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Gauss-Bonnet corrected string/black hole transition in large dimensions

Bum-Hoon Lee, Hocheol Lee, Somyadip Thakur

Abstract

We develop a unified analytic treatment of the Horowitz--Polchinski string/black hole correspondence that systematically incorporates higher-derivative corrections to gravity. Working in Euclidean signature -- where the Euclidean black hole and the thermal scalar arise as competing saddles of the same finite-temperature ensemble -- we include the Gauss--Bonnet term. The analysis is rendered tractable in this UV--sensitive regime by the large-\(D\) expansion, which sharply separates the geometry into a universal near-zone and an asymptotic far-zone. In the near-zone, the coupled large-\(D\) equations reduce the thermal-scalar sector to an exactly solvable Schrödinger problem, from which we extract the \(α'\)-corrected decay exponent and the corresponding shift of the Hagedorn temperature. In the far-zone, we construct closed-form Euclidean solutions of Einstein--Gauss--Bonnet theory at leading order in both \(1/D\) and \(α'\). Matching the two regions yields the complete corrected saddle -- fixing its temperature, horizon data, and on--shell action -- and permits a fully analytic comparison of free energies between the thermal-scalar and black hole phases. This provides a controlled derivation of the HP correspondence point with explicit higher-curvature corrections.

Gauss-Bonnet corrected string/black hole transition in large dimensions

Abstract

We develop a unified analytic treatment of the Horowitz--Polchinski string/black hole correspondence that systematically incorporates higher-derivative corrections to gravity. Working in Euclidean signature -- where the Euclidean black hole and the thermal scalar arise as competing saddles of the same finite-temperature ensemble -- we include the Gauss--Bonnet term. The analysis is rendered tractable in this UV--sensitive regime by the large- expansion, which sharply separates the geometry into a universal near-zone and an asymptotic far-zone. In the near-zone, the coupled large- equations reduce the thermal-scalar sector to an exactly solvable Schrödinger problem, from which we extract the -corrected decay exponent and the corresponding shift of the Hagedorn temperature. In the far-zone, we construct closed-form Euclidean solutions of Einstein--Gauss--Bonnet theory at leading order in both and . Matching the two regions yields the complete corrected saddle -- fixing its temperature, horizon data, and on--shell action -- and permits a fully analytic comparison of free energies between the thermal-scalar and black hole phases. This provides a controlled derivation of the HP correspondence point with explicit higher-curvature corrections.
Paper Structure (96 sections, 354 equations, 4 figures, 3 tables)

This paper contains 96 sections, 354 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Higher-Derivative Horowitz--Polchinski Model
  3. Euclidean EGB--Dilaton--Thermal Scalar Action
  4. Equations of Motion
  5. Metric Equation of Motion
  6. Dilaton Equation of Motion
  7. Thermal Scalar Equation of Motion
  8. Solutions in large number of dimensions
  9. Large-D Expansion: Methodology and Structure
  10. Near-Zone Coordinates and Leading-Order Fields
  11. Subleading Near-Zone Behaviour
  12. Far-Zone Expansion and Background Asymptotics
  13. Overlap Region and Matching Conditions
  14. Near-zone coordinate and leading-order expansion
  15. Static EGB black hole and basic notation
  16. ...and 81 more sections

Figures (4)

  • Figure 1: Quartic Landau coefficient $a_4(\lambda_{\rm GB})$ extracted from the NLO quartic contribution $\mathcal{F}^{(4)}$ to the dimensionally--reduced thermal-scalar free energy functional. The dependence on the Gauss--Bonnet coupling is U-shaped, with a pronounced minimum at finite $\lambda_{\rm GB}$ (marked by $\times$). In the vicinity of this minimum the leading nonlinearity in $\mathcal{F}[\chi_0]=\mathcal{F}_0+\mathcal{F}^{(2)}+\mathcal{F}^{(4)}+\cdots$ is parametrically suppressed, defining a quasi-tricritical window in which the system becomes especially sensitive to subleading corrections beyond the quartic Landau description.
  • Figure 2: Numerical solutions of second-order Eqs. \ref{['eq:EinsteinEq']}, \ref{['eq:DilatonEOM']} and \ref{['eq:ChiSchrodinger']}, where $\Lambda = 0$.
  • Figure 3: Numerical solutions of second-order Eqs. \ref{['eq:EinsteinEq']}, \ref{['eq:DilatonEOM']} and \ref{['eq:ChiSchrodinger']}, where $\ell = 20$.
  • Figure 4: Near-horizon solutions of reduced-order Eqs. \ref{['eq:reduced_eom_h']}, \ref{['eq:reduced_eom_g']}, \ref{['eq:reduced_eom_phi']} and \ref{['eq:reduced_eom_chi']}, where $\bar{\alpha} = 1$ and $\ell = 10^7$.