d remnant could be at the origin of some of the D-term contributions found here. 25

4.2

PV masses

Given a set10 of PV chiral superfields Z P with K¨ ahler metric KP Q¯ , a mass term with a well-defined modular weight can be constructed by coupling them to a set YP with with inverse K¨ ahler metric i multiplied by a light field-dependent function that is modular invariant up to factors eqi g : ¯ K Y = fY (Z p , S, VX )YP Y¯Q¯ K P Q ,

(4.49)

where VX is the vector supermultiplet11 of the anomalous U (1)X . The K¨ ahler potentials are modular invariant if under (3.10) for an appropriate choice of the matrix M P Q Z ′P = MQ Z ,

Y

YQ′ = e−qi

Fi

P NQ YP ,

N = M −1 ,

i Y ¯i f ′ = eqi (F +F ) f,

(4.50)

and the superpotential W1Z = µZ (T j )Z P YP ,

Z

i

W1′Z = e−Qi F W1Z ,

Z Y QZ i = −ωi + qi ,

P

µ′Z = µZ e

i

ωiZ F i

(4.51)

Q i i 2ni . A modular covariant has modular weights QZ i ; for example, ωi = ni if µ(T ) = µ i [η(T )] mass term W1X has QX i = 1. Note also that the squared mass matrix ¯

¯

¯

Y M −1 P δQ |µZ (T )|2 (m2Z )PQ = KZP R W1SR¯ KSM ¯ W1Q = f

(4.52)

is diagonal and commutes with the operators that appear in the one-loop effective action (2.2). The moduli-dependence of the PV masses we have introduced in (4.51) can be interpreted as a parameterization of the threshold corrections [9, 33] corrections from higher KK and string mode loops. This would be consistent with an interpretation of the PV masses as fully parameterizing Planck/string scale physics which provides the cancellations that restore the finiteness of the underlying theory, as well as restoring the perturbative symmetries of string theory, up to the terms that require including a field dependence in the (infinite) UV cut-off. The requirement of modular covariant mass terms entails the introduction of factors of Dedekind eta functions η(iT n ); these functions, and therefore PV masses [34] with ωi > 0, are exponentially suppressed in the (strong coupling) limit of large Ret. However we do not expect our perturbative treatment to be applicable in this strong coupling limit. ˙ Z, b Z; e see Appendix F. These may be, e.g., any of Z, In Yang-Mills superspace [5] only the gauge covariant superfield strengths Wαa are introduced and the chiral superfields Φb are covariantly chiral with a nonholomorphic gauge transformation superfield operator. In the presence of an anomalous gauge symmetry GX , it is necessary to introduce the corresponding gauge potential superfield VX . For U (1)X we use “partial” U (1)X superspace; see Appendix D.4. 10

11

26

If the fields Z P have the same K¨ ahler metric as the light fields Z p : KP Q¯ = Kp¯q , the Z P and YP loop contributions to terms linear in Γα in Φ1,2 and LQ cancel, while they give a double contribution to terms quadratic in Γα in Φ1 . Therefore we need to either 1) introduce other PV fields (with negative signature) with a simpler metric that can mimic the contribution of the light fields or 2) couple one set of (negative signature) fields Z P to fields ΦP with the opposite gauge charge P P ahler but trivial metric, e.g. K Φ = P eg |ΦP |2 . Option 1) is possible for sigma models whose K¨ potentials have the special property X n G(Z p ) = gn (Znp ), gpq = cn gpn gqn , (4.53) n

as shown in II for no-scale models that characterize the untwisted sector of orbifold compactifications. This allows full regularization of the theory in a simple way if the twisted sector fields are set to zero in the background. This was justified in II on the grounds that the K¨ ahler potential for orbifolds is not known beyond leading (quadratic) order in the twisted sector superfields ΦaT . As a consequence the O(|ΦT |2 ) loop corrections cannot be determined. However since in realistic models [30] the SM spectrum includes twisted sector fields, one would like to include them in the effective one-loop Lagrangian using a general modular invariant parametrization of the K¨ ahler potential. Therefore we adopt option 2), which entails a squared-mass matrix that is not of the form (4.52) and generally does not commute with other operators in the one-loop effective action. As shown in Appendices B–D, cancellation of on-shell UV divergences can be achieved with a PV sector such that the only masses that are not invariant under the anomalous symmetries are of the form (4.52), and indeed we argued above that the Z˙ mass matrix should be modular and U (1)X invariant. In fact we will adopt the simplest possible approach, with noninvariant masses generally present only for PV fields ΦC with a K¨ ahler metric of the form KC D¯ = fC δCD . σ ˙ To construct invariant masses for Z = Z˙ P , Z˙ N , we introduce fields Y˙ σ = Y˙ P , Y˙ N , with K¨ ahler potential i X Nh X A ˙ (4.54) ¯n¯¯ Y¯˙ N¯ + |Y˙ N |2 , eG gni¯ Y˙ I − χni Y˙ N Y¯˙ J¯ − χ eG |Y˙ A |2 + KY = A

N

P

n ahler potential for just the moduli with φa all gauge-charged where g = n g is either the K¨ matter, or it is the K¨ ahler potential for the untwisted sector with φa the twisted sector,12 , and X A,N m GA,N = qm g + ℓA,N , (4.55) gni¯ g¯nk = δki , m

12

In this case we have to slightly modify (4.54) and (4.57); see Appendix D.3.

27

where ℓA,N is a modular and gauge invariant function of the chiral superfields. The Y˙ A are PV counterparts of the light fields Z a not included in the gn ; they transform according to the gauge group representation conjugate to that of the Z a and have modular weights qnA . The K¨ ahler potential (4.54) is modular invariant provided under (3.10) j n ˙ ′ −F A ˙ ′ −F N j ′ ¯′ n ˙ ˙ ˙ ¯ Y + aF ˙ Y n , Y = e Y , Y = e χ′n (Z , Z ) = n χ (Z, Z) + aF ˙ J A q j N , A I i j i i X N N,A m Y˙ N′ = e−F Y˙ N , F N,A = qm F . (4.56) m

Nqp ,

nij

where is the submatrix of the terms in the superpotential ˙ W1Z

=

X

Wa +

a

Wa

X

defined in (3.12), that acts only on the fields Z i in

i

Wn ,

Wn = µ˙ n (T )

n

= µ˙ a (T i ) Z˙ A Y˙ A − a˙ −1 qna Z a Z˙ N Y˙ A ,

X

˙P

˙N

Z Y˙ P + Z Y˙ N

P ∈n

µ˙ a (T ′i ) = eF

!

,

a +F A −F

P

µ˙ n (T ′i ) = eF

µ˙ a (T i ),

ng

n.

N −F

Then

µ˙ n (T i ), (4.57)

are modular covariant. The UV-divergent contributions of the Y˙ to the loop corrections can be canceled by the fields U and V along with additional fields Φ with very simple transformation properties, K¨ ahler potential and superpotential W1 (U, V, Φ), that are given explicitly in Appendix D. b and Ye that To give masses to the fields introduced in Section 3.3, we introduce additional fields Z transform under the nonanomalous gauge group like Z p and its conjugate, respectively, as well as gauge singlets φˆr , with K¨ ahler potentials   ¯ X Q 0 b b¯ G + a2 G G + a ZbP Z b¯ b0 2  , ZbαP Z KαZ = fZbα  p¯ q α α α p q¯ α α Gp + h.c. + |Zα | P,Q=p,q

e

KαY

ˆ

Kαφ

= fYeα

X

P,Q=p,q

α

¯ Q¯ , Gp¯q YePα Ye

¯ S Sk K β0α k 0 β α 2 2k ˆS 2 0 2 k ˆ = e e 2e |φα | + |φˆα | + e φ¯α φˆα + h.c. + e α |φˆS | ,

(4.58)

that are modular and U (1)X invariant with the appropriate transformation properties.13 We thus modify the signature constraints (3.36) to read X X b b e e ˜ +N ˆ = 0, ηαZ = ηαY = ηˆα , ηαY = ηαZ = η˜α , η˜α + ηˆα ≡ N α

ˆr ηαφ

φr

= ηα ,

φr = (φS , φS , φ0 ),

α

(4.59)

bσ and Ybσ , respectively, with the substitution The K¨ ahler potential for φˆS,0 and φS,0 are the same as for Z ˆ p ¯p ¯ φ α ¯ G(Z , Z ) → k(S, S) and aα = −1, fφˆα = exp(K + β0 k). 13

28

and the first two constraints in (3.35) are modified to read X α

X

b

ηαY a ˆ2α = −1,

b

ηαY a ˆ4α = +1.

(4.60)

α

bI to ΦP V doubles that from Yb α . because the contribution from Z α 1 I The remaining PV masses can be generated, for example, by the superpotential ˙

b

e

W1 = W Z + W Y + W Z + W ϕ + W ϕ˜ + W S + W φ + W V + W Φ , X e X X b X e b bP Yb α , ZeαP YePα , Z µZ WZ = µYα WY = α α P α

W

ϕ

=

α

P =A,I,0

X

a a µϕ ˆα , α ϕα ϕ

α,a

WS =

X

µSα φrα φˆαr ,

P =A,I

1 ˜ a a W ϕ˜ = µϕ ϕ˜ ϕ˜ , 2 α α α

Wφ =

α,r

1X

2

W

V

=

X A,γ

µφαβ φα φβ ,

γ A µU γ UA Uγ

1 V γ 2 + µγ (VA ) , 2

ηαφ = ηβφ if µφαβ 6= 0,

α,β

(4.61)

˙

where W Z is given by (4.57), and W Φ is given in Appendix D. The K¨ ahler potential for the fields Φ, whose only role is to cancel unwanted curvature terms generated by the K¨ ahler potential (4.54), ˙ ϕ S Z ˙ Y˙ ) are U (1)X is also given in Appendix D. Note that W , W and W (for at least one set Z, invariant and modular covariant. The mass parameters µ can in general depend on the moduli: µ = µ(T i ). When all the above additional PV fields are included, the prefactors in their K¨ ahler potentials and their U (1)X charges are understood to be included in the sums in (3.37) and (3.38), and in (3.36) N ′ includes all PV chiral superfields. We further require X α

ηαY ln fαY −

X

Y = Yb , Ye ,

ηαZ ln fαZ = 0,

α

b Z. e Z = Z,

(4.62)

The simplest form of the prefactors needed for the cancellation of all UV divergences is f = eαK+βk+

P

n qn g

n +qV X

.

(4.63)

Although we are working generally in Yang-Mills superspace [5] in which the gauge potential superfields Va do not appear explicitly in the superpotential, as explained in Appendix D.4 fields with superpotential mass terms that are not U (1)X invariant must transform in such a way that the mass term remains holomorphic under a U (1)X transformation; this requires the presence of qVX in the exponent of the prefactor for these fields. As shown in Appendix D, all the anomalies arising from gauge-charged matter in the light sector can be absorbed into the mass terms of U, V, Φ. This 29

means in particular that we need to include factors eqVX only for these fields. These fields also have P A,N n in (4.54), that accommodate the charged matter prefactors of the form e n qn g , similar to eG and T -moduli contributions to the modular anomaly. For the other PV fields we take the simpler form (4.64) fC = eαC KC +βC kC . A similar prefactor may be included in the K¨ ahler potential for any chiral superfield Φ 6= θ that does not have other couplings except in W1 , provided the moduli-dependence of W1 assures modular covariance as needed; this includes additional fields with the same K¨ ahler metric, gauge charges ˙ b e ˙ b e and superpotential terms in W1 as Z, Z, Z, Y , Y , Y but no other couplings. All PV fields with couplings that contribute to the renormalization of the superpotential must have invariant mass terms. Here we will assume only the minimal number of these fields needed to regulate light field ˙ Y˙ only with the contributions to the renormalization of the K¨ ahler potential. Thus we include Z, K¨ ahler potential given in (3.16), (3.18) and (4.54). We take all Yb , Ze to have the K¨ ahler potentials b e given in (3.28), and Z, Y to have, respectively, the inverse K¨ ahler metrics with prefactors b Y e Z,

fZbα ,Yeα = eαα

K

,

(4.65)

b e

such that including the T i -dependence of µYα ,Z , the masses are modular invariant, and the constraint (4.62), which reduces to X e b (4.66) ηˆα αYα − αZ α = 0, α

is satisfied.

4.3

Quadratic PV mass terms

We have constructed the PV Lagrangian in such a way that PV fields that couple to one another in W1 have no other common coupling. The the term quadratic in masses takes the form, in a basis where the squared mass matrix is diagonal ! ! " 1 X X X X 1 PV θ 2 θ LQ = ℓC − ℓγ D α Xα | Vˆ + M − 3 ℓγ − ℓC 32π 2 2 γ γ C C # X X , (4.67) + ℓsγ D α kα | + ℓC D α ΓC Cα γ

C

30

where here C refers to all heavy chiral supermultiplets ΦC 6= θ, and ℓX = m2X ln(Λ2 /m2X )ηX ,

X = ΦC , θ.

Finiteness requires that the coefficient of ln Λ vanish: X X X X 0= m20γ ηγ0 = m2sγ ηγs = m2C ηC = m2C ηC D α ΓC Cα . γ

γ

C

(4.68)

(4.69)

C

If ΦC couples to ΦD in W1 , with

m2C = m2D = µ2C |wC (t)|2 eK−2(qC +qD )VX fC (z, z¯) ≡ µ2C Λ2C ,

(4.70)

D then Dα ΓC Cα + ΓDα is completely determined in terms of (qC + qD ) and scalar derivatives of fC , so the third equality in (4.69) insures the fourth, since the coefficient of each term of fixed Λ2C 6= constant must vanish separately. We have regulated the theory such that all masses are invariant except for some chiral superfields ΦCα with masses of the form (4.70). These give contributions to the renormalized K¨ ahler potential of the form KQ m2Xγ

∋

X λC Λ2 C

C 32π 2

= µ2Xγ Λ2X ,

−2

λ0 Λ20 , 32π 2

nX X

λX =

nX X

η Xγ µ2Xγ ln(µ2Xγ ),

γ=1

η Xγ µ2Xγ = 0.

(4.71)

γ=1

We require λC = 0 if Λ2C is not modular and U (1)X invariant. Note that in order to cancel the chiral anomaly and the logarithmic divergences, we have necessarily nC X

α=1

η Cα 6= 0

(4.72)

for at least some sets {C, D}. For these the constraint λC = 0 requires14 nC ≥ 5.

5

Superfield structure of the anomaly and anomaly cancellation

The bosonic part of the anomaly under infinitesimal modular and U (1)X transformations that arises from the noninvariance of PV masses is given in (C.70) of Appendix C. Referring to (C.72) 14

See Appendix C of [35].

31

and (E.11), when combined with the additional contributions described in Section 4.1–with the cancellation of (4.37) imposed–that expression is the bosonic part of an infinitesimal variation of the superfield operator Z Lanom = L0 + L1 + Lr = d4 θE (L0 + L1 + Lr ) , (5.1) Z 1 1 L0 = Trη ln M2 Ω0 + K ΩGB + Ω′D , Trη d ln MΩr . (5.2) Lr = − 2 8π 192π 2

L1 is defined by its variation: ∆L1 =

1 1 1 1 Trη∆ ln M2 Ω′L = 2 TrηHΩ′L + h.c., 2 8π 192 8π 192

(5.3)

and Ω′D is the “D-term” anomaly that arises from certain total derivatives with logarithmically divergent coefficients as discussed in Section 4.1. The superfield Ωr is defined by (E.11) and (E.12), ΩGB is defined in (4.31), the real superfield M2 is the PV squared mass matrix,

with H holomorphic, and

¯ , ∆ ln M2 = 2 H + H

1 1 1 1 ¯ 2R ¯ − 1 Ω′L ΩW + Ω0YM − ΩX + ΩL − 6φχ + D 2 R + D 3 36 192 48 192 1 1 = − ΩGB + Ω0YM − Ωφ , 24 48 ¯ 2 R, ¯ = ΩL − 16ΩX , Ωφ = 12φχ + D 2 R + D

(5.4)

Ω0 =

Ω′L

(5.5)

with the Chern-Simons superfields in (5.5) defined in (4.23), (4.32), and (E.7). The above results were also found [29] by performing a superspace calculation of the anomaly in PV regulated superconformal supergravity, and then gauge-fixing to K¨ ahler U (1)K superspace [5]. Note that the part ′ of ΩL that is independent of the U (1)X charges and modular weights, namely X C 2 TrηHΩ′L = 16 ϕ˜+ (5.6) C (1 − 2α ) − 1 ΩX + O(qX , qi ), C

drops out of (5.3) by virtue of (D.103) and (D.113) [with (D.114)]. In the present approach, the coefficient of the Chern-Simons superfield ΩX Y M comes from a combination of terms in ΩXm and ΩG in (C.72). Because the anomalous U (1)X is not incorporated into the superspace geometry, the superfield strength in the chiral projection of Ω0YM , defined in (C.72), does not include WαX . Instead, the PV masses depend on the the U (1)X vector field strength VX , giving a U (1)X -dependent 32

contribution from ΩL . We note here that the operator Ω0 in (5.2) has contributions that contain a dependence on the dilaton field, such as ΩX and ΩL ; these do not satisfy the condition (4.15) of Section 4.1. We show in Appendix E.2 that the linear/chiral multiplet duality outlined in Section 4.1 still holds in the presence of these operators; although some intermediate equations are modified the action for the dilaton coupling to Ω is the same in both the linear and chiral multiplet formulations, and still takes the form (4.9) or (4.14). There are additional U (1)X -dependent contributions from the “D-terms”in Ωr . Integrating (E.11) gives Z 1 1 1 α ˙ α 4 Lr = − 2 D Lα + 2D Xα (ln M)2 + Gαβ Dα ln MDβ˙ ln M d θETrη ln M 8π 24 2 1 2 ˙ − D ln MDβ˙ ln MD β ln M + h.c. 4 1 2 ¯2 α − ln M D D ln M + D (RDα ln M) + h.c. 8 1 α β˙ − D ln MDα ln MDβ˙ ln MD ln M , (5.7) 2 up to a linear superfield and a total derivative; the variation of Lr under an anomalous transformation is given in (E.16). In this section we evaluate the anomalies and discuss the counterterms needed to cancel them. Possible connections of these counterterms to the 10d GS term were discussed in Section 4.1. For simplicity we present explicit results only for the case without threshold corrections as in Z3 and Z7 orbifold compactifications; the result for any specific compactification with threshold corrections can be reconstructed from the results of Appendix D.4, by matching the parameters ωnC to the threshold corrections specific to the model.

5.1

Generalization of the 4d GS mechanism

The traces in L0 are completely determined, up to possible threshold corrections, in terms of the quantum numbers of the light spectrum. Taking the traces subject to the constraints given in (3.35)–(3.38), Appendix A and Appendix D.4, we obtain a contribution that can be written in the form (up to terms that are invariant under modular15 and U (1)X transformations and terms that 15

P

In the case that string threshold n A n 2 ¯n ΩA . n,A bn ln (T + T )|η(T )|

corrections

33

are

present,

these

include

terms

of

the

form

do not contribute to the variation of the action) 1 g L0 = bg − δX VX Ω0 + 2 Ω′D + Ωφ . 2 8π

(5.8)

In contrast, the traces in L1 depend on the details of the PV regularization procedure. For example if we use either (D.130) or (D.147) we get an expression of the form " X 1 1X l ′ TrηHΩL = Amnl Ωnm F CGS ΩX Y M + Al ΩWX X − 192 3 m,n l # X − (2Bnl ΩWX n − Anl ΩnX ) n

"

# X X 1 ′ ΩX amn Ωnm − (2bn ΩWX n − an ΩnX ) (5.9) , + Λ 3CGS Y M + aΩWX X − 3 m,n n

with ¯ 2 − 8R)ΩW X (D X

¯ 2 − 8R)ΩnX (D

¯ 2 − 8R)Ωnm = gnα gαm , (D

= WXα Xα ,

¯ 2 − 8R)ΩW n = WXα gαn , (D X

= gnα Xα ,

(5.10)

the expressions for the parameters A, B, a, b, c are given in (D.131)–(D.138) for the PV sectors of Appendix D.1 and Appendix D.2 with the choice (D.130), and CGS = 8π 2 b,

′ CGS = −4π 2 δX ,

(5.11)

In writing the above we used the known string theory constraints, which for orbifold compactifications with no threshold corrections read ! X 1 2 qnp − N + NG − 21 ∀ n 8π 2 b = 24 p X = Ca − CaM + 2 Cab qnb ∀ n, a, Ca = Tr(Ta )2adj , CaM = Tr(Ta )2mat , (5.12) b

1 1 (5.13) 2π 2 δX = − TrTX = − TrTX3 = −Tr(Ta2 TX ) ∀ a 6= X. 24 3 If the expression (5.9) were to factorize: ′ 1 ′ TrηHΩ′L = CGS F + CGS Ω0 , (5.14) 192 " # X X 2Bn′ ΩWX n − A′n ΩnX , (5.15) Ω′0 = ΩX A′mn Ωnm − Y M + AΩWX X − m,n

34

n

the full F-term contribution to the anomaly could be canceled provided the tree-level Lagrangian includes a contribution [15] (in the chiral formulation for the dilaton) Z Z 1 E 4 ¯ ¯ 2 − 8R)Ω, LS = − d θ(S + S)Ω = Φ = (D Ω = Ω0 + Ω′0 , (5.16) d4 θ SΦ + h.c., 8 R with S transforming under modular and U (1)X transformations as S→S−

δX Λ + bF, 2

(5.17)

so that the dilaton K¨ ahler potential δX KS = k(S + S¯ + VX − bG) 2

(5.18)

is invariant. Note that the coefficient of ΩX Y M in Ω is fixed, since the dilaton is known to couple universally to the gauge field strength (for affine level k = 1). The Lagrangian (5.16) and the K¨ ahler potential (5.18) can be obtained by a duality transformation from the formulation where the dilaton is the lowest component of a modified linear multiplet, as described in Section 4.1. The new couplings in (5.16) of course also contribute ultraviolet divergent contributions in the effective QFT. We expect that these can be regulated by PV fields with modular and U (1)X invariant masses, as we have shown to be the case for the dilaton coupling to ΦY M , so that they will not induce any further contributions to the anomaly. In the case that string loop corrections of the form 1 X Lthresh = ln |η(T n )|2 bA (5.19) n ΩA , 4π 2 n,A

∆Lthresh =

1 X F (T n ) + F¯ (T¯n¯ ) bA n ΩA , 2 8π

(5.20)

n,A

are present, the constraints (5.12) and (5.13) are modified as in (D.102) and (D.121), and the quantum anomaly arising from the F-term should be canceled by a the combination of (5.17) and (5.20). If we make a different choice than (D.130) or (D.147) for the invariant functions ℓ in (4.55), there will be additional terms on the right hand side of (5.9). However this model-dependence is removed if we evaluate it with only the K¨ ahler moduli nonvanishing among the background chiral superfields, such that X X X ℓ → 0, K→g= gn , ΩnX → Ωnm , ΩWX X → ΩWX n , (5.21) n

m

35

n

so that (5.9) reduces to 1 TrηHΩ′L = 192

# " X X 1X l (Al − 2Bnl ) ΩWX n − (Amnl − Anl ) Ωnm F CGS ΩX YM + 3 n m,n l # " X X 1 ′ X (5.22) (a − 2bn ) ΩWX n − (amn − an ) Ωnm . + Λ 3CGS ΩY M + 3 n m,n

One can check that this does not factorize for the models listed in Appendix D.5 if we use the results (D.131)–(D.138) that apply if the prescription of Appendix D.1 or Appendix D.2 is used. With this prescription, the modular weights and U (1)X charges of the PV fields that enter the sums in (D.131)–(D.138) are exactly those of the light chiral supermultiplets. They arise from the contributions of PV fields ΨC with, in the absence of threshold corrections (ωnC = 0 ∀ ΦC ) modular weights C qnΨ = 1 − qnC . (5.23) In contrast, the more constrained prescription of Appendix D.3 replaces the gauge singlet PV “moduli” ΦN , Φn with N N n , 0) (5.24) (qm , qm ) = (2δm by the same number of PV “moduli” but with N n N n (qm , qm ) = (δm , δm )

(5.25)

This leaves sums linear in the modular weights unchanged, but slightly modifies the nonlinear sums: ∆

X C

C

C

qnΨ1 . . . qnΨM = (2 − 2M )

M −1 Y

δnk nk+1 .

