Noncommutative Chiral Gravitational Anomalies in Two Dimensions

REVISTA MEXICANA DE FISICA´ S 53 (2) 120–124 FEBRERO 2007

Noncommutative chiral gravitational anomalies in two dimensions

H. Garc´ıa-Compean´ and C. Soto-Campos Departamento de F´ısica, Centro de Investigacion´ y de Estudios Avanzados del IPN Apartado Postal 14-740, 07000, Mexico´ D.F., Mexico.´ Recibido el 18 de julio de 2005; aceptado el 14 de marzo de 2005

Gravitational anomalies in a noncommutative space are examined. The analysis is generic and independent of a particular noncommutative theory of gravity, and it depends only on how gravity is noncommutatively coupled to chiral fermions. Delbourgo-Salam computation of the gravitational correction of the axial ABJ-anomaly is studied in detail in this context. Finally, we show that the two-dimensional gravitational anomaly does not permit noncommutative corrections in the parameter Θ.

Keywords: Gravitational anomalies; Delbourgo-Salam anomaly; noncommutative ﬁeld theory.

Se examinan las anomal´ıas gravitacionales en un espacio no conmutativo. El analisis´ es general e independiente de alguna teor´ıa de gravedad no conmutativa espec´ıﬁca y depende solo´ de como´ la gravedad se acople a los fermiones quirales. El calculo´ de Delbourgo-Salam de la correcccion´ gravitacional a la anomal´ıa axial ABJ se estudia en detalle en este contexto. Finalmente se muestra que la anomal´ıa gravitacional en dos dimensiones no admite correcciones no conmutativas en el parametro´ Θ.

Descriptores: Anomalias gravitacionales; anomal´ıa de Delbourgo-Salam; teor´ıas de campo nonconmutativas.

PACS: 04.50.+h; 11.30.Rd

ab 1. Introduction with ωµ (x) being the noncommutative spin connection as- a sociated with the tetrad eµ(x), and [A, B]∗ ≡ A ∗ B − B ∗ A An important effect in quantum field theory are the anoma- is the Moyal bracket. Here the ∗-product is defined by lies. Axial and gauge anomalies in various dimensions, in µ ¶ ¯ ¯ particular in four dimensions, have been discussed in the i µν ∂ ∂ ¯ F ∗ G(x) ≡ exp Θ µ ν F (y)G(z)¯ . context of noncommutative gauge theories by various au- 2 ∂y ∂z y=z=x thors [1–13]. From now on, in order to avoid causality problems, we will On the other hand, recently, various noncommutative the- take θ0ν = 0. ories of gravity have been proposed by a number of authors, Noncommutative perturbative gravity is defined by a per- providing different Moyal deformations of Einstein gravity turbative expansion I = I(0) + I(1) + I(2) + O(κ4) of the in four dimensions (for a recent review of noncommutative noncommutative Einstein-Hilbert action generated by a per- gravity, see [14]). In this context, a noncommutative proposal turbative expansion of the metric as follows: for a topological gravity generalizing Euler and the signature 2 α topological invariants was given in Ref. 15. gµν = ηµν − κhµν + κ hµ ∗ hαν However, at the present time there is not a definitive, well- − κ3hα ∗ h ∗ hβ + O(κ4). defined, realistic noncommutative theory of gravity. In this µ αβ ν note we will not deal with any specific noncommutative the- This note is organized as follows: In Sec. 2 we give ory of gravity. This is because at the end we will not con- the relevant Feynman rules for linear gravity coupled to chi- sider a specific theory of pure gravity, but we will be inter- ral fermions in a theory D=2k dimensions. Sec. 3 dis- ested only in the interactions of a linearized noncommutative cusses the noncommutative correction to the Delbourgo- gravitational field with chiral fermions. However, to be con- Salam anomaly. In Sec. 4 we describe the pure noncom- crete, we will briefly review a particular proposal of noncommutative anomaly in two dimensions. mutative Einstein gravity [16] given by the noncommutative Einstein-Hilbert action: Z 2. Coupling Gravity to Chiral Fermions b 1 4 a b µν IEH = − d x(−e) ∗ eµ(x) ∗ eν (x) ∗ Rab (x), Reference 17 gave the Feynman rules of this pure noncom- 16πGN X mutative gravity theory. Let us consider the theory in D = 2k dimensions. The coupling of the gravitational field with chi- where ral fermions is given as usual by Z g (x) = ea (x) ∗ eb (x)η , 1 µν µ ν ab I = d2kx e ∗ eµa(x) ∗ ψ(x) int 2 and µ ¶ ←→ 1 − Γ¯ ab ab ab ab ∗ iΓa D µ ψ(x), (1) Rµν (x) = ∂µων (x) − ∂ν ωµ (x) + [ωµ(x), ων (x)]∗ , 2 NONCOMMUTATIVE CHIRAL GRAVITATIONAL ANOMALIES IN TWO DIMENSIONS 121

