arXiv:0902.3805v4 [hep-th] 4 Dec 2009
Preprint typeset in JHEP style - HYPER VERSION
AEI-2009-020 NSF-KITP-09-17
On Integrable Backgrounds Self-dual under Fermionic T-duality
Ido Adam1 , Amit Dekel2 and Yaron Oz2 1
Max-Planck-Institut f¨ ur Gravitationsphysik Albert Einstein Institut Am M¨ ulehnberg 1, 14476 Golm, Germany and Kavli Institute for Theoretical Physics University of California Santa Barbara, CA 93106-4030, USA 2 Raymond and Beverly Sackler School of Physics and Astronomy Tel-Aviv University, Ramat-Aviv 69978, Israel E-mails:
[email protected],
[email protected],
[email protected]
Abstract: We study the fermionic T-duality symmetry of integrable Green-Schwarz sigma-models on AdS backgrounds with Ramond-Ramond fluxes in various dimensions. We show that sigma-models based on supercosets of PSU supergroups, such as AdS2 × S 2 and AdS3 × S 3 are self-dual under fermionic T-duality, while supercosets of OSp supergroups such as non-critical AdS2 and AdS4 models, and the critical AdS4 × CP 3 background are not. We present a general algebraic argument to when a supercoset is expected to have a fermionic T-duality symmetry, and when it will fail to have one. Keywords: Duality in Gauge Field Theories, String Duality.
Contents 1. Introduction and summary
1
2. AdSp × S p target-spaces 2.1 The AdS2 × S 2 target-space 2.2 The AdS3 × S 3 target-space
2 3 5
3. Models based on the ortho-symplectic supergroup 3.1 The Green-Schwarz sigma-model on AdS2 with four supersymmetries 3.2 The AdS4 target-space 3.3 The AdS4 × CP 3 target-space 3.3.1 Current expansion in fermions 3.4 The AdS2 target-space with eight supersymmetries
8 8 10 11 13 15
4. A general analysis
16
A. The osp(2|2) algebra
18
B. The osp(2|4) algebra
19
C. The osp(6|4) algebra in so(1, 2) ⊕ u(3) basis
20
D. The psu(1, 1|2) algebra
22
E. The psu(1, 1|2) ⊕ psu(1, 1|2) algebra
23
1. Introduction and summary Recently, Alday and Maldacena proposed [1] that planar N = 4 SU(N) SYM MHV gluon scattering amplitudes at leading order in the strong ’t Hooft coupling expansion can be calculated using the dual gravity (string) description. A crucial step in the calculation procedure is an application of an ordinary bosonic T-duality transformation to the four CFT coordinates of AdS5 × S 5 . This suggestion implies that such amplitudes possess a dual conformal symmetry at strong coupling originating from the fact that AdS5 is self-dual under the T-duality. This dual symmetry has been observed also in gluon scattering amplitudes calculated in the weakly-coupled gauge theory description [2, 3].
–1–
Using the AdS/CFT duality it was shown that such a symmetry is expected to be valid at all values of the ’t Hooft coupling [4, 5]. This was done by proving that the aforementioned T-duality together with a novel T-duality of Grassmann-odd coordinates of the target-superspace form an exact quantum duality under which the full AdS5 × S 5 superstring is self-dual. Since both the original background and its dual possess superconformal symmetry, it means that each of them has both the manifest superconformal symmetry and a dual one. Furthermore, [4, 5] have linked the duals of the superconformal Noether currents to the non-local currents implied by the integrability of the superstring on AdS5 × S 5 [6].
In view of these results it is natural to inquire how ubiquitous this dual superconformal symmetry is. The aim of this paper is to consider this question by analyzing the fermionic T-duality symmetry of integrable Green-Schwarz sigma-models on AdS backgrounds with Ramond-Ramond fluxes in various dimensions. Some of these sigma-models have been constructed in [7, 8, 9, 10]. We show that sigma-models based on supercosets of PSU supergroups, such as AdS2 ×S 2 and AdS3 ×S 3 are self-dual under fermionic T-duality. Supercosets of OSp supergroups such as non-critical AdS2 and AdS4 models, and the critical AdS4 ×CP 3 background (whose coset model was constructed and explored in [11, 12, 13, 14, 15]) are not self-dual. In the OSp models we find that the Buscher procedure [16, 17] fails due to a lack of appropriate quadratic terms. This is because the Cartan-Killing bilinear form of the ortho-symplectic group is non-zero only for products of different Grassmann-odd generators. Thus, one may expect this to imply that in those cases in which a dual theory exists, the theory does not have a dual superconformal symmetry. The paper is organized as follows. In section 2 we show that the AdSp × S p (p = 2, 3) target-spaces based on PSU cosets are self-dual under a combination of bosonic and fermionic T-duality. In section 3 we consider models based on supercosets of the ortho-symplectic supergroup, for which the Buscher procedure [16, 17] of gauging an isometry of the target-space in order to obtain the T-dual sigma-model fails. These include the non-critical superstring on AdS2 with four supersymmetries and AdS4 with eight supersymmetries, the supercoset construction of AdS4 × CP3 , and a model of AdS2 with eight supersymmetries. In section 4 we present a general algebraic argument to when a supercoset is expected to have a fermionic T-duality symmetry, and when it will fail to have one. In the appendices we provide details on the relevant superalgebras that are used in the paper.
2. AdSp × S p target-spaces In this section we show that the AdSp × S p (p = 2, 3) target-spaces based on PSU supercosets are self-dual under a combination of bosonic and fermionic T-duality.
–2–
2.1 The AdS2 × S 2 target-space The target superspace whose bosonic part is AdS2 × S 2 can be realized as the coset space PSU(1, 1|2)/(U(1) × U(1))1 . The Green-Schwarz sigma-model for supercosets with a Z4 automorphism is given by the action [7] Z 1 ¯ 1¯ R2 2 ¯ d zStr J2 J2 + J1 J3 − J1 J3 , (2.1) S= 4πα′ 2 2 where J = g −1 ∂g for g ∈ G and Ji is the current J restricted to the invariant subspace Hi of the Z4 automorphism of the algebra of the group G. Using the psu(1, 1|2) algebra given in appendix D, the sigma-model (2.1) takes the form Z h1 R2 1 1 1 2 d z S= (JP − JK )(J¯P − J¯K ) + JD J¯D + JR1 J¯R1 + JR2 J¯R2 − ′ 4πα 2 2 2 2 i i − ηαβ (JQα J¯Qβ − JQˆ α J¯Qˆ β + JSα J¯Sβ − JSˆα J¯Sˆβ ) , (2.2) 2 where η12 = η21 = 1 and zero otherwise. The analysis here will follow that of [4]. A general group element g ∈ PSU(1, 1|2) can be parameterized as ′
g = exP +x K+θ
αQ
α +ξ
αS
α
eB ,
ˆ α +ξˆα Sˆα D P y i Ri /y ˆα Q
eB ≡ eθ
y e
.
(2.3)
We partially fix the κ-symmetry such that ξ α = 0, and we use the U(1) × U(1) gauge symmetry generated by P + K and R3 to set x′ = 0, thus the coset representative is g = exP +θ
αQ
α
eB .
(2.4)
ˆ S, ˆ D, Ri } transforms P and Qα among Using the fact that the group generated by {Q, themselves, the components of the Maurer-Cartan 1-form are JP = [e−B (dxP + dθα Qα )eB ]P , JK = 0 ,
JQˆ α = [e−B deB ]Qˆ α ,
JD = [e−B deB ]D ,
JQα = [e−B (dxP + dθβ Qβ )eB ]Qα , JSα = 0 ,
JRi = [e−B deB ]Ri .
JSˆα = [e−B deB ]Sˆα , (2.5)
This sigma-model is T-dualized in the directions of the Abelian sub-algebra formed by the generators P and Qα according to the procedure of [16, 17], by introducing the gauge fields A, A¯ for the translation P and Aα , A¯α for the supercharges Qα and the corresponding Lagrange multipliers x˜ and θ˜α : Z h1 R2 2 ¯ + A¯α Qα )eB ]P − S= d z [e−B (AP + Aα Qα )eB ]P [e−B (AP 4πα′ 2 1
For this to be a superstring, this superspace has to be supplemented by an additional internal CFT with the appropriate central charge. It is not clear that such a description exists using such a partial Green-Schwarz action (the hybrid model discussed in [18] might be a better option for this kind of superstring construction).
–3–
i ¯ + A¯γ Qγ )eB ]Q + − ηαβ [e−B (AP + Aγ Qγ )eB ]Qα [e−B (AP β 2 1 1 1 i + JD J¯D + JR1 J¯R1 + JR2 J¯R2 + ηαβ (JQˆ α J¯Qˆ β + JSˆα J¯Sˆβ ) + 2 2 2 i 2 α α ¯ ˜ ¯ ¯ ¯ + x˜(∂A − ∂ A) + θα (∂A − ∂ A ) .