(5.26)

k=1

This does not achieve factorization, but suggests that an even more constrained PV sector, such as “option 1”, discussed in Section 4.2 and briefly outlined in the beginning of Appendix D, might allow a significantly different set of modular weights for the PV fields that have noninvariant masses. Note also that in the effective field theory for the Z7 case of Appendix D.5, there are mixed anomalies associated with TrTa qn 6= 0 and TrTa qn2 6= 0. No anomalies are present in the corresponding string theory [14] (which has no Wilson lines and therefore no anomalous U (1)). Therefore in the appropriately regulated effective field theory these anomalies should disappear. For the case where an anomalous U (1)X is present, we also need a modification of TrTX2 qn . In addition there may be larger symmetries, or partial symmetries–such as the “Heisenberg symmetry” of the untwisted 36

sector K¨ ahler potential that is used in Appendix D.3–that should be respected in the PV sector.16 For example, the K¨ ahler metric of the untwisted sector for Z3 models, including the more realistic FIQS model with Wilson lines and a U (1)X , have a larger symmetry than the [SL(2, Z)]3 group of modular transformations that we have been using. If that symmetry is preserved in the PV sector, this might also give a different result for terms nonlinear in modular and U (1)X charges. Cancellation of the remaining contributions to the anomaly, namely the the term proportional to g in (5.8)–that has not been considered previously in this context–and the D-term anomaly Lr in (5.7) appears to require additional counterterms, although possibly the former, and certainly the latter, will be modified by any modification of the PV sector that can provide factorization in (5.22). Because ln M contains a term ( 12 − α)K, it is not impossible that the the former term can be canceled by a contribution from the latter, in the same way that that the ΩX term in Ω′L is canceled in (5.6); this cancellation seems to be independent of the details of the PV sector.

6

Summary of Results

We used on-shell Pauli-Villars regularization of the one-loop ultra-violet divergences of supergravity to determine the anomaly structure of these theories. This regularization procedure requires constraints on the chiral matter representations of the gauge group that are satisfied in any supersymmetric extension of the Standard Model, and by hidden sectors that have been found in orbifold compactifications of the heterotic string. We showed that the logarithmic and quadratic one-loop divergences in the S-matrix can be canceled by a PV sector that respects the classical symmetries of the theory except for certain superpotential terms that generate large, noninvariant PV masses for a subset of PV chiral superfields that have a very simple K¨ ahler metric. A PV sector with this feature was explicitly constructed for effective supergravity theories with three K¨ ahler moduli T i obtained from orbifold compactifications of the weakly coupled heterotic string. These theories are classically invariant under a U (1)X gauge transformation and under the T-duality group of modular transformations that are anomalous at the quantum level. If all linear divergences were canceled in the regulated theory, it would be anomaly free, with 16

In writing the transformation properties (3.10) and (3.14) we neglected constant phases [36] and matrices [37] that mix chiral superfields with the same modular weights. These can be trivially incorporated and do not affect the conclusions.

37

noninvariance of the action arising only from that of the Pauli-Villars masses. Here we calculated their contribution to the bosonic part of the anomaly in the component Lagrangian. Because the regularization procedure is manifestly supersymmetric the full anomaly generated by PV loops necessarily forms a superfield. Since only chiral superfields contribute, and since they have a diagonal K¨ ahler metric, we were able to draw on the recent superfield calculation [29] of the chiral multiplet loop contribution to the UV divergences and anomalies in supergravity. This provided a check of our component results, and greatly simplified the identification of the superfield form of the anomaly. Pauli-Villars regulator fields allow for the cancellation of all quadratic and logarithmic divergences, but there remain residual linear divergences associated with nonrenormalizable gravitino/gaugino interactions. This result follows directly from the constraints imposed by cancellation of on-shell quadratic divergences, and is therefore independent of the details of the PV sector. These residual divergences result in additional contributions to the chiral anomaly. There is also an additional conformal anomaly that appears if the UV cut-off Λ is field dependent, because there are residual terms in the coefficient of ln Λ that are not canceled by the PV sector. These are total derivatives, so the S-matrix of the regulated theory is finite, but a field-dependent cut-off can induce finite terms that are not invariant under the classical symmetries. These new contributions, which are anomalous only under T-duality invariance, combine to form a supermultiplet provided the ultraviolet cut-off has the field dependence given in (1.1). When this contribution is combined with the PV contribution, we obtain results in agreement with string-loop calculations of the anomalous coefficients of the squared (nonanomalous) Yang-Mills and space-time curvatures. It is well known that these contributions to the anomaly can be canceled by a four-dimensional version of the Green-Schwarz mechanism. We showed that the remaining contributions depend on the choice of PV sector couplings. Provided that the F-term contributions factorize [14], they can be canceled by a generalization this mechanism, corresponding to a generalization of the modified linearity condition for the linear supermultiplet that is dual to the dilaton chiral supermultiplet. It may be that factorization is possible only with constraints on the superpotential of the type discussed at the beginning of Appendix D. Such constraints would be somewhat analogous to the constraint on gauge charges mentioned above, although the cancellation of UV divergences by itself requires no constraint on the K¨ ahler potential. Such a factorization condition could be used as a tool to probe higher order terms in the twisted sector K¨ ahler potential. This is of importance for phenomenology if some fields require large vacuum values as, for example, in U (1)X gauge symmetry breaking at the string scale. As an example, we found that if we impose a K¨ ahler 38

potential of the form (D.7) for the twisted sector of the Z7 model of Appendix D.5 one can find values of the parameters such that terms proprotional to X α gαn , with coefficients both linear and quadratic in the modular weights, either factorize or vanish, while the cancellation in (5.6) is unaffected. This approach will be pursued further elsewhere. There are additional contributions to the conformal anomaly that may require new counter-terms. These include terms that are nonlinear in the parameters of the anomalous transformations. On the other hand it is possible that some or all of these terms can be made to cancel. We briefly considered the possible connections between the 4-d counterterms that we find and the 10-d GS term; establishing a clear connection requires further work. Acknowledgments. We are happy to acknowledge the contributions of Andreas Birkedal, Choonseo Park, Matthijs Ransdorp and So-Jong Rey to various early stages of this work. We thank Dan Freedman, Joel Giedt, Brent Nelson, Claudio Scrucca, Marco Serone, Tom Taylor and Lewis Tunstall for discussions and helpful input. MKG would like to thank the Kavli Institute of Physics at U.C. Santa Barbara for hospitality during part of the completion of this work. This work was supported in part by the Director, Office of Science, Office of High Energy and Nuclear Physics, Division of High Energy Physics, of the U.S. Department of Energy under Contract DE-AC02-05CH11231, in part by the National Science Foundation under grants PHY-0457315 and PHY-1002399.

39

Appendix A

Linear divergences

The one-loop contribution to the bosonic part of the effective action is i i 2 ˆΦ Tr ln(D + HΦ ) − Tr ln(−i D 6 + MΘ ) 2 2 2 ˆ 2 + Hgh ) + HGh ) − iTr ln(D +iTr ln(DGh gh i ˆ 2 + H) − i T− , STr ln(D ≡ 2 2

L1 =

(A.1)

where Φ is a 2N + 4NG + 10 component boson, Θ is an N + NG + 5 component Majorana fermion, with N is the number of chiral multiplets and NG is the number of gauge multiplets, and the last two terms are the (4-component fermion) ghostino and (NG + 4 component boson) ghost contributions, respectively. The last equality is obtained after casting the fermionic determinant in the form i i i 2 ˆΘ − Tr ln(−i D 6 + MΘ ) = − Tr ln(D + HΘ ) − T− , 2 8 2

(A.2)

ˆ Θ and HΘ are 8 × 8 matrices in Dirac space that act on the vector-valued 4where in (A.2) D component fermion (ΘR , ΘL ); for example HΘ for PV the fields with noninvariant masses is given in (C.9) below. The first term is the helicity averaged part of the fermion contribution, and [17] n o −1 T− = Tr − D 6 2+iD 6 M iD 6 N (A.3)

is the helicity-odd part,17 where N is the helicity-odd part of −i D 6 + MΘ . T− is not invariant under chiral transformations because, unlike the the other operators appearing in (A.1), N cannot be recast in invariant form; it explicitly contains the axial current. If the linear divergences in T− cancel, the non-invariance can be shifted to the PV masses. This requires ˜ = 0, Trη(G · Gφ)

(A.4)

where the signature η = 1 for light fields, Gµν = [Dµ , Dν ] = Gµν + Zµν , 17

(A.5)

As discussed in [1] the separation of the fermion determinant into helicity odd and even parts is ambiguous; there is a unique choice that allows a PV regularization of the quadratic divergences. For the universal axion couplings to gauginos, this choice is also consistent with nonrenormalization [28] of the θ-parameter and with linearity constraints [12] in the dual linear formulation for the dilaton supermultiplet.

40

is the fermion field strength, the fermion covariant derivative is Dµ = ∂µ + Zµ + vµ + Jµ γ5 = dµ + Zµ ,

Gµν = [dµ , dν ],

(A.6)

with Zµ the spin connection: 1 Zµν = rµνρσ γ σ γ ρ , 4

1 Zµ = γν [∇µ , γ ν ], 4

(A.7)

and δJµ = −i∂µ φ

(A.8)

under a chiral transformation. In fact, the trace in (A.4) does not vanish because the gravitino connection contains the affine connection as well as the spin connection, whereas the PV fields are chiral fermions and (Abelian) gauginos, with only the spin connection contributing to space-time curvature dependent terms. In addition there is an off-diagonal gaugino-gravitino connection that is not reproduced in the PV sector. This implies that there is a residual linear divergence. When the PV fields are included all the other terms are reproduced, provided they satisfy condition (A.4). The first equality in (4.24), which follows from the cancellation of on-shell quadratic divergences, ensures the equality ! X 1 ˜ = ˜ TrηZ · Zφ = 0. (A.9) N + N ′ − NG − NG′ − 3 − 2 ηC αC Z · ZImF 2 C

To implement the condition (A.4) for the chiral fermions it is useful to use the relations G+ µν

Writing

n¯

m ¯ ¯ + n Dµ m ¯

q¯ n n ¯ − p + 2Zµν δm = −Kpm ¯ Gµν q K ¯, q¯ n n ¯ − p + K q¯n ∂µ Kqm = −Kpm ¯ + 2Zµ δm ¯ Dµ q K ¯,

Dµ± = ∂µ + Zµ + Jµ± ,

(Jµ )ij ≡ (Jµ− )ij ,

¯ + m ¯ (Jµ )m n ¯ ≡ (Jµ )n ¯ ,

(A.10)

(A.11)

for a chiral superfield with a general K¨ ahler metric K: Kim ¯ = Kim ¯ , the anomalous part of the one loop action under a chiral transformation takes the form [see (B.1)] 1 1 Tr ln(−i D 6 + R) − Tr ln(−i D 6 − L) = Tr ln(−i D 6 + R) − Tr ln(−iK D 6 − K−1 L) 2 2 1 Tr ln(−i{6 ∂ + Z 6 + 6 J + }R) − Tr ln(−iK{6 ∂ + Z 6 + 6 J − }K−1 L) = 2 1 = Tr ln(−i{6 ∂ + Z 6 + 6 J + }R) − Tr ln(−i{6 ∂ + Z 6 − 6 J + }L) , (A.12) 2 41

where in the last equality we used (A.10), so we can simply take Jµ+ (or −Jµ− ) to be the axial current: Jµ = Jµ± . Then GVµν = Zµν + [Jµ± , Jν± ],

Jµν = ∂µ Jν± − ∂ν Jµ± ,

(A.13)

and the condition (A.4) reduces to

This implies, in particular,

0 = ǫµνρσ Tr ∂µ Jν± ∂σ Jρ± φ .

(A.14)

0 = TrTa2 TX ,

(A.15)

and 0=

X C

X ηC −8α3C + 12α2C − 6αC + N + N ′ − NG − NG′ − 3 = −8 ηC α3C − 8, C

X C

ηC α3C = −1.

(A.16)

where αC is defined in (4.64) and we used the constraints (3.36)–(3.38). There are also mixed terms in the gauge and K¨ ahler U (1) connections; these require 0 = 2

X C

0 = 4

X C

ηC αC (Ta )2C − TrTa2 = 2 ηC α2C (TX )C

−4

X

X

ηC αC (Ta )2C ,

C

ηC αC (TX )C + TrTX = 4

(A.17) X

ηC α2C (TX )C ,

(A.18)

C

C

since cancellation of quadratic divergences (2.16) requires TrTX = 0, and cancellation of the logaP rithmic divergences (2.8) requires TrTa2 = C ηC αC (TX )C = 0 when PV contributions are included b Ye fermion loops are, respectively in the trace. The contributions to (A.17) and (A.18) from Z, X b 2 e 2 X b M Ye Z ηˆα αZ − αYα 4 (TrTX )M . (A.19) 2 ηˆα αα + αα Ca , α α

α

Since

P

ˆα αη

= 0, both terms in (A.19) as well as (4.66) vanish if b

b

e

Z αZ α =α ,

e

αYα = αY ,

(A.20) b

e

Y independent of α. For example, if there are no threshold corrections, µZ α and µα are constants b e b e in (4.61), and modular covariance of W Z , W Y requires αZ = αY = 1. More generally, (A.20)

42

b e

b e

requires that µαZ,Y = cα µZ,Y (T i ), which simplifies the constraints in Section 4.3. This constraint b Ye , Ze loops. There is no (A.20) also assures that there are no other contributions to (A.4) from Yb , Z, a a contribution to (A.17) and (A.18) from the adjoint fermions χα = Dα ϕ | which have vector gauge connections and no U (1)X charges. Then these constraints are trivially satisfied if the superfields φγ in (3.34) carry no gauge charge, which is natural since they were introduced to regulate gravitational couplings. The constraints involving the parameters βC are, using (3.37) and (3.38), ! X X X X βC2 αC − 1 = βC 4α2C − 4αC + 1 = 4 βC2 (2αC − 1) = 2 0= βC α2C , C

C

X

C

βC2 αC = 1,

C

X

βC α2C = 0,

C

C

X

βC3 = 0.

(A.21)

C

Finally, we consider the gravitino and gaugino sector. As discussed in Appendix B.3, cancellation of quadratic divergences arising from the axion connection in the gaugino covariant derivative, which is defined as Dµ λaL = (∂µ + Zµ − iΓµ + ℓµ ) λaL + i(Tc )ab Acµ λbL ,

Dµ λaR = (∂µ + Zµ + iΓµ + ℓµ ) λaR + i(Tc )ab Acµ λbR ,

(A.22)

where Γµ is the K¨ ahler U (1)K connection defined in (B.51), and ℓλ =

1 µνρσ ∂λ y ǫ γµ γν γρ γσ 24 2x

(A.23)

is the “vector” axion connection. Note that although the gauge representation is real, we have, instead of (A.10), G+ µν Dµ+

a b

a b

c da = −Kcb G− + 2 (Zµν + ℓµν ) δba , µν d K c = −Kcb Dµ− d K da + K da ∂µ Kdb + 2 (Zµ + ℓµ ) δba ,

(A.24)

Kbc (Te )cd K da = (Te )ba = (TeT )ab = −(Te )ab .

(A.25)

where ℓµν is the field strength associated with the connection ℓµ , and Kab = xδab , so

We therefore identify J ± in (A.4) as Jµ± = ±iΓµ + iT · Aµ . The contribution from Γµ alone to (A.4) is included in the sum rule (A.16). It remains to cancel the gaugino contribution proportional to

43

Γµ Ca , where Ca is the adjoint Casimir. From (3.39) and (B.51) we have the following contribution from ϕa , ϕâ , ϕã : # " X X X (A.26) ηγϕ (Γµ − ∂µ y/2x) + ηγϕˆ (∂µ y/2x − Γµ ) − ηγϕ˜ Γµ , Ca γ

γ

γ

which cancels the gaugino contribution provided X

ηγϕ =

γ

X

ηγϕˆ =

X

ηγϕˆ +

γ

X

ηγϕ˜ = 1,

(A.27)

X

ηγϕ˜ = 3,

(A.28)

γ

in accordance with the overall constraint X γ

ηγϕ +

γ

γ

needed for the cancellation of gauge and gaugino one-loop logarthmic divergences that arise from their gauge couplings [1, 3]. Additional constraints on the PV sector required to cancel the remaining contributions to (A.4) from PV chiral supermultiplets that regulate chiral matter loops are discussed in Appendix D.