(i) where e stands for det(e) and Dµ is the covariant derivative respectively. We have introduced εµα for the polarization ten- ab with respect to the spin connection ωµ given by sors from the graviton field. 1 D ψ(x)=∂ ψ(x)+ ω σcdψ(x), µ µ 2 µcd with 3. Noncommutative Delbourgo-Salam Gravi- 1 σcd= [Γc, Γd], Γ=Γ¯ ... Γ tational Anomaly 4 1 2k and the Γ’s are the Dirac matrices in euclidean 2k dimen- Gravitational anomalies in four dimensions were studied first sions. by Delbourgo and Salam [18] as a gravitational correction a Expanding eµ around flat space eµa=ηµa+(1/2)hµa, our to the violation of a global symmetry responsible for the de- 0 noncommutative action splits into two parts, Iint=I1 + I2, cay: π → γγ. This idea was further developed in Refs. 19 where and 20. Here we shall discuss the noncommutative counter- Z µ ¶ 1 ↔ 1 − Γ¯ part of Delbourgo and Salam work [18], which showed that in I = dx e∗eµa(x)∗ψ(x)∗iΓ ∂ ψ(x) (2) 1 2 a µ 2 addition to the fermion triangle diagram with three currents, the triangle diagram with one current J of a global symmetry and Z and two energy-momentum tensors T is also anomalous. The 1 µa cd I2= dx e ∗ e (x) ∗ ωµ (x) corresponding contribution from the anomalous Ward iden- 4 tity is given by µ ¶ 1−Γ¯ × ∗iψ(x) ∗ Γacd ψ(x), (3) 1 2 R Rρσ εκλµν . (8) 384π2 κλρσ µν where Γacd = 1/6(ΓaΓcΓd ±permutations). The lineariza- tion of our noncommutative action Iint given by Eq. (1) leads This is precisely proportional to the signature invariant σ(X) to the Moyal deformation of linear gravity given by the la- (or the first Pontrjagin class) which, with the Euler number grangians χ(X), are the classical topological invariants of the smooth µ ¶ 1 ↔ 1 − Γ¯ spacetime manifold X. L = − ihµν (x) ∗ ψ(x) ∗ Γ ∂ ψ(x), (4) 1 4 µ ν 2 Now we shall discuss in detail the derivation of the non- µ ¶ commutative counterpart of Eq.(8). The scattering amplitude 1 1 − Γ¯ L = − ih (x) ∗ ∂ h ∗ Γµλν ψ(x). (5) of the process in 4 dimensions is given by 2 16 λα µ να 2 Z ¡ ¢ The corresponding noncommutative Feynman rules can be exp − i Θρσ(p − k ) (p + k ) Tr d4p{Γ · p, Γ } 2 2 ρ 1 σ obtained from the lagrangians L1 and L2, giving κλµν µ ¶ µ ¶ [Γ · (p + k1) − M] i 1 − Γ¯ i − εµν Γ (2p + p0) exp − Θρσp p0 , (6) 4 µ 2 ν 2 ρ σ ¡ ¢ exp − i Θρσ(p + k ) p ρ1 σ1 2 1 ρ σ ρ2 σ2 and ×ερ1σ1 p Γ ερ2σ2 p Γ µ ¶ µ ¶ (Γ · p − M) i 1 − Γ¯ i − Γλµν ε(1)ε(2) exp Θρσp p0 να λα ρ σ ¡ ¢ 16 2 2 i ρσ · µ ¶ exp − 2 Θ pρ(p − k2)σ i × , (9) × k exp Θρσk k [Γ · (p − k2) − M] 1µ 2 1ρ 2σ µ ¶ ¸ where we have used the Feynman rule Eq. (6) in each vertex i ρσ −k2µ exp Θ k1ρk2σ , (7) of the triangle diagram and the corresponding fermion prop- 2 agators. In order to evaluate this amplitude we promote the integral from 4 to 2` dimensions

