One-loop Conformal Invariance of the Type II Pure Spinor Superstring

arXiv:hep-th/0609161v2 4 Oct 2006

IFT-P.033/2006

One-loop Conformal Invariance of the Type II Pure Spinor Superstring in a Curved Background Oscar A. Bedoya1 Instituto de F´ısica Te´ orica, Universidade Estadual Paulista Rua Pamplona 145, 01405-900, S˜ ao Paulo, SP, Brasil Osvaldo Chand´ıa2 Departamento de Ciencias F´ısicas, Universidad Andrés Bello Rep´ ublica 252, Santiago, Chile We compute the one-loop beta functions for the Type II superstring using the pure spinor formalism in a generic supergravity background. It is known that the classical pure spinor BRST symmetry puts the background fields on-shell. In this paper we show that the one-loop beta functions vanish as a consequence of the classical BRST symmetry of the action.

September 2006 1

e-mail: [email protected]

2

e-mail: [email protected]

1. Introduction It is well known that superstrings are not consistent on any background field. In the simplest case of a closed bosonic string coupled to a curved background, the quantum preservation of the Weyl symmetry implies that the background metric satisfies the Einstein equations plus α′ -corrections. The Weyl symmetry preservation can be seen as the absence of ultra-violet (UV) divergences in the quantum effective action [1]. In the case of superstrings in curved background, the preservation of Weyl symmetry at the quantum level is much more involved, although the case where only Neveu SchwarzNeveu Schwarz background fields are turned on is very similar to the bosonic string case [2]. Once we allow the full supersymmetric multiplet to be turned on, we need a manifestly supersymmetric space-time sigma model in order to study the corresponding quantum regime. The Green-Schwarz formalism provides us with a sigma model action which is manifestly space-time supersymmetric, nevertheless, one does not manifestly preserve this symmetry if one tries to quantize. There exists another formalism for the superstring which does not suffer of this disadvantage, known as the pure spinor formalism [3]. In this formalism the space-time supersymmetry is manifest and the quantization is straightforward by requiring a BRSTlike symmetry. We will briefly discuss the basics of this model in section 2, now let us mention what have been done to check the consistency of this formalism. The spectrum of the model is equivalent to the Green-Schwarz spectrum in the light-cone gauge [4] and it also allows to find the physical spectrum in a manifestly super-Poincaré covariant manner [5]. The pure spinor formalism is suitable to describe strings in background fields with Ramond-Ramond fields strengths turned on, as it happens on anti-de Sitter geometries. In this case, it has been checked the classical [6] and the quantum BRST invariance of the model [7] as well as its quantum conformal invariance [8] [7] (see also [9] and [10]). The superstring in the pure spinor formalism can be coupled to a generic supergravity background field, as it was shown in [11], where a sigma model action was written for the Heterotic and Type II superstrings. Here, the classical BRST invariance puts the background fields on-shell. In the heterotic string case, the background fields satisfy the tendimensional N=1 supergravity equations plus the super Yang-Mills equations in a curved background. In the type II case, the background fields satisfy the ten-dimensional type II supergravity equations. Of course, it could be very interesting to obtain α′ -corrections to these equations in this formalism by requiring the quantum preservation of some symmetries of the classical sigma-model action. Since the lowest order in α′ BRST symmetry puts 1

the background fields on shell, one expects that the equations of motion for the background fields derived from the beta function calculation are implied by the BRST symmetry. This property was verified for the heterotic string, in whose case, the classical BRST symmetry implies one-loop conformal invariance [12]. In this paper we show that the same property is also valid for the Type II superstring3 . In section 2 we review the type II sigma-model construction of [11]. In section 3 we expand the action by using a covariant background field expansion. In section 4 we determine the UV divergent part of the effective action at the one-loop level and finally, in section 5, we use the expanded action of section 3, to write from the UV divergent part found in section 4, the beta functions for the Type II superstring. Then we show that beta functions vanish after using the constraints on the background fields implied by the classical BRST invariance of the sigma-model action4 .

2. Classical BRST Constraints The pure spinor closed string action in flat space-time is defined by the superspace coordinates X m with m = 0, . . . , 9 and the conjugate pairs (pα , θ α ), (e pα , θeα ) with (α, α) = 1, . . . , 16. For the type IIA superstring the spinor indices α and α have the opposite

chirality while for the type IIB superstring they have the same chirality. In order to define a conformal invariant system we need to include a pair of pure spinor ghost variables eα , ω eα ). These ghosts are constrained to satisfy the pure spinor conditions (λα , ωα ) and (λ m m e λ) e = 0, where γ m and γ m are the 16 × 16 symmetric ten dimensional (λγ λ) = (λγ αβ

αβ

gamma matrices. Because of the pure spinor conditions, ω and ω e are defined up to δω = e m )Λ e m . The quantization of the model is performed after the (λγ m )Λm and δe ω = (λγ H H α e deα , here dα and deα are e = λ construction of the BRST-like charges Q = λα dα , Q the world-sheet variables corresponding to the N = 2 D = 10 space-time supersymmetric

derivatives and are supersymmetric combinations of the space-time superspace coordinates of conformal weights (1, 0) and (0, 1) respectively. The action in flat space is a free action involving the above fields, that is 3

There is a more recent development [13] with an even richer world-sheet structure, called non-

minimal pure spinor formalism, which is interpreted as a critical topological string. Nevertheless, in this paper we restrict to the so called minimal pure spinor formalism of [3]. 4

In a similar way, using the he hybrid formalism[14], in [15] and [16] were derived the type II

4D supergravity equations of motion in superspace by requiring superconformal invariance.

2

1 S= 2πα′

Z

1 d2 z ( ∂X m ∂Xm + pα ∂θ α + peα ∂ θeα ) + Spure , 2

(2.1)

where Spure is the action for the pure spinor ghosts.

In a curved background, the pure spinor sigma model action for the type II superstring is obtained by adding to the flat action of (2.1) the integrated vertex operator for supergravity massless states and then covariantizing respect to ten dimensional N = 2 super-reparameterization invariance. The result of doing this is

1 S= 2πα′

Z

+dα deβ P αβ

1 1 b B α eα ω e αβ e β )Ω d2 z ( Πa Π ηab + ΠA Π BBA + dα Π + deα Πα + (λα ωβ )Ωα β + (λ 2 2 (2.2) eα ω eα ω eα βγ + (λα ωβ )(λ e )dγ C e )Sαα ββ ) + Spure + SF T , + (λα ωβ )deγ Cα βγ + (λ β

A

where ΠA = ∂Z M EM A , Π

β

= ∂Z M EM A with EM A the supervielbein and Z M are the

curved superspace coordinates, BBA is the super two-form potential. The connections A e α β = ∂Z M Ω e M α β = ΠA Ω e Aα β . They are appears as Ωα β = ∂Z M ΩM α β = Π ΩAα β and Ω

independent since the action of (2.2) has two independent Lorentz symmetry transforma-

tions. One acts on the α-type indices and the other acts on the α-type indices. Spure is the action for the pure spinor ghosts and is the same as in the flat space case of (2.1). As was shown in [11], the gravitini and the dilatini fields are described by the lowest θeα βγ , while the Ramond-Ramond field strengths components of the superfields Cα βγ and C

are in the superfield P αβ . The dilaton is the theta independent part of the superfield Φ which defines the Fradkin-Tseytlin term SF T