B B.1

Chiral Anomalies Chiral anomaly analysis in the regulated theory

The chiral anomaly arises from the helicity odd part of the fermion determinant T− =

1 [Tr ln M(γ5 ) − Tr ln M(−γ5 )] , 2

M = −i D 6 + M,

(B.1)

in which the axial connection appears explicitly. In [1]–[3], explicit cancellation of ultraviolet quadratic and logarithmic UV divergences in (A.1) was shown to be possible by the introduction of the PV fields discussed in Section 3. In the previous appendix we found additional constraints that insure cancellation of most linear divergences. We defer to Appendix B.3 the discussion of residual linear divergences that are associated with the gauge-gravity-dilaton sector. In this section we outline a check that, in the absence of linear divergences, the anomaly appears only through the noncovariance of the PV masses, and show that the “consistent anomaly” of YM theory is recovered under appropriate assumptions. Under an infinitesimal chiral transformation Θ → e−iγ5 φ Θ,

δΘ = −iγ5 φΘ, 44

(B.2)

where Θ = (ψµ , λa , Lχp + Rχp¯, α) is a 2N + 4NG + 10 component Majorana spinor18 , we have M → M + δM, δTr ln M = TrM−1 δM, and (B.1) is shifted by δT− =

1 TrM−1 (γ5 )δM(γ5 ) − TrM−1 (−γ5 )δM(−γ5 ) . 2

(B.3)

Using the methods developed in Appendix A of [17] the corresponding shift in the Lagrangian can be cast in the form Z i d4 p δL ∋ − δT− = −i [T (γ5 ) − T (−γ5 )] , 2 4(2π)4 ∞ X T (γ5 ) ≡ T = Tr (−R)ℓ δR, (B.4) ℓ=0

where now the trace is over only Dirac indices and internal quantum numbers, and Lorentz indices for the gravitino, and iˆ 1 ν ν cµ 2 µν p − T ∆µ ∆ν − G · σ + X + Pµν (p + G ) M , R = −p2 2 ∆µ = pµ + Gµ + δµ 1 ∂ 1 ν ν bµ µ δR = P (p + G ) N = p N + O . (B.5) µν µ −p2 −p2 ∂p

The operators in these expression are given in Appendix F.4 as covariant derivative expansions. Specifically, T µν , X, Pµν , δµ and Gµ , Gˆµν are expansions in covariant derivatives of, respectively, the space-time curvature tensor19 and the field strength Gµν acting on fermions: L ± = [Dµ± , Dν± ] = G± Gµν µν + Zµν + iGµν γ5 ,

1 Zµν = − rρσµν γ ρ γ σ , 4

(B.6)

L where +(−) refers to right(left)-handed fermions, and G± µν , Gµν are unit matrices in Dirac space. GL contributes only through loops in the gaugino-gravitino-dilatino sector. In addition we defined Rγµ iδ D 6 + R −Rγµ δM L 0 Rγµ M L Nµ = − , Mµ = , (B.7) ¯ R Lγµ iδ D ¯R −Lγµ δM 6 −L Lγµ M 0

¯ ) → (D − , D + , M ¯ , M ), with and T (−γ5 ) is obtained from T (γ5 ) by the substitutions (D + , D − , M, M M = M0 + Mµν σ µν . 18

(B.8)

The chiral fermion α is an auxiliary field used to implement gravitino gauge fixing [16]. There is a sign error in the third and second terms, respectively, of the expressions for Tµν and X in Eq. (A.19) of [17]. 19

45

The matrix Mµν has nonvanishing elements only between the gauginos λ and either the dilatino χS or the auxiliary field α introduced to fix the gravitino gauge [16, 17]. Under the chiral transformation (B.2), ¯ = i{M ¯ , φ}, δDµ± = ±i Dµ± , φ , δM δM = −i{M, φ}. (B.9)

In order to evaluate the light quark contribution to the chiral anomaly, we must resum the derivative expansion of [17]. This resummation can be expressed in terms of the action of the operators in (B.1) on a function of momentum: Z 1 ∂ ˆ F f (p) = f (p − iD)F, Gµ f (p) = dλλ f (p − iλD)Gρµ , ∂pρ 0 Z 1 ∂2 dλλ(1 − λ) Tµν f (p) = gµν − 2 f (p − iλD)rµρσν + O(r 2 ), (B.10) ∂pρ ∂pσ 0

and, by partial integration f (p)Fˆ = f (p + iD)F, Tµν f (p) = gµν − 2

Z

f (p)Gµ = −

1 0

dλλ(1 − λ)

∂2

Z

∂pρ ∂pσ

1

dλλ

0

∂ f (p + iλD)Gρµ , ∂pρ

f (p + iλD)rµρσν + O(r 2 ),

(B.11)

where F is any local field operator. First neglecting GL and Mµν , the anomaly is mass-independent and arises from terms of order p−6 in (B.1). Writing Z Z 1 d4 p d4 p n n Tn = Tr {[R(γ )] δR(γ ) − [R(−γ )] δR(−γ )} = tn , (B.12) 5 5 5 5 2 (2π)4 (2π)4 we have T0 = 0, and we only need to evaluate t1 ∼ p−3 and t2 ∼ p−5 . Since T2 is finite we can evaluate it directly using (B.10) and (B.11). This is a local operator which is bilinear in the field strength Gµν . Since it is independent of momenta we can set one external momentum to zero. ′ we first set D G ′ ′ That is, in the product Gρσ Gµν λ ρσ = 0 and then Dλ Gµν = 0, and take the average. This trick considerably simplifies the calculation and, using standard Feynman parameterization techniques and the Bianchi identities, and dropping total derivatives: Tr(GGDφ) → −Tr[φD(GG)],

(B.13)

we find that the various contributions cancel: T2 = 0. 46

(B.14)

Since T1 has logarithmically divergent contributions, we must explicitly cancel them against one another before using integration by parts or shifts in the integration variable. To this end we first rewrite Tr(GDφ) → −Tr(φDG) (B.15) and

ˆ µν ] = i[G ˆν , ∗ G ˆ µν ] = i ǫµνρσ [G ˆν , G ˆ ρσ ], [D ν , ∗ G 2

(B.16)

where ˆ µ f (p) = G

Z

1 0

∂ f (p − iλD)Gρµ , dλ ∂pρ

ˆµ = − f (p)G

Z

1

dλ

0

∂ f (p + iλD)Gρµ . ∂pρ

(B.17)

ˆ one finds the contribution Using this result in the expression for T1 that contains a factor σ · G i i 1 e . − δT− = + T1 = G · Gφ (B.18) 2 2 16π 2

In the regulated theory, we have to subtract a contribution with −p2 → −(p2 − m2 ) in the denominators in (B.5), and drop the terms that vanish in the limit m2 → ∞. Using e′ gρ − G eµν G′ρν , e′ = 1 G · G Gρν G µ µν 2

eµν = 0, Dµ G

(B.19)

and assuming [m2 , G] = 0 we get a contribution # Z 1 " eµν Gρν pµ pρ pρ pµ G gµρ 2 ρν e dλ t1 (m ) − t1 (0) = −8Tr G Gµν + 2 − φ (p2 − m2 )3 (p − m2 )2 p2 − m2 2 0 Z 1 1 p2 e − G · Gφ dλ = −4Tr (p2 − m2 )3 (p2 − m2 )2 0 Z 1 m2 e dλ G · Gφ , = −4Tr (p2 − m2 )3 0 Z Z Z i d4 p e , t1 (m2 ) − t1 (0) = + 2 Tr G · Gφ ≡ (B.20) 8π (2π)4 p p and we recover (B.18)

i i − δT− = + 2 2

Z

p

t1 (0) − t1 (m2 ) =

Z 1 i e t1 (0), G · Gφ = 16π 2 2 p

(B.21)

which is the standard result for the contribution from renormalizable gauge couplings with G± µν → 2 ∓iT · Fµν . However there are additional contributions from the PV sector when [m , G] 6= 0, as we will see in Section B.2. 47

The space-time curvature term in T1 has two contributions. Those arising from Zµ Z µ and (T µν − gµν ){pµ , Zν } are UV finite and may be straightforwardly evaluated as described above. After dropping a total derivative as in (B.15), the remaining contribution reduces to 1 6p µ ν Tr {p , [D , Z ]} γ γ φ . (B.22) µ ν 5 p2 p2 To evaluate this we adopt the convention that all p-derivatives act to the left so that ∂ , pβ = −g αβ , ∂pα

(B.23)

and we define Cµν = [Dµ , Dν ] = [∇µ , ∇ν ] + Zµν .

(B.24)

{pµ , Zµ } = 2pµ Zµ + [Zµ , pµ ] ,

(B.25)

Then writing

the first term drops out of (B.22) because Z pµ 2pµ pρ − p2 δµρ X 1 ∂ m λ Zρµ = 0 Zµ = −iλD · dλ p2 p4 m! ∂p 0

(B.26)

m=0

by symmetry, and we obtain [Zµ , pµ ] = i

Z

1

dλλD µ Zµ (λ) +

0

Z

1

dλλ 0

Z

λ

dηη [C µ (η), Zµ (λ)] ,

(B.27)

0

where if G = Z, C ∂ f (p − iλD)Gρµ , ∂pρ ∂ f (p − iλD)Gρµ . f (p)Gµ (λ) = − ∂pρ Gµ (λ)f (p) =

(B.28)

Finally, using (B.17) we obtain µ

[Dν , [Zµ , p ]] = − +

Z

1

dλλ

2

0

Z

0

1

dλλ

Z

Z

λ

µ

dη [Cν (η), D Zµ (λ)] + i 0 λ

Z

1

dλλDν D µ Zµ (λ)

0

dηη ([C µ (η), Dν Zµ (λ)] + [Dν C µ (η), Zµ (λ)]) .

0

48

(B.29)

Evaluating this contribution by the above procedure and combining all three contributions gives Z i 1 i δT = − t1 = r · r˜Trφ, − δT = r · r˜Trφ 2 192π 2 384π 2 p 1 µνρσ r · r˜ = ǫ rρστ λ r τ λµν . (B.30) 2 It is straightforward to check that there is no contribution to this term when p2 → p2 − m2 in the denominators of R, δR in (B.5). In the following section we will see that the remaining terms of the “consistent anomaly” [26] will emerge when we include all the contributions from the PV sector. B.2

Full PV contribution

In this section we consider PV fields with no spin-dependent mass terms. Then the only relevant mass is the PV mass m, and MP V is given by (A.7) of [17] with L = 0, M = 0, m = mP V . To evaluate (B.1) we follow (A.13) of [17], but take M0 = MP V (−m, −~σ ). Then we have h i 1 2 2 µν 2 ˆ d ˆ p − m − T ∆ ∆ + m + h + X − i D 6 m , RP V = µ ν −p2 + m2 ∂ 1 µ (p N + N ) + O δRP V = , (B.31) µ −p2 + m2 ∂p with δM → δm in Nµ , defined in (B.7), and 0 R[6D + m − m D 6 − ]L 0 RD 6 mL D 6 m = ≡ , L[6D − m ¯ −m ¯ D 6 + ]R 0 LD 6 mR ¯ 0 RmδmR ¯ −iRmδ D 6 −L RmmR ¯ 0 2 N = − , m = . −iLmδ ¯ D 6 +R LmδmL ¯ 0 LmmL ¯

(B.32) (B.33)

First consider chiral fermion contributions to the space-time curvature term. The only contribution from the PV sector that is not suppressed by m−2 comes from the replacement 6 piδ D 6 φ → mδm in µ the term involving Zµ Z , giving Z r · r˜ 1 −1 −1 PV Trη m ¯ δ m ¯ − m δm , δT = − tP1 V = 192π 2 4 p i r · r˜ i − δT P V = − Trη m ¯ −1 δm ¯ − m−1 δm , (B.34) 2 2 384π 4 and the total contribution from light and heavy modes is i i 1 + r · r˜ 1 −1 −1 PV − − δT + δT Tr ¯ δm ¯ − m δm , =− φ −φ + η m (B.35) 2 16π 2 24 2 4 49

where for a chiral transformation defined by (B.2) φ+ = −φ− = −φ. If the PV masses were modular covariant they would satisfy δm ¯ = i(φ− m ¯ − mφ ¯ + ) → i{m, ¯ φ},

δm = i(φ+ m − mφ− ) → −i{m, φ},

(B.36)

and the regulated theory would be anomaly free, provided (A.9) is satisfied. If we define dm ¯ = δm ¯ − i{m, ¯ φ},

dm = δm + i{m, φ}, ,

(B.37)

the total contribution from light and heavy modes reads, setting η = −1, −

i i r · r˜ 1 Trη m δT + δT P V = − ¯ −1 dm ¯ − m−1 dm . 2 2 16π 24 4

(B.38)

In other words, we may write −

i i PV δT + δT P V = − δT + δTm=0 − 2 2 i PV = − δT + δTm=0 − 2

i PV δT P V − δTm=0 2 i r · r˜ 1 −1 −1 Trη m ¯ . d m ¯ − m dm 16π 2 24 4

(B.39)

The first term on the right hand side of (B.39) vanishes by virtue of (A.9), which is consistent with the absence of linear divergences associated with the spin connection. To evaluate the remaining contributions we have to take into account that m is a priori matrixvalued and field dependent. Then (B.20) is replaced by 1 m2 1 m2 µν e µν e G + G G G φ, τ1 = −2Trη µν µν 2 (p2 − m2 )2 p − m2 p 2 − m2 (p2 − m2 )2 Z Z ∞ i 2 τ1 (p ) = − dp2 p2 τ1 (−p2 ) 16π 2 0 p Z ∞ i m2 m2 1 m2 2 e e = dp Trη − 2 G · G, 2 φ G · G, 2 8π 2 (p + m2 )2 p + m2 p + m2 p 2 + m2 0 Z ∞ i m2 m2 2 ∂ e e = dp Trη 2G · G + G·G 2 φ 8π 2 ∂p2 p2 + m2 p + m2 0 i e , = Trη G · Gφ (B.40) 8π 2 where here and below we use a shorthand notation with φ = −φ+ = φ− , and, for example, i 1 h e e+ + G− · G e− φ . (B.41) Trη G · Gφ ≡ Trη G+ · G 2 50

That is, we average over helicities with the convention that Oφ ≡ −

1 1 O+ φ+ − O− φ− = O+ + O− φ. 2 2

If δm 6= 0, there is another contribution with 6 piδ D 6 φ → mδm: 2 σ µν 1 1 Trη G mδmγ5 σ2′ = µν 2 2 2 4 p −m p − m2 1 1 1 eµν = 2iTrη 2 G Gµν 2 mdm 2 2 2 p −m p −m p − m2 1 1 1 µν e −2Trη m, 2 Gµν 2 G mφ. p − m2 p − m2 p 2 − m2

(B.42)

(B.43)

These are the only contributions if [Dµ , m] = 0; in this case [G, m] = 0, and if also dm = 0 this reduces to Z Z ′ PV T1 − T2 → 0, (B.44) σ2 → τ1 = T1P V , T2 → p

p

as expected. The remaining contributions involve covariant derivatives on the mass matrix. Writing Z Z 1 n n PV (B.45) Trη {[R(γ5 )] δR(γ5 ) − [R(−γ5 )] δR(−γ5 )}P V = (τn + σn ) , Tn = − 2 p p where τn and σn are the contributions without and with, respectively, a factor δm. They take the general form τn = Trη (Onτ D 6 φ) → −Trη ([6D , Onτ ] φ) ,

σn = Trη (Onσ δm) = Trη (Onσ dm) + iTrη ({m, Onσ }φ) .

(B.46)

Using relations such as i D 6 2 m = D 2 m − [σ · G, m], 2 2 [6D , m ] = {m, D 6 m}, [6D , P ] = {m, P [6D , m]P }, ∂ ∂ ∂ P · Dm2 P = · [D, P ] + 2P [p · D, P ] = [D, P ] · − 2[p · D, P ]P, ∂p ∂p ∂p 1 , {m, P [G, m]P } = [G, P ], P = 2 p − m2 it is possible to show that

X n

τn +

X n

σn =

X n

51

Trη (Onσ dm) .

(B.47)

(B.48)

As a check of our anomaly calculation, consider the case with constant PV masses such that [D, m2 ] = [G, m2 ] = δm = 0. Then σi = 0 and h i2 1 τ2 = −Trη P iD 6d m P 6 p[6D φ]γ5 + Trη {P [6D , m], P σ · G} P m[6D , φ]γ5 2 o n 3 2 µν eµν , m], Gµν mφ e = −2TrηP m P [G , m][Gµν , m]φ − 2TrηP 3 [G o n eµν [D ν , m]φ eµν , [D µ , m]][D ν , m]φ + 4TrηP 2 [D µ , m], G −2Trηp2 P 4 [G +2TrηP 3 [D µ , m][Gµν , [Dν , m]] p2 P − 2 φ, τ3 = Trη [P i D 6 m]3 P m[6D , φ]γ5

eµν , m], Dµ mDν m}mφ + 4TrηP 4 Dµ m[G eµν , m]Dν mmφ = −4TrηP 4 {[G +4ǫµνρσ TrηP 4 Dµ mDν mDρ Dσ mφ.

(B.49)

The covariant derivative on chiral fermions is given in (A.6) with ± Jµ± = Γ± µ ∓ iΓµ ∓ iT · Aµ ,

T + = (T − )T = (T − )∗ ,

(B.50)

where Aµ is a gauge field and Γµ =

i ¯ Ki Dµ z i − Km ¯m , ¯ Dµ z 4

Γ− µ

p q

= Γpqk Dµ z k =

h

Γ+ µ

p¯i† q¯

.

(B.51)

are the K¨ ahler U (1) and reparameterization connections, respectively. It follows from gauge invariance of the K¨ ahler potential that + m ¯ n ¯j Kim = (∂µ + Zµ )δij − (Jµ− )ji . ¯ (Dµ )n ¯ K

(B.52)

For chiral fermions the field strength Gµν can be expressed in terms of the general two-forms Tµν defined20 in (E.2); explicitly: p 1 a Xµν δqp + iFµν (Ta )pq − Γpqµν , G− µν q = 2 n¯ q¯ n − p G+ = −K K G , Xµν = Kµν . p m ¯ µν m µν ¯ q

(B.53)

For constant masses we have

[Dµ , m] = [Jµ , m] = [aµ , m],

Jµ = jµ + aµ =

20

Jµ+ 0

0 Jµ−

,

(B.54)

The two-forms Tµν used here and below should not confused with the tensor T µν introduced in the derivative expansion in (B.5) and defined in (F.27).

52

where aµ is the connection associated with the anomalous symmetry: [jµ , m] = {aµ , m} = 0,

Gµν = gµν + aµν ,

[gµν , m] = {aµν , m} = 0.

(B.55)

For the helicity components of these matrices, this gives + ma− µν = −aµν m,

a− ¯ = −ma ¯ + µν m µν ,

− + mgµν = gµν m,

− + gµν m ¯ = mg ¯ µν ,

(B.56)

and we obtain in this case i Trη (˜ aµν aµν φ) 12π 2 i + 2 Trη [({˜ gµν , aµ aν } + aµ g˜µν aν + {˜ aµν , aµ aν } − aµ a ˜µν aν ) φ] 3π i T3 = Trη [({˜ aµν , aµ aν } − aµ a ˜µν aν ) φ] 3π 2 2i (B.57) + 2 ǫµνρσ Trη (aµ aν aρ aσ φ) . 3π P Terms linear and cubic in a drop out of δT = 3n=1 (−1)n Tn , and, provided there is a PV particle with negative signature, η = −1, for every light particle, we recover the standard [26] result: 1 1 r · r˜ i Trφ g · g˜ + a · a ˜+ − δT = 2 2 16π 3 24 8 16 − [{˜ gµν , aµ aν } + aµ g˜µν aν ] + ǫµνρσ aµ aν aρ aσ . (B.58) 3 3 T2 =

B.3

Gauge, gravity and dilaton sector: nonrenormalizable operators

In the preceeding subsections the cancellation of linear divergences was implicitly assumed, that is, we assumed PV e light = −Tr(ηφG · G) e PV , δT = −δTm=0 , Tr(φG · G) (B.59) However contributions from nonrenormalizable interactions to the gaugino and gravitino connections, including an additional curvature term in the gravitino connection and [17] an off-diagonal gravitino-gaugino connection, do not have counterparts in the PV sector. These have to be treated separately. The contribution from the gravitino field strength: Tr (g · g˜)ψ ∋ −r˜ r

(B.60)

leads to the contribution (4.26) to the chiral anomaly. There is a an additional contribution from the off-diagonal gaugino-gravitino connection; the corresponding field strength gµν has terms linear 53

and quadratic in the Yang-Mills field Fµν . The terms quartic in Fµν cancel between gaugino and gravitino loops: λ x2 σ bβµ αρ a ν g · g˜ ∋ ± ǫµνρσ Fa Fβ Fbα F . (B.61) for 2 ψ The remaining contribution takes the form h √ √ ρν √ ρν µ √ a i a Tr (g · g˜)λ+ψ ∋ 4 D µ − Dρ xFa Dρ xF˜µν xFa D xF˜µν a a ˜ µν c +2xFãµν Fρσ rµν ρσ − xrFµν Fa − 4xcabc F˜µν F aρµ Fρb = 4D µ xF ρν Dρ F˜µν .

ν

(B.62)

The second expression on the right hand side of (B.62) is obtained from the first by using the identities (B.19) and [17] 1 a a a a a F˜µν [D µ , Dρ ]Faρν = −Fµν [D µ , Dρ ]Fãρν = cabc F˜µν F bµρ F cνρ + rνµ F˜µρ Faνρ − rµν ρσ Fãµν Fρσ , 2

(B.63)

The contribution from (B.62) to the chiral anomaly is given in (4.34). As described in Section 4.1, these anomalies arise from uncanceled linear divergences that have counterparts in uncanceled logarithmic divergences which are total divergences, and so do not affect the finiteness of the Smatrix, but the resulting anomalies form superfields provided the the cut-off has the form (1.1). The field strength GL in (B.6) arises from a term in the gaugino connection, iγ5 Lµ = −iγ5 ∂µ y/2x, that, as was shown in [1], must be defined as a “vector” (rather than an “axial vector”) connection, through the use of the identity γ5 = (i/24)ǫµνρσ γµ γν γρ γσ , (B.64) in order to allow Pauli-Villars regularization of the quadratic divergences. This choice further insures that the nonrenormalization [28] of the topological charge, θ = 8π 2 y, is consistent with linear-chiral multiplet duality [38] for the dilaton supermultiplet, and preserves [12] the modified linearity condition in the linear multiplet formulation. Since BRST invariance requires (and supersymmetry allows) the regulation of nonabelian multiplet gauge loops by PV chiral multiplets, it is clear that only the chiral multiplet axial connections can appear in the anomalies associated with the regulated pure Yang-Mills sector. As a result, contributions from δLµ are absent21 from δR. The operator T− is defined in such a way that the contribution to the anomaly involving only GL , 21

In Feynman diagram language the gaugino-loop contribution to the Lµ anomaly arising from a shift in the axion y is canceled by a gauge vector loop contribution [39].

54

and G is proportional to o 2 i 2 1 nh γ5 φ Tr [G+ + iγ 5 GL ] · σ + [G− + iγ 5 GL ] · σ 2 = Trφ (g · σ)2 + (a · σ)2 − (GL · σ)2 γ5 + i{(g · σ), (GL · σ)} .

(B.65)

From the expressions for the field strengths given in Eq. (C.18) of [17], it is easy to see that the traces involving GL vanish identically in the above expression. Those involving only G have already been taken into account. In particular, the dilatino χS , gauginos λ and gravitino ψ transform only under modular transformations: i

χS → e 2 ImF χS ,

i

λ → e− 2 ImF λ,

i

ψ → e− 2 ImF ψ.