Z µ ¶ (Γ · (p + k1)+M) i (Γ · p + M) d2`p £ ¤ · exp − Θρσ(p − k ) (p + k ) ε pρ1 Γσ1 £ ¤ 2 2 2 ρ 1 σ ρ1σ1 2 2 (p + k1) −M 2 p −M µ ¶ µ ¶ i (Γ · (p − k2)+M) i × exp − Θρσ(p + k ) p £ ¤ ε pρ2 Γσ2 exp − Θρσp (p−k ) . (10) 1 ρ σ 2 2 ρ2σ2 ρ 2 σ 2 (p − k2) −M 2

Rev. Mex. F´ıs. S 53 (2) (2007) 120–124 122 H. GARCIA-COMPE´ AN´ AND C. SOTO-CAMPOS

To calculate this integral, we introduce Feynman’s parame- gravitational anomalies. In this section, we are going into the ters x, y and z, in the usual way, integrating out the variable z, details of the computation of the pure gravitational anomaly keeping only the divergent terms and integrating out the mo- in two dimensions. We shall follow the notation and conven- mentum variable p. We finally obtain tions of Ref. [21] (for further work, see [22]). We shall not consider global gravitational anomalies [23] here. `+1 ρ1 ρ2 σ1σ2αβ 2 (` − 2)k2 k1 ε εκλαβk1αk2β In two dimensions, the noncommutative action for a µ ¶ i Majorana-WeylR fermion in a gravitational field is given by ρσ −` 2 µa 1 × exp − Θ k1ρk2σ (4π) Γ(2 − `) I = d x e∗e (x)∗ ψ¯(x)∗iΓ ∂ ψ(x). At the linearized 2 2 a µ Z level, the lagrangian is 2 2 `−2 × (k3xy − M ) ixy θ(1 − x − y)dxdy + ..., (11) 1 L = − hµν (x) ∗ iψ¯(x) ∗ Γ ∂ ψ(x). (16) int 4 µ ν where θ(z) is the usual Heaviside function. In the last expres- The corresponding energy-momentum tensor is given by sion we have used the trace identity given by: ¡ ¢ 1 ¯ σ1 σ2 α β ` [σ1 σ2 α β] ` σ1σ2αβ Tµν (x) = iψ(x) ∗ Γµ∂ν ψ(x). (17) Tr Γκλµν Γ Γ Γ Γ =2 δ[κ δλ δµ δν] =2 ε εκλµν . 2 Performing the expansion of the gamma function Γ(ε) for In order to facilitate the computation, as usual, we intro- small values of ε with ε = 2 − `, taking the limit ` → 2 and duce light-cone coordinates evaluating the integral in x and y, we finally get 1 x± = √ (x0 ± x1). kρ1 kρ2 2 2 1 σ1σ2αβ −i ε εκλαβk1αk2β 12π2 Dirac matrices are decomposed into ¡ i ρσ ¢ × exp − Θ k1ρk2σ . (12) 1 2 Γ± = √ (Γ0 ± Γ1), 2 Now, taking into account the most general Lorentz invariant amplitude, the last expression becomes with (Γ±)2 = 0 and Γ+Γ− + Γ−Γ+ = 2. In these coordinates, the energy-momentum tensor takes the form i ¡ − ε (k )ε (k )k k εσ1σ2αβ ηρ1ρ2 k · k 192π2 ρ1σ1 1 ρ2σ2 2 1α 2β 1 2 1 T (x) = iψ¯(x) ∗ Γ ∂ ψ(x), (18) µ ¶ ++ 2 + + ¢ i −kρ2 kρ1 ε exp − Θρσk k . (13) 1 2 κλαβ 2 2ρ 1σ while the interaction action (16) of the gravitational field with fermions in the light-cone coordinates reduces to In the coordinate space, this equation can be rewritten as 1 σ1σ2αβ γ ρ1 ρ2 ρ1 ¯ ε (∂α∂γ hρ σ ∗ ∂β∂ h −∂α∂ hρ σ ∗ ∂β∂ hρ σ ) Lint = − ih−−(x) ∗ ψ(x) ∗ Γ+∂+ψ(x). (19) 1 1 σ2 1 1 2 2 4