(2.6)
It is convenient to change variables to A′ = [e−B (AP + Aα Qα )eB ]P ,
A′α = [e−B (AP + Aβ Qβ )eB ]Qα
(2.7)
and similarly for the right-moving gauge fields. Using the inverted relations A = [eB (A′ P + A′α Qα )e−B ]P ,
Aα = [eB (A′ P + A′β Qβ )e−B ]Qα ,
the action in terms of the new variables reads Z h1 i R2 2 ′ ¯′ d z A A − ηαβ A′α A¯′β + . . . − S= ′ 4πα 2 2 ′ B −B ¯ − ∂ x˜(A [e P e ]P + A′α [eB Qα e−B ]P ) + + ∂ x˜ A¯′ [eB P e−B ]P + A¯′α [eB Qα e−B ]P − − ∂¯θ˜α A′ [eB P e−B ]Qα + A′β [eB Qβ e−B ]Qα + i ′β B −B ′ B −B ˜ ¯ ¯ , + ∂ θα A [e P e ]Qα + A [e Qβ e ]Qα
(2.8)
(2.9)
where . . . denotes the spectator terms. Since the gauge fields appear quadratically in the action, one can integrate them out by substituting their equations of motion A′ = −2[eB ∂ x˜P e−B ]P − 2[eB ∂ θ˜α P e−B ]Qα = −2[e−B (∂ x˜K + i∂ θ˜α S α )eB ]K , A¯′ = 2[eB ∂¯x˜P e−B ]P + 2[eB ∂¯θ˜α P e−B ]Qα = 2[e−B (∂¯x˜K + i∂¯θ˜α S α )eB ]K , A′α = 2iη αβ ([eB ∂ x˜Qβ e−B ]P − [eB ∂ θ˜γ Qβ e−B ]Qγ ) = −2η αβ [e−B (∂ x˜K + i∂ θ˜γ S γ )eB ]S β ,
A¯′α = 2iη αβ ([eB ∂¯x˜Qβ e−B ]P − [eB ∂¯θ˜γ Qβ e−B ]Qγ ) = −2η αβ [e−B (∂¯x˜K + i∂¯θ˜γ S γ )eB ]S β .
(2.10)
and obtain after rescaling x˜ → 12 x˜ and θ˜α → 21 θ˜α Z h1 R2 2 S= d z [e−B (∂ x˜K + i∂ θ˜α S α )eB ]K [e−B (∂¯x˜K + i∂¯θ˜α S α )eB ]K − 4πα′ 2 i i −B ¯ γ B −B γ B ¯ ˜ ˜ − ηαβ [e (∂ x˜K + i∂ θγ S )e ]Sα [e (∂ x˜K + i∂ θγ S )e ]Sβ + . . . ,(2.11) 2 where we used ǫαβ to lower the spinor indices of η αβ . In order to show the self-duality of this background under the above T-duality, the original action has to be brought to the same form as (2.11). One can easily check by using the ǫβγ σαijγ = ǫαγ σβijγ that the psu(1, 1|2) algebra admits the automorphism P ↔K ,
D → −D ,
Qα ↔ S α ,
–4–
ˆ α ↔ Sˆα , Q
(2.12)
with the rest of the generators unchanged. Applying this automorphism combined with the change of variables x → x˜ ,
θα → iθ˜α ,
θˆα ↔ ξˆα ,
yi →
yi , y2
(2.13)
to (2.5) one obtains (2.11). In order to complete the proof of quantum mechanical equivalence we also have to show that the Jacobian functional determinant from the change of variables (2.7) is the identity. The transformation of variables was done using eB . Since it is in a unitary subgroup of the PSU(1, 1|2) group, its super-determinant is equal to one and hence the Jacobian of the transformation is trivial and does not induce any shift of the dilaton. In order to see this explicitly, one can treat (A, Aα ) as a vector and ˆ α and Sˆα as matrices acting on write the adjoint action of the generators D, Rij , Q this vector 1 0 0 0 [adD ] = , [adRij ] = , 0 12 δα β 0 − 12 σijα β 0 0 0 −ǫαβ , (2.14) , [adSˆα ] = [adQˆ α ] = −iδα β 0 0 0 which evidently have a vanishing supertrace. Hence, the super-determinant of similarity transformations with elements of the group generated by these generators is the identity. 2.2 The AdS3 × S 3 target-space We construct the Green-Schwarz sigma-model on AdS3 × S 3 using the supercoset manifold (PSU(1, 1|2)×PSU(1, 1|2))/(SU(1, 1)×SU(2)) with 16 supersymmetry generators. Using the Z4 structure (E.4) we have 1 J2 = Ja− (Pa − Ka ) + JD D + Jµ Rµ = (JPa − JKa )(Pa − Ka ) + JD D + Jµ Rµ (2.15) 2 1 I + aIJ JQJααˆ )(SαI αˆ + aIK QK J1 = JSαI αˆ +aIJ QJααˆ (SαI αˆ + aIK QK αα ˆ) αα ˆ ) = (JSα α ˆ 2 1 I − aIJ JQJααˆ )(SαI αˆ − aIK QK J3 = JSαI αˆ −aIJ QJααˆ (SαI αˆ − aIK QK αα ˆ) . αα ˆ ) = (JSα α ˆ 2 Thus, 1 Str(J2 J¯2 ) = − (JPa − JKa )(J¯Pb − J¯Kb )η ab + JD J¯D + JRµ J¯Rµ , 2
(2.16)
1 Str(J1 J¯3 ) = (σ 3 )IK ǫαβ ǫαˆ βˆ(JSαI αˆ + aIJ JQJααˆ )(J¯S Kˆ − aKLJ¯QLˆ ) , ββ ββ 2
(2.17)
–5–
and the action reads Z h 1 R2 2 d z − (JPa − JKa )(J¯Pb − J¯Kb )ηab + JD J¯D + JRµ J¯Rµ S= 4πα′ 2 i 1 + (σ 3 )IK ǫαˆ βˆǫαβ JSαI αˆ J¯S Kˆ + JQIααˆ J¯QKˆ . ββ ββ 2 Next we parameterize the supergroup element ˆ ˆ ˆ g = exp(xa Pa + θαα1 Q1ααˆ ) exp(θαα2 Q2ααˆ + ξ αα1 Sα1 αˆ )y D exp(yµ Rµ /y)
(2.18)
(2.19)
ˆ ≡ exp(xa Pa + θαα1 Q1ααˆ )eB .
Define the currents J = g −1 dg = j + j,
ˆ j = e−B (dxa Pa + dθαα1 Q1ααˆ )eB ,
j = e−B deB .
(2.20)
Using these definitions and the algebra (E.1) we get the currents JPa = jPa ,
JQ1ααˆ = jQ1ααˆ , JKa = 0,
JQ2ααˆ = jQ2ααˆ , J D = jD ,
JSα1 αˆ = jSα1 αˆ ,
JSα2 αˆ = 0 (2.21)
JRµ = jRµ ,
the action reads R2 S= 4πα′
h 1 d2 z − jPa ¯jPb ηab + jD¯jD + jRµ¯jRµ 2 i 1 ¯ ¯ ¯ + ǫαβ ǫαˆ βˆ jSα1 αˆ jS 1 ˆ + jQ1ααˆ jQ1 ˆ − jQ2ααˆ jQ2 ˆ . ββ ββ ββ 2 Introducing gauge fields and Lagrange multipliers we get the action Z h 1 R2 2 S= d z − A′Pa A¯′Pb ηab + jD¯jD + jRµ¯jRµ 4πα′ 2 1 ′ ′ ¯ ¯ ¯ + ǫαβ ǫαˆ βˆ jSα1 αˆ jS 1 ˆ + AQ1 AQ1 − jQ2ααˆ jQ2 ˆ ˆ αα ˆ ββ ββ ββ 2 i a a 1 ¯ ˜ ¯ ¯ ¯ +˜ xa (∂A − ∂ A ) + θααˆ (∂AQ1ααˆ − ∂ AQ1ααˆ ) , Z
(2.22)
(2.23)
where
A′ = e−B (Aa Pa + A1ααˆ Q1ααˆ )eB ,
A1ααˆ ≡ AQ1ααˆ ,
APa ≡ Aa ,
(2.24)
A = eB (A′a Pa + A′1ααˆ Q1ααˆ )e−B . Solving for A′ we get A′a = 2(∂ x˜b [eB Pa e−B ]Pb + ∂ θ˜α1 αˆ [eB Pa e−B ]Q1ααˆ )
–6–
(2.25)
= 2ηac [e−B (∂ x˜b Kb + ∂ θ˜1ααˆ Sα2 αˆ )eB ]Kc A¯′a = −2(∂¯x˜b [eB Pa e−B ]Pb + ∂¯θ˜α1 αˆ [eB Pa e−B ]Q1ααˆ )
(2.26)
= −2ηac [e−B (∂¯x˜b Kb + ∂¯θ˜1ααˆ Sα2 αˆ )eB ]Kc
1 ǫαβ ǫαˆ βˆA′Q1 = −(∂ x˜b [eB Q1ααˆ e−B ]Pb − ∂ θ˜β1 βˆ[eB Q1ααˆ e−B ]Q1 ˆ ) ˆ ββ ββ 2 2 B = −ǫαβ ǫαˆβˆ[e−B (∂ x˜b Kb + ∂ θ˜1γˆγ Sγˆ γ )e ]S 2 ˆ
(2.27)
ββ
1 ǫαβ ǫαˆ βˆA¯′Q1 = −(∂¯x˜b [eB Q1ααˆ e−B ]Pb − ∂¯θ˜β1 βˆ[eB Q1ααˆ e−B ]Q1 ˆ ) ˆ ββ ββ 2 −B ¯ b 1γˆ γ 2 B = −ǫαβ ǫαˆ βˆ[e (∂ x˜ Kb + ∂¯θ˜ Sγˆγ )e ]S 2 ˆ .