(B.66)

To fix the gravitino gauge, we follow [16, 17] and introduce an auxiliary field α that transforms like i a chiral fermion: α → e 2 ImF α. With this procedure, the ghostino makes no contribution to the chiral anomaly, but we must include the contribution of the auxiliary field. The fields in this sector are U (1)X -neutral, and λ transforms under modular transformations with the opposite phase from the phase of χS , α. Therefore the masses (B.8) are modular invariant, and terms quadratic or quartic in M ·σ and linear in φ cannot contribute to (B.1); the only nonvanishing contributions involve the additional, modular invariant, connections iγ5 Lµ and (Γ± )SS . The λ-α,λχS matrix elements (B.8) satisfy [17] M0 = M0T ,

µν 0 ¯ µν ¯0 , =M Mλα = −M Mλα λα λα µν µν µν 0 ˜ ¯0 . ¯ ¯ Mλχ = −M = iMλχ , M λχ = −iMλχ , λχ

T Mµν = −Mµν ,

µν ¯ λχ 0 = Mλχ Mµν ,

fµν M λχ

(B.67)

The contributions to the anomaly that may arise from these masses can be determined by identifying local operators that could be obtained from the terms of order p−6 in the expansion (B.1). The masses in (B.67) are gauge invariant and modular covariant: ¯ =i M ¯ φ + φM ¯ =0 δM

(B.68)

because φχ = φα = −φλ = 21 ImF . Then since only even powers of M can occur, the a priori possibilities are T2,3,4 with Z d4 p 1 Tr (ti + h.c.) , Ti = 2 (2π)4 1 c1 c 1 c c 1 (6 p+ 6G ) M ¯1D ¯1D 6 φγ5 ≡ t∞ + t′2 , t∞ = − 2 6 p M 6 pM 6 φγ5 , t2 = − 2 (6 p+ 6G ) M 2 2 p p 6p p p 6p 55

t3 t4

1 1 1 c1 c iˆ µ ¯ = − 2 {Gµ , p } + G · σ D 6 φγ5 , M M + permutations p 2 6p 6p 6p 2 1 1 c1 c ¯ D 6 φγ5 . M M = − 6p 6p 6p

(B.69)

µν = cα(χ) Faµν , no Lorentz and gauge invariant operator can be constructed from Because Maα(χ) 2 G φ or from three factors of M Mµν ρσ µν in the trace. The only invariant involving the space-time Riemann tensor has two factors of M µν and vanishes due to (B.67). Then from power counting in pµ ∼ Dµ , we can get the following local operators:

¯ MM ¯ + h.c. φγ5 = 0, T4 ∝ Tr M M

+ − ¯ µν ¯ 0 + Gµν T3 ∼ t′2 ∼ Tr Gµν M µν M M M0 + permutations + h.c. φ.

(B.70)

Terms even Mµν cancel between λ and α, χS . For the odd terms, from (B.67) we have µν ¯ αλ ¯ µν M αλ = 0, Mλα M0 + M λα 0

and ¯ 0 + M0 M ¯ µν M µν M

λλ

¯ 0 + M0 M ¯ µν + h.c. = M µν M

(B.71)

χχ

+ h.c. = 0.

(B.72)

It follows that T4 vanishes, as does any term with Gµν real. A part that is not real is Xµν which L L ± cancels between λ and α, χS . There are also terms with G± λ → iγ5 G , Gχ → ∓iG , where GL µν =

1 (∂µ x∂ν y − ∂ν x∂µ y) . 2x

(B.73)

Since for these contributions + G+ χ R = −Gλ R,

− G− χ L = −Gλ L,

(B.74)

if we keep explicit the helicity projection operators in the expressions (B.5), it becomes clear that these also cancel. The divergent piece t∞ in (B.69) is proportional to a total derivative, as we will explicitly display below.22 Next consider the PV sector. The λ-θ PV mass that is generated by the supersymmetric Higgs mechanism, satisfies δm = 0, 22

1 φλ = −φχ = − ImF, 2

See (B.79) for m2 = 0; (B.81) vanishes identically in this case.

56

dm = −i{φ, m} = 0,

(B.75)

and there are no linear divergences or residual anomalies generated by λ,θ loops. The divergent contribution t∞ is regulated by the coupling of λ0 to ϕˆ defined by (3.33), giving rise to masses Mλ0 ϕâ− of the form (B.8) which have the same properties (B.67) as Mλχ . Nµ is still defined as in (B.31) since δM = 0, but RP V contains an additional term c ˆ + (pν + Gν ) P µν γµ } M {m

(B.76)

inside the square brackets. Under a chiral modular transformation the phases satisfy (B.75) provided the PV masses m are constant. Since M mM = δm = 0, for constant masses the only potentially nonvanishing contribution analogous to (B.69) is 1 1 6 p 6 p c c c 6p + m c ¯m ¯ t2 = − 2 M ¯M M D 6 φγ5 . (B.77) 6 pM 2 2 2 2 2 2 p −m p −m p −m p − m2

Terms in (B.77) with no factor of σ · M vanish identically; evaluation of the terms with one of σ · M gives Z d4 p 1 p2 1 ¯0 + M fµν ¯0 fµν M Dµ M T2 = 4 iTrη Dµ M (2π)4 (p2 − m2 ) p 2 − m2 p 2 − m2 µν µν 1 1 f f ¯ ¯ −Dµ M0 2 Dµ M M − M0 2 + h.c. Dν ImF, (B.78) p − m2 p − m2 T2

since

µν 1 d4 p p2 1 µν f ¯ ¯ f Dµ iTrη M M0 − M0 2 M + h.c. Dν ImF ∋ 4 (2π)4 p2 − m2 p 2 − m2 p − m2 = total derivative, (B.79) Z

¯ 0 = G+ , M µν M ¯ 0 = 0. 2Dν Dµ M µν M νµ

(B.80)

Note that the coefficient of the divergent integral in (B.78) is just the variation of the first line P 2 of the expression for TrR2 R5 in (B.29) of [17]. The condition on ηe in (3.35) assures that the divergence is canceled. Finally, taking into account the symmetry properties (B.67), terms containing two factors of σ · M vanish in the limit of equal PV masses: m2λ0 θ0 = m2χχˆ , and we get a contribution ImF i ρν µ ˜µν , f (1) = 0. (B.81) r = m2λ0 θ0 /m2χχˆ , xF D F f (r)D − T2 = ρ 2 16π 2 57

The function f (r) takes all possible values −∞ ≤ f (r) ≤ ∞ over the range 0 ≤ r ≤ ∞; therefore the PV mass ratio can be chosen so that (B.81) exactly cancels the contribution (B.62) from the offdiagonal gaugino-gravitino connection. Since the PV regularization procedure respects supersymmetry, the conformal anomaly necessarily includes the supersymmetric completion, proportional to (4.35), of (B.81), and the contribution (4.36) can thus be cancelled. Therefore in the following section we include only PV fields that have noninvariant PV masses through superpotential couplings to one another.

C

The full anomaly

In this appendix we calculate the full contribution to the anomaly that is generated by superpotential couplings of PV chiral superfields resulting in noninvariant masses. The one-loop effective action from chiral multiplet loops is given by i ˆ 2 + HΦ (m) − i Trη ln [−i D S1 = Trη ln D 6 Θ + MΘ (m)] , Φ 2 2

(C.1)

where the subscripts Φ and Θ denote scalar and fermion loops, respectively. We require that under an infinitesimal transformation on superfields Z(θ) → Z ′ (θ ′ ) = g(θ)Z(θ), such that δΦ = φΦ Φ, δΘ = φΘ Θ, 4

X i i δS1 = S1 (m + dm) − S1 (m) = STrη[1 + (1 + R)−1 dR] = STrη (−R)n dR, 2 2

(C.2)

n=0

where ˜ −m dmΘ = e−φΘ m(z ′ , VX′ )eφΘ − m(z, VX ) = δm − [φΘ , m] = m − + 0 0 (δm − φΘ m + mφΘ )R = = + − 0 ¯ + mφ ¯ Θ )L (δm ¯ − φΘ m dmR ¯ dmΦ = e−φΦ m(z ′ , VX′ )eφΦ − m(z, VX ) = δm − [φΦ , m] − 0 (δm − φ+ Φ m + mφΦ ) . = 0 ¯ + mφ ¯ + (δm ¯ − φ− Φ) Φm

dmL 0

(C.3)

(C.4)

Here we define, using the notation introduced in (B.5) and (B.31), with now δ D 6 → 0, δm → dm, STrF dR = Tr(F dR)Φ − Tr(F dR)Θ , i h 1 µν 2 2 ˆ RΦ,Θ = T ∆ ∆ − H − X − p + m , µ ν p2 − m2 Φ,Θ 58

(C.5) (C.6)

1 [− 6 p + MΘ (m)] dm, p2 − m2 i h 1 ′ ′ φΦ −φΦ = − 2 − H (z, V ) . H (z , V )e e Φ X Φ X p − m2

dRΘ = −

(C.7)

dRΦ

(C.8)

We argued in Section 4.1 that fields ΦP with couplings in W2 , or with K¨ ahler curvature terms ˆ RP mQ¯ ¯ n 6= 0, must have modular and U (1)X invariant masses. Then MΘ (m) = m, and H, Gµ are given by the derivative expansions of Appendix A of [17] in terms of the following operators:23 ¯ ¯ ¯ h + ∆H h′ hR 0 i σ · GΘ − i[6D , m], (C.9) HΦ = , H = − Θ ′ ¯ 2 h h + ∆H 0 hL 1 Q Q Q ˆ −K ¯k m ¯ k µ m ¯ 2 + Da DP (T a z)Q , (C.10) e A A + Dµ z D z¯ (∆H)P = δP V + M + RP mk ¯ x K/2 (C.11) h = mm, ¯ mP Q′ = δP Q e µP , 1 Q P h′P Q = e−K A¯k Dk − A¯ eK/2 mP Q − (qX + qX )FX mP Q , FX = − D 2 VX ,(C.12) 4 i h Q′ Q q Dµ mP P ′ = Dµ z mP P ′ (Kq + ∂q ln µP ) − mQP ′ ΓP q − mP Q′ ΓP ′ q A P′ (C.13) q + q +iAX X X mP P ′ , µ 0 D 6 mL Θ [6D , m] = , Gµν = GΦ (C.14) µν + Zµν − γ5 Γµν , D 6 mR ¯ 0 where indices are raised and lowered with the K¨ ahler metric, the operator A and its covariant derivatives are defined in Section 2 [see (2.3)], Zµν is defined in (B.6), and 1 1 ¯ ¯ (Dν z¯m Dµ z i − Dµ z¯m Dν z i )Kim Γµν = ¯ = Xµν , 2 2 ¯ ¯ Φ− P a P P PM Φ+ N Gµν Q = iFµν (Ta )Q − ΓQµν = −K Gµν M¯ KN¯ Q ,

(C.15) a 6= X,

(C.16)

are the field strengths associated with the K¨ ahler and reparameterization + gauge connections, respectively. The expression for dRΘ is obtained using the methods of Appendix A of [17], with R5 → dR, M0 → M4 (−M, −~σ ). Since dR appears only on the far right in (C.2), we can drop all momentum derivatives in the resulting operator. 23

In the interest of simplification we have evaluated the component Lagrangian in WZ gauge for VX : VX | = FX = 0, since the regulated theory, including counterterms, must be U (1)X gauge invariant. However we have left explicit the ¯ This term FX term in (C.12) because it gives a contribution to dmΦ through FX → FΛ = − 14 D2 Λ if VX → Λ + Λ. P P qX VX ¯ P |2 . arises from mixed F FX terms in the Lagrangian when K¨ ahler potential terms take the form e f (Z, Z)|Φ

59

± ahler transformation that is induced by a For a gauge transformation φ± Φ = φΘ . Now consider a K¨ chiral field redefinition:

Z ′p (θ) = f p [Z(θ ′ )],

W ′ = e−F W,

K ′ = K + F + F¯ ,

i

θ ′ = e 2 ImF θ.

(C.17)

For an infinitesimal transformation dZ ′p =

∂f p q dZ ≈ dZ p + (φ− )pq dZ q . ∂Z q

(C.18)

Then for example if K(Z p ) is the light field K¨ ahler potential, the corresponding light loop diverP gences can be canceled by PV fields Z with K¨ ahler potential ¯

KP V = Z P Z¯ Q Kp¯q ,

(C.19)

δZ P = (φ− )pq Z Q

(C.20)

which is modular invariant provided under C.17. This gives δz P = (φ− )pq z Q ,

i δχP = (φ− )pq χQ + ImF χP + (∂k φ− )pq z Q χk . 2

(C.21)

Since the last term does not contribute to the effective bosonic Lagrangian that we are evaluating, from now on we will set + i ¯ . (C.22) ¯ = φ+ φΦ , mm φΘ = φΦ − γ5 ImF = φ, Θ , mm 2

Then, since mm ¯ = mm ¯ ≡ m2 for the chiral PV fields that contribute to the anomaly, we obtain ¯ ¯ dhΦ dh′ dhΘ R − 6 pdmL 1 1 dRΦ = − 2 , (C.23) dRΘ = − 2 ¯ ′ dhΦ , p − m2 dh p − m2 − 6 pdmR ¯ dhΘ L dhΘ = mdm, ¯ dhΦ = dmm ¯ + mdm, ¯ (C.24) o iP¯ hn ¯ ¯ Q Q P P + qX )FΛ mPQ dh′QP = e−K A¯k Dk − A¯ eK/2 − (qX + qX )FX dm − (qX Q i h 1 2 Q P¯ + ′P¯ k + P FΛ = − D Λ , = ϕ˜ hQ − F Dk ϕ˜ − (qX + qX )FΛ mQ , (C.25) 4

where F k = e−K/2 A¯k = − 41 D 2 Z k at lowest order in the loop expansion. A priori the expansion (C.2) contains ultraviolet divergences which must vanish if the regulated theory is truly finite. The quadratic divergences are contained in STrdR = −Trη

p2

1 (dhΦ − 2dhΘ ) + h.c. = 0, − m2 60

(C.26)

where the trace is over internal indices only. The logarithmic divergences occur in 1 rˆ 1 1 − µν − d STrRdR = Trη 2 6 pdm + T GΘµ GΘν − dhΘ L −iD 6 m ¯ 2 p − m2 p − m2 4 p 2 − m2 1 1 1 ′ ˆ′ ¯ d − T µν G − G − +Trη 2 ∆H dh + h dh Φ Φµ Φν p2 − m2 p − m2 p 2 − m2 Φ 1 1 ˆ − m2 (dhΦ − dhΘ L) + h.c., (C.27) h +Trη 2 2 2 p −m p − m2

where the traces are over both Dirac and internal indices. Since PV fields with dm 6= 0 have no V = 0, there are no terms odd in m, and the logarithmically superpotential couplings and have KPP Q divergent part of this expression is proportional to 1 2 r ′ ′ ¯ ¯ − (dmm ¯ + mdm) ¯ ∆H − h dh − D mdm ¯ + h.c. (C.28) Trη mdm 2 2

where here the traces are over internal indices only. The expression (C.28) is just the variation of the logarithmically divergent m-dependent part of the one loop action,24 which, under the above assumptions, is proportional to [2] r 1 µ ′¯′ Trη {mm, ¯ ∆H} − mm ¯ − Dµ mD m ¯ +hh . (C.29) 2 2 The ultraviolet divergent terms that are independent of m have been constructed to cancel the light loop divergences. We also require that the logarithmically divergent terms proportional to m2 vanish, which means that for a given functional form of m2C (z, z¯, cX ), where cX = VX |,25 we have to introduce a set of PV fields with masses X 2 ¯, cX ), ηγC ρC (C.30) m2Cγ = ρC γ mC (z, z γ = 0. γ

This condition assures the vanishing of (C.29) and (C.28) as well as the finite terms proportional to m2 that arise from 2i TrR2 dR. For the masses that are not modular and U (1)X invariant, we need also to eliminate the residual finite terms proportional to m2 ln m2 , which requires the additional constraint X C ηγC ρC (C.31) γ ln(ργ ) = 0. γ

24

There is no quadratically divergent m-dependent contribution to the one loop action S1 . Here and throughout this appendix Φ| is the θ = θ¯ = 0 component of the superfield Φ with all fermion fields also set to zero. 25

61

for these masses. Both the PV masses and the functions H are controlled by the choice of K¨ ahler metric for the PV fields, which is constrained by the requirement of cancellation of quadratic divergences. Assuming that D 2n m2 commutes with other operators if dm 6= 0, the last line drops out in (C.27). Dropping terms that vanish as m2 → ∞, the finite part of (C.27) is ( i 1 1 − µν − STrηRdR|finite = −Trη G− ¯ − mdm) ¯ − G− ¯ µν X mdm Φ · GΦ (dmm 2 2 2 192π m 2 1 2 4 µν µν µν − D m ¯ − Gµν G m ¯ − ([Dµ , G ] + Dµ r ) Dν m ¯ dm 2 3 1 + rµνρσ r µνρσ − ir · r˜ + 2Xµν X µν − 4∇2 r mdm ¯ 8 ) ¯ ′ dh′ + h.c., (C.32) + D 2 ∆H (dmm ¯ + mdm) ¯ + D2 h

where we define

Dµ m ¯ = Dµ− m ¯ − mD ¯ µ+ ,

Gµν m ¯ = [Dµ , Dν ]m ¯ = G− ¯ − mG ¯ + µν m µν ,

etc.,

(C.33)

± 1 and G± µν = (GΦ )µν ∓ 2 Xµν is defined as in (B.6). If Lm is the part of the Lagrangian that contains the mass matrix m for PV chiral fermions f P = fLP : ¯

¯

QR (Lm )RP ′ , mQ P ′ = −K

(Lm )RP ′ =

∂2 ∂f R ∂f P ′

Lm ,

(C.34)

under a transformation Z p → Z ′p , φP → gφP , f P → eiα gf P , we have ¯

′

¯

¯

′

Q MR (L′m )RN ′ (g−1 )N m′PQ′ = −e−i(α+α ) gM ¯K P′ ,

(C.35) ′

since KP′ V = KP V by construction. For superpotential fermion mass terms Lm = −eK/2 µP Q′ χP χQ , PQ i α = 12 ImF , and if under (C.17) µP Q (Z ′ ) = eωi F µP Q (Z), we have L′m = −eK ¯

′ /2+iImF

¯

′

P M′ F i

′

P M ′ gR gQ′ µP M ′ χR χQ = −eK/2+F +ωi ¯

P M′ F i

′

Q NR ReF +ωi (L′m )RM ′ (g−1 )M m′PQ′ = −e−iImF gN ¯K P′ = e ¯

¯

¯

′

P M′ F i

iImF −1 Q ′N M m ˜Q (g )N¯ mM ′ gP ′ = eF +ωi P′ = e

¯

′

′

P M gR gQ′ µP M ′ χR χQ , ¯

¯

¯

Q NR M gR KM S¯ mSP ′ , gN ¯K ¯

′

M K QR gR KM S¯ mSN ′ gPN′ ,

(C.36)

For the chiral PV fields ΨPγ , Ψ′Q γ , Ψ = U, V, Φ, with couplings defined by (D.19)–(D.21) in Appendix D, we have ¯

µγP Q′ = δP Q µPγ (T ),

¯

mPQ′ = eK/2 K P Q µPγ (T ), 62

′

KP ′ Q¯ ′ = δP Q fγP ,

KP Q¯ = δP Q fγP , φ− Rγ

R (gγ )R Q = δQ e

φ− R′

′

R (gγ )R Q′ = δQ e

,

γ

.

(C.37)

Explicitly ln fγP

=

X

P

P

qn γ gn + qXγ cX ,

φ− Pγ = −

n

cX

=

VX | ,

λ = Λ| .

X i

P

P

qn γ F n − qXγ λ, (C.38)

where Λ is the U (1)X gauge transformation superfield introduced in (4.6) and (D.91), and qn and qX are modular weights and U (1)X charges, respectively. Then we obtain X

′

− ϕ− Pγ +ϕ ′

R R gP ′ = e gQ

Pγ

δP Q ,

P

m ˜ Pγ = e

P

i

− F i (1+ωi γ )+ϕ− Pγ +ϕ ′

Pγ

R

ϕ ˜+ Pγ

mγ ≡ e

mPγ ,

(C.39)

To evaluate (C.2) we need only consider fields with noninvariant masses: m ˜ 6= m. In addition to the PV superfields U, V, Φ, considered above, these include the gauge singlets φγ for which we take X γ i φ φ + µφγγ ′ 6= 0, (C.40) = 1 − 2α + ω = −α F, ϕ ˜ ln fγφ = αφγ K, ϕ− γ γ i F , φγ φγ i

and the adjoint chiral multiplet ϕãα with fαϕ˜ = 1,

ϕ˜+ γ =

ϕ− γ = 0,

X

(1 + ωiγ ) F i .

(C.41)

i

Let us first consider the coefficient of r · r˜ in (C.32). To evaluate this we use the conditions (3.36), (3.37), (D.97) and (D.98). These include contributions from chiral PV fields that have covariant masses: m ˜ = m, ϕ˜± = 0; we can also include them in the sums here, since their net contribution vanishes. However we must exclude the fields θ γ ; their masses arise from D-terms rather than F′ terms. Each pair ΦP , Φ′P gives gives an identical contribution, and we remove from Trη = N ′ the P contribution NG′ = γ ηγθ . On the other hand, the moduli-dependent prefactors in (C.38), with, referring to (D.19), UA

A qn γ = αA γ qn ,

etc.,

(C.42)

that have been chosen to cancel loop contributions from the PV fields Y˙ are not included; to include these we use (D.22)–(D.28) and the identifications (C.42), giving the result in (D.99).