1 = 2π

Z

d2 z r Φ,

(2.3)

where r is the world-sheet curvature. Because of the pure spinor constraints, the superfields in (2.2) cannot be arbitrary. In fact, it is necessary that e A δα β + 1 Ω e Aab (γ ab )α β , e Aα β = Ω Ω 4

1 ΩAα β = ΩA δα β + ΩAab (γ ab )α β , 4 1 Cα βγ = C γ δα β + Cab γ (γ ab )α β , 4

eab γ (γ ab )α β , eα βγ = C eγ δα β + 1 C C 4

(2.4)

1 1 1 Sαα ββ = Sδα β δα β + Sab (γ ab )α β δα β + Seab (γ ab )α β δα β + Sabcd (γ ab )α β (γ cd )α β . 4 4 16 3

The action of (2.2) is BRST invariant if the background fields satisfy suitable constraints. As was shown in [11], these constraints imply that the background field satisfy the type II supergravity equations. The BRST invariance is obtained by requiring that eα deα are conserved. Besides, the BRST charges the BRST currents jB = λα dα and e jB = λ H H e= e Q = jB and Q jB are nilpotent and anticommute. Let us review these properties

now.

2.1. Nilpotency As was shown in [11] (see also [17]), nilpotency is obtained after defining momentum variables in (2.2) and then using the canonical Poisson brackets. The only momentum variable that does not appear in (2.2) is the conjugate momentum of Z M which is defined as PM = (2πα′ )δS/δ(∂0 Z M ) where ∂0 = 12 (∂ + ∂). It is not difficult to see that ωα is the eα . Nilpotence of Q determines conjugate momentum to λα and that ω eα is the one for λ the constraints

eαβγ δ = 0, λα λβ HαβA = λα λβ λγ Rαβγ δ = λα λβ R

(2.5)

λα λβ Tαβ a = λα λβ Tαβ γ = λα λβ Tαβ γ = 0,

where H = dB, the torsion TAB α and RABγ δ are the torsion and the curvature constructed eABγ δ are the torsion and the curvature using ΩAβ γ as connection. Similarly, TAB γ and R e γ as connection. using Ω Aβ

e leads to the constraints The nilpotence of the BRST charge Q δ δ eβ H eα eβ e α eβ eγ e eα λ λ αβA = λ λ Rαβγ = λ λ λ Rαβγ = 0,

(2.6)

eα λ eβ T a = λ eα λ eβ T γ = λ eα λ eβ Te γ = 0. λ αβ αβ αβ

e determines Finally, the anticommutation between Q and Q

δ eβ R eα λ e HαβA = Tαβ a = Tαβ γ = Tαβ γ = λα λβ Rγαβ δ = λ γαβ = 0.

(2.7)

Note that given the decomposition (2.4) for the connections, we can respectively write 1 RDCα β = RDC δα β + RDCef (γ ef )α β , 4 eDC δα β + 1 R eDCef (γ ef )α β . eDCα β = R R 4 4

(2.8)

2.2. Holomorphicity The holomorphicity of jB and the antiholomorphicity of e jB constraints are determined

after the use of the equations of motion derived from the action (2.2). The equation for the pure spinor ghosts are

eα ω eβ Sβα αβ ) = 0, ∇λα + λβ (deγ Cβ αγ + λ

eα ω eβ Sαα ββ )ωβ = 0, ∇ωα − (deγ Cα βγ + λ

eβ (dγ C eα + λ e αγ + λα ωβ S βα ) = 0, ∇λ β αβ

eα βγ + λα ωβ Sαα ββ )e ∇e ωα − (dγ C ωβ = 0, (2.10)

and

(2.9)

e connections according to the where ∇ is a covariant derivative which acts with Ω or Ω index structure of the fields it is acting on. For example,

eγβ. ∇P αβ = ∂P αβ + P γβ Ωγ α + P αγ Ω

The variations respect to dα and deα provide the equations α eα ω eα βα = 0, eβ C Π + deβ P αβ + λ

Πα − dβ P βα + λα ωβ Cα βα = 0.

(2.11)

The most difficult equations to obtain are those coming from the variation of the superspace coordinates. Let us define σ A = δZ M EM A , then it is not difficult to obtain δΠA = ∂σ A − σ B ΠC EB M EC N ∂[N EM ] A (−1)C(B+M ) . Here we can express this variation in terms of the connection Ω . In fact, δΠA = ∇σ A − σ B ΠC (TCB A + ΩBC A (−1)BC ). There is a point about our notation for the torsion that we should make clear. Using tangent superspace indices, the torsion can be written as TBC A = −EB N (∂N EC M )EM A + (−)BC EC N (∂N EB M )EM A + ΩBC A − (−)BC ΩCB A . (2.12) 5

In our notation, TBC α will mean that the connection in (2.12) is ΩCβ α while TBC α means e α . Since we also have two connections with bosonic that the connection in (2.12) is Ω Cβ

a

e Cb a , we use TBC a to denote the torsion when we use the and Ω

tangent space index ΩCb first and TeBC a to denote the torsion when we use the second.

We vary the action (2.2) under these transformations and, after using the equations

(2.10), (2.11) and some of the nilpotence constraints, we obtain

1 1 b a a ∇dα = − Πa Π (Tα(ba) + Hαba ) + Πβ Π (Tβαa − Hβαa ) − dβ Π Taα β 2 2

(2.13)

1 a γ 1 e γδ eβ ω e e γ Πa (R −deβ Πa (Taα β + P γβ (Tγαa +Hγαa ))+λβ ωγ Π Raαβ γ +λ − Cβ (Tδαa +Hδαa )) aαβ 2 2 1 δ eβ ω e γρ Hρδα ) e γ + 1C e γ Πδ (R −deβ Πγ (Tγα β + P δβ Hδγα ) + λβ ωγ Π Rδαβ γ + λ δαβ 2 2 β eβ ω e γρ Tρα δ + P δρ R e γ) e γδ + C eγ dδ (∇α C +dβ deγ (P δγ Tδα β − ∇α P βγ ) + λ β β ραβ

eδ ω e ρ+C e ρσ Rσαβ γ ), eρ (∇α Sβδ γρ + Cβ γσ R +λβ ωγ deδ (∇α Cβ γδ − P ρδ Rραβ γ ) − λβ ωγ λ σαδ δ

and

1 1 b β ∇deα = − Πa Π (Tα(ba) + Hαba ) + Πa Π (Tβαa + Hβαa ) − deβ Πa Taα β 2 2

(2.14)

1 1 a a γ eβ ω e e γ Πa R +λβ ωγ Π (Raαβ γ − Cβ γδ (Tδαa −Hδαa )) −dβ Π (Taα β − P βγ (Tγαa −Hγαa ))+λ aαβ 2 2 1 γ eβ ω e γ + λβ ωγ Πδ (R γ + 1 Cβ γρ H ) e γ Πδ R −dβ Π (Tγα β + P βδ Hγδα ) + λ δαβ δαβ δρα 2 2 +dβ deγ (P βδ Tδα γ − ∇α P βγ ) + λβ ωγ deδ (∇α Cβ γδ + Cβ γρ Teρα δ − P ρδ Rραβ γ )

eδ ω eβ ω e γ ) − λβ ωγ λ e ρ+C e ρσ Rσαβ γ ). e γδ + P δρ R eγ dδ (∇α C eρ (∇α Sβδ γρ + Cβ γσ R +λ β ραβ σαδ δ

