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The Black Hole Interior and a Curious Sum Rule

Amit Giveon, Nissan Itzhaki, Jan Troost

TL;DR

This work analyzes the Euclidean sections of the near-horizon NS5-brane geometry, mapping exterior, interior, and beyond-singularity regions to exact CFTs (cigar, negative-level minimal model, and trumpet) and demonstrates a nontrivial sum rule among their worldsheet elliptic genera. By decomposing and comparing holomorphic and nonholomorphic pieces with Jacobi form techniques, the authors prove that the combined elliptic genus vanishes: χ_I + χ_II + χ_III = 0. The result hints at a cylinder-like topological structure underlying the three regions and potentially reveals insights into black hole interior physics within string theory, with connections to T-duality and compactifications of the cigar. The finding, also noted to have an independent contemporary derivation, invites further exploration of Euclidean black hole interiors and their CFT descriptions.

Abstract

We analyze the Euclidean geometry near non-extremal NS5-branes in string theory, including regions beyond the horizon and beyond the singularity of the black brane. The various regions have an exact description in string theory, in terms of cigar, trumpet and negative level minimal model conformal field theories. We study the worldsheet elliptic genera of these three superconformal theories, and show that their sum vanishes. We speculate on the significance of this curious sum rule for black hole physics.

The Black Hole Interior and a Curious Sum Rule

TL;DR

This work analyzes the Euclidean sections of the near-horizon NS5-brane geometry, mapping exterior, interior, and beyond-singularity regions to exact CFTs (cigar, negative-level minimal model, and trumpet) and demonstrates a nontrivial sum rule among their worldsheet elliptic genera. By decomposing and comparing holomorphic and nonholomorphic pieces with Jacobi form techniques, the authors prove that the combined elliptic genus vanishes: χ_I + χ_II + χ_III = 0. The result hints at a cylinder-like topological structure underlying the three regions and potentially reveals insights into black hole interior physics within string theory, with connections to T-duality and compactifications of the cigar. The finding, also noted to have an independent contemporary derivation, invites further exploration of Euclidean black hole interiors and their CFT descriptions.

Abstract

We analyze the Euclidean geometry near non-extremal NS5-branes in string theory, including regions beyond the horizon and beyond the singularity of the black brane. The various regions have an exact description in string theory, in terms of cigar, trumpet and negative level minimal model conformal field theories. We study the worldsheet elliptic genera of these three superconformal theories, and show that their sum vanishes. We speculate on the significance of this curious sum rule for black hole physics.

Paper Structure

This paper contains 4 sections, 8 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The NS5-brane geometry
  3. The Sum Rule
  4. Interpretations

Figures (2)

  • Figure 1: The relationship between the Lorentzian black hole in Kruskal coordinates and the different regions obtained after Wick rotation. Regions $I$ and $III$ are Euclidean. Region $II$ includes two time directions.
  • Figure 2: Equation (\ref{['ty']}) suggests that the topology of the combined regions is that of a cylinder.