63

˙ Y˙ , Ψ from N ′ where However we also have to exclude the net contribution of all the fields26 Z, NΨ = −NY˙ = −NZ˙ = N + 2. That is, we have to add a factor N + 2 to N ′ , giving 16π 2 i 1 1 √ δSr = − r · r˜Trη(ϕ˜− − ϕ˜+ ) = − r · r˜TrηφP V = r · r˜Trφ g 96 24 24 # ! " X X 1 = − r · r˜ N ′ − NG′ − 2α1 ImF + 2TrTX aX + ωn + 2 qnp ImF n 48 n p # ! " X X 1 (C.43) = N − NG − 3 − ω n − 2 qnp ImF n − 2TrTX aX , r · r˜ 48 n p where [see (3.37)], α1 =

X C

ηC αC = −10,

ωn =

X

ηC ωnC ,

aX = Imλ.

(C.44)

C

Apart from the “threshold corrections” ωn (T i ), this is precisely the result in (B.30) (with the P i identification φ = −ηφP V = ηφ+ ˜− ) for N chiral fermions with φp = n ( 12 − qnp )ImF n − P V = − 2 ηϕ P p qX aX , the auxiliary fermion needed for gravitino gauge fixing [16] with φα = 12 n ImF n , and P NG + 4 gaugino and gravitino degrees of freedom with φ = − 21 n ImF n . To evaluate the full anomaly, we can simplify further by setting µφαβ = µφα δαβ

(C.45)

U = q UA . in (4.61), and imposing (D.38) and (D.118) with qX X Then for the fields with noninvariant PV masses introduced in Appendix D.4 we have A

φˆ0 , φˆ± : ΨP :

ˆ

ln fφˆγ = α ˆ φγ K, α ˆ 0γ = α ˆ+ α ˆ− γ = 0, γ = 1, X P qnP g n + qX VX , qnP = 1 − qnp + ω˙ n , ln fΨP =

(C.46) N I qX = qX = 0, (C.47)

n

a

ϕ˜ :

f

ϕ ã

= 1,

(C.48)

A are subject to the constraints given in (D.40), (D.41) and (D.43). Here P = N, I, A, and the qX and we identify [see (D.19)]

ΨN = Φn ,

I N ΨI = δN Φ ,

ΨA = U A , UA , V A ,

26

n qm = 0,

qni = 2δni .

(C.49)

Equivalently, we remove NΨ which is already included in (D.99), and include the reparameterization + threshold ˙ Y˙ which exactly cancels their K¨ contributions from Z, ahler connection contributions NZ˙ + NY˙ .

64

The PV superfields ΨP , ϕã and φˆγ , γ = 1, . . . , 5, have η = +1 and φˆ± γ , γ = 1, . . . 2N −4 has η = −1. ˆ Using (C.45), and denoting the fields U, V, Φ, φ, ϕ˜ collectively by ΦC , the covariant derivative in (C.14) reduces to Dµ mCC ′

Dµ z i [Ki + ∂i ln(µC − 2 ln fC )] − 2qX (iAX µ + ∂µ cX ) mCC ′ 1 − † + VµC + iAC µ mCC ′ ≡ Vµ mCC ′ = (Vµ ) mCC ′ . 2

= ≡

(C.50)

¯

Since the K¨ ahler metric is covariantly constant: Dq K C C = 0, we also have ¯

Dµ mC C′

¯

= K CC Dµ mCC ′ ,

∂i ln µC = δitn ωnC ζ(tn ),

ζ(t) = ∂t η(t)/η(t).

(C.51)

The vectors Vµ , Aµ satisfy C C = i Dµ AC ν − Dν Aµ = 2fµν − Xµν ,

iAC µν

Dµ VνC − Dν VµC = 0,

i ¯ ¯ a C X ¯m − Dν z i Dµ z¯m − iFµν (Ta z)i Di ln fC − 2iqX Fµν , = ∂i ∂m ¯ ln fC Dµ z Dν z

C fµν

(C.52)

and from (C.14)–(C.16) and (C.33) we have G± Φ G± Θ

C

µν

C

µν

Gµν mC

C = ± fµν − iT C · Fµν ,

1 C ≡ (G± Φ )µν ∓ Xµν = ±i 2 = iAC µν mC ,

X Fµν 6= Fµν ,

1 C A − T C · Fµν 2 µν

¯ C, Gµν m ¯ C = −iAC µν m

(C.53) X Fµν 6= Fµν

,

Gµν mm ¯ =0

(C.54) (C.55)

We are interested only in the variation of the on shell action. Therefore we may drop terms proportional to 1 r r 1 α ∂Ltree 2 ˆ g− 2 gµν = − 2V + Dµ z¯m D µ z i Kim = − D X − 3 . (C.56) V + M ¯ α ∂gµν 2 2 2 Then, defining

1 ¯2 − 8R)Dα ln f, fα = PDα ln f = − (D 8 X ˆ φˆ P fαP = qnP gαn + qX WαX , fα γ = αφγ Kα , n

we have C ∆HD

=

C δD

1 α C 1 2 ˆ V + M + D fα + Da (T a )C D 2 x 65

a

fαϕ˜ = 0,

(C.57)

1 1 α C 1 α = (r − D Xα |) + D fα + Da (T a )C D, 6 2 x a 6= X, (TrTa )C = 0, h i ¯′ −1 ′ C −K/2 ¯k m h C = e A (Kk − 2∂k ln fC + ∂k ln µC ) − A¯ − 2q C FX C δD

¯′

(m−1 dh′ )C C

¯ ≡ y¯, ¯ = 1 D 2 ln M2 − 8R = −F i Di ln m2 + 2qX FX − M 4 X 1 − 2qnC + ωnC e−K/2 A¯k Fkn − 2q C FΛ = y¯ϕ˜+ + n

+

= y¯ϕ˜ + Fϕ˜+ C

where M2 = eK−2f |µ|2 ,

1 , Fϕ˜+ = − D 2 ϕ˜+ C C 4 M2 = m2 ,

(C.58)

(C.59)

(C.60)

(C.61)

is a real superfield, and the identifications on the right in (C.59) and (C.60) hold to the order we are working in because to that order the auxiliary fields can be interchanged with their tree-level values27 F k = −e−K/2 A¯k , M = eK/2 W = e−K/2 A = 2 R| . (C.62) Since the relevant PV masses have a diagonal K¨ ahler metric, it follows from (B.53) that G+ = −G− , so the first term in (C.32) drops out, and, from, (C.33), Gm ¯ = {G− , m}, ¯ etc. Then writing X i T = (−)k Tk , Tk = STrηRk dR , (C.63) 2 finite k

we obtain, imposing (C.30), " ( √ g µ − 2 + 1 µνρσ µν µν µ − Trη ϕ ˜ (r r − ir · r ˜ + 2X X + 4iA X ) − ∇ V + V Vµ T1 = + µνρσ µν µν µ − 192π 2 8 1 1 α µ − µ − α +✷ D fα | − D Xα | − ∇ Vµ − V− Vµ − r 3 6 1 2i −2Vν− ∇ν ∇µ + Vµ− V−µ − Aµν Aµν − Vν− Dµ Aµν 2) 3 2 2 + ∇µ r µν Vν− + r µν Vµ− Vν− 3 3 # (C.64) + y¯ϕ˜+ + Fφ˜+ ✷y + 2Vµ− ∂ µ y + y∇µ Vµ− + yVµ− V−µ + h.c. 27

Our normalization of the auxiliary field M differs by a factor −3 from that of Binétruy, Girardi and Grimm [5]: M = − 31 MBGG .

66

T4

g 1 1 1 1 − µ µ 2 2 µ ν + 1 − µ 2 = − (y y¯) − y y¯Vµ V + Trη ϕ˜ Vµ V− − V V + Vµ V− Vν V 32π 2 6 2 3 6 µ − 3 +h.c., (C.65) √

The expressions for T2 and T3 are more complicated; they combine to give " ( √ g 1 α 1 α 1 2 1 1 α 1 µ 1 α + D fα | − D Xα | + r − Vµ ∂ D fα | − D Xα | + r Trη ϕ˜ T2 − T3 = − 32π 2 2 6 6 3 2 6 6 1 1 α 1 1 − (∇µ V µ + 3y y¯ + r) D fα | − D α Xα | + r 3 2 6 6 2 r 4i 1 + V−ν D µ Aµν + ∇µ (Vµ r) + 12 48 9 1 − y✷¯ y + y¯✷y + ∂µ y∂ µ y¯ + y¯(Vµ− − Vµ )∂ µ y + y(Vµ+ − Vµ )∂ µ y¯ 6 −y y¯ ∇µ V µ + 3Vµ V µ + Vµ+ V−µ 1 1 a b 1 a b a b ˜ F · F − iF · F A · A + iA˜ · A Ta Tb + + 2D D − x 2 24 1 ✷ V−µ Vµ− + V µ ∇µ ∇ν Vν− + (∇µ Vµ ) ∇ν Vν− − ∇µ V−ν ∇µ Vν− 6 1 + ∇µ Vµ Vν− V−ν + Vµ− V−ν Vν− 6 1 − Vµ V µ V−ν Vν− + (Vµ− V µ )2 − (Vµ− V−µ )2 3 1 1 − − µν µ − ν − µ 10Vµ Vν r + ∇ 3rVµ − 2rµν V− + 3rVµ V− + ry y¯ − 36 6 1 2 µ + µ α α µ − y y¯ + Vµ ∂ y + Vµ V− y + y∇ Vµ + y D Xα | − 3y D fα | Fφ˜+ + y¯ϕ˜ 6 +h.c., (C.66) +

In writing (C.66) we used (C.52), the Bianchi identities: eµν = 2∇µ rµν − ∇ν r = ǫλµνρ rµνρσ , 0 = DµG

Vµ− Vν ∇µ V−ν = Vµ Vν− ∇µ V−ν + iAµν Vµ+ Vν− , (C.67)

the identities

σ σ 0 = [Dµ , Dν ]Vρ + rρνµ Vσ = [Dµ , Dν ]Vρ+ + rρνµ Vσ+ − iAµν Vρ+ m = ([Dµ , Dν ] − iAµν ) m, (C.68)

and the conjugate relations. Combining these contributions using, from (C.52) Aµν Aµν

= 2iDµ (Vν− Aµν ) − 2iVν− Dµ Aµν , 67

V−µ ✷Vµ− = gives i T 2

=

1 1 µ V {∇µ , ∇ν }Vν− − Vν− Vµ− r µν − iVν− Dµ Aµν . 2 − 2

(C.69)

" ( g 2 a b 1 a b a b + ˜·A ˜ 3A · A − i A F · F − i F · F − Trη ϕ ˜ D D T T + a b 64π 2 x2 12 √

r2 i 1 (rµνρσ r µνρσ − ir · r˜ + 2Xµν X µν ) + − Aµν X µν 24 72 6 1 µ 1 α 1 α ν − ν − + ∇ (∇µ − Vµ ) D Xα | − D fα | + r + ∇ Vν + V− Vν 3 3 6 2 1 1 1 D α fα | − D α Xα | + ∇µ y ∂µ − Vµ− y¯ − ∇ν Vµ− Vν− − 2 3 3 ) 2 µ 1 − ν − 1 µ − ν − ν − + ∇ Vµ ∇ Vν + Vµ V− Vν + ∇ rVµ − 2rµν V− 3 2 6 1 − Fϕ˜+ ✷y − Dµ yV µ + y D µ Vµ− + V−µ Vµ− − Dµ V µ − V µ Vµ− 3 # − µ 2 α α + h.c. +2Vµ D y − y y¯ + y (3D fα − D Xα ) −

(C.70)

This expression is the bosonic part of the superfield expression Z i h 1 4 e + (T n , ΛX )Ω1 + h.c. (C.71) d θTr η Φ δL1 = 8π 2 1 1 1 Ω1 = − Ωm + ΩW + Ω0YM − ΩX m , Ωm = ΩD − 4ΩG + 8ΩR , (C.72) 48 3 36 where the operators in (C.72) are defined in (4.23) and (4.39)–(4.41), Ω0Y M is the Chern-Simons e + = ϕ˜+ . The expression (C.71) is the resuperfield for the nonanomalous gauge group, and Φ sult of an infinitesimal transformation, and must be integrated to give the expression for a finite transformation; in particular the modular transformations are discrete and therefore finite. This is e + contains only 1) operators such as ΩW , ΩYM and ΩX m whose chiral possible if the coefficient of Φ projections are invariant under U (1)X and modular transformations, 2) linear multiplets that drop out of the superspace integral and 3) derivatives of ln M. That this is indeed the case is shown in Section E.1.

D

Orbifold compactification: PV sector for matter

We argued in Section 4.1 that PV mass terms must have well-defined modular weights, with invariant masses for those fields that contribute to the renormalization of the K¨ ahler potential. One way 68

to achieve this would be to couple in the PV mass superpotential all PV fields Z σ with metric KσZρ¯ to fields Yσ with a metric proportional to its inverse, as in (4.52). However each Z, Y pair gives no contribution linear in the scalar curvature, and doubles the contribution quadratic in scalar curvature. This requires an even number of pairs with signatures that sum to zero, and the introduction of other fields that reproduce the curvature terms from the light fields. This was done in [3] for the untwisted sector with K¨ ahler potential: # " ! NX NX n −1 n −1 X i ai 2 i g n ai 2 n i n n i i |Φ | ) , = δi g − ln(1 − e |Φ | Gu = G (Zn ), G = −δi ln T + T¯ − n

g

i

a=1

a=1

= − ln(T + T¯i ), i

(D.1)

where the special property Gnij = Gni Gnj was exploited to mimic their curvature terms by other fields. The twisted sector K¨ ahler potential is not known beyond leading order: ¯ G = Gu + f (Z, Z),

f = X a + O(Φ3 ),

a

X a = eg |Φa |2 ,

ga =

X

(D.2)

i

Under a nonlinear transformation Zni → Zn′i (Znj ) such that Gu (Z ′ ) = Gu (Z) + F + F¯ ,

qia g i .

Gn (Zn′ ) = Gn (Zn ) + F n + F¯ n ,

F =

X

F n,

(D.3)

n

X a is invariant provided a

Φa → Φ′a = e−F Φa ,

Fa =

X

qna F n .

(D.4)

n

To use the same trick including the twisted sector fields requires a constraint on the overall K¨ ahler potential analogous to the constraint on gauge charges discussed in Section 3.2. For example this is possible with a K¨ ahler potential of the form ¯ Z), ¯ G = g + f (X a ) + H(Z) + H(

g=

X n

gn (Zni ),

ga =

X

qna gn ,

(D.5)

n

where gn transforms like Gn under a modular transformation, for example gn = δin g i or gn = Gn . The modular invariant holomorphic function H is constructed from operators of the form 3 Y Y a η 2qni (iT i )], [Φa a

i=1

69

(D.6)

and the fields Φa are separated into groups Φm ∋ Φam , a = 1, . . . Nm with the function f (X a ) in (D.5) restricted to the form ! Nm X X a m m −1 (D.7) Xm . f= k , k = −λm ln 1 − λm m

a=1

This is option 1) of Section 4.2; its implementation requires the introduction of a number of additional PV fields. Here we will focus on option 2) which entails no restriction on the twisted sector K¨ ahler potential. In Section D.3 we will consider a hybrid case where option 1) is implemented for the untwisted sector in order to include the possibility of maximal Heisenberg invariance. In order to make the PV K¨ ahler potential and superpotential fully modular invariant we introduced σ ˙ ˙ a set of PV fields Z = Z N , Z˙ P , with negative signature, that regulate the UV divergent contributions to the renormalization of the light field K¨ ahler potential and to the operator φ3 in (2.9). Their K¨ ahler potential is given by (3.1) and (3.18) with ζm¯n = δm¯n : ˙ Z˙ ˙ ρ ¯ ˙ σ¯ + 1 K Z˙ σ Z˙ ρ + h.c. , K(Z˙ σ ) = K Z + K(Z˙ N ) = Kρ¯ Z Z σρ σ 2 σ = P, N, P = I, A, (D.8)

which is modular invariant provided Z˙ σ transforms as in (3.15) under (3.3). These fields couple in ˙ the PV superpotential W1Z given in (4.57) to superfields Y˙ σ = Y˙ N , Y˙ P , with the K¨ ahler potential given in (4.54). To calculate the contributions of these fields to LQ , Φ1 , Φ2 , we need the affine connection derived from the PV K¨ ahler metric. For Z˙ α we have Γ˙ PN r = K p¯q ∂r χ ¯nq¯¯ ,

n˙Q Γ˙ N N ′ r = −χq ΓN ′ r ,

Γ˙ PQr = Γpqr + χnq Γ˙ PN r ,

n n n′ ˙ Q Γ˙ N P r = +Dr χp − χq χp ΓN ′ r ,

(D.9)

¯ z )n¯ = Γ˙ P = 0, χn drops out of the traces of If χnp = a∂ ˙ p hn (Z), with hn (Z) holomorphic, ∂r h(¯ q¯ p Nr q n ˙ products of Γ and its derivatives. In addition, since (Ta )p χq = 0, it also drops out of the traces of Γ˙ with gauge generators. Alternatively, since χnp is proportional to the parameter a˙ by virtue of the condition (3.18), we can cancel the UV divergent terms involving χnp by including several copies of the Z˙ σ → Z˙ λσ , λ = 1, . . . , 2nZ˙ + 1 with signatures ηλZ and parameters a˙ Z λ such that X ˙ X ˙ X ˙ (D.10) ηλZ a˙ 4λ = 0. ηλZ a˙ 2λ = n˙ = ηλZ = −1, λ

λ

λ

We also need to impose the condition (A.4). This is again automatically satisfied if χn is holomorphic because it also drops out of the trace of products of Jµ± with φ because φρN = 0, where † (Jµ− )ρσ = Γ˙ ρσr Dµ z r + iTσρ · Aµ + iδσρ Γµ = (Jµ+ )ρσ , 70

Γµ =

i ¯ Ki Dµ z i − Km ¯m . ¯ Dµ z 4

(D.11)

For the more general case, (A.4) requires the conditions on a˙ 2 in (D.10) and the additional constraint X ˙ (D.12) ηλZ a˙ 6λ = 0. λ

˙ Only one set of the Z˙ λ , say Z˙ 0 , with negative signature, η0Z = −1, need have couplings in the superpotential (3.16) and in the additional terms Kσρ in the K¨ ahler potential (D.8). Gauge and modular invariant mass terms for all the above fields may be constructed as in (4.57) by introducing a superfield Y˙ σλ , with the K¨ ahler potential (4.54), for each Z˙ λσ . Rather than use the most general parameterization, we will simply set

χnp = ∂p χn ,

(D.13)

and consider three choices for the pair of functions gn in (4.54) and χn that we take to be the same in (D.8) and in (4.54). In our second and third scenarios we make some assumptions on the form of the K¨ ahler potential for the light fields. D.1

χn holomorphic

We set gn = δin gn (T i + T¯¯ı ) = −δin ln(T i + T¯¯ı ), i, n = 1, 2, 3, i ∂η(T ) n i n χn = 2aδ ˙ in ln η(T i ), χni = 2a˙ δ ≡ 2aζ(T ˙ )δi , ∂T i i

(D.14)

in (3.18) and (4.57). Then for Z˙ σ : Γ˙ PQr = Γ˙ pqr ,

n Γ˙ N Qr = Dr χq ,

Γ˙ PN r = Γ˙ N N r = 0,

(D.15)

and the UV divergent contributions from Z˙ simply cancel the UV divergences from the light fields Z. The affine connections for Y˙ I,N are n Γ˙ IIj = GN j − 2gj ,

n n Γ˙ N Ij = − (∂j − 2gi ) χi ,

Γ˙ IN j = 0,

N Γ˙ N N j = Gj ,

P and the UV divergent contributions from Y˙ I,N , with λ η˙ λ = −1, involve the operators X n η˙ λ (ΓnY˙ )σσα = −2 GN α − gα , X λ

λ

n η˙ λ (ΓnY˙ )σρα (ΓnY˙ )ρσα = − GN α − 2gα

71

n N N GN β − 2gβ − Gα Gβ .

(D.16)

(D.17) (D.18)

All the UV divergent contributions from Y˙ can be canceled by additional PV fields Ψ = U, V, Φ with K¨ ahler and superpotential " X X A A A A A A eαγ G |UγA |2 + eαγ G |UAγ |2 + eγγ G |VγA |2 K(Ψ) = γ

A

# X N N n n N 2 ǫγ G |Φnγ |2 , + eδγ (G −2g ) |ΦN γ | +e

(D.19)

N =n

" X X

W (Ψ) =

γ

A

X

1 A γ µU γ Uγ UA + 2

Ta (UγA ) = −TaT (UAγ ),

µVγ (VγA )2 +

A

X

N 2 µN γ (Φγ ) +

µnγ (Φnγ )2

n

N

Ta (V ) = −TaT (V ),

a 6= X;

X

Ta (Φ) = 0 ∀ a,

!#

, (D.20) (D.21)

if we require X

ηγΨ = 6 + dM = N + 2,

Ψ,γ

a CM

=

X γ

1 =

X γ

(TrTX )M (TrTb )M a dM a

= − = − =

X

X γ

X

ηγn ǫnγ =

ηγN (δγN )2 =

γ

γ

ηγU

X

(TrTX )Uγ + (TrTX )U γ ,

h i A γ , + (TrT ) (TrT ) ηγU αA A b b Uγ U γ A

X A VγA A 2ηγU αA + η d δ d A A γ V γ γ Uγ γ

=

ηγU (TrTa2 )Uγ + (TrTa2 )U γ + ηγV (TrTa2 )Vγ ,

ηγN δγN =

γ

(D.22)

Xh γ

i A VA A 2 2 2ηγU (αA γ ) dUγA + ηγ (δγ ) dVγA ,

X

ηγn (ǫnγ )2 ,

(D.23) (D.24)

γ

(D.25) (D.26) (D.27) (D.28)

where the subscript M stands for the gauge-charged light sector, and dM a is the dimension of the (generally reducible) gauge group representation in the matter sector with modular weights qna . In addition to (D.10) and (D.12), the constraint (A.4) imposes further conditions on the parameters in K(Ψ). The Y˙ contributions to the first condition in (A.4), namely  ! X X X A   qX ∂µ ti GA ∂ν tj GA Imλ, (D.29) i + iTA · Aµ + iΓµ j + iTA · Aν + iΓν λ,A

i

j

72

and 

ImF m (tm )  +

X λ,N

+

X λ,N

require

X X



 

(TrTa2 TX )M

i

λ,A



X j

X j

 ! X A   qm ∂ν tj GA ∂µ ti GA j + iTA · Aν + iΓν i + iTA · Aµ + iΓµ 

n n · ∂µ tj GN j − 2∂µ t gn + iΓµ



 ∂µ tj GN j + iΓµ

= −

1 =

X

X

γ

dM a

k

n n ∂µ tk GN k − 2∂µ t gn + iΓµ

!

