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
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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)