(2.28)
ββ
Thus,
Z h1 R2 2 S= (2.29) d z A′ A¯′ ηab + jD¯jD + jRµ¯jRµ 4πα′ 2 Pa Pb i 1 + ǫαβ ǫαˆβˆ jSα1 αˆ¯jS 1 ˆ − A′Q1 A¯′Q1 − jQ2ααˆ¯jQ2 ˆ ˆ αα ˆ ββ ββ ββ 2 h 2 Z R 1 −B 2 = d z − 4[e (∂ x˜b Kb + ∂ θ˜1ααˆ Sα2 αˆ )eB ]Ka [e−B (∂¯x˜b Kb + ∂¯θ˜1ααˆ Sα2 αˆ )eB ]Kb η ab 4πα′ 2 1 +jD¯jD + jRµ¯jRµ + ǫαβ ǫαˆ βˆ jSα1 αˆ¯jS 1 ˆ − jQ2ααˆ¯jQ2 ˆ ββ ββ 2 i . −4[e−B (∂ x˜b Kb + ∂ θ˜1γˆγ S 2 )eB ]S 2 [e−B (∂¯x˜b Kb + ∂¯θ˜1γˆγ S 2 )eB ]S 2 γˆ γ
γˆ γ
αα ˆ
ˆ ββ
We can define J ’s ′
2 B J ′ = e−B (2∂ x˜b Kb + 2∂ θ˜1γˆγ Sγˆ γ )e ,
(2.30)
and the action reads R2 S= 4πα′
1 d z − JP′ a J¯P′ b ηab + jD¯jD + jRµ¯jRµ 2 i 1 + ǫαβ ǫαˆ βˆ jSα1 αˆ¯jS 1 ˆ − JS′ 2 J¯S′ 2 − jQ2ααˆ¯jQS 2 ˆ . αα ˆ ˆ ββ ββ ββ 2 Applying the following automorphism of the algebra Z
2
3 Ka ↔ −σab Pb ,
D → −D, R1 → R1 ,
h
N12 → −N12 ,
R2 → −R2 ,
N23 → N23 ,
Jab → −Jab ,
(2.31)
(2.32)
R3 → −R3 ,
N31 → −N31 ,
1 QIααˆ ↔ σIJ σα1ˆ βˆSαJβˆ ,
followed by the change of variables θα1 αˆ → 2σα1ˆβˆθ˜α2 βˆ,
3 xa → −2σab x˜b ,
θα2 αˆ ↔ σα1ˆ βˆξα1 βˆ,
y1 → y1 /y 2 ,
y2 → −y2 /y 2 ,
the action (2.31) is mapped to the original one (2.22).
–7–
(2.33) y3 → −y3 /y 2 ,
3. Models based on the ortho-symplectic supergroup In this section we consider models based on supercosets of the ortho-symplectic supergroup, for which the Buscher procedure [16, 17] of gauging an isometry of the target-space in order to obtain the T-dual sigma-model fails. These include the non-critical superstring on AdS2 with four supersymmetries and AdS4 with eight supersymmetries, the supercoset construction of AdS4 × CP3 which is conjectured to be dual to superconformal Chern-Simons theory in three dimensions [19], and a model of AdS2 with eight supersymmetries. 3.1 The Green-Schwarz sigma-model on AdS2 with four supersymmetries The non-critical AdS2 background with RR-flux can be realized as the supercoset OSp(2|2)/(SO(1, 1) × SO(2)). The Green-Schwarz action is of the form (2.1) (see Appendix A for the details of the algebra and the conventions that are used). Fixing the SO(1, 1) × SO(2) gauge symmetry and κ-symmetry one can choose the coset representative ¯ ¯ ¯¯ (3.1) g = exP +θQ eθQ+ξS y D . The generators P and Q form an Abelian subalgebra, which we will attempt to T-dualize. In this parameterization the Maurer-Cartan current J takes the form J =
1 1 1 ¯ + y 1/2 ∂ ξ¯S¯ + ∂x − 2∂θθ¯ P + 1/2 ∂θQ + 1/2 2∂θθ¯ − ∂x ξ¯ + ∂ θ¯ Q y y y ∂y ¯ ¯ . + − 2∂θξ D + i∂θξR (3.2) y
The sigma-model can be written explicitly as Z h 1 R2 1 2 S= d z − (JP + JK )(J¯P + J¯K ) + JD J¯D + JQ J¯Q¯ + JQ¯ J¯Q − ′ 4πα 2 2 i (3.3) − JS J¯S¯ − JS¯ J¯S . The action in terms of the coordinates is then given by Z ¯ + ∂y ∂y ¯ −∂x∂x 1¯ R2 2 ¯ + ∂θ∂x) ¯ + 1 ξ(∂y ¯ ∂θ ¯ + ∂θ∂y) ¯ + d z − 2 θ(∂x ∂θ S= ′ 2 4πα 2y y y 1¯ ¯ 1 4 ¯¯ ¯ ¯ ¯ ¯ ¯ ¯ (3.4) + θξ∂θ∂θ + ξ(∂θ∂x − ∂x∂θ) + (∂θ∂ θ + ∂ θ ∂θ) . y y y Note that the action is indeed quadratic in θ so naively one should be able to Tdualize it along that coordinate. An equivalent action can be written using two gauge fields [16, 17] Z ¯ R2 1¯ 1¯ −Ax A¯x + ∂y ∂y 2 ¯ ¯ ¯ ¯ d z S= − 2 θ(A x Aθ + Aθ Ax ) + ξ(∂y Aθ + Aθ ∂y) + ′ 2 4πα 2y y y
–8–
4 ¯ ¯ 1¯ 1 ¯¯ ¯¯ ¯ x − ∂ A¯x ) + ¯ ¯ + θ¯ξA ˜(∂A θ Aθ + ξ(Aθ Ax − Ax Aθ ) + (Aθ ∂ θ + ∂ θ Aθ ) + x y y y ˜ ∂A ¯ θ − ∂ A¯θ ) . + θ( (3.5) The classical equations of motion for the gauge fields are 1 1¯ 1¯ ξAθ − ∂ x˜ = A + θA − x θ 2y 2 y2 y 1 1 1 ¯ Ax + 2 θ¯A¯θ + ξ¯A¯θ + ∂¯x˜ = 2 2y y y 1¯ 4 ¯¯ 1¯ 1 ¯ 1¯ ξ∂y − θ ξA + ξA − θA − ∂ θ − ∂ θ˜ = θ x x y2 y y y y 1 ¯ 4 1 1 1 ¯¯ θAx − ξ¯∂y + θ¯ξ¯A¯θ − ξ¯A¯x + ∂¯θ¯ + ∂¯θ˜ = 2 y y y y y
0,
(3.6)
0,
(3.7)
0,
(3.8)
0.
(3.9)
Since Grassmann variables such as θ¯ and ξ¯ have non-trivial kernels, one cannot solve for Aθ and for A¯θ . Solving for Ax and A¯x one gets ¯ θ + 2y ξA ¯ θ + 2y 2 ∂ x˜ , Ax = −2θA A¯x = −2θ¯A¯θ − 2y ξ¯A¯θ − 2y 2 ∂¯x˜ .
(3.10) (3.11)
Substituting (3.10) and (3.11) in the action (this can be done since they appear only quadratically in the action, but the classical procedure should be supplemented by a functional determinant coming from the Gaussian path integration of Ax and A¯x ) yields Z h R2 2 ¯ x˜A¯θ − 2y ξA ¯ θ ∂¯x˜ − 2θ∂ ¯ x˜A¯θ + 2θA ¯ θ ∂¯x˜ + S= d z − 2y 2∂ x˜∂¯x˜ − 2y ξ∂ 4πα′ i ¯ 1¯ ∂y ∂y ¯ + 1 (Aθ ∂¯θ¯ + ∂ θ¯A¯θ ) − ∂¯θA ˜ θ + ∂ θ˜A¯θ . (3.12) ¯θ + Aθ ∂y) + ξ(∂y A + 2y 2 y y Thus, after integrating out Ax and A¯x the remaining fermionic gauge fields Aθ and A¯θ serve as Lagrange multipliers forcing in the path integration over θ˜ that ¯ x˜ − 1 ξ∂y ¯ + 2y ξ∂ ¯ x˜ − 2θ∂ y 1 ¯ − 2y ξ¯∂¯x˜ − 2θ¯∂¯x˜ + ξ¯∂y y
1 ¯ ∂ θ − ∂ θ˜ = 0 , y 1 ¯ ¯ ¯˜ ∂θ − ∂θ = 0 , y
(3.13)
which express the non-zero modes of θ˜ in terms of the other fields, effectively reducing the path integration over θ˜ only to its zero-modes. This appears rather strange as the original action did include a term quadratic in θ. A possible explanation is that the quadratic term is κ-symmetry-exact, and hence does not influence the equations of motion.