!

N , ∂ν tk GN k qm + iΓµ

(D.30)

ηγN (δγN )3 =

X

(D.31)

ηγn (ǫnγ )3 ,

(D.32)

γ

o h i Xn A A VγA γ , + (TrT T ) (TrT T ) δ + η (TrT T ) = η Uγ αA A A b c U b c Uγ b c Vγ γ γ

(D.33)

= −

(D.34)

γ

(TrTb )M a

k

X

ηγU (TrTa2 TX )Uγ + (TrTa2 TX )U γ

γ

(TrTb Tc )M a

X

j

=

A

X γ

Xh γ

i h A 2 γ , + (TrT ) (TrT ) ηγU (αA ) A b U b Uγ γ A

o A VA A 3 3 . + η (δ ) d 2ηγU (αA ) d A A γ γ Vγ γ Uγ

(D.35)

N n n − Note that, for example, since the right hand side of (D.18) is equal to −2 GN − g G − g α α β β 2gαn gβn , we can cancel (D.17) and (D.18) if we replace the last term in (D.19) and the condition (D.24) by, respectively X N N n n 2 ǫn γ g |Φn |2 (D.36) eδγ (G −g ) |ΦN | + e γ γ N =n

and

2=

X γ

ηγN δγN =

X γ

ηγN (δγN )2 =

X γ

ηγn (ǫnγ )2 ,

0=

X

ηγn ǫnγ .

(D.37)

γ

Imposing the additional constraints (A.4) removes this ambiguity. However we will see in Appendix D.3 that there is at least one other choice of PV sector with this (and the following) choice ˙ Y˙ K¨ of Z, ahler potentials.

73

A simple solution to the constraints (D.22)–(D.28) and (D.31)–(D.35) is28 δN = ǫn = αA = δA = 1,

(D.38)

in which case they reduce to 1 =

X γ

(TrTb )M a (TrTb Tc )M a

= − = =

γ

Xh γ

(TrTa2 TX )M

X

Xn γ

dM a

ηγN =

= −

X

ηγn ,

(D.39)

γ

h i A η Uγ (TrTb )UγA + (TrTb )U γ ,

(D.40)

A

o h i A A η Uγ (TrTb Tc )UγA + (TrTb Tc )U γ + η Vγ (TrTb Tc )VγA , A

i A A 2η Uγ dUγA + η Vγ dV A γ ,

X

ηU

Aγ

A,γ

h

(D.41) (D.42)

i (TrTa2 TX )UγA + (TrTa2 TX )UAγ .

(D.43)

Note that (D.39) and (D.42) are equivalent to (D.22). D.2

Preserving shift symmetry

The shift symmetries ImtI → ImtI +αI , Ims → Ims+β, αI , β ∈ R, of the classical K¨ ahler potential I I a ¯ ¯ are preserved if K = K[(T + T ), (S + S), Φ ]. Then shift symmetry and modular covariance of the light particle K¨ ahler potential is preserved if it takes the form X X ¯ + g + f (X a ), X a = ega |Φa |2 , ga = qna gn , f (X) = X a + O(X 2 ). (D.44) K = k(S + S) n

a

The metric is a

Ka¯b = eg δab fa + φb φ¯a¯ fab eg

a +g b

,

Ka¯ı =

X

¯

g¯ıb φ¯b Ka¯b ,

b

Kj¯ı = gj¯ı fni +

X

¯ gja g¯ıb φa φ¯b Ka¯b ,

f ni =

1+

a

a,b

where fa = 28

X

∂f , ∂X a

fab =

!

qnai X a fa ,

∂f . ∂X a ∂X b

The constraints (D.26), and (D.34) are nontrivial only if Tb is a U (1) generator; V A is a U (1) singlet.

74

(D.45)

(D.46)

χn = ag ˙ n = aδ ˙ in g(T i + T¯ i ).

(D.47)

If we define ka¯b = ∂a ∂¯b f (X a ) = Ka¯b ,

¯

ka¯b kbc = δca ,

g˜i¯ = fni gi¯ = fni gi2 δij ,

g˜k¯ g˜i¯ = δik ,

(D.48)

the inverse metric is: ¯

¯

¯ X a b ¯ıj gj g¯ı g˜ , ¯ıj

K ab = kab + φa φ¯b

K j¯ı = g˜¯ıj = fn−1 gi−2 δij , i

K a¯ı = −φa

X

gja g˜j¯ı .

(D.49)

j

Using the properties K¯ıj = δij , gi = g¯ı = egni and (Ta )ij gi = 0, one can see that only (D.10) is ˙ Y˙ is still required for (A.4) to be satisfied in this case; however a minimum of 3 sets of the Z, required. To maintain invariance of the PV K¨ ahler potential we take χn = ag ˙ n = aδ ˙ in g(T i + T¯ i ).

(D.50)

For Y˙ we now have ΓIN j = −aδ ˙ ji δni ,

ΓIN α =

a˙ i ¯ 2 δ (D − 8R)Dα T i = 0, 8 n

(D.51)

because T i is gauge invariant, and once (D.10) is imposed the contributions from Y˙ involve the same operators as in (D.18), and are canceled by contributions from the fields Ψ = U, V, Φ introduced in (D.19)–(D.21), subject to the conditions (D.22)–(D.28) and (D.31)–(D.35). D.3

Heisenberg invariant untwisted sector K¨ ahler potential

Here we set χn = ag ˙ n = aG ˙ n , where Gn is defined in (D.1). Once the conditions (D.10) and (D.12) are imposed, the Z˙ cancel the UV divergences from the light fields Z p . The contributions from Y˙ I decouple from those from Y˙ N in the relevant sums. Those from Y˙ I reduce to those from fields with metric equal to the inverse of the untwisted sector metric derived from (D.1), with an extra overall N N factor eG , and those from Y˙ N are the same as the contributions from a field with metric eG . We may cancel their contributions to the UV divergences with the method used in [3] to cancel the UV divergences from the light untwisted sector. That is, the Y˙ contributions to the UV divergences can be canceled with Ψ = U, V, ΦN , ΦnI , ΦIn where U, V, have the same K¨ ahler potential as in (D.19). N The Φ , which are no longer all gauge singlets, have K¨ ahler potential XX N N n 2 K(ΦN ) = eδγ (G −g ) |ΦN (D.52) γ | , γ N =n

75

and the constraints (D.22)–(D.28) are appropriately modified. The fields ΦnI , I = 0, . . . , Nn , have a N K¨ ahler potential of the same form as Y˙ I,N , but without the prefactor eG in (4.54), and the fields ΦI have the inverse K¨ ahler metric: I

¯

KIΦJ¯ KΦJK = δJK , I

I, J, K = 0, . . . , Nn .

(D.53)

This set cancels the remaining Y˙ I,N UV contributions if we impose X X X 1 I 2 4 ηγΦ = ηγΦI ≡ ηγΦ , ηγΦ = 0, ηγΦ (aΦ ηγΦ (aΦ γ) = − γ) = . 2 γ γ γ

(D.54)

However, it is not possible to cancel all the contributions from Y˙ to (A.4) because terms odd in Φn Φn γ I γ I Φn I I Y Y (Γµ )J , (Ta )J and (φ γ )J cancel between Φ and ΦI . Terms proportional to Tr Γ˙ µ φ˙ are indeed canceled by virtue of (D.54), but there is no contribution from the Φ sector to cancel terms P ˙ like Gi Dµ z j (T˙aY φ˙ Y )ij , for example. These terms could be made to vanish with λ ηλZ a˙ 2λ = −1, but this contradicts the condition in (D.10). So in this case we need to modify (4.54)–(4.57) as follows: i X Nh X A ˙ (D.55) eG gni¯ Y˙ I − bGni Y˙ N Y¯˙ J¯ − bGn¯ Y¯˙ N¯ + |Y˙ N |2 , KY = eG |Y˙ A |2 + N

A

which is modular invariant provided under (3.10) A N Y˙ A′ = e−F Y˙ A , Y˙ I′ = e−F mji Y˙ J + bFjn Y˙ N ,

N Y˙ N′ = e−F Y˙ N ,

Wn = µ˙ n (T )

X

˙P

Z Y˙ P + a˙

−1

bZ Y˙ N

P ∈n

Now we have (see Appendix D of [3]) X n η˙ λ (ΓnY˙ )σσα = −(Nn + 1) GN α − Gα , X λ

λ

n η˙ λ (ΓnY˙ )σρβ (ΓnY˙ )ρσα = −(Nn + 1) GN α − Gα

X λ

η˙ λ (ΓnY˙ )JIα (T a )IJ

˙N

!

N,A m F , qm

.

(D.58) n nα ¯ i GN β − Gβ + 2(b2 + b4 )Gi Zα

(D.59) (D.60)

ˆ n )ijα = P Gnj Dα Zni , (G

ˆ n )iα = P Gni Gnj Dα Znj , (G X 1 ¯2 − 8R), bp = − η˙ λ bpλ P = − (D 8 λ

76

(D.56)

(D.57)

i h ˆ n )ijα (G ˆ n )j , −(1 + 2b2 + b4 ) Gnα Gnβ + (G iβ j n a ˆn i = (TrT a )Zn GN α − Gα − (1 + b2 )(T )i (G )jα ,

Zαi = PDα Z i ,

X m

and the superpotential is given by (4.57), except that i

F N,A =

(D.61) (D.62)

If bp = 0, we recover the inverse of the untwisted sector K¨ ahler potential. However if b4 = −b2 = 1,

(D.63)

we eliminate both the terms arising from this contribution that cannot be eliminated by U, V, ΦIN , with Nn XX N I (D.64) eδγ (GN − Gn )|ΦIN |2 , K(ΦN ) = γ,N I=0

as well as the new term involving the last two operators in (D.61). A simple solution to (D.63) with three sets Y˙ γ is, for example (η, b) = (1, 3), (−1, 2), (−1, 2).

(D.65)

The constraints (D.22)–(D.28) now read X

ηγΨ = 3 +

X

Nn + dM T = N + 2,

(D.66)

n

Ψ,γ

a CM

=

X γ

1 =

X γ

(TrTX )M (TrTb )M a dM a

= − = − =

ηγU (TrTa2 )Uγ + (TrTa2 )U γ + ηγV (TrTa2 )Vγ +

ηγN δγN =

X

ηγN (δγN )2 ,

(TrTX )Zn = −

γ

X

X

ηγN (TrTa2 )ΦN γ

N

N

ηγN δγN TX γ ,

(D.68) (D.69)

h i A γ + (TrT ) (TrT ) ηγU αA , A b U b Uγ γ

(D.70)

i A VA A 2 2 2ηγU (αA + η ) d (δ ) d A A Uγ Vγ , γ γ γ

(D.71)

X γ

, (D.67)

γ

X , ηγU (TrTX )Uγ + (TrTX )U γ + ηγN (TX )ΦN γ γ

)

A

X A VA A + η δ d 2ηγU αA d A A γ γ Vγ γ Uγ γ

=

(

Xh γ

where M a now includes only twisted sector matter. For the relevant Y˙ contributions to (A.4) we have # " i h X X n i = − GN (Nn + 1)(F − FN ) + qX λX η˙ λ Tr (ΓnY˙ )µν ΦY˙ µν − Gµν i

λ

+(1 + b2 )Yµν ,

77

(D.72)

X λ

# " X i ˜ n˙ · Γn˙ Φ ˙ eN − G en · GN − Gn (Nn + 1)(F − F N ) + qX λX η˙ λ Tr Γ = − G Y Y Y i

+(1 + b2 )Y + (b2 + b4 )Z, i h X n = GN η˙ λ Tr (ΓnY˙ )µν (T a )Y˙ ΦY˙ (TrT a )Zn (F − F N ) + (TrT a TX )Zn λX µν − Gµν

(D.73)

λ

X λ

h

η˙ λ Tr (T a )Y˙ (ΓnY˙ )µν ΦY˙

a +(1 + b2 )Yµν , n = GN (TrT a )Zn (F − F N ) + (TrT a TX )Zn λX µν − Gµν

i

a +(1 + b2 )Zµν .

(D.74)

(D.75)

The precise form of the various operators Y, Z in the above is unimportant, since they drop out when (D.63) is imposed, and the remaining contribution is canceled by U, V, ΦN , with the additional conditions X X N (TrTa2 TX )M = − ηγU (TrTa2 TX )Uγ + (TrTa2 TX )U γ − (D.76) ηγ (TrTa2 TX )ΦN γ γ

(TrTX )Zn

=

X

N,γ

γ

(TrTb Tc )M a

=

Xn γ

(TrTb )M a

= −

1 =

X

=

o h i A A VγA γ , + (TrT T ) δ (TrT T ) + η (TrT T ) η Uγ αA A A c c c b b b Uγ Vγ γ U γ

(D.78)

(TrTb Tc )Zn =

γ

A

h i A 2 γ , + (TrT ) ηγU (αA (TrT ) ) A b b U U γ γ

(D.79)

A

ηγN (δγN )3 ,

γ

dM a

(D.77)

X γ

X

δN , ηγN (TrTb Tc )ΦN γ γ

(δγN )2 , ηγN (TrTX )ΦN γ

Xh γ

(D.80)

i A 3 VA A 3 2ηγU (αA ) + η (δ ) d γ γ γ VγA .

(D.81)

As before these constraints have a simple straightforward solution: X 1 = αA = γ N = δN , 1= ηγN ,

(D.82)

γ

a CM b

=

o i Xn B h B ηγU (TrTa2 )UγB + (TrTa2 )U γ + ηγV (TrTa2 )VγB , γ

(TrTb )Ma dM a

= −

(D.83)

B

X γ

ηγU

A

h

(TrTb )UγA + (TrTb )U γ

A

X A A = 2ηγU dUγA + ηγV dVγA , γ

78

i

CZa n

(TrTb )Zn = − =

X γ

X

N

ηγN Tb γ ,

(D.84)

γ

ηγN (TrTa2 )Nγ .

(D.85)

Note that the K¨ ahler potential gn used in the previous two subsections is just the limiting case n ΦA n → 0 of G , so this gives an alternative PV sector for those cases. They differ in the resulting sum rules that are quadratic and cubic in the modular weights. D.4

Anomalous masses

The conditions (D.22)–(D.28) and (D.31)–(D.35) imply that the the Ψ masses are not modular invariant (at least in the absence of moduli-dependent mass factors) and are not in general U (1)X invariant. Their K¨ ahler potential is invariant under K¨ ahler transformations arising from nonlinear transformations on the light fields: Z p → Z ′p (Z),

dZ ′p = Mqp dZ p ,

K(Z ′ ) = K(Z) + F (Z),

W (Z ′ ) = e−F (Z) W (Z),

(D.86)

provided A

A

A

Φ′N γ

−δγN (F N −F n )

= e

A

A

UA′γ = e−αγ F UAγ ,

Uγ′A = e−αγ F UγA ,

ΦN γ ,

Φ′n γ

n −ǫn γF

=e

A

Vγ′A = e−γγ F VγA , Φnγ .

(D.87) (D.88)

The K¨ ahler potential is also gauge invariant; in Yang-Mills superspace [5] the superfields Z p are defined to be covariantly chiral. In particular, under U (1)X Z a → g qa Z a ,

Z a¯ → g −qa Z a¯ ,

AM → AM + g−1 DM g

(D.89)

with the gauge covariant superspace derivative DM given (neglecting other connections) by DM Z p = (DM + q p AM )Z p .

(D.90)

For the PV fields ΨA = U A , UA we take instead, with (4.6), ΨA → e−q

AΛ

ΨA ,

¯ A¯ . ¯ A¯ → e−qA Λ¯ Ψ Ψ

(D.91)

It is necessary to introduce the vector superfield VX in order that the U (1)X -violating terms in the PV superpotential remain holomorphic under a U (1)X transformation. In addition, in the regulated theory noninvariance under U (1)X arises from the PV masses; this must cancel the noninvariance of the GS term (4.3) that explicitly involves VX . Working in “partial” U (1)X superspace allows us to meet these conditions while keeping the notation from becoming too cumbersome. With this modification the K¨ ahler potential in (D.21) for U is replaced by X A A A A A A (D.92) K(U ) = eqγ VX +αγ G |UγA |2 + epγ VX +αγ G |UAγ |2 . γ,A

79

In this way we have put all the anomaly associated with the T-moduli and gauge charged chiral fields in the superpotential W (Ψ) in (D.20). We define n N ω˙ m = qm − 1,

a A a ω˙ m = qm + qm − 1,

(D.93)

as the (negative of the) modular weights of µ˙ n,a in the superpotential (4.57). If, for illustration, we use the PV sector of Appendix D.1 or Appendix D.2, the Ψ have modular weights ΦN

qmγ

UA qmγ VA qmγ

N n n n = qm − 2δm = ω˙ m − 2δm + 1,

Uγ qmA

A a a = qm = ω˙ m − qm + 1,

A a a + 1, = qm = ω˙ m − qm

Φn

N n qmγ = qm = ω˙ m + 1, A a a = qm = ω˙ m − qm + 1,

(D.94) (D.95) (D.96)

Ψ

where we have assumed29 (D.38). Then, if we further assume that ωn γ is independent of γ, using the sum rules (D.22)–(D.28) we have X X X Ψ a a n n ηγΨ qmγ = (ω˙ m − qm + 1) + 2 (ω˙ m − δm + 1) a

Ψ,γ

n

= N +2−

X

η˙ λ

2

X

n ω˙ m

+

a

n

λ

X

a ω˙ m

!

−

X

p , qm

(D.97)

p

where p refers to t-moduli and all gauge-charged matter, with, for the untwisted sector charged P ai = δ i , and for t-moduli q i = 2δ i . In writing the last line of (D.97) we used matter qm m m m λ η˙ λ = −1 and, including the dilaton, X N = dM + 4 = ηγΨ − 2. (D.98) Ψ,γ

Then m ˜Ψ γ is given by (C.39) with, using also the constraint (D.25) on (TrTX )Ψ , " # X X X Ψγ Ψγ Ψγ m Ψ Ψ + 1 − 2qm + ωm F − 2qX λ ηγ ηγ ϕ˜Ψγ = Ψ,γ

Ψ,γ

=

X m



2

m

X p

p qm −N −2+

X

P =Ψ,Y˙ ,Z˙



P m η P ωm F + 2TrTX λ.

(D.99)

When combined with other PV loop contributions, we get the general result (C.43). Similarly, there is a contribution proportional to (for a 6= X) X X X a a 2 Ψ + + C (T ) + h.c. = η ϕ ˜ η C ϕ˜+ η ϕγ ϕ˜+ Ψ C γ ϕγ Ca + h.c. Ψγ C γ C

29

Ψ,γ

ϕγ

The assumption does not affect the results below, except for the case with p = 4 in (D.145).

80

X

=

n



 Ca 1 + a 2

X

ϕ ã γ

ω ñ

γ

!

+2Tr(T ) TX λ + h.c.,

−

X b



a  CM 1 − 2qnb − b

X

P =Ψ,Y˙ ,Z˙



B B η P ωnP  F n

(D.100)

where we used (D.31), (D.33) and (A.27). This result is again completely general, and agrees with b Imλ . The conditions that (B.58) when multiplied by 1/64π 2 with φb = i qnb ImF n − 12 ImF + qX the anomalies be canceled places constraints on the threshold factors as functions of charges; the coefficients of F n must be independent of n in (C.43), and independent of n and a in (D.100). To insure this, we set    X X X a a 1  B B ϕ ˜ a  ω ˜ nϕ˜ ≡ ω ñ γ = CGS + CM 1 − 2qnb − η P ωnP  − 1, (D.101) b C a γ P =Ψ,Y˙ ,Z˙

b

where CGS = 8π 2 b is the Green-Schwarz coefficient, and and we identify the coefficients ban of the threshold corrections as X X X B B a P P a a b ban = Ca ω ˜ nϕ˜ + ω = C − C + η (D.102) CM C 1 − 2q b GS a n n . Mb P

b

b

For the anomalous coefficient of r · r˜, the expression in square brackets in (C.43) is also equal to the explicit contribution from just the fields with noninvariant masses:   X X X X a X ˆ φ p P ϕ ˜ 2 − αγ + ω ˆ mγ  F m qm − N − 2 + NG + η P ωm + ω ˜m + ηˆγ 1 − 2ˆ m

p

−2TrTX λ = −

X C

a

P =Ψ,Y˙ ,Z˙

′ ηC ϕ˜+ C = −24 CGS F + CGS λ − 24F,

γ

(D.103)

where φˆγ is the subset of neutral chiral multiplets φγ that has noninvariant masses, and the coef′ ficient CGS of the U (1)X GS term is defined in (5.11) and (5.13). In writing the last equality in (D.103) we used string theory results for the anomalous coefficient of ΩW in the full result (4.33), including the contributions (4.26) and (4.30) that are not included in the PV contribution. String results require that the coefficient of F m in (D.103) be independent of m. ˜ µν following the arguments leading to (C.43), using (3.38) If we calculate the coefficient of Xµν X and (A.16) as well as (3.36) and (3.37), we obtain −2 times the RHS of (C.43) with ωn in the

81

expression in brackets replaced by30 X X ωnC η C (1 − αC )2 =

η P ωnP +

a

ω ˜ nϕ˜ +

a

P =Ψ,Y˙ ,Z˙

C

X

X γ

φ

ηγφ ωnγ (1 − αγ )2 .

