Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

How Does a Fundamental String Stretch its Horizon?

Ashoke Sen

TL;DR

The work investigates how higher derivative corrections in heterotic string theory modify the black hole solution representing a fundamental string wrapped on a circle and whether the corrected geometric entropy reproduces the microstate entropy. It develops an $N=2$ supergravity framework with supersymmetric curvature-squared terms, derives a corrected near-horizon geometry, and reveals modified T-duality rules that connect moduli to the physical radius in the presence of these corrections. For a specific curvature-squared completion with $f(u)= -\frac{C}{64}u$ (and $C=1$), the corrected black hole entropy matches the microscopic $S_{stat}\sim 4\pi\sqrt{nw}$, illustrating a precise macroscopic-microscopic agreement at leading order and highlighting universality in the near-horizon regime. The paper also analyzes quantum corrections and holomorphic anomaly, showing logarithmic subleading terms and emphasizing that entropy comparisons depend on ensemble choices and potential non-generic compactifications, thereby outlining important open questions for nonleading corrections and broader string theories.

Abstract

It has recently been shown that if we take into account a class of higher derivative corrections to the effective action of heterotic string theory, the entropy of the black hole solution representing elementary string states correctly reproduces the statistical entropy computed from the degeneracy of elementary string states. So far the form of the solution has been analyzed at distance scales large and small compared to the string scale. We analyze the solution that interpolates between these two limits and point out a subtlety in constructing such a solution due to the presence of higher derivative terms in the effective action. We also study the T-duality transformation rules to relate the moduli fields of the effective field theory to the physical compactification radius in the presence of higher derivative corrections and use these results to find the physical radius of compactification near the horizon of the black hole. The radius approaches a finite value even though the corresponding modulus field vanishes. Finally we discuss the non-leading contribution to the black hole entropy due to space-time quantum corrections to the effective action and the ambiguity involved in comparing this result to the statistical entropy.

How Does a Fundamental String Stretch its Horizon?

TL;DR

The work investigates how higher derivative corrections in heterotic string theory modify the black hole solution representing a fundamental string wrapped on a circle and whether the corrected geometric entropy reproduces the microstate entropy. It develops an supergravity framework with supersymmetric curvature-squared terms, derives a corrected near-horizon geometry, and reveals modified T-duality rules that connect moduli to the physical radius in the presence of these corrections. For a specific curvature-squared completion with (and ), the corrected black hole entropy matches the microscopic , illustrating a precise macroscopic-microscopic agreement at leading order and highlighting universality in the near-horizon regime. The paper also analyzes quantum corrections and holomorphic anomaly, showing logarithmic subleading terms and emphasizing that entropy comparisons depend on ensemble choices and potential non-generic compactifications, thereby outlining important open questions for nonleading corrections and broader string theories.

Abstract

It has recently been shown that if we take into account a class of higher derivative corrections to the effective action of heterotic string theory, the entropy of the black hole solution representing elementary string states correctly reproduces the statistical entropy computed from the degeneracy of elementary string states. So far the form of the solution has been analyzed at distance scales large and small compared to the string scale. We analyze the solution that interpolates between these two limits and point out a subtlety in constructing such a solution due to the presence of higher derivative terms in the effective action. We also study the T-duality transformation rules to relate the moduli fields of the effective field theory to the physical compactification radius in the presence of higher derivative corrections and use these results to find the physical radius of compactification near the horizon of the black hole. The radius approaches a finite value even though the corresponding modulus field vanishes. Finally we discuss the non-leading contribution to the black hole entropy due to space-time quantum corrections to the effective action and the ambiguity involved in comparing this result to the statistical entropy.

Paper Structure

This paper contains 11 sections, 119 equations, 2 figures.

Table of Contents

  1. Introduction and Summary
  2. Supergravity Solution for Two Charge Black Holes and its Near Horizon Limit
  3. Modification of the Solution by Higher Derivative Terms and its Near Horizon Limit
  4. The low energy effective action as $N=2$ supergravity
  5. Higher derivative corrections
  6. Near horizon geometry and entropy
  7. T-duality
  8. Tree level heterotic string theory and universality
  9. The analysis of the interpolating solution
  10. Effect of quantum corrections and holomorphic anomaly
  11. Generalizations and Open Questions

Figures (2)

  • Figure 1: Trajectories neighbouring $h=2 \ln{r\over 2}$ for small $r$.
  • Figure 2: Numerical result for the solution to (\ref{['e7.3']}) satisfying the boundary condition $h = 2\ln{r\over 2}$ for small $r$. The smooth curve represents $h=\ln{r\over 2}$.