Boundary conformal anomalies on hyperbolic spaces and Euclidean balls

Journal of High Energy Physics, Dec 2017

We compute conformal anomalies for conformal field theories with free conformal scalars and massless spin 1/2 fields in hyperbolic space ℍ d and in the ball \( {\mathbb{B}}^d \), for 2≤d≤7. These spaces are related by a conformal transformation. In even dimensional spaces, the conformal anomalies on ℍ2n and \( {\mathbb{B}}^{2n} \) are shown to be identical. In odd dimensional spaces, the conformal anomaly on \( {\mathbb{B}}^{2n+1} \) comes from a boundary contribution, which exactly coincides with that of ℍ2n + 1 provided one identifies the UV short-distance cutoff on \( {\mathbb{B}}^{2n+1} \) with the inverse large distance IR cutoff on ℍ2n + 1, just as prescribed by the conformal map. As an application, we determine, for the first time, the conformal anomaly coefficients multiplying the Euler characteristic of the boundary for scalars and half-spin fields with various boundary conditions in d = 5 and d = 7.

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Boundary conformal anomalies on hyperbolic spaces and Euclidean balls

JHE Boundary conformal anomalies on hyperbolic spaces Diego Rodriguez-Gomez 0 1 2 3 Jorge G. Russo 0 1 2 0 Universitat de Barcelona , Mart ́ı Franqu`es, 1, Barcelona, 08028 Spain 1 Pg. Lluis Companys , 23, Barcelona, 08010 Spain 2 Avda. Calvo Sotelo 18 , Oviedo, 33007 Spain 3 Department of Physics, Universidad de Oviedo We compute conformal anomalies for conformal field theories with free conformal scalars and massless spin 1/2 fields in hyperbolic space Hd and in the ball Bd, for 2 ≤ d ≤ 7. These spaces are related by a conformal transformation. In even dimensional spaces, the conformal anomalies on H2n and B2n are shown to be identical. In odd dimensional spaces, the conformal anomaly on B2n+1 comes from a boundary contribution, which exactly coincides with that of H2n+1 provided one identifies the UV short-distance cutoff on B2n+1 with the inverse large distance IR cutoff on H2n+1, just as prescribed by the conformal map. As an application, we determine, for the first time, the conformal anomaly coefficients multiplying the Euler characteristic of the boundary for scalars and half-spin fields with various boundary conditions in d = 5 and d = 7. Conformal Field Theory; Anomalies in Field and String Theories - HJEP12(07)6 1 Introduction and conclusions 2 3 Setting the computation Spaces of even dimension Coefficients of boundary anomalies in d = 3, 5, 7 dimensions in [11], where d dimensional conformally flat spaces of the form Sa considered. These spaces generalize the celebrated S1 covers only the causal development of an Sd−1 and hence allows for the computation of the entanglement entropy across the sphere in terms of a standard thermal partition function. × Hd−1 space, whose mapping to Rd × Hd−a have been Among the spaces introduced in [11], an interesting class is the case of odd d and even anomaly arises through the regularization of the divergent volume of the odd-dimensional hyperbolic space, which may lead to the somewhat surprising conclusion that the anomaly is an IR effect. Note that in fact the same puzzle arises in the S1 × Hd−1 case for even d. In this paper we will clarify this issue by considering the particular case of the Hd spaces. These can be conformally mapped into the ball Bd with no topology changing, so that the partition functions, as we show below, agree. However, in the Bd case, the divergence leading to the anomaly is due to short distance (to the boundary) effects, and it is therefore a UV effect. This shows that also in the Hd case the boundary anomalies can be regarded as a UV effect, both for even and odd d. The fact that the naively-looking IR regulator in Hd is secretly a UV regulator in the ball Bd can be directly understood considering the mapping among the two spaces. To be explicit, the Hd space of radius R can be conformally mapped into Bd of the same radius as In particular, the dictionary between radial coordinates is tanh y2 = Rr and Note that from here it follows that r ∈ [0, R].2 Moreover, from (1.2) it also follows that the boundary of the Hd at large ρ is mapped to the boundary of the Bd at r ∼ R. An IR cutoff at ρ = L corresponds to a UV cutoff at R − r = δ, with becomes ds2 = R2 d̺2 + ̺2 dΩ2d−1 . Thus L is to be interpreted as the large volume (IR) cut-off for the Hd while δ = R − r is to be interpreted as a short distance (UV) cut-off for the Bd (in fact, we will use the same notation δ for UV short-distance cutoffs in all spaces). Then, (1.3) implements explicitly the relation between the IR regulator in the Hd and the UV regulator for the Bd. It is interesting to note that this UV/IR mixing is very reminiscent to the AdS/CF T correspondence, and may suggest that a sort of “rigid holography” is at play. A second reason to consider spaces of the form Hd or Bd is that these spaces have as boundary Sd−1. As discussed below, one would expect that the only possible (boundary) anomaly term which can be constructed is proportional to the Euler number of the sphere. 2One may also consider introducing a dimensionless coordinate ̺ = Rr ∈ [ 0, 1 ], so that the Bn metric ρ = 2r 1 − R2 4 [5, 7, 8], one can separate contributions to the boundary anomaly proportional to the Euler number of the boundary (“A-type” anomaly) from contributions constructed in terms of the Weyl tensor and the traceless part of the extrinsic curvature of the boundary [7]. From our computation, we can read off the boundary a central charges for scalars and spin 1/2 fields in different dimensions. We stress that the new anomaly coefficients found here (multiplying the Euler characteristic of the boundary) should be universal and apply to any conformal field theory on a manifold with boundary (i.e. not only to H2n+1 or B2n+1). The remainder of the paper is structured as follows. In section 2 we briefly describe a standard method to compute the conformal anomaly that will be used here. In section 3 we compute the logarithmic part of the free energy for scalars and fermions on (...truncated)


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Diego Rodriguez-Gomez, Jorge G. Russo. Boundary conformal anomalies on hyperbolic spaces and Euclidean balls, Journal of High Energy Physics, 2017, pp. 66, Volume 2017, Issue 12, DOI: 10.1007/JHEP12(2017)066