From these equations, (2.9), (2.10) and also two equations in (2.7) we obtain the holomorphicity constraints. In fact, ∇jB = 0 implies

Tα(ab) = Hαab = Tαβa − Hαβa = Taα β = Taα β + P γβ Tγαa = λα λβ Raαβ γ = 0,

(2.15)

γ β 1 δβ e e γρ Hρδα = P δγ Tδα β −∇α P βγ −Cα βγ = 0, e γδ e γ+1 C R Hδγα = R aαβ −Cβ Tδαa = Tγα + 2 P δαβ 2 β

6

e γρ Tρα δ + P δρ R e γ − S δγ = λα λβ (∇α Cβ γδ − P ρδ Rραβ γ ) = 0, e γδ + C ∇α C αβ β β ραβ e ρ+C e ρσ Rσαβ γ ) = 0, λα λβ (∇α Sβδ γρ + Cβ γσ R σαδ δ

and ∇e jB = 0 implies

γ eβ R eα λ e Tα(ab) = Hαab = Tαβa + Hαβa = Taα β = Taα β − P βγ Tγαa = λ aαβ = 0,

(2.16)

1 1 eα γβ = 0, Raαβ γ −Cβ γδ Tδαa = Tγα β + P βδ Hγδα = Rδαβ γ + Cβ γρ Hδρα = P βδ Tδα γ −∇α P βγ +C 2 2 eα λ eβ (∇α C e γ ) = 0, e γδ + P δρ R ∇α Cβ γδ + Cβ γρ Tρα δ − P ρδ Rραβ γ − Sβα γδ = λ β ραβ γ eα λ eβ (∇α S ργ + Cδ ρσ R e e γσ Rσαδ ρ ) = 0. λ σαβ + Cβ δβ

2.3. Solving the Bianchi identities We can gauge-fix some of the torsion components and determine others through the use of Bianchi identities. It is not necessary but it will simplify the computation of the one-loop beta functions. As in [11], we can set Hαβγ = Hαβγ = Hαβγ = Hαβγ = 0 since there is no such ten-dimensional superfields satisfying the constrains of (2.15) and (2.16). a a , therefore We can use the Lorentz rotations to gauge fix Tαβ a = γαβ and Tαβ a = γαβ

the above constraints imply Hαβa = (γa )αβ and Hαβa = −(γa )αβ . We can use the shift symmetry of the action (2.2)

δdα = δΩαβ γ λβ ωγ ,

eβ ω e γλ δ deα = δ Ω eγ , αβ

δCα βγ = P δγ δΩδα β ,

e δ +C e δρ δΩρα γ , δSαβ γδ = Cα γρ δ Ω ρβ β

eα βγ = −P γδ δ Ω e β, δC δα

to gauge-fix Tαβ γ = Tαβ γ = 0.

The Bianchi identity for the torsion is (∇T )ABC D ≡ ∇[A TBC] D + T[AB E TEC] D − R[ABC] D = 0,

(2.17)

where brackets in (2.17) mean (anti-)symmetrization respect to the ABC indices. The e if the upper index D is δ or δ respectively. When D = d, we use curvature will be R or R

e Bc a ; then the the notation (∇T )ABC d or (∇Te)ABC d , if we use the connection ΩBc a or Ω e curvatures in each case will be R or R. 7

The Bianchi identity (∇T )αβγ a = 0 implies Tαab = 2(γab )α β Ωβ . Similarly, the Bianchi a e . The Bianchi identity (∇T ) identity (∇Te)αβγ a = 0 implies Teαab = 2(γab )α β Ω β αβγ = 0 e α = Teαa b = 0. Similarly, the Bianchi identity (∇T )αβγ a = 0 implies Ωα = Tαa b = implies Ω e a = 0. 0. It is not difficult to show that the constraints Taα α = Taα α = 0 imply Ωa = Ω

We can write two sets of Bianchi identities for H depending on what is the connection

we choose in the covariant derivative. Note that the components of the superfield H do e = 0 and not depend on such choice. The Bianchi identities come from ∇H = 0 and ∇H it is not difficult to check that both sets are equivalent. Let us write only one of them

3 (∇H)ABCD ≡ ∇[A HBCD] + T[AB E HECD] = 0. 2 There is one more Bianchi identity involving a derivative of the curvature (∇R)ABCD E ≡ ∇R[ABC]D E + T[AB F RF C]D E = 0.

(2.18)

(2.19)

The identities (∇H)αβγδ , (∇H)αβγδ , (∇H)αβγδ , (∇H)αβγδ , (∇H)αβγδ are easily satisa a fied if we recall the identities for gamma matrices γ(αβ (γa )γ)δ = γ(αβ (γa )γ)δ = 0. The iden-

tities (∇H)aαβγ , (∇H)aαβγ , (∇H)aαβγ , (∇H)aαβγ are satisfied after using the dimension1 2

constraints. The identity (∇H)abαβ = 0 implies Tabc + Habc = 0 and the identity e (∇H)abαβ = 0 implies Teabc − Habc = 0. The identity (∇H)abαβ = 0 is satisfied if we use the constraints involving the superfield P αβ in the first lines of (2.15) and (2.16). 2.4. The remaining equation of motion In the computation of the one-loop beta function we will need to know the equation a

a

of motion for Πa and Π . Since we know that the difference ∇Π − ∇Πa is given by the a

torsion components, then we only need to determine ∇Π + ∇Πa which is determined by the varying the action respect to σ a = δZ M EM a . To make life simpler we will write this equation using the above results for torsion and H components. The equation turns out to be 1 e 1 1 c b b b (∇Πa + ∇Πa ) = Πb Π Hcba − Πα Π Tαab + dα Π Tab α + λα ωβ Π Rabα β (2.20) 2 2 2 1 eα ω eα ω eα βγ Tγab )+ 1 deα Πβ Taβ α + 1 λ eabα β + 1 C eaγα β +deα Πb (Tab α + P βα Tβab )+ λ e β Πb (R e β Πγ R 2 2 2 2 1 1 β γ + dα Π Taβ α + λα ωβ Π Raγα β + dα deβ ∇a P αβ + λα ωβ deγ (∇a Cα βγ − P δγ Raδα β ) 2 2 eα ω eγ ω eα βγ + P γδ R e β ) + λα ωβ λ eγ δρ Raρα β − Cα βρ R eaργ δ ). +λ e dγ (∇a C e (∇a Sαγ βδ − C β

aδα

δ

8

2.5. Ghost number conservation As it was shown in [11], the vanishing of the ghost number anomaly determines that the spinorial derivatives of the dilaton superfield Φ are proportional to the connection Ω. This relation is crucial to cancel the beta function in heterotic string case [12] and will be equally essential in our computation. Let us recall how this relation is obtained. Consider the coupling between ghost number currents and the connections in the action (2.2). Namely Z 1 e d2 z (JΩ + JΩ). ′ 2πα The BRST variation on this term contains the term Z 1 e αΩ e α ). − d2 z (∂Jλα Ωα + ∂ Jλ 2πα′ The anomaly in the ghost number current conservation turns out to be proportional to the two dimensional Ricci scalar, as noted by dimensional grounds. The proportionality can be determined by performing a Weyl transformation, around the flat world-sheet, of the anomaly equation. In this way, the triple-pole in the OPE between the current and the corresponding stress tensor yields e α, ∇α Φ = 4 Ω

∇α Φ = 4Ωα ,

(2.21)

which will be used in section 5 to cancel the UV divergent part of the effective action.

