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

Abstract

We develop new techniques to efficiently evaluate heat kernel coefficients for the Laplacian in the short-time expansion on spheres and hyperboloids with conical singularities. We then apply these techniques to explicitly compute the logarithmic contribution to black hole entropy from an N = 4 vector multiplet about a Z N orbifold of the nearhorizon geometry of quarter-BPS black holes in N = 4 supergravity. We find that this vanishes, matching perfectly with the prediction from the microstate counting. We also discuss possible generalisations of our heat kernel results to higher-spin fields over Z N orbifolds of higher-dimensional spheres and hyperboloids.

Heat kernels on cone of AdS2andk-wound circular Wilson loop in AdS5× S5superstring

“…This will be our main new technical result. As a check, we shall match the small t expansion of the spinor kernel with the general form of asymptotic expansion of spinor heat kernel on a conical singularity in [24] and also show that in the special case of κ = 1/n corresponding to AdS 2 /Z n orbifold it agrees with the result in [18].…”

“…The special case of κ = 1/n corresponds to the Z n orbifold of AdS 2 which is obtained by identification φ = φ + 2π n . The analog of the Sommerfeld formula for orbifolds is equivalent to the summation over images expression discussed in [18]. There it was found that the small t expansion of the heat kernel for a massless scalar Laplacian on AdS 2 /Z n is given by…”

“…Then if we take κ to be equal to 1/n with an integer n the metric (1.5) becomes that of an orbifold AdS 2 /Z n (with a conical singularity of positive deficit) and the corresponding scalar or fermion heat kernel can be found as a sum over images [16,17,18]. This special case of κ = 1/n will thus allow us to check the general κ expressions for heat kernels.…”

Logarithmic corrections toN≥2black hole entropy

“…Gauge symmetries (including diffeomorphism invariance) further complicate the situation by introducing ghost sectors that can be quite nontrivial. The logarithmic corrections to black holes were developed in many recent works including [7,8,9,10,11].…”

Supersymmetric localization for BPS black hole entropy: 1-loop partition function from vector multiplets

“…By computing the path integration, the proposal of the quantum entropy function has been tested. Perturbative calculation on a classical saddle point and comparing it with the similar expansion on the microscopic side has led to perfect match of logarithmic correction in case of BPS black hole in N = 4 and N = 8 supergravities [21][22][23][24] in 4-dimensions and BMPV black hole in 5-dimensions [25].…”