× εκλαβ, (14) Only the component h−−(x) of the graviton is coupled to chiral matter described by the component T++(x) of the which ﬁnally we recognize as the invariant energy-momentum tensor. The effective action to the second order in the metric perturbation h is encoded in the two-point 1 ρσ κλµν 2 Rκλρσ ∗ Rµν ε . (15) correlation function 384π Z 2 ¡ ¢ This is precisely the noncommutative signature invariant U(p) = d x exp ip·x hΩ|T (T++(x)∗T++(0)|Ωi. (20) Z τb(X) = R ∗ Re d4x, The naive Ward identity is given by p−U(p) = 0. This should imply U(p) = 0 for all p−; thus, it should be an where the tilde over R stands for the Hodge dual with respect anomaly. We are going to compute U(p) by evaluating the to the tangent space indices. Compare this with the noncom- corresponding one-loop diagram with two external gravitons; mutative signature σb(X) of Ref. 15, where the Hodge duality this yields was associated with the tetrad indices. Z 1 dk+dk− 2 1 U(p) = − 2 (2k + p)+ 4 (2π) k− + iε/k+ 4. Noncommutative Pure Gravitational ¡ ¢ i ρσ 0 exp − Θ p pσ Anomaly in Two Dimensions × 2 ρ δ(p + p0) (k + p)− + iε/(k + p)+ In Sec. 2, we introduced Feynman rules for noncommutative ¡ 0 ¢ perturbative quantum gravity relevant to computing the chiral × exp i(p + p )x , (21)