3. Covariant Background Field Expansion We use the method explained in [18] and [12]. Here, we need to define a straight-line geodesic which joins a point in superspace to neighbor ones and allows us to perform an expansion in superspace. It is given by Y A which satisfies the geodesic equation ∆Y A = Y B ∇B Y A = 0. The connection we choose to define this covariant derivative has the none Aα β . These same connections are defined in the vanishing components ΩAa b , ΩAα β and Ω

action (2.2). In this way, the covariant expansions of the different objects in (2.2) are determined by

∆ΠA = ∇Y A − Y B ΠC TCB A ,

B

∆Ωα β = −Y A Π RBAα β ,

e α β = −Y A ΠB R eBAα β . ∆Ω

(3.1)

9

Any superfield Ψ is expanded as ∆Ψ = Y A ∇A Ψ. As in [12], we see that dα , deα and the pure spinor ghosts are treated as fundamental

fields, then we expand them according to dα = dα0 + dbα ,

b deα = deα0 + deα ,

bα λα = λα 0 +λ ,

α b eα + λ e , eα = λ λ 0

ωα = ωα0 + ω bα ,

(3.2)

b eα0 + ω e α0 , ω eα = ω

where the subindex 0 means the background value of the corresponding field which will dropped in the subsequent discussion. The quadratic part of the expansion of (2.2), excluding the Fradkin-Tseytlin term, has the form

1 S2 = Sp + 2πα′

Z

d2 z (Y A Y B EBA + Y A ∇Y B CBA + Y A ∇Y B C BA + dbα Y A DA α (3.3)

α α b b b b b bα ω bα ωβ +λα ω eα ω e ω e ω eβ + λ e β )Hα β +(λ +deα Y A DA α +(λ bβ )H α β +(λ bβ )Y A I Aα β +(λ e β )Y A IAα β

α γ b b b b b b eα ω eγ ω bα ωβ +λα ω e ω bα ωβ +λα ω e ω eα βγ +(λ eβ +λ e β )dbγ C e δ )Sαγ βδ ), +dbα deβ P αβ +(λ eδ +λ bβ )deγ Cα βγ +(λ bβ )(λ

where EBA , CBA , . . . are background superfields given by

1 C D Π Π (TCB E HEDA (−1)D(C+B) − TDB E HECA (−1)BC (3.4) 4 1 C) +∇B HDCA (−1)B(C+D) + 2TCB a TDAa (−1)D(C+B) ) − Π(a Π (RCBAa − TCB D TDAa 4 1 C +∇B TCAa (−1)BC ) + dα Π (−1)A+B (−RCBA α + TCB D TDA α − ∇B TCA α (−1)BC ) 2 1 1 C + deα ΠC (−1)A+B (−RCBA α + TCB D TDA α − ∇B TCA α (−1)BC ) + λα ωβ Π (TCB D RDAα β 2 2 1 eα 1 −∇B RCAα β (−1)BC ) + λ ω eβ ΠC (TCB D RDAα β − ∇B RCAα β (−1)BC ) + dα deβ ∇B ∇A P αβ 2 2 1 1 eα eγ ω eα βγ (−1)A+B + 1 λα ωβ λ ω e β d γ ∇ B ∇A C eδ ∇B ∇A Sαγ βδ , + λα ωβ deγ ∇B ∇A Cα βγ (−1)A+B + λ 2 2 2 EBA =

1 1 1 1 1 a − ΠA HCBA − dα TBA α (−1)A+B − λα ωβ RBAα β , CBA = − Πa TBAa − ΠC TCAa δB 4 2 4 2 2 (3.5) 10

1 1 1 C 1 1 eα e a ω eβ RBAα β , C BA = − Πb TBAa − Π TCAa δB + ΠA HCBA − deα TeBA α (−1)A+B − λ 4 2 4 2 2 (3.6) B eβ ω e γα , e γ ∇A C DA α = −Π TBA α + deβ ∇A P αβ (−1)A + λ β

(3.7)

DA α = −ΠB TeBA α − dβ ∇A P βα (−1)A + λβ ωγ ∇A Cβ γα ,

(3.8)

eγ ω eδ Sαγ βδ , H α β = Ωα β + deγ Cα βγ λ

(3.9)

e α β + dγ C eα βγ + λγ ωδ Sγα δβ , Hα β = Ω

(3.10)

A eγ ω eδ ∇A Sαγ βγ , I Aα β = −Π RBAα β + deγ ∇A Cα βγ (−1)A + λ

eBAα β + dγ ∇A C eα βγ (−1)A + λγ ωδ ∇A Sγα δβ . IAα β = −ΠB R

(3.11)

(3.12)

In (3.3) Sp provides the propagators for the quantum fields and is given by 1 Sp = 2πα′

Z

1 b d2 z ( ∇Y a ∇Ya + dbα ∇Y α + deα ∇Y α ) + Lpure , 2

(3.13)

where Lpure is the Lagrangian for the pure spinor ghosts.

4. The one-loop UV divergent Part of the Effective Action The effective action is given by −Sef f

e

=

Z

DQ e−S ,

(4.1)

where Q represents the quantum fluctuations. To compute the one-loop beta functions we need to expand (2.2) up to second order in the quantum fields. In this way, we will obtain the UV divergent part of the effective action, SΛ . Here Λ is UV scale. Note that the Fradkin-Tseytlin term is evaluated on a 11

sphere with metric Λdzd¯ z . Finally, the complete UV divergent part of the effective action becomes Z 1 A A d2 z (∇Π ∇A Φ + Π ΠB ∇B ∇A Φ) log Λ. SΛ + (4.2) 2π The computation of SΛ is performed by contracting the quantum fields. From (3.13) we read

a

b

′ ab

Y (z, z¯)Y (w, w) ¯ → −α η

2

log |z−w| ,

α′ δα β , dbα (z)Y β (w) → (z − w)

be dα (¯ ¯ → z )Y β (w)

α′ δαβ . (¯ z − w) ¯ (4.3)

For the pure spinor ghosts we note that, because of (2.4), they enter in the combinations N ab =

1 (λγ ab ω), 2

e ab ω e ab = 1 (λγ N e ), 2

J = λα ωα ,

eα ω eα . Je = λ

e N ab and N e ab . We can expand each of these combinations as J + J1 + J2 , similarly for J,

As in [12], the only relevant OPE’s involving the pure spinor ghosts and contributing to SΛ are N1ab (z)N1cd (w) →

