Revista Mexicana de Física ISSN: 0035-001X [email protected] Sociedad Mexicana de Física A.C. México García-Compeán, H.; Soto-Campos, C. Noncommutative chiral gravitational anomalies in two dimensions Revista Mexicana de Física, vol. 53, núm. 2, febrero, 2007, pp. 120-124 Sociedad Mexicana de Física A.C. Distrito Federal, México Available in: http://www.redalyc.org/articulo.oa?id=57066217 How to cite Complete issue Scientific Information System More information about this article Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Journal's homepage in redalyc.org Non-profit academic project, developed under the open access initiative REVISTA MEXICANA DE F ISICA´ S 53 (2) 120–124 FEBRERO 2007 Noncommutative chiral gravitational anomalies in two dimensions H. Garc ´ıa-Compe an´ and C. Soto-Campos Departamento de F ´ısica, Centro de Investigaci on´ y de Estudios Avanzados del IPN Apartado Postal 14-740, 07000, M exico´ D.F., M exico.´ 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 an alisis´ es general e independiente de alguna teor ´ıa de gravedad no conmutativa espec ´ıfica y depende s olo´ de c omo´ la gravedad se acople a los fermiones quirales. El c alculo´ de Delbourgo-Salam de la correccci on´ 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 par ametro´ 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 i µν ∂ ∂ particular in four dimensions, have been discussed in the F ∗ G(x) ≡ exp Θ F (y)G(z) . 2 ∂y µ ∂z ν context of noncommutative gauge theories by various au- 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 − κh µν + 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=2 k 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 noncom- mutative anomaly in two dimensions. mutative Einstein gravity [16] given by the noncommutative Einstein-Hilbert a ction: 2. Coupling Gravity to Chiral Fermions 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- 16 πG N X mutative gravity theory. Let us consider the theory in D = 2 k dimensions. The coupling of the gravitational field with chi- where ral fermions is given as usual by g (x) = ea (x) ∗ eb (x)η , 1 µν µ ν ab I = d2kx e ∗ eµa (x) ∗ ψ(x) int 2 and ←→ 1 − Γ¯ ab ab ab ab ∗ iΓa D µ ψ(x), (1) Rµν (x) = ∂µων (x) − ∂ν ωµ (x) + [ ωµ(x), ω ν (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 σcd ψ(x), µ µ 2 µcd with 3. Noncommutative Delbourgo-Salam Gravi- 1 σcd = [Γ c, 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- 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 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 1−Γ¯ × ∗ iψ(x) ∗ Γacd ψ(x), (3) 1 2 R Rρσ εκλµν . (8) 384 π2 κλρσ µν where Γacd = 1 /6(Γ aΓcΓd ±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 1 ↔ 1 − Γ¯ spacetime manifold X. L = − ih µν (x) ∗ ψ(x) ∗ Γ ∂ ψ(x), (4) 1 4 µ ν 2 Now we shall discuss in detail the derivation of the non- 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 i ρσ The corresponding noncommutative Feynman rules can be exp − Θ (p − k2)ρ(p + k1)σ Tr d4p{Γ · p, Γ } 2 obtained from the lagrangians L1 and L2, giving κλµν [Γ · (p + k1) − M] ¯ i µν 1 − Γ i ρσ − ε Γµ (2 p + p )ν exp − Θ pρpσ , (6) 4 2 2 i ρσ exp − Θ (p + k1)ρpσ ×ε pρ1 Γσ1 2 ε pρ2 Γσ2 and ρ1σ1 (Γ · p − M) ρ2σ2 i 1 − Γ¯ i − Γλµν ε(1) ε(2) exp Θρσ p p 16 2 να λα 2 ρ σ − i ρσ − exp 2 Θ pρ(p k2)σ i × , (9) ×k exp Θρσ k k [Γ · (p − k2) − M] 1µ 2 1ρ 2σ 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 2 (Γ · (p + k1)+ M) i ρσ ρ σ (Γ · p + M) d p · exp − Θ (p − k ) (p + k ) ε p 1 Γ 1 2 2 2 ρ 1 σ ρ1σ1 2 2 (p + k1) −M 2 p −M i ρσ (Γ · (p − k2)+ M) ρ σ i ρσ × exp − Θ (p + k ) p ε p 2 Γ 2 exp − Θ p (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. GARC IA-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 εκλαβ k1αk2β In two dimensions, the noncommutative action for a i Majorana-Weyl fermion in a gravitational field is given by ρσ − 2 µa 1 × exp − Θ k1ρk2σ (4 π) Γ(2 − ) I = d x e ∗e (x)∗ ψ¯(x)∗iΓ ∂ ψ(x).