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From Horowitz -- Polchinski to Thirring and Back

Jinwei Chu, David Kutasov

TL;DR

The authors address the problem of understanding $d+1$-dimensional Euclidean Schwarzschild black holes near the Hagedorn regime by exploiting the enhanced $SU(2)_L\times SU(2)_R$ worldsheet symmetry at the Hagedorn radius and mapping to a generalized non-abelian Thirring model. They implement a large-$k$ limit, rendering the problem tractable and yielding an effective spacetime action with up to two derivatives, whose kinetic term and potential are computed exactly in the fields and shown to reproduce Horowitz–Polchinski results in the small-field limit while providing a controlled large-field extension. The resulting action clarifies how non-geometric HP features become geometric on a large $S^3$ in the $k\to\infty$ limit and establishes connections to Thirring beta-functions, enabling exact expressions for the metric on field space and the potential expressed in $SU(2)_L\times SU(2)_R$ invariants. The work also outlines two classes of backgrounds—HP-like and more general EBH-like solutions—that could interpolate as the temperature is varied, and discusses the role of higher spherical harmonics and NS5-brane analogies for future exploration. Overall, the approach provides a controlled bridge between HP EFT, worldsheet CFT, and semiclassical black-hole physics, with implications for black-hole thermodynamics and microstate structure in string theory.

Abstract

We propose a new approach for studying $d+1$ dimensional Euclidean Schwarzschild black holes with Hawking temperature near the Hagedorn temperature and Horowitz-Polchinski solutions. The worldsheet theory that describes some of these backgrounds is strongly coupled. We use its underlying affine $SU(2)_L\times SU(2)_R$ symmetry to continue to weak coupling, by varying the level of the current algebra from the small value relevant for black holes and HP solutions to a large value. In this limit, one can describe the dynamics by a solvable effective field theory, and the non-geometric features of the original problem are geometrized. The resulting construction is closely related to previous work on the non-abelian Thirring model, and sheds light on both problems.

From Horowitz -- Polchinski to Thirring and Back

TL;DR

The authors address the problem of understanding -dimensional Euclidean Schwarzschild black holes near the Hagedorn regime by exploiting the enhanced worldsheet symmetry at the Hagedorn radius and mapping to a generalized non-abelian Thirring model. They implement a large- limit, rendering the problem tractable and yielding an effective spacetime action with up to two derivatives, whose kinetic term and potential are computed exactly in the fields and shown to reproduce Horowitz–Polchinski results in the small-field limit while providing a controlled large-field extension. The resulting action clarifies how non-geometric HP features become geometric on a large in the limit and establishes connections to Thirring beta-functions, enabling exact expressions for the metric on field space and the potential expressed in invariants. The work also outlines two classes of backgrounds—HP-like and more general EBH-like solutions—that could interpolate as the temperature is varied, and discusses the role of higher spherical harmonics and NS5-brane analogies for future exploration. Overall, the approach provides a controlled bridge between HP EFT, worldsheet CFT, and semiclassical black-hole physics, with implications for black-hole thermodynamics and microstate structure in string theory.

Abstract

We propose a new approach for studying dimensional Euclidean Schwarzschild black holes with Hawking temperature near the Hagedorn temperature and Horowitz-Polchinski solutions. The worldsheet theory that describes some of these backgrounds is strongly coupled. We use its underlying affine symmetry to continue to weak coupling, by varying the level of the current algebra from the small value relevant for black holes and HP solutions to a large value. In this limit, one can describe the dynamics by a solvable effective field theory, and the non-geometric features of the original problem are geometrized. The resulting construction is closely related to previous work on the non-abelian Thirring model, and sheds light on both problems.

Paper Structure

This paper contains 13 sections, 137 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Review
  3. HP EFT
  4. Higher order contributions to the effective action
  5. $SU(2)_L\times SU(2)_R$
  6. The large $k$ limit
  7. The metric
  8. The potential
  9. Beyond the large $k$ limit
  10. Summary and discussion
  11. Some useful integrals
  12. Derivation of $F_{c\bar{d}\bar{f}}(\phi_{i\bar{j}})$
  13. Derivation of $C$

Figures (2)

  • Figure 1: The profiles of $\chi$ and $\phi$ for $d=3$ and $\chi(0)=0.01$.
  • Figure 2: The potential $V$ (\ref{['Vchiphi']}) for $\chi=-\sqrt{2}\phi$ as a function of (a) $\phi$, and (b) $\tilde{\phi}$ (\ref{['defphitilde']}).