1 (−η a[c N d]b (w) + η b[c N d]a (w)), (z − w)

e1ab (¯ e1cd (w) N z )N ¯ →

1 e d]b (w) e d]a (w)). (−η a[c N ¯ + η b[c N ¯ (¯ z − w) ¯

(4.4)

(4.5)

The one-loop contributions to SΛ come from self-contraction of Y A ’s in the term with EBA in (3.3) and a series of double contractions in (3.3). These come from products between the term involving CBA with the one involving C BA , CBA with D A β , C BA with eα βγ and Sαγ βδ with DA β , D A β with DA β , EBA with P αβ , I Cα β with Cα βγ , ICα β with C

itself. After adding up all these contributions, the one-loop UV divergent part of the effective action is proportional to Z

d2 z [−η ab Eba +η a[c η d]b Cba C dc +η ab C[aα] D b α +η ab C [aα] Db α +Dα β Dβ α +E[αβ] P αβ (4.6)

e ab Iαa c C ecb α + 1 N ab N e cd Sa e c f Sbedf + ∇ΠA ∇A Φ + ΠA ΠB ∇B ∇A Φ] log Λ, +N ab I αa c Ccb α + N 2 where we used the expressions (2.4). Now it will be shown that (4.6) vanishes as consequence of the classical BRST constraints. 12

5. One-loop Conformal Invariance To write the equations derived from the vanishing of (4.6), we need to determine ∇Π

A

from the classical equations of motion from (2.2). In order to do this, we need to know A

C

∇ΠA − ∇Π = ΠB Π TCB A .

(5.1)

Note that we are using here the connection ΩA B to calculate the covariant derivatives and the torsion components. The equation for ∇Πa is c b b eα ω eabα β + λα ωβ Πb Rabα β e β Πb R ∇Πa = Πb Π Tabc − Πα Π Tαab + deα Πb Tab α + dα Π Tab α + λ

(5.2)

eα ω eaγα β + dα de ∇a P αβ + λα ωβ deγ (∇a Cα βγ − P δγ Raδα β ) +deα Πβ Taβ α + λ e β Πγ R β

eα ω eγ ω eα βγ + P γδ R e β ) + λα ωβ λ eγ δρ Raρα β − Cα βρ R eaργ δ ). +λ eβ dγ (∇a C eδ (∇a Sαγ βδ − C aδα α

Now we compute the equation for Π . We start by noting that this world-sheet field is determined from the equation of motion (2.11), then α

eβ ω e γα ). eγ C ∇Π = −∇(deβ P αβ + λ β

eα βγ acts with Ωα β on α-indices Remember that the covariant derivative on P αβ and C e α β on α-indices. Now we can use the equations (2.10) and (2.14) to obtain and with Ω α e γβ P αδ + P βδ ∇ P αγ ) + λβ ωγ de (−Sβρ γδ P αρ + Cβ γρ ∇ρ P αδ ) ∇Π = dβ deγ (C δ δ δ

(5.3)

eβ ω eρ γα − C eρ γδ C e ρα − P δρ ∇ρ C e ρδ C e γα −deβ Πa ∇a P αβ − deβ Πγ ∇γ P αβ + λ e γ dδ (C β β β

eδ ω e γ ))+λβ ωγ λ eσ ρα −Sβσ γρ C e σα +Cβ γσ ∇σ C e γδ +P δǫ R e ρα +P αǫ (∇ǫ S γρ −P αδ (∇δ C eρ (Sβδ γσ C β ǫδβ δ δ βδ eβ ω eβ ω e ρ+C e ρσ Rσǫβ γ )) − λ e γ P αδ ) − λ e γα + R +Cβ γσ R eγ Πa (∇a C eγ Πδ Sδβ αγ . σǫδ δ β aδβ α

To obtain the equation for Π we can use (5.1). After all this we get

α eβ ω e γρ ∇ρ P δα ) ∇Π = dβ deγ (Cδ βγ P δα − P δγ ∇δ P βα ) + λ eγ dδ (Sρβ δγ P ρα − C β a

(5.4)

γ

+dβ Π ∇a P βα + dβ Π ∇γ P βα + λβ ωγ deδ (Cβ ρδ Cρ γα − Cρ γδ Cβ ρα + P ρδ ∇ρ Cβ γα

eδ ω e ρσ ∇σ Cβ γα +P δα (∇δ Cβ γδ − P ǫδ Rǫδβ γ )) + λβ ωγ λ eρ (Sβδ σρ Cσ γα − Sσδ γρ Cβ σα + C δ e ρ+C e ρσ Rσǫβ γ )) − λβ ωγ Πa (∇a Cβ γα − Raǫβ γ P ǫα ) +P αǫ (∇ǫ Sβδ γρ + Cβ γγ R γǫδ δ

b a δ eβ ω e γδ Taδ α . e γ Πa C −λβ ωγ Π Sβδ γα + Πa Π Tab α − Πβ Π Taβ α − deβ Πa P γβ Taγ α − λ β

13

5.1. Beta functions Now we can obtain the equations for the background fields implied by the vanishing of the beta functions. These are the background dependent expressions for the conformal weights (1, 1) independent couplings in (4.6). That is, all the independent combinations α a eα ω eβ ) because Πα formed from the products between (Πa , Πα , dα , λα ωβ ) and (Π , Π , deα , λ α

and Π are determined from the equations of motion (2.11). Let us first concentrate on B B the beta functions coming from the couplings to ΠA Π , dα Π and ΠA de fields. After β

using the results for the expansion (3.4)-(3.12) and the equations (5.2)-(5.4) in (4.6), the β

b

b

β

couplings Πα Π , Πα Π , Πa Π and Πa Π lead respectively to a first set of equations Tcβ δ Tδα c − Tcα δ Tδβ c + 4∇α ∇β Φ = 0,

(5.5)

∇d Tαdb + Rαdeb η de + Tbc δ Tδα c + 4∇b ∇α Φ = 0,

(5.6)

Rβdea η de + Tac δ Tδβ c − Tcβ δ Tδa c + 4∇a ∇β Φ = 0,

(5.7)

η cd (Racdb +Rbcda )−∇c Tabc +Tc(a α Tb)α c +8Taα β Tbβ α +4Tab c ∇c Φ+4Tab α ∇α Φ+4∇a ∇b Φ = 0. (5.8) We wrote them by increasing their dimensions, that is, if X a has dimension −1 and each θ α , θeα have dimension − 1 , then the first has dimension 1, the second and third 2

dimension

3 2

β

b

and the fourth dimension 2. The couplings to dα Π , Πα deβ , dα Π and Πa deβ

lead respectively to a second set of equations

∇c Tcβ α − 2∇β P αγ ∇γ Φ + 2P αγ ∇γ ∇β Φ = 0,

(5.9)

∇c Tcα β + 2∇α P γβ ∇γ Φ − 2P γβ ∇γ ∇α Φ = 0,

(5.10)

∇c Tcb α − Tcd α Tb cd + (Tδb c Tcγ α − Rbγδ α )P δγ + Tbγ δ (3∇δ P αγ − 2P αγ ∇δ Φ) +2Tbc α ∇c Φ − 2∇b P αγ ∇γ Φ = 0, 14

(5.11)

e β P γδ − Taγ δ (3∇ P γβ − 2P γβ ∇ Φ) ∇c Tca β − 2Tcd β Ta cd + P γβ Tα de Tade + R aγδ δ δ +2Tac β ∇c Φ + 2∇a P γβ ∇γ Φ = 0.

