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The Heat Kernel on $AdS$

Rajesh Gopakumar, Rajesh Kumar Gupta, Shailesh Lal

TL;DR

This work develops explicit heat-kernel methods for arbitrary spin tensors on the thermal quotient of odd-dimensional Euclidean AdS, using harmonic analysis on coset spaces and analytic continuation from spheres to AdS. By employing the method of images and Harish-Chandra characters, it yields closed-form traced heat kernels and one-loop determinants, with detailed results for AdS$_5$ scalars and STT fields. The findings reproduce and extend previous three-dimensional results, connect to known scalar partition functions, and provide a framework for higher-spin theories and potential string-theory applications in AdS backgrounds. The approach and formulas pave the way for systematic one-loop analyses in both odd and even AdS spacetimes and offer a concrete tool for testing AdS/CFT in the quantum regime.

Abstract

We explicitly evaluate the heat kernel for the Laplacian of arbitrary spin tensor fields on the thermal quotient of (Euclidean) $AdS_N$ for $N\geq 3$ 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.

The Heat Kernel on $AdS$

TL;DR

This work develops explicit heat-kernel methods for arbitrary spin tensors on the thermal quotient of odd-dimensional Euclidean AdS, using harmonic analysis on coset spaces and analytic continuation from spheres to AdS. By employing the method of images and Harish-Chandra characters, it yields closed-form traced heat kernels and one-loop determinants, with detailed results for AdS scalars and STT fields. The findings reproduce and extend previous three-dimensional results, connect to known scalar partition functions, and provide a framework for higher-spin theories and potential string-theory applications in AdS backgrounds. The approach and formulas pave the way for systematic one-loop analyses in both odd and even AdS spacetimes and offer a concrete tool for testing AdS/CFT in the quantum regime.

Abstract

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

Paper Structure

This paper contains 21 sections, 107 equations.

Table of Contents

  1. Introduction
  2. The Heat Kernel on Homogeneous Spaces
  3. The Heat Kernel on $S^{2n+1}$
  4. The Heat Kernel on Quotients of Symmetric Spaces
  5. The Thermal Quotient of $S^{5}$
  6. The Method of Images
  7. The Traced Heat Kernel on thermal $S^{2n+1}$
  8. The Heat Kernel on $AdS_{2n+1}$
  9. Preliminaries
  10. Harmonic Analysis on ${\mathbb{H}}_{2n+1}$
  11. The coincident Heat Kernel on $AdS_{2n+1}$
  12. The Heat Kernel on Thermal $AdS_{2n+1}$
  13. The Thermal Quotient of $AdS$
  14. The Heat Kernel
  15. The one-loop Partition Function
  16. ...and 6 more sections