–9–
In order to support the above claim we do the same computation with a different gauge choice for the κ-symmetry. The coset representative in this gauge can be parameterized as ¯¯ (3.14) g = exP +θQ eθQ+ξS y D , and the Maurer-Cartan current is J =
1 1 1 ¯ ¯ ¯ + y 1/2 ∂ξS + (∂x − 2∂θθ¯ − ∂xθξ)P + 1/2 (∂θ − ∂xξ + ∂θθξ)Q + 1/2 ∂ θ¯Q y y y ∂y i ¯ ¯ ¯ ¯ . + − ∂ θξ − ∂ξ θ D − (∂ θξ + ∂ξ θ)R (3.15) y 2
In this gauge the action is Z ¯ ¯ + ∂y ∂y ¯ R2 (−1 + θξ)∂x ∂x 1¯ 2 ¯ + ∂θ∂x) ¯ − S= d z − 2 θ(∂x ∂θ ′ 2 4πα 2y y 1 ¯ + 1 ξ(∂y ∂¯θ¯ + ∂ θ¯∂y) ¯ + 1 θ(∂y ¯ ∂ξ ¯ + ∂ξ ∂y) ¯ + − ξ(∂x∂¯θ¯ − ∂ θ¯∂x) y 2y 2y 1 1 ¯ ¯¯ ¯ + (1 + θξ)(∂θ ¯ ¯ ∂¯θ¯ + ∂ θ¯∂θ) . (3.16) + θξ(∂ θ∂ξ − ∂ξ ∂¯θ) 2 y This action has no quadratic term in the Grassmann coordinate θ so the usual procedure introduced by Buscher [16, 17] cannot be applied. This lends credence to the explanation that in the different gauge above the procedure failed because the quadratic term is κ-symmetry-exact. 3.2 The AdS4 target-space The non-critical superstring on AdS4 with eight supersymmetries can be constructed as the supercoset OSp(2|4)/(SO(3, 1) × SO(2)). Taking g ∈ OSp(2|4) in (2.1) one obtains the sigma-model Z h R2 2 S=− d z ηmn (JPm + JKm )(J¯Pn + J¯Kn ) + JD J¯D + 4πα′ (3.17) + 4iǫαβ JQα J¯Qˆ β − JQˆ α J¯Qβ + JSα J¯Sˆβ − JSˆα J¯Sβ , where the conventions for the algebra are given in Appendix B. The general group element can be parameterized as g = ex
mP
m +w
mK
m +θ
αQ
α
eB ,
ˆ α +ξ α Sα +ξˆα Sˆα D φR+ω mn Mmn ˆα Q
eB = eθ
y e
.
(3.18)
We assume that we can partially gauge-fix the κ-symmetry such that ξˆα = 0 and we will also fix the SO(3, 1) × SO(2) gauge symmetry by setting w m = 0, φ = 0 and ω mn = 0 essentially picking the specific coset representative g = ex
mP
m +θ
αQ
α
eB ,
ˆα Q ˆ α +ξ α Sα D
eB = eθ
– 10 –
y
.
(3.19)
One can check that the needed components of the Maurer-Cartan 1-form are JPm = e−B (dxn Pn + dθα Qα )eB Pm , JKm = 0 , JQα = e−B (dxm Pm + dθβ Qβ )eB Qα , JSα = e−B deB Sα , JQˆ α = e−B deB Qˆ α , JSˆα = 0 , JD = e−B deB D .
(3.20)
The action then becomes Z h −B −B R2 2 l α B ¯ p Pp + ∂θ ¯ β Qβ )eB e ( ∂x + d z η e (∂x P + ∂θ Q )e S=− mn l α Pn Pm 4πα′ −B B ¯ ¯ B + 4iǫαβ e−B (∂xm Pm + ∂θγ Qγ )eB e ∂e + e−B ∂eB D e−B ∂e ˆα D Q Qα i −B B −B ¯ m Pm + ∂θ ¯ γ Qγ )eB − e ∂e Qˆ α e (∂x . (3.21) Q β
Replacing the partial derivatives of xm and of θα by the gauge fields Am , A¯m , Aα and A¯α and then performing the field redefinition A′m = e−B (An Pn + Aα Qα )eB Pm , A′α = e−B (Am Pm + Aβ Qβ )eB Qα
(3.22)
with similar expressions for the right-moving sector and introducing the Lagrange multiplier terms forcing the gauge fields to be flat, the new equivalent action takes the form Z h −B B R2 2 ′m ¯′n ′α ¯ ¯ e ∂e d z η A A + J J + 4iǫ S=− mn D D αβ A ˆβ − ′ Q 4πα i ¯ m − ∂ A¯m ) + θ˜α (∂A ¯ α − ∂ A¯α ) . (3.23) − e−B ∂eB Qˆ α A¯′β + x˜m (∂A As can be immediately seen, just as for the case of the AdS2 space discussed in section 3.1 for the (physically trivial) AdS2 the fermionic gauge fields A′α and A¯′α appear only linearly in the action and hence integration over them will yield constraints rather than equations of motion. 3.3 The AdS4 × CP 3 target-space Using the same procedure as for the AdSn × S n models we construct the GreenSchwarz sigma model action (2.1) using the bilinear forms of (C.21), Z h R2 2 S= (3.24) d z − 2ηab (JPa + JKa )(J¯Pb + J¯Kb ) 4πα′ −JD J¯D − 2(JRkl J¯Rp˙q˙ + J¯Rkl JRp˙q˙ )(δ kp˙ δ lq˙ − δ kq˙ δ lp˙ ) i ˙ . −2iδ kl Cαβ JQlα J¯Qk˙ − J¯Qlα JQk˙ − JSαl J¯S k˙ + J¯Sαl JS k˙ β
β
– 11 –
β
β
(This coset model is a Green-Schwarz superstring with kappa-symmetry partially fixed so that the broken supersymmetries are gauged away, leaving only the fermionic coordinates corresponding to unbroken supersymmetries. This partial gauge-fixing is not compatible with all the string solutions so a general analysis would seem to require the use of the complete unfixed sigma-model derived in [20]. However, as the broken supersymmetries are not isometries of the background, they cannot participate in the T-duality transformation so we expect the OSp(6|4) coset model to suffice for this purpose.) We proceed by taking the gauge-fixed coset representative g = ex
aP
α l a +θl Qα
eB ,
α
l˙
α
l
eB ≡ eθl˙ Qα +ξl Sα y D e(Σy
kl R +Σy k˙ l˙R )/y ˙ l˙ kl k
,
(3.25)
where we also fixed six out of the eight κ-symmetry degrees of freedom by setting ξlα˙ = 0. Pa and θlα Qlα form an Abelian subalgebra. Next, we would like to write the currents in terms of this parameterization, but ¯ and {Q, S} ∼ R prevent us from in this case the commutation relations [Q, R] ∼ Q getting two kinds of currents as in (2.20), so JT = [e−B (dxa Pa + dθlα Qlα )eB ]T + [e−B deB ]T ≡ jT + jT .