(5.12)

The first two with have dimension 2 and the second two have dimension 25 . Now we will prove that these equations are implied by the classical BRST constraints, the Bianchi identities (2.17) and the relations (2.21). Firstly, it is important to know the expression for the scale curvature in terms of the scale connection. This are found to be Rαβ = ∇(α Ωβ) , Rab = Tab γ Ωγ ,

Raβ = ∇a Ωβ ,

e , e = ∇(α Ω R αβ β)

eγ, eab = Tab γ Ω R

Rαβ = ∇β Ωα ,

Raβ = Taβ γ Ωγ .

e = ∇α Ω e , R αβ β

e , e = ∇a Ω R β aβ

Rαβ = 0,

eαβ = 0, R

eaβ = Taβ γ Ω eγ. R

(5.13)

(5.14)

Secondly, let us write some expressions useful for later use. We note that the Bianchi identity (∇T )αab c = 0, using (5.13) can be written as

Rα[ab]c = ∇α Tabc − 2(γc[a)α β Rb]β + (γc )αβ Tab β − Tαdc Tab d − Tα[a d Tb]dc ,

(5.15)

now, we can use the identity 2Rαabc = Rα[ab]c + Rα[ca]b − Rα[bc]a,

(5.16)

and the Bianchi identity (∇H)αabc = 0 to write (5.15) as Rαabc = Ta[b β (γc] )βα − 2(γbc )α β Raβ .

(5.17)

An identical procedure starting with (∇Te)αab c = 0 allows us to find e . eαabc = Ta[b β (γc] ) − 2(γbc )α β R R αβ aβ 15

(5.18)

Then, replacing (5.17) and (5.18) respectively in (∇T )aαβ β = 0 and (∇T )aαβ β = 0, we find b eaα . γαβ Tba β = 8R

b γαβ Tba β = 8Raα ,

(5.19)

We have enough information to show that the equations (5.5) , (5.6) and (5.7) are satisfied. From the Bianchi identity (∇T )αβγ β = 0 we obtain 1 Tαβ d Tdγ β = 17Rαγ + Rγβcd (γ cd)α β . 4

(5.20)

Since we need an expression for Rγβcd , we can use (∇T )αβa b = 0, finding Rγβcd = 2(γcd )β δ ∇γ Ωδ + Tcγ ǫ (γd )ǫβ + Tcβ ǫ (γd )ǫγ .

(5.21)

Replacing (5.21) in (5.20) ,using the second equation in (5.13) , ∇α Φ = 4Ωα and the constraints coming from holomorphicity-antiholomorphicity of the BRST current Taβ γ = −(γa )βδ P δγ , Taβ γ = (γa )βδ P γδ we can verify the equation (5.5). To verify (5.6) and (5.7), we must contract the a and b indices using η ab in (5.17) and (5.18), and use (5.19) together with the relations (2.21). b

For deriving the remaining equation of the first set, the coupling to Πa Π , it is useful to find an expression for Rabcd , which can be found from the Bianchi identity (∇T )abα β 1 Rabcd = − (γcd )β α (∇α Tab β − Tα[a e Tb]e β − Tα[a γ Tb]γ β ), 8

(5.22)

from this equation we construct η cd (Racdb + Rbcda ): 1 η cd (Racdb + Rbcda ) = − η cd [(γdb )β α ∇α Tac β + (γda )β α ∇α Tbc β ] 8 1 + η cd [(γdb )β α Tα[a e Tc]e β + (γda )β α Tα[b e Tc]e β ] 8 1 + η cd [(γdb )β α Tα[a ǫ Tc]ǫ β + (γda )β α Tα[b ǫ Tc]ǫ β ]. 8

(5.23)

Let us consider the right hand side of (5.23) line by line. We can use (5.19) , the Bianchi identity (∇R)αaδβ γ to write

(γb )δα ∇α Raδ = −2∇a ∇b Φ − 2Tab γ ∇γ Φ − 2(γb γae )δβ Ωβ Re δ − (γa γb )β δ P βǫ Rǫδ , 16

(5.24)

and the beta function with dimension 1 (5.5) to find the following expression for the first line in the right hand side of (5.23)

1 −4∇b ∇a Φ+2Tab C ∇C Φ−4ηab (γ e )δβ Ωβ Reδ +4(γb )δβ Ωβ Raδ +4(γa )δβ Ωβ Rbδ + ηab η cd Tcβ δ T β . dδ 4 (5.25) Finding an expression for the second line is a matter of gamma matrices algebra, once we use (5.13) .

For this line we find

1 η T T cdβ 4 ab βcd

−

3 T β Tb)β c . 4 c(a

Using Taβ γ =

−(γa )βδ P δγ and some gamma matrices algebra, it is straightforward to find Tβ(a γ Tb)γ b − 1 η η cd Tdβ γ Tcγ β 4 ab

for the third line. So, adding the results for the three lines and using

(5.19) we find

η cd (Racdb + Rbcda ) = −4∇b ∇a Φ − Tc(a β Tb)β c + 2Tab E ∇E Φ + Tβ(a γ Tb)γ β ,

(5.26)

which contains some of the terms in (5.8) . It is also needed to use (∇T )abc c = 0 in order to generate the term ∇c Tabc . This Bianchi identity gives ∇c Tabc − Tc[a e Tb]e c − Tc[a ǫ Tb]ǫ c − η cd (Racdb − Rbcda ) = 0.

(5.27)

Finding an expression for η cd (Racdb − Rbcda ) is not difficult following the description given to compute (5.26) . After we compute it and replace it in (5.27) we find ∇c Tabc + Tβ[a δ Tb]δ β − 2Tab c ∇c Φ + 2Tab γ ∇γ Φ − 2Tab γ ∇γ Φ = 0.

(5.28)

Combining (5.26) and (5.28) gives the desired beta function equation (5.8). A similar procedure, but with more steps, is performed to prove the equations of the c ∇c P γβ + second group. To probe (5.9) one can start by computing {∇α , ∇β }P γβ = −γγβ

e β P γδ . Then we split the curvature as a scale curvature plus a Lorentz curvature. For R αβδ the latter, use (∇Te)αβc d = 0 to obtain cd β cd β cd e e e e e e cd e R αβcd (γ )δ = −180∇α Ωδ + (γ )δ ∇β Tαcd + 16Tδ Tαcd + (γ γ )δα Tecd ,

so on one hand we will have 17

(5.29)

e P γδ − 45∇α Ω e P γδ + 1 (γ cd ) β ∇ Teαcd P γδ {∇α , ∇β }P γβ = −∇c Tcα γ + R αδ δ δ β 4 1 −4Teαcd Teδ cd P γδ + (γ cdγ e )δα Teecd P γδ . 4

(5.30)

eα γβ , C e γ = −P γδ Ω e and C ecd γ = 1/10(γ a)γα R eaαcd , On the other hand, we can use ∇α P βγ = C δ

which come from antiholomorphicity of the BRST current, to write

e − 17P γδ ∇α Ω e + {∇α , ∇β }P γβ = −17∇α P γδ Ω δ δ

1 a γα eaαcd (γ cd )α β ). (γ ) ∇β (R 40

(5.31)

e αbcd = 0 it is straightforward to find Using (∇Te)αbcd = 0 and (∇H) eaαcd = 10Tcd γ − 10P γǫ Teǫcd . (γ a )γα R

(5.32)

(γ cd )α β ∇β Tcd γ = −18∇d Tdα γ + (γ cd γ e )αδ Teecd P γδ + 16Teαcd Tdc γ .