(D.104)

Alternatively, if we calculate directly using only the noninvariant PV fields, we obtain (−2 times) (D.103) with the last term replaced by i X h φˆ (D.105) αγ )3 + (1 − 2ˆ αγ )2 ω ˆ mγ F m . ηˆγ (1 − 2ˆ γ,m

Comparing (D.103) with(C.43) and (D.105) with (D.104), consistency requires31 X X φ φˆ = 5+ ηγφ ωmγ , αγ + ω ˆ mγ ηˆγ 1 − 2ˆ X γ

(D.106)

γ

γ

i h X φ φˆ = 5+ ηγφ (1 − 2αγ )2 ωmγ , αγ )3 + (1 − 2ˆ αγ )2 ω ˆ mγ ηˆγ (1 − 2ˆ

(D.107)

γ

in conformity with the sum rules X ηγφ = X ηγφ αγ = X ηγφ α2γ = X ηγφ α3γ =

N ′ − NG′ + N + 2 − 3NG = −15 − 2NG ,

(D.108)

−10 − NG ,

(D.109)

−4 − NG ,

(D.110)

−1 − NG ,

(D.111)

˙ ϕa , ϕâ , ϕã from N ′ and subtracting which are obtained by subtracting the contributions of θγ , Z, a the contribution of ϕa , with αϕ = 1 from the sum rules for αn . The constraints (D.108)–(D.111) imply in particular X X ηγφ (1 − 2αγ ) = ηγφ (1 − 2αγ )3 = 5, (D.112) γ

and we obtain for the coefficient of  X X p 2 qm − N + 3 + NG + m

p

γ

X α Xα X

in ΦL [see (E.9)] P η P ωm +

X a

P =Ψ,Y˙ ,Z˙

a

ϕ ˜ ω ˜m +

X γ

ηˆγ (1 −

30



φˆ 2ˆ αγ )2 ω ˆ mγ  F m

+ 2TrTX λ

The superpotential mass term µϕ is constant with ω ϕ = 0 for ϕa , ϕ â as required by modular covariance for this term. 31

e b

The constraints (3.36) and (A.20) assure that ω Z,Y drops out of these equations.

82

=

X C

2 ηC ϕ˜+ C (1 − 2αC )

i ˆ X h φ ′ αγ )2 − 1 ω ˆ mγ F m . = 24 CGS F + CGS λ + 24F + ηˆγ (1 − 2ˆ

(D.113)

γ,m

The last term in(D.113) vanishes in the absence of threshold corrections, or more generally if we impose X X φˆ φˆ ηˆγ (1 − 2ˆ αγ )2 ω ˆ mγ = ηˆγ ω ˆ mγ . (D.114) γ

γ

φ˜ If we define φγ = φˆγ , φ˜γ , where φ˜γ has a modular invariant mass term: 1 − 2˜ αγ + ω ˜ mγ = 0, so that φ˜

˜

ω ˜ mγ ≡ ω ˜ γφ

(D.115)

is independent of m, it is straightforward to show that (D.112) is equivalent to (D.106)–(D.107). A third constraint reads X X X ˜2 (D.116) ˜ γφ = −2NG + 9, ηγφ (1 − 2αγ )2 = ηˆγ (1 − 2ˆ αγ )2 + η˜γ ω γ

γ

γ

˜

A simple solution to these constraints, that satisfies (D.114) is to set ω ˜ γφ = (1 − 2˜ αγ ) = 0 with X

η˜γ = −1,

η˜γ = −24,

1 α ˜γ = , 2

(D.117)

1 − 2ˆ α± γ = ±1,

X

(D.118)

γ

and to take φˆγ = (φˆ0 , φˆ± ) with X α ˆ 0γ = 0, ηˆγ0 = 5, γ

γ

ηˆγ± = −NG + 2. φˆ

In this case the coefficient of F m in (D.103) is independent of m provided ω ˆ nγ satisfies ω ˆn ≡

X

φˆ

ω ˆ nγ

γ

= 24CGS −

X

P =Ψ,Y˙ ,Z˙

η P ωnP −

X a

a

ω ˜ nϕ˜ + N − NG + 21 − 2

X

qnp , (D.119)

p

where ωnp =

X

P =Ψ,Y˙ ,Z˙

p

p

η P ωnP =

X λ

p

p

ηλΨ ωnΨ − 2ω˙ np ,

ωn =

X

P =Ψ,Y˙ ,Z˙

83

η P ωnP =

X λ

ηλΨ ωnΨ − 2

X

ω˙ np .

p

(D.120)

We identify the corresponding threshold corrections as X X a X brn = ωnP + ω ˜ nϕ˜ + ω ˆ n = 24CGS − NG + N + 21 − 2 qnp . a

P

(D.121)

p

The “F-term operator ΩX m and the “D-term” operator Ωm also contain terms linear in the PV sector parameters; the linear terms in (E.5)–(E.14) have a single factor 1 ln M = K − 2 ln f + ln |µ2 | . (D.122) 2 The sum rule (D.116) assures consistency between the coefficient of derivatives of K as calculated by summing over all PV states as in (C.43) and (D.103), and by summing over only states with noninvariant masses as in (D.104) and (D.105). However in the former case, when using the sum rules (3.37) and (3.38), we have to subtract a spurious contribution32 i Xh a a a a (D.123) αϕˆ = 0, αϕ = 1, (1 − 2αϕ )2 + (1 − 2αϕˆ )2 = 2NG , a

giving

X n

= −



N + NG − 7 −

X m



2

X p

+ = −

X C

X

P =Ψ,Y˙ ,Z˙

η P ωnP −

p qm − N − 2 + NG +

X γ

X a

X

a

ω ˜ nϕ˜ −

γ

P η P ωm +

P =Ψ,Y˙ ,Z˙

φˆ ηˆγ (1 − 2ˆ αγ ) 1 − 2ˆ αγ + ω ˆ mγ

ηC ϕ˜+ C (1 − 2αC ),

X

#

φ

ηγφ ωnγ (1 − 2αγ ) − 2

X

X p



qnp  F n − 2TrTX λ

a

ϕ ˜ ω ˜m

a

F m − 2TrTX λ (D.124)

which is consistent with (D.116). The chiral projection ΦL of ΩL is given by ¯ 2 − 8R)ΩL = 16 (X α − 2f α )(Xα − 2fα ) . ΦL = (D Using the above results we obtain X 1 X Ψ + α Ψγ 1 ˜ + ΦL = 1 (1 − 2αC )2 X α Xα + ηC Φ+ ηγ ΦΨγ fΨγ fα − Xα Tr η Φ C 192 12 3 C

32

Ψ,γ

P a a The actual contribution from these fields is 2 a (1 − αϕ − αϕˆ )2 = 0.

84

(D.125)

(D.126)

+ is a chiral superfield with Φ ˜+ where Φ+ C =ϕ C, C X X p n fαn = (1 + ω˙ m )gαm − ℓnα , fαP = (1 + ω˙ np − qnp )gαn − qX WαX − ℓPα m

Φ+ n = −

X n (1 − ωm )F m , m

(D.127)

n

Φ+ p =−

X p (1 − ωnp − 2qnp )F n + 2qX ΛX .

(D.128)

n

The term in brackets can be evaluated using the conditions (D.93)–(D.96), together with the relation X p p 2 CGS = (ωl + 2qlp − 1)(qX ) ∀ l, (D.129) p

which, like (D.102), is known to be satisfied in orbifold compactifications of the heterotic string. If we take, again for illustration, ¯ = 0, ℓC (Z, Z) C = n, P, (D.130) when (D.126) is combined with the other terms in Ω′L , defined in (5.5), we obtain the result given in (5.9) with X k Amnl = ) (ωlk − 1)(1 + ω˙ nk )(1 + ω˙ m k

+

X p p p (ωl + 2qlp − 1)(1 + ω˙ np − qnp )(1 + ω˙ m − qm )

(D.131)

p

Anl =

X X p (ωl + 2qlp − 1)(1 + ω˙ np − qnp ) (ωlk − 1)(1 + ω˙ nk ) +

Al =

(D.132)

p

k

X p (1 − ωlp − 2qlp )qX

(D.133)

p

Bnl

X p = (1 − ωlp − 2qlp )(1 + ω˙ np − qnp )qX

(D.134)

p

amn = 2

X p

an = 2

X p

a = −2 bn = −2

p p p qX (1 + ω˙ np − qnp )(1 + ω˙ m − qm )

(D.135)

p qX (1 + ω˙ np − qnp )

(D.136)

X p M (qX )2 = −2CX ,

(D.137)

p

X p (qX )2 (1 + ω˙ np − qnp ).

(D.138)

p

The anomalous part of the Lagrangian also contains terms quartic in the PV parameters; these are not constrained by the requirement of finiteness alone. As an example we can assume the simple 85

φˆ0,±

solutions defined by (D.38), (D.39), and (D.117)–(D.118). If we further assume ω ˆ nγ independent of γ, The real superfields MφˆC take the form M0,± = eK/2 µ0,± γ γ

Y n

0,±

|η(T n )|2ˆωn .

= ω ˆ n0,± is

(D.139)

The constant parameters ln µ drop out in the variation of the action and in derivatives of ln M, and we obtain !p p X X 5 0 n ηˆγ O ln Mφˆγ → p OK + 4 ω ˆ n O|η(T )| 2 γ n " !p !p # X X 2 − NG OK + 4 ω ˆ n+ O|η(T n )| + 4 ω ˆ n− O|η(T n )| − OK , (D.140) + 2p n n !p X X a NG p ϕ ˜ n ηϕã (O ln Mϕã ) → p OK + 4 ω ˜ n O|η(T )| (D.141) 2 a n where O stands for any operator, and

ω ˆ n = 5ˆ ω 0 + (2 − NG ) ω ˆ+ + ω ˆ− ,

ˆbφ = (NG − 2)ˆ ω− , n

(D.142)

with ω ˆ n given by (D.119). If in addition to (D.39)–(D.43) we assume i X Ah η Uγ (TrTb Tc Td )UγA + (TrTb Tc Td )U γ (TrTb Tc Td )M a = − A

γ

A

(TrTa2 TX2 )M

+η Vγ (TrTb Tc Td )VγA , i X A h = η U γ (TrTa2 TX2 )UγA + (TrTa2 TX2 )U γ , A

(D.143) (D.144)

A,γ

we obtain

X P

with

ηPΨ (O ln MΨP )p →

ln Mn = K/2 − g −

X m

n m ω˙ m g ,

3 X

(O ln Mn )p +

n=1

ln Mp = K/2 − g +

N −1 X q=1

X n

(O ln Mq )p ,

p (qnp − ω˙ np )g n + qX VX ,

(D.145)

(D.146)

m = 0 for the moduli, and a denotes the gauge-charged where q = m = 1, 2, 3; a with qnm = 2δnm , qX matter fields.

86

As mentioned above, the terms quadratic and higher in weights are modified if use instead the PV sector of Appendix D.3. They are also modified if ℓp 6= 0; a natural choice might be ℓP = K − g. D.5

(D.147)

Orbifold models

Here we list the quantum numbers needed to evaluate the anomaly coefficients for three orbifold compactification models for which all the modular weights and gauge charges are known. These are C Z7 and Z3 models with no string threshold corrections: ba,r n = ωn = 0, so the conditions (D.102) and (D.121) reduce to (5.12). One can check that these are satisfied33 from the tables given below, where we group the chiral multiplets according to their representation (rep) under the semi-simple part of the gauge group, and their modular weights qn , and give the multiplicity (ns ) of each set P with fixed quantum numbers, as well as the total multiplicity n for each set: N = n. For the two i¯  Z3 models the superfields T are the K¨ ahler moduli of which only the diagonal elements T i = T i¯ı are assumed to have nonvanishing vacuum values. In all the tables S is the dilaton superfield, and the indices i, j = 1, 2, 3. The first two models were studied at the string level in [14]. They have no Wilson lines and no anomalous U (1) so they have 8π 2 b = CGS = 30,

′ δX = CGS = 0,

(D.148)

and all chiral multiplets are E8 singlets. For the third (FIQS) model,34 which is a Z3 orbifold model with Wilson lines, we also list the U (1)X charges qX . Z3 : The gauge group is E8 ⊗ E6 ⊗ SU (3) with NG = 334. name Ui T Yi T i¯ S

rep

qn

ns

n/3

(27,3) (27,1) (1,¯3) (1,1) (1,1)

δni 2 3

1 27 27 1 1

81 243 81 3

2 i 3 + δn δni + δnj

0

33

1 3

They are also satisfied for Ta ∈ U (1)a for the nonanomalous U (1) factors whose charges are not listed. The fields in a given set are not degenerate under under these U (1)’s. 34 See the last reference in [30].

87

From the table we have

X

N = 1225,

qnp = 816,

(D.149)

p

and X

X

p qnp qm = 542 + 168δmn ,

(D.150)

p

p p qnp qm ql = 396 + 56 (δmn + δnl + δlm ) + 168δmn δnl .

(D.151)

p

Z7 : The gauge group is E8 ⊗ E6 ⊗ U (1)2 with NG = 328. name U1i U2i ti Y1i Y2i Y3i Y4i Y5i Y6i Y7i Y8i Ti S where35 qn1 and

=

6 5 3 , , 7 7 7

,

rep

qn

ns

n/3

27 1 27 1 1 1 1 1 1 1 1 1 1

δni δni qni qni + δni qni + 2δni qni + 4δni i qni + δnM i qni + δnm i qni + 2δnM i qni + 2δni + δnM i qni + δni − δnm 2δni 0

1 1 7 7 7 7 7 7 7 7 7 1 1

27 1 189 7 7 7 7 7 7 7 7 1

qn2

=

3 6 5 , , 7 7 7

M i = i + 1 mod 3, We have N = 823,

,

qn3

mi = i − 1 X p

35

These are respectively ρ1 , ρ4 , ρ2 of [14].

88

qnp = 618,

1 3

=

5 3 6 , , 7 7 7

mod 3.

,

(D.152)

(D.153)

(D.154)

and X

p qnp qm = 438 + 342δmn ,

(D.155)

p

X

− p p qnp qm ql = 291 + 1425δmn δnl + 279∆+ nml + 191∆nml ,

(D.156)

p

where 3 1 2 3 1 2 ∆+ ijk = δij δi δk + δi δk + δi δk + cyclic(ijk), 1 3 3 2 2 1 ∆− ijk = δij δi δk + δi δk + δi δk + cyclic(ijk).

(D.157)

The FIQS model: The gauge group is SO(10) ⊗ SU (3) ⊗ SU (2) ⊗ [U (1)]7 ⊗ U (1)X and NG = 64. √ name rep 6qX qn ns n/3 Qi ui Li Φ D d L1 L2 T1 T2 Yi T i¯ S

(1,3,2) (1,¯3,1) (1,1,2) (16,1,1) (1,3,1) (1,¯3,1) (1,1,2) (1,1,2) (1,1,1) (1,1,1) (1,1,1) (1,1,1) (1,1,1)

We have N = 415,

δni δni δni δni

0 0 0 3 2 2 3 2 3 2 3 − 34 2 3 − 34 2 3

2 3 2 3 2 3 2 3 2 3 2 3 2 i 3 + δn δni + δnj

0 0

0

X

1 1 1 1 12 15 33 3 114 30 9 1 1

6 3 2 16 12 15 22 2 38 10 9 3 1 3

qnp = 258,

(D.158)

p

and X

X

p qnp qm = 158 + 42δmn ,

p

p p qnp qm qX

=

√

X p

6 (12 + 5δmn ) ,

p

p 2 (qX ) = 50,

X p (qX )2 qnp = 28, p

89

X p

√ p qnp qX = 21 6,

(D.159)

X

p p qnp qm ql = 108 + 8 (δmn + δnl + δlm ) + 42δmn δnl .

(D.160)

p

E

Construction of the GS Term

In this appendix we detail the steps in the construction of the GS term. As in Appendix C, the notation Φ| will denote the bosonic part of the lowest component φ of the superfield Φ. E.1

Anomaly superfields

Part of the anomaly can be expressed in term of supersymmetric field operators of the form36 Z E 1 d4 θ T α Tα′ H(Z) + h.c. L(T, T ′ , H) = 2 R √ g ′ α β H(z) Dα Tβ D T + h.c. + fermions = 2 √ ′ ′ µν ′ µν ˜ = + fermions, (E.1) g 2ReHT0 T0 + ReHTµν T + ImH Tµν T

where H(Z) is a holomorphic function of the chiral fields, with Tα defined by (2.12) and [3]: 1 α −K ¯i m ¯ D Tα | = −Dm A A ¯ + Dµ z i D µ z¯m + x−1 Da Ti (T a z)i , ¯ Ti e 2 ¯ ¯ a i = Dµ z i Dν z¯m − Dν z i Dµ z¯m Dm ¯ − iFµν (Ta z) Ti .

T0 = Tµν

(E.2)

In particular we have contributions with Tα = Xα and Xαm ; Xαm is defined in (4.39). In addition we have the supersymmetric operators Z 1 E L(YM, H) = d4 θ H(Z)Waα Wαa + h.c. 2 R √ g H(Z)Dα Wβa D α Waβ + h.c. + fermions = 2 2 √ µν a a a ˜ = − g ReH Fa Fµν − 2 Da D + ImHFa · F + fermions. (E.3) x and (Xµν = Kµν ) L(W, H) =

1 2

Z

d4 θ

E H(Z)W αβγ Wαβγ + h.c. R

36

The sign on the RHS of (A12) in I is incorrect. As a consequece the signs of the Lα terms should be flipped ˜ 0 and L ˜ G in (2.11). The coefficient of Lα in (2.11) should be −1/18, and the in (A7) and in the expressions for L α coefficient of X Xα in (A8) should be +1/6.

90

√ g H(Z)Dα Wβγδ D α W βγδ + h.c. + fermions = √2 g 1 2 µνρσ µν = ReH r rµνρσ − 2rµν r + r + ImHr · r˜ 8 3 √ g eµν X µν + fermions. + ReHXµν X µν + ImH X 12

(E.4)

For the full cancellation of the anomalies we will also need the operators introduced in Section 4.1, equations (4.40) and (4.41). The “D-term” combination Ωm defined in (C.72) can be written 1 ¯2 2 2 β˙ 2 α ¯ D D ln M + [D , Dβ˙ ] ln MD ln M − D ln MD ln MDα ln M + h.c. Ωm = − 4 3 ¯ 2 ln M − 1 {Dα , D ˙ } ln M{D α , D β˙ } ln M + 2D α ln MD β˙ ln M[Dα , D ˙ ] ln M + D 2 ln MD β β 2 2 α 2 α 2 + 4RD ln MDα ln M − 2RD ln M − 4D ln MDα R + D R + h.c. ¯ + 2G ˙ 2D α ln MD β˙ ln M − [D α , D β˙ ] ln M + Gαβ˙ −8RR αβ ¯ + D2R + D ¯ 2R ¯ + 2G ˙ Gαβ˙ + ln M 1 D α Lα + 2 ln MD α Xα = −ΩL − 2ΩLX − 8RR αβ 2 ∂ 1 ˙ ˙ − D 2 ln MDβ˙ ln MD β ln M + h.c. − 2Gαβ˙ D α ln MD β ln M ∂ ln M 4 1 ¯2 2 + ln M D D ln M + D α (RDα ln M) + h.c. 8 1 α β˙ + D ln MDα ln MDβ˙ ln MD ln M 2 ∂ ¯ + D2R + D ¯ 2R ¯ + 2G ˙ Gαβ˙ − ≡ −ΩL − 2ΩLX − 8RR f (ln M). (E.5) αβ ∂ ln M The result (E.5) is determined only up to a linear supermultiplet L0 ¯ 2 − 8R)L0 = (D 2 − 8R)L ¯ 0 = 0. (D

(E.6)

which does not contribute to the variation of the action, and the argument of the ln M derivative is determined up to a total spinorial derivative. The real superfields ˙

˙

ΩL = Lα Dα ln M + Dβ˙ (ln MLβ ) = Lβ˙ D β ln M + D α (ln MLα ), ΩLX

˙

˙

= X α Dα ln M + Dβ˙ (ln MX β ) = Xβ˙ D β ln M + D α (ln MXα )

(E.7) (E.8)

are the Chern-Simons superfields [40], respectively, for the chiral superfields Φ L = Lα Lα ,

ΦLX = X α Lα , 91

¯ 2 − 8R)Dα ln M, Lα = (D

(E.9)

as discussed in the next subsection. If we define Ω′m by 1 1 1 1 Ωm + ΩXm = Ω′m + ΩX , 48 36 48 36

3 Ω′m = Ωm + ΩL + 2ΩLX , 4

(E.10)

the mixed term ΩLX drops out and we obtain 1 ¯ + 2G ˙ Gαβ˙ + D 2 R + D ¯2R ¯ Ω′m = − ΩL − 8RR αβ 4 ∂ 1 ˙ ˙ − D 2 ln MDβ˙ ln MD β ln M + h.c. − 2Gαβ˙ D α ln MD β ln M ∂ ln M 4 1 ¯2 2 α D D ln M + D (RDα ln M) + h.c. + ln M 8 1 ˙ + D α ln MDα ln MDβ˙ ln MD β ln M 2 2 1 α α −(ln M) D Lα + ln MD Xα . 4

(E.11)

The first five terms on the right hand side of (E.11) are invariant under modular and U (1)X transformations and have the correct chiral and Weyl weights for some part of them be included in the chiral projection of the modified linear superfield L; for example the second and third terms complete the PV contribution to the anomalous coefficent of ΩGB . The remainder may require additional counterterms for anomaly cancellation. Defining Ωr by 1 ¯ + 2G ˙ Gαβ˙ + Ωr + D 2 R + D ¯ 2 R, ¯ Ω′m = − ΩL − 8RR αβ 4

(E.12)

it is the coefficient of ¯ , δ ln M2 = 2δ ln M = 2 δH + δH

¯ = 0, Dβ˙ H = Dα H

under an infinitesimal modular and/or U (1)X transformation on the corresponding part Z Z 1 1 1 1 4 L r = − 2 ℓr = − 2 d θETrη 2d ln MΩr (ln M) 8π 48 8π 48

(E.13)

(E.14)

of the Lagrangian. Under a finite transformation

¯ ∆ ln M = H + H, we have37 ∆ℓr = 2 37

Z

4

d θETrη

1 α α ¯ +2 H +H ¯ ln M D Lα + 2D Xα H H 2

The integrated anomaly for pure supergravity was given in ref. [41].