(3.26)
j = jPa Pa + jQαl Qαl + jQα˙ Qαl˙ + jRkl Rkl + jRk˙ l˙ Rk˙ l˙ + jλκl˙ λκl˙ ,
(3.27)
We have l
and j = jQαl Qαl +jQα˙ Qαl˙ +jSlα Slα +jS α˙ Slα˙ +jRkl Rkl +jRk˙ l˙ Rk˙ l˙+jλκl˙ λκl˙+jD D+jMmn Mmn . (3.28) l
l
The action in terms of these currents reads Z h R2 2 S= d z − 2η ab jPa ¯jPb − jD¯jD (3.29) 4πα′ −2((j + j)Rkl (¯j + ¯j)Rp˙q˙ + (¯j + ¯j)Rkl (j + j)Rp˙q˙ )(δ kp˙ δ lq˙ − δ kq˙ δ lp˙ ) i ˙ kl ¯ ¯ ¯ ¯ . −2iδ Cαβ (j + j)Qlα (j + j)Qk˙ − (j + j)Qlα (j + j)Qk˙ − jSαl jS k˙ + jSαl jS k˙ β
β
β
β
We add gauge fields instead of the xa and θlα derivatives and a suitable Lagrange multiplier term. We define A′ = e−B (Aa Pa + Aαl Qlα )eB ,
(3.30)
(where Aa ≡ APa and Aαl ≡ AQlα ) and the action becomes Z h R2 2 S= d z − 2ηab A′Pa A¯′Pb − jD¯jD (3.31) 4πα′ −2((A′ + j)Rkl (A¯′ + ¯j)Rp˙q˙ + (A¯′ + ¯j)Rkl (A′ + j)Rp˙q˙ )(δ kp˙ δ lq˙ − δ kq˙ δ lp˙ ) ˙ kl ′ ′ ′ ′ ¯ ¯ ¯ ¯ −2iδ Cαβ (A + j)Qlα (A + j)Qk˙ − (A + j)Qlα (A + j)Qk˙ − jSαl jS k˙ + jSαl jS k˙ β β β β i a α l l ¯ a ) + θ˜ (∂ A¯ − ∂A ¯ ) . +˜ x (∂ A¯a − ∂A l α α
– 12 –
3.3.1 Current expansion in fermions To zeroth order in the fermions (θlα , θlα˙ , ξlα ) we get a bosonic sigma model with no fermions, which we can T-dualize, Z h R2 2 S0 = (3.32) d z − 2y 2ηab APa A¯Pb − jD ¯jD ′ 4πα n o i ¯ a) , −2 jRkl¯jRp˙q˙ + ¯jRkl jRp˙q˙ (δ kp˙ δ lq˙ − δ kq˙ δ lp˙ ) + x˜a (∂ A¯a − ∂A kl
k˙ l˙
where j = e−B deB , eB = y D e(Σy Rkl +Σy Rk˙ l˙)/y . Next, leaving only θlα terms we find mn mn y m˙ n˙ l y m˙ n˙ l˙ k y −B a α l B a 1/2 α k y , )Qα + gl˙ ( , )Qα , e (A Pa + Aλ Qα )e = yA Pa + y Ak fl ( y y y y (3.33) ′ thus the A ’s in terms of the A’s are A′a = yAa ,
1/2 α k A′α Ak fl ( l = y
y mn y m˙ n˙ , ), y y
A′α = y 1/2 Aαk glk˙ ( l˙
y mn y m˙ n˙ , ). y y
(3.34)
Plugging these in the action we have Z n h o R2 2 2 k p˙ lq˙ k q˙ lp˙ ¯ ¯ ¯ ¯ S1 = j + j j A −j j −2 j d z −2y η A Rkl Rp˙ q˙ (δ δ −δ δ ) (3.35) D D Rkl Rp˙ q˙ ab Pa Pb ′ 4πα i ˙ a α l l kl α n ¯β m α n β m ¯ ˜ ¯ ¯ ¯ ¯ −2iyδ Cαβ An fl Am gk˙ − An fl Am gk˙ + x˜ (∂ Aa − ∂Aa ) + θl (∂ Aα − ∂Aα ) Z n h o R2 2 2 ¯ ¯ ¯ ¯ = d z − 2y ηab APa APb − jD jD − 2 jRkl jRp˙q˙ + jRkl jRp˙q˙ (δ kp˙ δ lq˙ − δ kq˙ δ lp˙ ) ′ 4πα i ˙ m n a α l l ¯ ˜ ¯ ¯ ¯ − f g −2iyδ kl Cαβ Aαn A¯βm fln gkm + x ˜ (∂ A − ∂A ) + θ (∂ A − ∂A ) . a a ˙ l l α α k˙
The bosonic T-duality works as before, but now we also have a quadratic term for the fermions. The equation of motion for the fermions is
i αβ n ˙ m n C ∂θβ . (3.36) Aαm δ lk fln gkm ˙ − fl gk˙ = − 2y n m m n mn lk˙ The matrix M ≡ δ fl gk˙ − fl gk˙ = −4iy n˙ m˙ /y + O(y n˙ m˙ y kl /y 2), is an antisymmetric three-dimensional matrix and hence has a vanishing determinant, so we cannot solve for Aαl . Next, to first order in ξlα we have nontrivial jSαl and jSαl˙ , but these terms do not mix with the A’s so we will ignore them. The A’s change as follows, Aβk → Aβk − iAa (γa )α β ξkα Aa → Aa
– 13 –
(3.37)
Akl → Aβk Cβα ξlα ,
β ij α A′rs → hrs kl (y /y)Ak Cβα ξl .
Thus the change in (3.36) goes like i αβ n ˙ β m n C ∂θβ + k αn (ξ, jRkl , y, ...) , (3.38) (Aαm − iAa (γa )β α ξm )δ lk fln gkm ˙ − fl gk˙ = − 2y and again, we have the same singular matrix M mn multiplying the gauge fields A. The second equation, which we can think of as the first order correction to the bosonic T-duality equation, is √ ˙ β 2y 2Aa + 2δ kl ξlγ (γ a C)γβ ( yAβm gkm ˜a . ˙ + jk˙ ) = −∂ x
(3.39)
We can plug the solution for Am in (3.38), but we will just get an expression of the form l Aβl (δβα δm + φαβ Fml )M mn = −
i αβ n C ∂θβ + k αn (ξ, jRkl , y, ...) + ... 2y
(3.40)
where φ, F , k and ... are functions of the coordinates but not of A, so again we cannot solve for Aβl . Going next to first order in θlα˙ (but leaving out terms of order θlα˙ ξkβ ) we have nontrivial jQlα and jQlα˙ , and also ˙
Aa → Aa + Aβk δ lk (γ a C)βα θlα˙
(3.41)
with respect to (3.35), so the action we get is Z h R2 ˙ ˙ 2 (γ b C)γδ θnδ˙ ) (3.42) d z − 2y 2ηab (Aa + Aβk δ lk (γ a C)βα θlα˙ )(A¯b + A¯γm δ nm S2 = ′ 4πα o n α β ¯jp˙q˙ + (hkl A¯β Cβα ξ α + ¯jkl )jp˙q˙ (δ kp˙ δ lq˙ − δ kq˙ δ lp˙ ) C ξ + j ) A −jD¯jD − 2 (hkl βα kl n mn m s rs r ˙ −2iδ kl Cαβ y(f m(Aα − iAa (γa )δ α ξ δ ) + jα )(g n˙ (A¯β − iA¯b (γb )γ β ξ γ ) + ¯jβ ) l
−y(flm (A¯αm
m
l
m
k
n
n
k˙
δ − iA¯a (γa )δ α ξm ) + ¯jαl )(gkn˙ (Aβn − iAb (γb )γ β ξnγ ) + jβk˙ ) − jSαl jS k˙ + jSαl jS k˙ β β i ¯ a ) + θ˜α (∂ A¯l − ∂A ¯ l) . +˜ xa (∂ A¯a − ∂A l α α
The equations of motion (to first order in ξ or θ) are, ˙
− 2yAa + Aβk δ lk (γa C)βα θlα˙
(3.43)
h i ˙ δ −2δ kl yCαβ gkn˙ (flm Aαm + jαl )(γa )γ β ξnγ − flm (γa )δ α ξm (gkn˙ Aβn + jβk˙ ) = ∂ x˜a n o ˙ kl m α a α δ α n n m α a α γ α 2iδ Cαβ y (fl (Am −iA (γa )δ ξm )+jl )gk˙ −fl (gk˙ (Am −iA (γa )γ ξm )+jk˙ ) (3.44)
– 14 –
δ n −2yAa δ nm˙ (γa C)βδ θm ˙ + Fβ = 0 .
Plugging (3.43) in (3.44) we find o n ∂ x˜a ∂ x˜a ˙ δ α α γ α (γa )δ α ξm ) + jαl )gkn˙ − fln (gkm (A + i (γ ) ξ ) + j ) 2iδ kl Cαβ y (flm (Aαm + i a γ ˙ m m k˙ 2y 2y (3.45) a nm ˙ δ n +∂ x˜ δ (γa C)βδ θm˙ + Fβ = 0 . Thus again the Aαl will multiply the same singular matrix. More generally, the bosonic singular matrix M will get corrections with even powers of the fermionic variables, s 2 4 Mmn = Mmn + Mmn + Mmn + ... ,
(3.46)
2n where Mmn is a matrix function with a power of 2n of the fermionic variables θkα˙ s and/or ξkα , and Mmn is the purely bosonic singular matrix. In order to solve the equations of motion we will need the inverse of Mmn , but such an inverse does not exists, because all the matrices involving fermions should cancel each other order by order and we should get, −1 s s δmn = Mmk Mkn = Mmk (M −1 )skn + ... = Mmk (M −1 )skn ,
(3.47)
s but Mmn is singular.
3.4 The AdS2 target-space with eight supersymmetries Similarly to section 2.1 we construct a Green-Schwarz sigma-model action on AdS2 , but using a different supercoset, namely PSU(1, 1|2)/(U(1) × SU(2)), so this time we will have eight supersymmetries rather then four. Using the Z4 structure of this super-algebra given in (D.7) we construct the Green-Schwarz action Z R2 i 2 ¯ ¯ ¯ ¯ ¯ ¯ S= d z −JP JK − JK JP + ǫαβ (JQα JSβ − JQˆ α JSˆβ − JSα JQβ + JSˆα JQˆ β . 4πα′ 2 (3.48) We parameterize the coset representatives such that g = exP +θ
αQ
α
eB ,
ˆα Q ˆ α +ξ α Sα +ξˆα Sˆα yK
eB = eθ
e
.