(5.33)

(γ ab )(α β (γab )γ) δ = −10δ(α β δγ) δ + 8(γ a )αγ (γa )βδ ,

(5.34)

Since there is a derivative acting on this terms in (5.31) , we make use of (∇Te)βcd γ = 0 to

find

We can now replace the last two equations in (5.31) and equate it to (5.30) . The identity

which can be proved using (γ a )(αβ (γa )γ)δ = 0, will be of help to find (5.9). A completely analog procedure allows us to arrive to (5.10). To prove (5.11) we make use of the Bianchi identities (∇R)αabβ γ = 0, (∇T )cαβ γ = 0 and the identity (γa )αβ Rαβγ δ = −2(γa )αβ Rγαβ δ , which follows from (∇T )αβγ δ = 0, to arrive to

(γ)αβ (∇α Rabβ γ − 2Tα[a e Rb]eβ γ − Tα[a ǫ Rb]ǫβ γ ) − 8Tb ac Tac γ + 8∇a Tab γ + 2Tab γ ∇a Φ 1 − (γ)αβ (γ cd )ǫ γ Rαβcd Tab ǫ + Tab ǫ (γ a )αβ Rǫαβ γ = 0. 8

(5.35)

The last term in this equation is zero as can easily seen using (∇T )ǫαβ γ = 0. The first term can be worked out using (5.22) and (∇T )aǫβ γ = 0, the curvature in the first term 18

of the second line can be rewritten using (∇T )αβa b = 0. The use of (∇T )cdb δ = 0 will be also needed to generate (5.11). Again, an analog procedure will allow at arrive to (5.12). So far, we concentrated on a specific set of beta functions. The remaining ones can be classified in a third and fourth sets. The third set involves first order derivatives of the curvatures. We present it again as the dimension increases. β

β

e N ac Π and At dimension 5/2 we find respectively from the couplings to JΠ , Πα J, e bc Πα N ∇a Raβ + ∇(ǫ Rβ)δ P δǫ + 2(∇β C α − Rβγ P γα )∇α Φ + 2C α ∇α ∇β Φ = 0,

(5.36)

ebα − ∇(δ R eα)ǫ P δǫ + 2(∇α C eβ + R eαγ P βγ )∇β Φ + 2C e β ∇β ∇α Φ = 0, ∇b R

(5.37)

∇d Rdβac + ∇(ǫ Rβδ)ac P δǫ + 2(∇β Cac α − Rβγac P γα )∇α Φ + 2Cac δ ∇δ ∇β Φ = 0,

(5.38)

γδ edαbc − ∇(δ R eα)ǫbc P δǫ + 2(∇α C ebc γ + R e e γ ∇d R αδbc P )∇γ Φ + 2Cbc ∇γ ∇α Φ = 0.

(5.39)

b

b

e N ac Π and While at dimension 3 we find respectively from the couplings to JΠ , Πa J, e bc Πa N ∇a Rab − Tba c Ra c + Tba γ Ra γ + 3Tbγ α ∇α C γ + 2Rbc ∇c Φ + 2Rbα P γα ∇γ Φ +2(∇b C α − Rbγ P γα )∇α Φ + P δǫ (∇ǫ Rδb + Tbδ c Rǫc + Tbǫ γ Rδγ ) = 0,

eb γ + Tabc C e γ + 2R eab ∇b Φ − 2R eaγ P γβ ∇ Φ eba + Tab γ R eδ Tδ bc + 3Taγ β ∇ C ∇b R β β eβ + R eaγ P βγ )∇β Φ − P δǫ (∇δ R eǫa + Tδa c R eǫγ ) = 0, eǫc + Taδ γ R +2(∇a C

(5.40)

(5.41)

∇d Rdbac − Tb de Rdeac + Tb dǫ Rdǫac + 3Tbδ γ ∇γ Cac δ + 2Rbdac ∇d Φ + 2Rbδac P ǫδ ∇ǫ Φ 19

+2(∇b Cac δ − Rbǫac P ǫδ )∇δ Φ + 2Rbδea Cc eδ + P δǫ (∇ǫ Rδbac + Tbδ f Rǫf ac + Tbǫ γ Rδγac ) = 0, (5.42)

ebc ǫ Tǫ df + 3Taδ ǫ ∇ǫ C ebc δ + 2R eadbc ∇d Φ − 2R eaδbc P δǫ ∇ǫ Φ edabc + Ta dǫ R edǫbc + Tadf C ∇d R

eǫf bc + Taδ γ R ebc δ + R eaǫbc P δǫ )∇δ Φ + 2R eaδeb C ec eδ − P δǫ (∇δ R eǫabc + Taδ f R eǫγbc ) = 0. +2(∇a C

(5.43)

The fourth set involves second order derivatives of the background fields P αβ , Cα βγ , eα βγ and S γδ . There is an equation at dimension 3, coming from the coupling to dα de C αβ β ∇2 P αβ − 2P γδ Sγδ αβ + Tde α T deβ − 2∇γ P δβ ∇δ P αγ − 2∇c P αβ ∇c Φ −2(P γδ ∇δ P αβ + P αδ ∇δ P γβ )∇γ Φ + 2(P δγ ∇δ P αβ + P δβ ∇δ P αγ )∇γ Φ = 0.

(5.44)

e N ac de and dα N e bc At dimension 7/2 we find respectively from the couplings to J deβ , dα J, β eαδǫ β P δǫ ∇2 C β − P αγ ∇[α ∇γ] C β − Tac β Rac + 2Rγ a ∇a P γβ + 2∇γ P αβ ∇α C γ − C α R

+P αβ (∇c Rα c − ∇[δ Rγ]α P δγ ) − 2(∇a C β − P γβ Raγ )∇a Φ − 2(P αβ ∇α C γ + P αγ ∇α C β 1 +P αβ Rαγ P γγ )∇γ Φ + 2(SP αβ + Secd (γ cd )ǫ β P αǫ − C γ ∇γ P αβ )∇α Φ = 0, 4

(5.45)

eα − Tbc α R ebc − 2R eγ b ∇b P αγ − 2∇γ C eβ Rβǫδ α P δǫ eα − P βγ ∇[β ∇γ] C e β ∇β P αγ + C ∇2 C

e c + ∇[δ R e P δγ ) − 2(∇b C eα + P αγ R ebγ )∇b Φ + 2(P αβ ∇ C eγ + P γβ ∇ C eα −P αβ (∇c R β γ]β β β eǫγ P γγ )∇γ Φ − 2(SP αβ + 1 Scd (γ cd )ǫ α P ǫβ − C e γ ∇γ P αβ )∇ Φ = 0, +P αǫ R β 4

