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Thermal spectral function asymptotics and black hole singularity in holography

Hewei Frederic Jia, Mukund Rangamani

TL;DR

Addressing how thermal spectral functions in holographic CFTs encode black hole interior data, the paper develops a transseries description of non-perturbative corrections at large momentum. It demonstrates a factorization of the spectral function for special integer-dimension operators into a vacuum piece and a non-perturbative piece, whose asymptotics are governed by a momentum-dependent transseries derived from exact WKB and Virasoro-block monodromy. The analysis yields explicit formulae for the non-perturbative piece, including a leading exponential structure and Voros-symbol-based monodromy, connecting to singularities in the complex time plane of thermofield correlators. The results extend previous radial-limit insights to finite momentum and clarify how the black hole singularity may be reflected in real-time correlators, while highlighting the need for a deeper geometric interpretation beyond Δ ~ O(1).

Abstract

We investigate the analytic structure of thermal spectral function of holographic CFTs, synthesizing recent developments into a set of observations about its asymptotics. Specifically, for a class of scalar primaries with integral dimension, we demonstrate factorization of the exact spectral function into a polynomial piece, which captures the vacuum dynamics, and a non-perturbative piece, which controls its asymptotics. Using exact WKB techniques, we derive a transseries expression for the latter. We use this information to deduce the singular loci of a spatially averaged thermofield double correlator in the complex time plane. Such singularities have been argued to encode information regarding the black hole singularity in the dual spacetime. Our results give a refinement of these statements by capturing the momentum dependence.

Thermal spectral function asymptotics and black hole singularity in holography

TL;DR

Addressing how thermal spectral functions in holographic CFTs encode black hole interior data, the paper develops a transseries description of non-perturbative corrections at large momentum. It demonstrates a factorization of the spectral function for special integer-dimension operators into a vacuum piece and a non-perturbative piece, whose asymptotics are governed by a momentum-dependent transseries derived from exact WKB and Virasoro-block monodromy. The analysis yields explicit formulae for the non-perturbative piece, including a leading exponential structure and Voros-symbol-based monodromy, connecting to singularities in the complex time plane of thermofield correlators. The results extend previous radial-limit insights to finite momentum and clarify how the black hole singularity may be reflected in real-time correlators, while highlighting the need for a deeper geometric interpretation beyond Δ ~ O(1).

Abstract

We investigate the analytic structure of thermal spectral function of holographic CFTs, synthesizing recent developments into a set of observations about its asymptotics. Specifically, for a class of scalar primaries with integral dimension, we demonstrate factorization of the exact spectral function into a polynomial piece, which captures the vacuum dynamics, and a non-perturbative piece, which controls its asymptotics. Using exact WKB techniques, we derive a transseries expression for the latter. We use this information to deduce the singular loci of a spatially averaged thermofield double correlator in the complex time plane. Such singularities have been argued to encode information regarding the black hole singularity in the dual spacetime. Our results give a refinement of these statements by capturing the momentum dependence.

Paper Structure

This paper contains 6 sections, 21 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Thermal spectral function in CFTs
  3. Holographic correlators from Virasoro block
  4. Exact WKB of black hole wave equations
  5. Inklings of the black hole singularity?
  6. Discussion

Figures (1)

  • Figure 1: The Stokes graphs for \ref{['eq:Qpot']} in the timelike regime $0<\zeta <1$ with $\vartheta = \arg(\mathfrak{w}) = \arg(\mathfrak{q})$. The thick black lines are Stokes lines that demarcate the different Stokes regions on the $u$ plane. The turning points, singularities, placement of branch cuts, monodromy contours $\mathcal{C}_{0,1}$ centered at base point $u_0$, and the A and B cycles are indicated for reference.