(3.49)
If we define the following currents J = g −1 dg = j + j,
j = e−B (dxP + dθα Qα )eB ,
j = e−B deB ,
(3.50)
Expanding j to zeroth order in the fermions in B we get, j(0) = dxP + sθα Qα + 2ydxD + idθα Sˆα − y 2 dxK .
– 15 –
(3.51)
Thus by using the Buscher procedure of introducing gauge fields as in the examples above, to this order the action will not have quadratic terms in the fermions, so all the quadratic dθ (or more precisely Aα ) terms in the action, coming from higher orders in the fermions in B, will multiply some fermions, and we will not be able to solve the equations of motion for the fermions. The reason again is that we will have to multiply the equations of motion by a singular matrix.
4. A general analysis In this section we present a general algebraic argument to when a supercoset is expected to have a fermionic T-duality symmetry, and when it will fail to have one. Let us assume a superconformal algebra G with a Z4 automorphism structure with the zero grading subalgebra H, with the usual conformal bosonic generators Pµ , ¯ superconformal generators S, S¯ and R-symmetry Kµ , D, Jµν , supercharges Q, Q, generators Ra . We can find an Abelian subalgebra A composed of a subset of the P ’s and Q’s if we can find an anti-commuting combination of supercharges. We denote by A, B, .. the indices of A. Uppercase letters denote both bosonic and fermionic indices, lowercase letters bosonic ones and Greek letters fermionic ones, A = {a, α}. We gauge fix the H symmetry and parameterize the coset element as follows, A
g = exA T eB ,
xA T A ∈ A,
B ∈G⊖A .
(4.1)
¯ + ξS + ξ¯S)y ¯ D exp(y µRµ /y) , eB = exp(θ¯Q
(4.2)
More specifically we parameterize
where contraction of the fermions is understood (Rµ includes only the generators in the supercoset, Rµ ∈ G/H). The left-invariant one-form current will split into two pieces, J = e−B dxa T a eB + e−B deB ≡ j + j , (4.3) where j is independent of the coordinates associated with the Abelian subalgebra xa . Generally, j and j can take any value in G. We give special indices to these generators as follows, j = jI T I , j = jW T W (4.4) (in general a is both in I and W and T I ∩ T W 6= ∅), so we can write the general Green-Schwarz sigma model action as, Z h (2) (2) (2) (2) (2) (2) (2) (2) S = d2 z jI ¯jJ η IJ + jW ¯jJ η W J + jI ¯jX η IX + jW ¯jX η W X (4.5) +
1 (1) ¯(3) IJ (1) (3) (1) (3) (1) (3) jI jJ η + jW ¯jJ η W J + jI ¯jX η IX + jW ¯jX η W X 2
– 16 –
i 1 (1) (3) (1) (3) (1) (3) (1) (3) − ¯jI jJ η IJ + ¯jW jJ η W J + ¯jI jX η IX + ¯jW jX η W X 2 Z h ′(2) ′(2) (2) ′(2) ′(2) (2) (2) (2) = d2 z AI A¯J η IJ + jW A¯J η W J + AI ¯jX η IX + jW ¯jX η W X 1 ′(1) ¯′(3) IJ (1) ′(3) ′(1) (3) (1) (3) AI AJ η + jW A¯J η W J + AI ¯jX η IX + jW ¯jX η W X 2 1 ′(1) ′(3) (1) ′(3) ′(1) (3) (1) (3) − A¯I AJ η IJ + ¯jW AJ η W J + A¯I jX η IX + ¯jW jX η W X 2 i ¯ A) , +˜ xA (∂ A¯A − ∂A +
where A′ equals j when replacing the coordinate derivatives dxA with the gauge field AA , and x˜A is the Lagrange multiplier. By the structure of j, (4.3), we see that A′ depends linearly on A, so quadratic terms will rise only from A′ A¯′ interactions in the action. We also know that ′(1) I ′(1) ¯I ′(1) ¯ I A′(1) A′(1) = QI AQI + Q ¯ I + S AS I + S AS¯I , Q
(4.6)
and similarly for A′(3) . Generally, we encountered two cases where JQI is coupled to J¯QI or to J¯Q¯ I , namely we have terms in the action of the form or cη IJ (JQI J¯Q¯ J − J¯QI JQ¯ J ) ,
cη IJ (JQI J¯QJ − JQ¯ I J¯Q¯ J ),
(4.7)
where c is some constant. Let us call these two cases, case I and case II respectively. Case I appeared usually when considering PSU based model and case II when considering ortho-symplectic based model. To zeroth order in the fermions of B we have A′i = f (y)Ai,
A′Qι = g(y)hκι(y µ /y)AQκ ,
˜ κ(y µ /y)AQ¯ κ , A′Q¯ ι = g(y)h ι
(4.8)
¯ so Q might have more where the ι, κ indices are the R-symmetry indices of Q, Q, indices of transformation under the bosonic conformal group, which are the same on both sides of the equations (4.8). To this order, the equations of motion for the bosons are (2) AI η IJ f 2 = F J (j, x˜A ) , (4.9) while for the fermions we have for the two cases (1)
cη ις g 2 hκς hλι Aλ = F κ (j, x ˜A ),
˜ κ hλ − h ˜ λ hκ )A(1) = Gκ (j, x˜A ) , and cη ις g 2(h ς ι ς ι λ
(4.10)
[κ λ]
˜ ς hι in the second equation is singular when the respectively. The matrix mκλ ≡ η ις h dimension of the representation of Q under R-symmetry is odd. That is the case for the AdS4 × CP 3 action. The matrix m will get corrections at higher orders in the fermions of B, but these will contribute additively with even number of fermions to keep m bosonic. Let us call the full matrix M so, M = m + O(χ2 ) ,
– 17 –
(4.11)
¯ ξ, ξ}. ¯ The inverse matrix should have the form M −1 = m−1 + O(χ2 ), where χ = {θ, since all the fermions should cancel, so when m is singular so is M. ¯ (as in the AdS4 case), we In case II when the R-symmetry does not mix Q with Q don’t get a quadratic term to this order. Actually in this case we will get quadratic terms in AQ when going to higher orders, so we will get equations of motion for AQ , but these will always multiply terms of order O(χ2 ) or higher in the fermions, so again we will have a singular matrix multiplying AQ , and we will not be able to solve for them. The argument above does not rely on fixing any fermionic degrees of freedom (specifically we don’t use κ-symmetry). We saw that there can also be a case III where the WZ term gives an interaction of the form JQ J¯S + ... , (4.12) and this is the case of AdS2 with eight supersymmetries realized by the supercoset PSU(1, 1|2)/(U(1)×SU(2)). In this case, like for AdS4 , AS does not get corrections to ¯ again will be multiplied by zeroth order in the fermions, so the quadratic term ∂θ∂θ a singular fermionic matrix. This case has a different structure since the R-symmetry does not play any role (its generators are in H), instead we have K (special conformal ¯ with S. transformations) that mixes Q with S¯ and Q
Acknowledgments We would like to thank N. Berkovits, P. A. Grassi, M. Magro, L. Mazzucato, V. Schomerus, D. Sorokin and S. Theisen for valuable discussions. I.A. would like to thank the KITP, UCSB for hospitality during the Fundamental Aspects of String Theory workshop. This research was supported by the German-Israeli Project cooperation (DIP H.52), the German-Israeli Fund (GIF) and supported in part by the National Science Foundation under Grant No. NSF PHY05-51164.
A. The osp(2|2) algebra The osp(2|2) algebra is [P, K] = −2D ,
[D, P ] = P ,
[R, K] = 0 , [P, Q] = 0 , ¯ = [D, Q]
¯ =0, [P, Q]
1¯ Q, 2
[P, S] = −Q ,
¯ = − 1 S¯ , [D, S] 2
[D, K] = −K ,
[K, Q] = S ,
[R, Q] = −iQ , ¯ = −Q ¯, [P, S] [R, S] = −iS ,
[R, D] = 0 ,
¯ = S¯ , [K, Q]
[R, P ] = 0 ,
1 [D, Q] = Q , 2
¯ = iQ ¯, [R, Q] [K, S] = 0 ,
¯ =0, [K, S]
¯ = iS¯ , [R, S]
– 18 –
1 [D, S] = − S , 2
¯ Q} ¯ = 0 , {Q, Q} ¯ = 2P , {Q, Q} = 0 , {Q, ¯ = 2K , {Q, S} = 0 , {Q, ¯ S} ¯ =0, {S, S} ¯ = 2D − iR , {Q, ¯ S} = 2D + iR . {Q, S}
{S, S} = 0 ,
¯ S} ¯ =0, {S, (A.1)
This algebra admits a Z4 automorphism. The automorphism relevant to our needs is the one implemented by ΩXΩ−1 for X ∈ osp(2|2), where Ω=
−iσ1 0 0 1
.