(5.46)

eγδǫ β P δǫ ∇2 Cac β − P δǫ ∇[δ ∇ǫ] Cac β − Rdeac T deβ − 2Rdǫac ∇d P ǫβ + 2∇δ P ǫβ ∇ǫ Cac δ − Cac γ R

−P ββ (∇d Rdβac −∇[δ Rǫ]βac P δǫ +2Rβδea Cc eδ )+2∇δ Cea β Cc eδ −2(∇d Cac β −P ǫβ Rdǫac )∇d Φ 1 −2(Cac δ ∇δ P γβ − Sac P γβ − Sacbd (γ bd )δ β P γδ )∇γ Φ 4 20

−2(P γβ ∇γ Cac δ + P γδ ∇γ Cac β − P ǫβ Rǫγac P γδ )∇δ Φ = 0,

(5.47)

ebc α − P δǫ ∇[δ ∇ǫ] C ebc α − R edebc T deα + 2R edǫbc ∇d P αǫ − 2∇ C ebc γ Rγǫδ α P δǫ e ǫ ∇ǫ P αδ + C ∇2 C δ bc

δǫ ǫβ e d e e e e eδ e α e eδ e α −P αβ (∇d R dβbc − ∇[δ Rǫ]βbc P + 2Rβδeb Cc ) + 2∇δ Ceb Cc − 2(∇d Cbc + P Rdβbc )∇ Φ

ebc δ ∇δ P αγ − Sebc P αγ − 1 Sadbc (γ ad )δ α P δγ )∇γ Φ +2(C 4

αǫ e δγ ebc δ + P δβ ∇ C e α +2(P αβ ∇β C β bc + P Rǫγbc P )∇δ Φ = 0.

(5.48)

e JN e ac , N ab Je and N ab N e cd Finally, at dimension 4 we find from the couplings to J J,

respectively

eab + 2R e ∇a C β + 2Raβ ∇a C eβ − 2∇α C e β ∇β C α ∇2 S − P δǫ ∇[δ ∇ǫ] S − Rab R aβ

eβ (∇a Raβ − P δǫ ∇[δ Rǫ]β ) − C β (∇a R e − P δǫ ∇[δ R e ) + 2(C eα Rbα + C α R ebα )∇b Φ −C aβ ǫ]β eγα + C eβ + P βα (∇α S + C γ R e γ Rγα ))∇β Φ − 2(C eα ∇α C β − P αβ (∇α S −2(C α ∇α C eγ Rγα ))∇ Φ = 0, eγα + C +C γ R β

(5.49)

b δ be δ β eedac +2R e e δ e e bδ ∇2 Seac −P δǫ ∇[δ ∇ǫ] Seac −Red R bδac ∇ C +2Rbδ ∇ Cac −2∇β Cac ∇δ C −2∇δ Sba Cc δǫ δǫ e e eδ e β d e e −C β (∇d R dβac − P ∇[δ Rǫ]β + 2Rβδea Cc ) − Cac (∇ Rdβ − P ∇[δ Rǫ]β )

d δ γ eac β Rdβ + C β R e e γ e e bγ e β +2(C dβac )∇ Φ − 2C ∇δ Cac ∇γ Φ + 4Sab Cc ∇γ Φ − 2Cac ∇β C ∇γ Φ = 0,

(5.50)

ecd Rcdab +2Rcδab ∇c C e δ +2R e ∇c Cab δ −2∇γ C eδ ∇δ Cab γ −2∇ Sca Cb cδ ∇2 Sab −P δǫ ∇[δ ∇ǫ] Sab −R cδ δ γ de δǫ e eδ eγ (∇d Rdγab − P δǫ ∇[δ Rǫ]γab + 2R −C γδea Cb ) − Cab (∇ Rdγ − P ∇[δ Rǫ]γ )

edγab )∇d Φ − 2Cab γ ∇γ C eδ ∇δ Φ + 4Sac Cb cδ ∇ Φ − 2C eγ ∇γ Cab δ ∇ Φ = 0, eγ Rdγab + Cab γ R +2(C δ δ (5.51)

21

eef cd Ref ab +2R ef ǫcd ∇f Cab ǫ +2Rf ǫab ∇f C ecd γ ∇γ Cab ǫ ecd ǫ −2∇ǫ C ∇2 Sabcd −P δǫ ∇[δ ∇ǫ] Sabcd − R

eǫδec C ed f ǫ −Cab ǫ (∇e R eeǫcd −P δγ ∇[δ R eγ]ǫcd +2R ed eδ )−C ecd ǫ (∇e Reǫab +2∇ǫ Saf cd Cb f ǫ +2∇ǫ Sabcd C ecd ǫ Reǫab + Cab ǫ R ecd ǫ ∇ǫ Φ + 4Sabcf C ed f ǫ ∇ǫ Φ eeǫcd )∇e Φ − 2Cab γ ∇γ C −P δγ ∇[δ Rγ]ǫab ) + 2(C ecd γ ∇γ Cab ǫ ∇ǫ Φ + 4Saf cd Cb f ǫ ∇eb Φ = 0. −2C

(5.52)

Since the Bianchi identities allow to write the curvature components in terms of the torsion components, we expect that the beta functions of the third set will be implied by the eight beta functions already proven, i.e first and second set. In the same way we expect that the beta functions of the fourth set will also be implied by the first two sets of beta functions since the constraints coming from holomorphicity and antiholomorphicity of the BRST current allows to relate the background fields to some components of the torsion. This is not too hard to check in the case of lower dimension, for example, at dimension β

5/2 consider the beta functions coming from the coupling to JΠ

∇a Raβ + ∇(ǫ Rβ)δ P δǫ + 2(∇β C α − Rβγ P γα )∇α Φ + 2C α ∇α ∇β Φ = 0.

(5.53)

By using Raβ = Taβ γ Ωγ and Rβδ = ∇β Ωδ , which follow from the definition of the curvature, and C β = P αβ Ωα , which follows from the antiholomorphicity constraints, we find that (5.53) can be written as (∇c Tcβ α − 2∇β P αγ ∇γ Φ + 2P αγ ∇γ ∇β Φ)Ωα = 0,

(5.54)

so, the beta function (5.9) with dimension 2 implies (5.53) . Similarly we checked that (5.10) implies (5.37) and that the beta functions with dimension 5/2 (5.11) and (5.12) imply respectively the beta functions with dimension 3 (5.40) and (5.41) . We have not found proofs for the vanishing of the remaining beta functions, since this task becomes clumsy as the dimension increases. Nevertheless, given the above explanation, we consider our work sufficient to assure that the beta functions vanish as a consequence of the classical BRST symmetry of the action for the Type II superstring in a generic supergravity background. Acknowledgements: We would like to thank Nathan Berkovits, Brenno Carlini Vallilo and Daniel L. Nedel. The work of OC is supported Fundación Andes, FONDECYT grant 1061050 and Proyecto Interno 27-05/R from UNAB. The work of O.B is supported by CAPES, grant 33015015001P7. 22

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