The heat kernel on AdS

Abstract

We explicitly evaluate the heat kernel for the Laplacian of arbitrary spin tensor fields on the thermal quotient of (Euclidean) AdS N for N ≥ 3 using the group theoretic techniques employed for AdS 3 in arXiv:0911.5085. Our approach is general and can be used, in principle, for other quotients as well as other symmetric spaces.

Heat kernels on the AdS2 cone and logarithmic corrections to extremal black hole entropy

“…These are coset spaces G/H where G and H are Lie groups, and H is a subgroup of G. Harmonic analysis on such spaces has a group theoretic structure which has been exploited to evaluate the traced integrated heat kernel, on quotients of symmetric spaces where the quotient group acts freely on the sphere or hyperboloid [40,41]. In this section, we draw an interesting parallel between the results obtained previously for freely acting quotient groups and the results obtained here.…”

“…This will serve as a useful warm-up for the final computation of section 6, where the effects of the graviphoton coupling to the N = 4 vector multiplet fields is accounted for. Additionally, we will find a particularly suggestive form of the answer closely related to those obtained previously for freely acting quotients of spheres and hyperboloids in [40,41]. We begin with defining the geometry and the orbifold projection that will be imposed.…”

“…The integrated heat kernel on AdS 2 and AdS 2 /Z N suffers from a volume divergence which needs to be regulated carefully to capture the finite N -dependent terms in the heat kernel expansion. This is analogous to the problem encountered for the heat kernel on hyperboloids quotiented by freely acting orbifolds [40,41]. Even in those cases the heat kernel suffers from a volume divergence, but the regulated answer turns out to have an…”

“…To obtain this N dependence, we will adopt a strategy very similar to [40,41]; we will carry out the computation on the sphere, and analytically continue the result to the hyperboloid by a prescription we develop below.…”

“…15 Such differences have already been encountered in the cases where the orbifold group acts freely on the sphere and hyperboloid [40,41]. The analytic continuation acress the quotient spaces should account for these differences, but once that is done, completely consistent answers are obtained.…”