(A.2)
Using this automorphism the algebra can be decomposed into Z4 -invariant subspaces Hk (k = 0 . . . , 3) such that Hk = {X ∈ osp(2|2)|ΩXΩ−1 = ik X} . These subspaces are H0 = {P − K, R} , ¯ + S} ¯ , H1 = {Q + S, Q
(A.3) (A.4)
H2 = {P + K, D} , ¯ − S} ¯ . H3 = {Q − S, Q
(A.5) (A.6)
The Cartan-Killing bilinear form is defined by gXY = Str(XY ). Its non-trivial elements are Str(P K) = Str(KP ) = −1 ,
Str(DD) =
¯ = Str(QS) ¯ =2. Str(QS)
1 , 2
Str(RR) = 2 , (A.7)
B. The osp(2|4) algebra The definition of the OSp(2|4) group is that of [21]. The non-trivial brackets of the osp(2|4) algebra are [Mmn , Mpq ] = ηmp Mnq + ηnq Mmp − ηmq Mnp − ηnp Mmq , [Mmn , Pp ] = ηmp Pn − ηnp Pm ,
[Mmn , Kp ] = ηmp Kn − ηnp Km ,
[D, Pm ] = Pm , [D, Km ] = −Km , [Pm , Kn ] = 2ηmn D − 2Mmn , 1 ˆ α , [D, Sα ] = − 1 Sα , [D, Sˆα ] = − 1 Sˆα , ˆ α] = 1 Q [D, Qα ] = Qα , [D, Q 2 2 2 2 1 1 β ˆ α ] = (Cγmn C −1 )α β Q ˆβ , [Mmn , Qα ] = (Cγmn C −1 )α Qβ , [Mmn , Q 2 2 1 1 β β [Mmn , Sα ] = (Cγmn C −1 )α Sβ , [Mmn , Sˆα ] = (Cγmn C −1 )α Sˆβ , 2 2
– 19 –
β
β ˆ [Pm , Sˆα ] = i(Cγm C −1 )α Q β , −1 β −1 ˆ α ] = −i(Cγm C )α β Sˆβ , [Km , Qα ] = −i(Cγm C )α Sβ , [Km , Q ˆ α ] = −iQ ˆ α , [R, Sα ] = iSα , [R, Sˆα ] = −iSˆα [R, Qα ] = iQα , [R, Q ˆ β } = 4(Cγ m )αβ Pm , {Sα , Sˆβ } = −4(Cγ m )αβ Km , {Qα , Q
[Pm , Sα ] = i(Cγm C −1 )α Qβ ,
{Qα , Sˆβ } = −2i(Cγ mn )αβ Mmn + 4iǫαβ D − 4ǫαβ R , ˆ α , Sβ } = −2i(Cγ mn )αβ Mmn + 4iǫαβ D + 4ǫαβ R , {Q
(B.1)
where ǫ12 = −ǫ21 = 1, ǫ21 = −ǫ12 = 1, Cαβ = ǫαβ and the Dirac matrices have the index structure γ mα β and indices are lowered and raised using C. The antisymmetrized Dirac-matrices are γ mn = 21 [γ m , γ n ]. The metric is η = diag(−1, 1, 1), m, n, p, q = 0, . . . , 2 are space-time indices, α, β = 1, 2 are spinor indices and I, J = 1, 2 are SO(2) R-symmetry indices. The non-trivial elements of the Cartan-Killing bilinear form are Str(Mmn Mpq ) = ηmp ηnq − ηmq ηnp , Str(Pm Kn ) = −2ηmn ,
Str(DD) = −1 ,
Str(RR) = −2 , ˆ α Sβ ) = −8iǫαβ .(B.2) Str(Q
Str(Qα Sˆβ ) = −8iǫαβ ,
The Z4 automorphism invariant subspaces are H0 = {Mmn , Pm − Km , R} , ˆ α + Sˆα } , H1 = {Qα − Sα , Q
H2 = {Pm + Km , D} , ˆ α − Sˆα } . H3 = {Qα + Sα , Q
(B.3)
C. The osp(6|4) algebra in so(1, 2) ⊕ u(3) basis The osp(6|4) algebra’s commutation relations are given by [λkl˙, λmn˙ ] = 2i(δml˙λkn˙ − δkn˙ λml˙)
(C.1)
[λkl˙, Rmn ] = 2i(δml˙Rkn − δnl˙Rkm )
(C.2)
i [Rmn , Rk˙ l˙] = (δmk˙ λnl˙ − δml˙λnk˙ − δnk˙ λml˙ + δnl˙λmk˙ ) 2 [Pa , Pb ] = 0, [Ka , Kb ] = 0, [Pa , Kb ] = 2ηab D − 2Mab
[Rmn , Rkl ] = 0,
[Mab , Mcd ] = ηac Mbd + ηbd Mac − ηad Mbc − ηbc Mad [Mab , Pc ] = ηac Pb − ηbc Pa , [D, Pa ] = Pa ,
[Mab , Kc ] = ηac Kb − ηbc Ka
[D, Ka ] = −Ka ,
1 [D, Qlα ] = Qlα , 2
[D, Mab ] = 0
1 [D, Sαl ] = − Sαl 2
– 20 –
(C.3) (C.4) (C.5) (C.6) (C.7) (C.8)
[Pa , Qlα ] = 0,
[Ka , Sαl ] = 0
[Pa , Sαl ] = −i(γa )α β Qlβ , i [Mab , Qlα ] = − (γab )α β Qlβ , 2 ˙ [Rkl , Qpα˙ ] = i(δ pl˙ Qkα − δ pk Qlα ), ˙
˙
˙
˙
(C.9)
[Ka , Qlα ] = i(γa )α β Sβl
(C.10)
i [Mab , Sαl ] = − (γab )α β Sβl 2
(C.11)
˙ [Rkl , Sαp˙ ] = −i(δ pl˙ Sαk − δ pk Sαl )
(C.12)
[Rk˙ l˙, Sαp ] = i(δ pl Sαk − δ pk Sαl )
(C.13)
˙
[Rk˙ l˙, Qpα ] = −i(δ pl Qkα − δ pk Qlα ), ˙
˙
˙
˙
[λkl˙, Qpα ] = 2iδ pl Qkα ,
[λkl˙, Sαp ] = 2iδ pl Sαk
˙
(C.14) ˙
˙ [λkl˙, Qpα˙ ] = −2iδ pk Qlα ,
˙
˙ [λkl˙, Sαp˙ ] = −2iδ pk Sαl
(C.15)
{Qlα , Qkβ } = 0,
{Qlα , Qkβ } = −δ lk (γ a C)αβ Pa
(C.16)
{Sαl , Sβk } = 0,
{Sαl , Sβk } = −δ lk (γ a C)αβ Ka
˙
˙
˙
{Qlα , Sβk } = −Cαβ Rlk ,
˙
˙
˙
{Qlα , Sβk } = −Cαβ Rl˙k˙
(C.17) (C.18)
1 1 ˙ ˙ {Qlα , Sβk } = −iδ lk (Cαβ D + i (γ ab C)αβ Mab ) + Cαβ λlk˙ (C.19) 2 2 1 1 ˙ ˙ {Qlα , Sβk } = iδ lk (Cαβ D − i (γ ab C)αβ Mab ) + Cαβ λkl˙ (C.20) 2 2 The indices take the values k, l = 1, 2, 3 and the same for the dotted ones — the 3 ¯ of u(3), a, b = 0, 1, 2 are the 3 of so(1, 2) and α, β, .. = 1, 2 are the so(2, 1) and 3 ∗ spinors, and η = diag(−, +, +). The generators satisfy Rkl = Rk˙ l˙ and λkl˙ = λ∗lk ˙ and ˙l Qα = (Qlα )∗ . The (γa )α β are the Dirac matrices of so(1, 2), and γab = 2i [γa , γb ]. We raise and lower spinor indices as explained in appendix E. The bilinear forms are given by Str(Rkl Rp˙q˙ ) = −2(δ kp˙ δ lq˙ − δ kq˙ δ lp˙ ) ˙
˙
(C.21)
Str(Qlα Sβk ) = 2iδ lk Cαβ
(C.22)
Str(Pa Kb ) = −2ηab
(C.23)
Str(DD) = −1
(C.24)
Str(Mab Mcd ) = ηac ηbd − ηad ηbc
(C.25)
The Z4 automorphism matrix is given by iσ2 0 0 0 iσ 0 2 Ω = 0 0 iσ2 0 0 0 0 0 0
– 21 –
0 0 0 0 0 0 . σ2 0 0 −σ2
(C.26)
The Z4 invariant subspaces of the algebra are H0 = {Pa − Ka , Mab , λlk˙ } , ˙
˙
H1 = {Qlα − Sαl , Qlα − Sαl } ,
H2 = {Pa + Ka , D, Rkl, Rk˙ l˙} , ˙
˙
H3 = {Qlα + Sαl , Qlα + Sαl } .