Rev. Mex. F´ıs. S 53 (2) (2007) 120–124 NONCOMMUTATIVE CHIRAL GRAVITATIONAL ANOMALIES IN TWO DIMENSIONS 123 where we have again used the Feynman rule Eq. (6) to com- Similarly to the usual commutative case, there is no way to eff eff eff pute U(p). In light-cone coordinates, the Moyal product is add generic counterterms ∆L+ such that L+ + ∆L+ is given by invariant under general coordinate transformations. µ ¶ µ ¶ Thus, let us consider a Dirac fermion in 1 + 1 dimen- i ρσ 0 1 +− 0 0 eff exp − Θ p pσ = exp − Θ (p p−−p p+) . ρ + − sions. Then we have the corresponding action LD , which is 2 2 eff the superposition of L+ , and its corresponding parity con- Thus, by analytic methods, the computation of the inte- eff jugate L resulting in grals gives − µ ¶ Z 3 1 i p+ i ρσ 0 eff 2 2 0 U(p) = exp − Θ pρpσ LD (hµν ) = − d pd p 24π p− 2 192π · µ ¶ ¡ 0 ¢ 0 3 × exp i(p + p )x δ(p + p ). (22) p+ i ρσ 0 0 × h−−(p) exp − Θ pρpσ h−−(p ) p− 2 Then the anomalous gravitational Ward identity is given by µ ¶ ¸ µ ¶ p3 i i − i ρσ 0 0 p U(p) = p3 exp − Θρσp0 p + h++(p) exp − Θ pρpσ h++(p ) − 24π + 2 ρ σ p+ 2 ¡ ¢ ¡ ¢ × exp i(p + p0)x δ(p + p0). × exp i(p + p0)x δ(p + p0). (23) (25) The computation of the two-graviton diagram coupled This action is not invariant under inﬁnitesimal general µ µ with chiral fermions in the noncommutative theory is given coordinate transformations δx =ε , hµν transforms as by the effective action δhµν (x)= − ∂µεν (x) − ∂ν εµ(x), or in the momentum space Z 3 eff 1 2 2 0 p+ L+ (hµν ) = − d pd p h−−(p) δh++(p) = −2ip+ε+, δh+−(p) = −ip−ε+ − ip+ε−, 192π p− µ ¶ i × exp − Θρσp0 p h (p0) δh−−(p) = −2ip−ε−. 2 ρ σ −− ¡ ¢ eff 0 0 However, in this case there exists a counterterm ∆LD × exp i(p + p )x δ(p + p ). (24) eff which can be added to LD so that it becomes invariant under general coordinate transformations

Z · 3 µ ¶ eff 1 2 2 0 p+ i ρσ 0 0 ∆LD = − d pd p h−−(p) exp − Θ pρpσ h−−(p ) 192π p− 2 3 µ ¶ µ ¶ p− i ρσ 0 0 i ρσ 0 0 + h++(p) exp − Θ pρpσ h++(p ) + 2p+p−h++(p) exp − Θ pρpσ h−−(p ) p+ 2 2 µ ¶ µ ¶ i i − 4p2 h (p) exp − Θρσp0 p h (p0) − 4p2 h (p) exp − Θρσp0 p h (p0) + −− 2 ρ σ +− − ++ 2 ρ σ +− µ ¶ ¸ i + 4p p h (p) exp − Θρσp0 p h (p0) δ(p + p0). (26) + − +− 2 ρ σ +− It is easy to see that this action can be rewritten in a compact form as the following: ¡ ¢ Z i ρσ 0 0 eff 1 2 2 0 R(p) exp − 2 Θ pρpσ R(p ) 0 ∆LD = − d pd p δ(p + p ), (27) 192π p+p− which after integration in the variable p0 gives precisely the usual correction to the commutative counterpart of [21], Z eff 1 2 R(p)R(−p) ∆LD = − d p , (28) 192π p+p− where R(p) is the linearized term of the noncommutative scalar cuvature, which is given by 2 2 R(p) = p+h−− + p−h++ − 2p+p−h+−.

There is a quantum correction to T+−(p) = 0 which holds classically due to the introduction of h+− in the counterterm eff lagrangian ∆LD and we have an expectation value of T+− different form zero, which gives rise to the gravitational anomaly eff δ∆LD 1 h2T+−(p)i = −2 = − R(p). (29) δh+−(−p) 24π

Rev. Mex. F´ıs. S 53 (2) (2007) 120–124 124 H. GARCIA-COMPE´ AN´ AND C. SOTO-CAMPOS

By momentum conservation, we have in the above analy- Acknowledgments 0 0 sis that¡ p = −p through¢ the δ(p + p ), and the phase factor ρσ 0 exp − iΘ pρpσ is equal to one, and therefore there is no This work was supported in part by CONACyT Mexico´ grant modiﬁcation to the gravitational anomaly in two dimensions No. 33951E. C.S.-C. is supported by a CONACyT (Mexico)´ in a noncommutative space. graduate fellowship.

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