92

(E.15)

1 2 ¯ β˙ H ¯ + 2D 2 HD ˙ HD ¯ β˙ ln M + D 2 ln MD ˙ ln HD ¯ β˙ H ¯ + h.c. D HDβ˙ HD β β 4 1 2 ˙ ¯ β˙ ln M + h.c. − D HDβ˙ ln MD β ln M + 2D 2 ln MDβ˙ HD 4 ˙ ¯ + Dα HD ˙ ln M + Dα ln MD ˙ H ¯ +Gαβ Dα HDβ˙ H β β 1 2 ¯2 ¯ α D D H + D (RDα H) + h.c. − {H + ln M} 8 1 2 ¯2 α − H D D ln M + D (RDα ln M) + h.c. 8 1 ¯ β˙ H ¯ − D α HDα HD ˙ HD ¯ β˙ ln M − D α HDα HDβ˙ HD β 2 ¯ β˙ H ¯ − 2D α HDα ln MD ˙ HD ¯ β˙ ln M −D α HDα ln MDβ˙ HD β 1 1 ˙ ¯ β˙ H ¯ − D α HDα HDβ˙ ln MD β ln M − D α ln MDα ln MDβ˙ HD 2 2 o ˙ ¯ β˙ ln M . (E.16) −D α HDα ln MD ˙ ln MD β ln M − D α ln MDα ln MD ˙ HD −

β

β

Note that since D α Lα , D α Xα are linear supermultiplets subject to the condition (E.6), it is obvious that the constant ln µ in the expression (D.122) for ln M drops out of (E.16). E.2

Chern-Simons superfields

We wish to impose the modified linearity condition ¯ 2 − 8R)(L + Ω) = (D 2 − 8R)(L ¯ (D + Ω) = 0,

(E.17)

where Ω is a superfield with U (1)K chiral weight wχ (Ω) = 0 and Weyl weight wW (Ω) = 2 such that √ Z g L(Ω) = d4 θEΩ, Ω = eK/3 Ω0 , (E.18) 32π 2

is invariant under superweyl transformations. For example, the standard case has Ω = ΩY M . In this case the Lagrangian (E.18) reads in component form √ √ g µν a g0 µν a L(ΩY M ) = − F F + · · · = − F F + ···, (E.19) a µν 2 32π 32π 2 a µν where we dropped fermions and auxiliary fields; these have nontrivial transformations under the superweyl transformation [5]. Girardi and Grimm [22] constructed an explicit expression for the

93

˙

Yang-Mills CS superfield in terms of prepotential superfields ϕα , ϕ¯β , defined as the (ordinary) spinorial derivatives of a hermetian superfield U (called Υ in the notation of [22]) such that the (covariant) chiral(anti-chiral) projection of ϕ(ϕ) ¯ is the chiral(anti-chiral) Yang-Mills superfield strength: ϕα = −U −1 Dα U,

˙

˙

ϕ¯β = D β U U −1 ,

¯ 2 − 8R)ϕα = −8Wα , (D

˙

˙

¯ ϕ¯β = 8W β . (D 2 − 8R)

(E.20)

The CS superfield is constructed from these prepotentials and their covariant spinorial derivatives: i ˙ 1 ΩY M = − Tr (ϕα Wα ) − D β χβ˙ + σβ˙ , 8 32 i i h χβ˙ = Tr Yβ˙ D α Yα + 2Y α Dα ϕ¯β˙ , 2 Z h i iZ 1 h i 1 α α dtYt Y , Yαβ˙ + dtYt D Yα , Yβ˙ , σβ˙ = Tr 2 0 0 YA = DA U U −1 ,

Yt = ∂t U (t)U −1 (t),

(E.21)

and the interpolating superfield U (t) is defined to satisfy: U (0) = 1,

U (1) = U.

(E.22)

The above construction is quite general. In particular we can take U to be a real function ¯ VX ) of the chiral superfields and the U (1)X vector superfield. The above construction U (Z, Z, with U → UK = e−K/8 (E.23) is the CS form ΩX for X α Xα . Similarly, U → Um = M

(E.24)

gives the CS form ΩL for Lα Lα . For these Abelian cases, the integration is trivial, giving, up to a linear superfield, the expression (E.7) for ΩL , and an analogous expression with ln M → − 81 K for ΩX . The CS forms ΩX and ΩL are explicitly dependent on the dilaton through the presence of the K¨ ahler potential K and therefore do not satisfy the condition (4.15). In the remainder of this section we show that the linear/chiral multiplet duality outlined in (4.8)–(4.18) of Section 4.1 still holds. The action is Z ¯ S = − E 3 − 2Ls(L) + (S + S)(L + Ω) , K = k(L) + G(T + T¯, Φ), (E.25) 94

with Ω = eK/3 Ω0 ,

¯ 2 − 8R)Σ, S = (D

(E.26)

and L, Σ unconstrained. To assure a canonical Einstein term in the chiral formulation we require ¯ = 0. 2Ls(L) − L(S + S)

(E.27)

∂ ∂ Ω0 = Ω0 = 0. ∂L ∂S

(E.28)

1 δS ¯ 2 − 8R)(L + Ω) = 0, = −(D E δΣ

(E.29)

First consider the standard case:

The equation of motion for Σ gives

which is just the modified linearity condition for L and which implies Z ES(L + Ω) + h.c. = 0, so the action (E.25) reduces to S=−

Z

E [3 − 2Ls(L)] .

(E.30)

(E.31)

On the other hand the equation of motion for L is 1 δS E δL

∂K = 2s(L) + 2Ls′ (L) − S − S¯ + ∂L 1 ∂K ¯ 2Ls(L) − (S + S)L − 3 ∂L = 2Ls′ (L) + K ′ (L) = 0,

(E.32)

where the last line, which assures a canonical Einstein term in the linear formulation, was obtained using (E.27). Integrating this using (E.27) gives Z dL ∂K S + S¯ = 2s(L) = − . (E.33) L ∂L So for example if K = ln L, S + S¯ = 1/L. Using (E.27), the action expressed in terms of S reduces to Z ¯ S = − E 3 + (S + S)Ω . (E.34) 95

To cancel the anomaly we can add a constant of integration V = V (Z), Z 6= S, L to the right hand side of (E.33): Z dL ∂K S + S¯ = 2s(L) = − + V. (E.35) L ∂L Or, equivalently, we can add a term Z −

ELV

(E.36)

to the action (E.25). ¯ Although the superfield σ is no longer Now consider instead the case Ω0 = Ω0 (σ), σ = S + S. simply a Lagrange multiplier, it is still a nonpropagating field that can be removed by its equation of motion. In this case Ω still drops out of the equation of motion for L, leaving (E.27) and (E.32)–(E.34) unchanged. However the equation of motion for Σ now reads ∂ 1 δS 2 ¯ ¯ = −(D − 8R) L + Ω + (S + S) Ω = 0, (E.37) E δΣ ∂S which gives a different modified linearity condition for L and the action (E.25) gets an extra term Z ∂ ¯ S = − E 3 − 2Ls(L) + (S + S) S Ω + h.c. . (E.38) ∂S

¯ 2 − 8R) and integratWhen we calculate the component Lagrangian, after inserting the operator (D R ing by parts and neglecting the Einstein term −3 E, we obtain Z 1 E ∂ 2 2 ¯ ¯ ¯ S ∋ − s(L)(D − 8R)L − S(S + S)(D − 8R) Ω + h.c. 8 R ∂S Z 1 E ∂ 2 2 ¯ ¯ ¯ = (E.39) s(L)(D − 8R)Ω + {s(L) − S}(S + S)(D − 8R) Ω + h.c.. 8 R ∂σ

Using (E.33) the last term cancels out and the first term is just the counterpart of the second term in (E.34) in the linear formulation. Finally if we take Ω0 = Ω0 (L), the equation of motion for Σ just gives (E.29)–(E.31). but using (E.27) the equation of motion for L gives ∂Ω0 1 δS = 2Ls′ (L) + k′ (L) − 2s(L)eK/3 = 0, E δL ∂L so now we get a different differential equation for s(L) but we still get the same action.

F

Notations and conventions

In this Appendix we summarize our notation and conventions. 96

(E.40)

F.1

Sign conventions

Our Dirac matrices and space-time metric signature (+ − −−) are those of Bjorken and Drell or Itzykson and Zuber. We use upper case notation (R, Γ) for derivatives of the K¨ aher metric Kim ¯ , and lower case (r, γ) for derivatives of the space-time metric gµν . Our sign conventions for, respectively, the Riemann tensor, Ricci tensor, and curvature scalar are µ rνρσ = gµλ rλνρσ = ∂σ Γµνρ − ∂ρ Γµνσ + Γµστ Γτνρ − Γµρτ Γτνσ ,

ρ rµν = rµρν ,

r = g µν rµν . (F.1)

The general coordinate covariant derivative is defined by ρ ∇µ Aν = ∂µ Aν − γµν Aρ ,

etc.

(F.2)

We use identical definitions for the K¨ ahler curvature and scalar field reparameterization derivatives with gµν → Kim γ → Γ, r → R, ∇µ → Di , Dm (F.3) ¯ = Kmi ¯ , ¯. We use the Yang-Mills sign conventions of [4] and [42]: Dµ φb = ∇µ φb + iAaµ (Ta )bc φc , ¯ ¯ ¯ Dµ φ¯b = ∇µ φ¯b − iAaµ (Ta )bc¯φ¯c¯,

(F.4) ¯ (Ta )bc¯

=

(Ta∗ )bc

=

(Ta )cb .

(F.5)

Note that Dµ is used for general coordinate and Yang-Mills covariant derivatives, while Dµ is fully covariant. Thus for a fermion ψ, Dµ ψ includes the spin, Yang-Mills, K¨ ahler and reparameterization ¯, (as well as the affine connection for the graviton ψµ ), and for a function of scalar fields z i , z¯m ¯ Dµ f (z, z¯) = Dµ z i Di f (z, z¯) + Dµ z m Dm ¯), ¯ f (z, z

(F.6)

with repeated indices summed. F.2

Supergravity conventions

We work in K¨ ahler U (1) superspace [5]. The tree level Lagrangian we are working with is defined by the K¨ ahler potential K, superpotential W and gauge kinetic function fab given in (2.1). A generic chiral superfield is denoted by Z i : Z i = S, T, Φ = (z i , χi , F i ),

97

(F.7)

with components zi = Z i ,

1 χiα = √ Dα Z i , 2

1 1 F i = − D α Dα Z i ≡ − D 2 Z i , 4 4

(F.8)

where Z| denotes the lowest component, and the antichiral superfields are ¯ ¯ ¯ ¯m Z¯ m = (Z m )† = (¯ zm ,χ ¯m , F ¯ ).

(F.9)

We also use the notation Z p = T i , Φa = (z p , χp , F p ),

i = 1, 2, 3,

(F.10)

for the moduli i

i

T i = (ti , χt , F t )

(F.11)

Φa = (φa , χa , F a ).

(F.12)

and the gauge charged chiral multiplets

The dilaton chiral supermultiplet decomposes as S = (s, χs , F s ),

s = x + iy.

(F.13)

The Yang-Mills superfield strengths are a Wαa = (λaα , Fµν , D a ),

(F.14)

and the supergravity supermultiplet includes the vierbein eαµ , the gravitino ψµ and the auxiliary ¯ Solution of the tree level equations of motion determine the auxiliary fields as fields Gαβ˙ , R, R. ¯ ¯m ¯ F i = −e−K/2 Ai = −eK/2 K im W ¯ + Km ¯W , R| =

1 ¯ † , M = e−K/2 A = eK/2 W (z) = R 2

1 1 D a | = D a = Ki (T a z)i , x x Gαβ˙ = 0,

(F.15) (F.16)

where here (and throughout the text) the notation O(Ψ)| denotes the bosonic component of the functional O of the superfields Ψ, that is, the θ = θ¯ = 0 component with all fermions set to zero. Our definition of M is such that the vacuum value hM i is the gravitino mass; it differs by a factor −1/3 from that of [5]. The covariant derivatives Ai1 ···in = Di1 · · · Din A 98

(F.17)

of the modular covariant operator A = eK W,

¯

¯

A → eF A,

Ai → eF Ai ,

etc.

(F.18)

are defined in (2.5). Note that for the K¨ ahler potential and superpotential W we use the conventional notation for ordinary derivatives Ki1 ···in = ∂i1 · · · ∂in K, F.3

Wi1 ···in = ∂i1 · · · ∂in W.

(F.19)

PV superfields

To regulate the superpotential couplings we introduce modular covariant PV fields ˙A Z˙ P = T˙ I , Φ

(F.20)

that transform like dZ p under gauge and modular transformations. To make the K¨ ahler potential for these fields modular invariant and their superpotential modular covariant, we need to introduce three additional fields Z˙ N , N = 1, 2, 3, and we also use the notation Z˙ ρ = Z˙ N , Z˙ P .

(F.21)

These fields acquire mass through invariant couplings to the PV superfields Y˙ ρ = Y˙ N , Y˙ P ,

(F.22)

where Y˙ P transforms like Kp¯q dZ¯ q¯ under YM transformations. These have no other superpotential couplings; their divergent contributions to field strength terms are canceled by the PV fields, introduced in Appendix D.1, Ψ C = U A , UA , V A , Φ N , Φ n ,

N, n = 1, 2, 3,

(F.23)

where ΦN,n are gauge singlets, and U A , UA , V A form a real (reducible) representation of the gauge group such that (3.25) is satisfied. Their masses not invariant under U (1)X and modular transformations; they reflect chiral matter contributions to the anomaly. To regulate all the nonrenormalizable terms in the superpotential and YM couplings, we also need fields ZeP ; they acquire invariant masses through coupling to YeP . These transform, respectively, like dZ p and Kp¯q dZ¯ q¯ under YM and modular transformations. 99

To regulate the renormalizable Yang-Mills couplings we introduce chiral PV fields YbP that transform like Kp¯q dZ¯ q¯ under YM and modular transformations, and acquire invariant masses through coupling bP that transform like Z˙ P . We also need chiral superfields ϕa , ϕâ , ϕã in the to chiral superfields Z adjoint representation of the YM gauge group. The fields ϕa , ϕâ , regulate gauge couplings to matter and to the gravity sector, respectively, and couple to one another in an invariant mass term; ϕã has no couplings to light matter and its noninvariant mass term reflects the gaugino/gauge contribution to the modular anomaly. To regulate nonrenormalizable dilaton/YM couplings we introduce chiral superfields φS , that acquire invariant masses through superpotential couplings to chiral fields φS , and chiral superfields θs and Abelian vector superfields Vs that acquire invariant masses through a superhiggs mechanism. Finally, to regulated additional nonrenormalizable gravity couplings we need chiral superfields φC , with noninvariant masses that reflect the gravity sector contribution to the modular anomaly, and chiral superfields θ0 and Abelian vector superfields V0 that acquire invariant masses through a superhiggs mechanism. Unless otherwise specified, ΦC denotes any chiral PV superfield with PV signature η C = ±1. F.4

(F.24)

The covariant derivative expansion

Here we collect the operators [17] that appear in the covariant derivative expansion used in the evaluation of the variation of the action. If F (x) is a scalar field operator, we define Fˆ =

∞ X (−i)n

n=0

n!

D·

∂ ∂p

n

F (x),

D·

∂ ∂ F (x) ≡ [Dµ , F (x)] . ∂p ∂pµ

(F.25)

In addition we define ∞ X n+1 ∂ ∂ n Gµ = , Gνµ −iD · (n + 2)! ∂p ∂p ν n=0

Gνµ = [Dµ , Dν ].

The full formal expansions of the following space-time curvature-dependent operators 4 ∂ ∂2 ∂3 i 1 , T µν = gµν − r µρσν ρ σ + ∇τ r µρσν ρ σ τ + O 3 ∂p ∂p 6 ∂p ∂p ∂p ∂p4 3 1 µρσν ∂ ∂2 µν µ µ P γν = P = γ − r γν ρ σ + O , 6 ∂p ∂p ∂p3 100

(F.26)

(F.27) (F.28)

i ∂2 δµ = (∇µ rρν − ∇ν rρµ ) +O 9 ∂pρ ∂pν 2 ∂ i ∂ r +O , X = − + ∇µ r 3 3 ∂pµ ∂p2

∂3 ∂p3

,

(F.29) (F.30)

are given38 in Appendix A of [35].

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There is a sign error in the third and second terms, respectively, of the expressions for Tµν and X in Eq. (A.19) of [17].

101

[14] C. A. Scrucca and M. Serone, JHEP 0102, 019 (2001). [15] D. Butter and M. K. Gaillard, Phys. Lett. B 679, 519 (2009). [16] M.K. Gaillard and V. Jain, Phys. Rev. D 49, 1951 (1994). [17] M.K. Gaillard, V. Jain and K. Saririan, Phys. Rev. D 55, 833 (1997). [18] M.K. Gaillard, V. Jain and K. Saririan, Phys. Lett. B 387, 520 (1996). [19] D.Z. Freedman and B. Kors, JHEP 0611, 067 (2006). [20] I. Antoniadis, E. Gava and K. S. Narain, Phys. Lett. B 283, 209 (1992) and Nucl. Phys. B 383, 93 (1992). [21] G.L. Cardoso and B.A. Ovrut, Nucl. Phys. 392, 315 (1993), ibid. B 418, 535 (1993). [22] G. Girardi and R. Grimm, Annals Phys. 272, 49 (1999). [23] P. Binetruy, G. Dvali, R. Kallosh and A. Van Proeyen, Class. Quant. Grav. 21 3137 (2004). [24] H. Elvang, D. Z. Freedman and B. Kors, JHEP 0611, 068 (2006). [25] W. A. Bardeen, Phys. Rev. 184, 1848 (1969). [26] J. Wess and B. Zumino Phys. Lett. B 37, 95 (1971). [27] W. A. Bardeen and B. Zumino, Nucl. Phys. B 244, 421 (1984). [28] M.A. Shifman, Nucl. Phys. B 352, 87 (1991); M.A. Shifman, and A.I. Vainshtein Nucl. Phys. B 365, 312 (1991); A.A. Iogansen, Sov. J. Nucl. Phys. 54 (1991). [29] D. Butter, arXiv:0911.5426 [hep-th]. [30] L.E. Ibàn ˜ez, H.-P. Nilles and F. Quevedo, Phys. Lett. 187, (1987) 25; A. Font, L. Ibàn ˜ez, D. Lust and F. Quevedo, Phys. Lett. B 245, (1990) 401; A. Font, L. E. Ibàn ˜ez, F. Quevedo and A. Sierra, Nucl. Phys. B 331, (1990) 421. [31] Joel Giedt, Annals Phys. 289, 251 (2001) and 297, 67 (2002).

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103

The Anomaly Structure of Regularized Supergravity

The Anomaly Structure of Regularized Supergravity

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The origin of electronic band structure anomaly in ... - CyberLeninka

The anomaly structure of theories with external gravity

Electronic Structure, Phonons and Dielectric Anomaly in ...

Yukawa structure, flavour and CP violation in Supergravity

Anomaly candidates and invariants of D= 4, N= 1 supergravity theories

The homogeneity conjecture for supergravity

Supergravity duals of gauge theories from F (4) gauged supergravity ...

Comparing Regularized and Non-Regularized ... - Springer Link

Structure Regularized Neural Network for Entity Relation Classification ...

Investigation of Hot Flow Anomaly Structure Observed ... - Springer Link

The Gaugings of Maximal D= 6 Supergravity

Cosmological Implications of the Supergravity Tracking Potential

BPS Branes in Supergravity

MATROID THEORY AND SUPERGRAVITY