(C.27)
D. The psu(1, 1|2) algebra The algebra of the PSU(1, 1|2) group was developed by using the definition given in [18] (which follows the definition given in [22] applied to matrices whose elements are all Grassmann-even). The su(1, 1|2) algebra in a basis oriented towards the SU(2) R-symmetry is [D, P ] = P , [D, K] = −K , [P, K] = −2D , [Ri , Rj ] = −ǫijk Rk , 1 ˆ α , [D, Sα ] = − 1 Sα , [D, Sˆα ] = − 1 Sˆα , ˆ α] = 1 Q [D, Qα ] = Qα , [D, Q 2 2 2 2 ˆ α ] = 0 , [P, Sα ] = iQ ˆ α , [P, Sˆα ] = iQα , [P, Qα ] = [P, Q ˆ α ] = iSα , [K, Sα ] = [K, Sˆα ] = 0 , [K, Qα ] = iSˆα , [K, Q 1 ˆ α ] = − 1 σijα β Q ˆβ , [Rij , Qα ] = − σijα β Qβ , [Rij , Q 2 2 1 1 [Rij , Sα ] = − σijα β Sβ , [Rij , Sˆα ] = − σijα β Sˆβ , 2 2 ˆ β } = ǫαβ P , {Sα , Sˆβ } = ǫαβ K , {Qα , Sβ } = − i ǫβγ σαijγ Rij − iǫαβ D , {Qα , Q 2 i i ˆ α , Sβ } = − ǫαβ 1 , {Q ˆ α , Sˆβ } = i ǫβγ σ ijγ Rij + iǫαβ D , {Qα , Sˆβ } = ǫαβ 1 , {Q α 2 2 2 (D.1) where a, b = 1, 2 are AdS2 indices, α, β, γ = 1, 2 are SU(2) R-symmetry indices, Ri (i = 1, . . . , 3) are the generators of the SU(2) R-symmetry and Rij = ǫijk Rk are defined for convenience. The antisymmetric tensors are defined such that ǫ12 = 1 β 1, ǫ123 = 1 and the generators of SU(2) are σijα = 2 σiαγ σjγβ − σjαγ σiγβ with σi being the Pauli matrices. In order to get the psu(1, 1|2) algebra, one has to divide by the U(1) generator 1. The Grassmann-odd generators in the algebra above are not Hermitian but are formed as linear combinations of the original Hermitian matrices with complex coefficients. These combinations correspond to multiplying the original Hermitian generators by the complex Killing spinors found by requiring the supercharges and superconformal transformations to form Abelian subalgebras. The non-trivial elements of the Cartan-Killing bilinear form are 1 1 Str(P K) = −1 , Str(DD) = , Str(Ri Rj ) = δij , 2 2 ˆ α Sˆβ ) = iǫαβ Str(Qα Sβ ) = −iǫαβ , Str(Q (D.2)
– 22 –
The Z4 automorphism is the same one as given in [18], σ3 0 . Ω= 0 iσ3 The Z4 invariant subspaces of the algebra defined as Hk = X ∈ psu(1, 1|2)|ΩXΩ−1 = ik X
(D.3)
(D.4)
are
H0 = {P + K , R3 } , ˆ 2 + Sˆ2 , Q1 − S1 , Q ˆ 1 − Sˆ1 } , H1 = {Q2 + S2 , Q
H2 = {P − K, D, R1, R2 } , ˆ 1 + Sˆ1 , Q2 − S2 , Q ˆ 2 − Sˆ2 } . H3 = {Q1 + S1 , Q
(D.5)
Another Z4 automorphism is given by Ω=
σ1 0 0 iI
(D.6)
yields the following invariant subspaces of the algebra H0 = {D , Ri } , ˆ α} , H1 = {Qα , Q H2 = {P , K} , H3 = {Sα , Sˆα } .
(D.7)
This will give us the supercoset space PSU(1, 1|2)/(U(1)×SU(2)) which is isomorphic to AdS2 with eight supersymmetries.
E. The psu(1, 1|2) ⊕ psu(1, 1|2) algebra We build the super-algebra by taking the direct sum of two psu(1, 1|2) algebras (where we take complex combinations of the odd generators), so the bosonic part is su(2) ⊕ su(2) ≃ so(4) and su(1, 1) ⊕ su(1, 1) ≃ so(2, 2). The non-vanishing part of the psu(1, 1|2) ⊕ psu(1, 1|2) algebra, involving odd generators, is given by [D, Pa ] = Pa , [Pa , Ka ] = 2(ηab D + Jab ),
[D, Ka ] = −Ka ,
[D, Jab ] = 0,
[Pc , Jab ] = ηac Pb − ηbc Pa ,
[Rµ , Rν ] = −Nµν ,
[Kc , Jab ] = ηac Pb − ηbc Ka ,
[Rρ , Nµν ] = δρµ Rν − δρν Rµ ,
[Nρσ , Nµν ] = δρµ Nσν + δσν Nρµ − δρν Nσµ − δσµ Nρν
– 23 –
(E.1)
[Pa , QIααˆ ] = 0,
[Ka , SαI αˆ ] = 0,
1 [D, QIααˆ ] = QIααˆ , 2
ˆ
1 [D, SαI αˆ ] = − SαI αˆ , 2 ˆ
[Ka , QIααˆ ] = i(ˆ γa )αˆ β SαI βˆ,
[Pa , SαI αˆ ] = i(ˆ γa )αˆ β QIαβˆ,
i i ˆ ˆ γ01 )αˆ β QIαβˆ, [J01 , SαI αˆ ] = − (ˆ γ01 )αˆ β SαI βˆ, [J01 , QIααˆ ] = − (ˆ 2 2 i i ˆ ˆ [Rµ , QIααˆ ] = − (iˆ γ01 )αˆ β (γµ )α β SβI βˆ, γ01 )αˆ β (γµ )α β QIβ βˆ, [Rµ , SαI αˆ ] = (iˆ 2 2 i ˆ i ˆ [Nµν , QIααˆ ] = Cˆαˆ β (γµν )α β QIβ βˆ, [Nµν , SαI αˆ ] = Cˆαˆ β (γµν )α β SβI βˆ, 2 2 i i ˆ ˆ Ka , ˆ ˆ Pa , {SαI αˆ , SβJβˆ} = ǫIJ Cαβ (ˆ γ a C) {QIααˆ , QJβ βˆ} = ǫIJ Cαβ (ˆ γ a C) α ˆβ α ˆβ 2 2 1 i {SαI αˆ , QJβ βˆ} = ǫIJ (Cαβ (iCˆαˆβˆD − (Cˆ γˆ ab )αˆ βˆJab ) 2 2 1 ˆ ˆRµ (γ µ C)αβ ) γ01 C) + CˆαˆβˆNµν (γ µν C)αβ − i(ˆ α ˆβ 2 where the indices go as I = 1, 2, α ˆ = 1, 2, α = 1, 2,a = 1, 2, µ = 1, 2, 3 and Cαβ = ǫαβ , ˆ ˆ ˆ Cαˆ βˆ = ǫαˆ βˆ. The gamma matrices are defined by (ˆ γa )αˆ β = {iσ2 , −σ 1 }, (ˆ γ01 )αˆ β = ˆ i [ˆ γ , γˆ ] ˆ β 2 0 1 α
= {−iσ 3 }, (γµ )α β = {σ1 , σ3 , σ2 }, (γµν )α β = 2i [γµ , γν ]α β = {σ2 , σ1 , σ3 }, where σ are the Pauli matrices, and ηab = diag(+, −). We raise and lower spinor indices using ψα = ψ β ǫβα , ψ α = ǫαβ ψβ , where ǫ12 = −ǫ21 = ǫ12 = −ǫ21 = 1 and the same for hatted objects. The bilinear Cartan-Killing forms can be rescaled to give Str(Pa Kb ) = ηab ,
Str(DD) = 1,
Str(Rµ Rν ) = δµν ,
Str(Jab Jcd ) = δac δbd − δad δbc
(E.2)
Str(Nµν Nρσ ) = δµρ δνσ − δµσ δνρ
Str(QIααˆ SβJβˆ) = ǫIJ ǫαβ ǫαˆ βˆ The super-algebra has Z4 grading structure using the automorphism matrix 0 0 σ3 0 0 12×2 0 0 Ω= 0 0 −iσ3 0 0 i12×2 0 0
(E.3)
so H0 = {J01 , Nµν , P0 − K0 , P1 − K1 }
(E.4)
H1 = {SαI αˆ + aIJ QJααˆ }
(E.5)
H2 = {Rµ , D, P0 + K0 , P1 + K1 }
(E.6)
H3 = {SαI αˆ − aIJ QJααˆ }
(E.7)
– 24 –
where aIJ = σ1 . So we can get the supercoset space PSU(1, 1|2)2/(SU(1, 1) × SU(2)), whose bosonic part is AdS3 × S 3 . We also have another Z4 grading structure with the same number of supersymmetry generators H0 = {D, J01 , Nµν , Rµ }
(E.8)
H1 = {QIααˆ }
(E.9)
H2 = {Pa , Ka }
(E.10)
H3 = {SαI αˆ }
(E.11)
which is the four-dimensional space BDI(2, 2) ≃ AdS2 × AdS2 , given by the supercoset PSU(1, 1|2)2/(SO(2)2 × SO(4)).
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– 26 –
