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Black hole microstates and supersymmetric localization

Seyed Morteza Hosseini

TL;DR

This work develops a unified, localization-based framework to count microstates of BPS black holes in AdS via the topologically twisted index of supersymmetric gauge theories. By reducing partition functions on curved manifolds to matrix models, it derives large-$N$ saddle points and Bethe Ansatz equations that encode the microstate degeneracy and reproduce Bekenstein-Hawking entropy across 4D and 5D AdS backgrounds, including AdS$_4$ and AdS$_5$ cases, with ABJM, D2$_k$ and various quiver theories. A key finding is the index theorem, relating the large-$N$ twisted index to the extremal twisted superpotential, and its parallel with the $S^3$ free energy, suggesting deep links between field theory extremization and supergravity attractor mechanisms. The results support a holographic microstate counting program, extend it to massive IIA duals, and reveal universal structures, such as long-range force cancellations and the mapping between matrix-model data and Sasaki-Einstein volumes. Collectively, the work provides exact, large-$N$ tools for matching microscopic states with macroscopic black hole entropy in diverse AdS/CFT contexts and highlights a broader thermodynamic/combinatorial dictionary between localization data and gravitational charges.

Abstract

This thesis focuses mainly on understanding the origin of the Bekenstein-Hawking entropy for a class of four- and five-dimensional BPS black holes in string/M-theory. To this aim, important ingredients are holography and supersymmetric localization. Using the method of supersymmetric localization, the Euclidean path integrals for supersymmetric field theories on $Σ_\mathfrak{g} \times T^{n}$ $(n=1,2)$, with at least four real supercharges, can be reduced to a matrix integral that depends on background magnetic fluxes and chemical potentials for the global symmetries of the theory. This defines the topologically twisted index which, upon extremization with respect to the chemical potentials, is conjectured to reproduce the entropy of magnetically charged static BPS AdS$_{4/5}$ black holes/strings. We solve a number of such matrix models both in three and four dimensions and provide general formulae in the large $N$ limit. We then use these results to provide the microscopic realization of the entropy of a class of BPS black holes in $\mathcal{N}=2$ gauged supergravity. Finally, inspired by our previous results, we put forward an extremization principle for reproducing the Bekenstein-Hawking entropy of a class of BPS electrically charged rotating black holes in AdS$_5\times S^5$.

Black hole microstates and supersymmetric localization

TL;DR

This work develops a unified, localization-based framework to count microstates of BPS black holes in AdS via the topologically twisted index of supersymmetric gauge theories. By reducing partition functions on curved manifolds to matrix models, it derives large- saddle points and Bethe Ansatz equations that encode the microstate degeneracy and reproduce Bekenstein-Hawking entropy across 4D and 5D AdS backgrounds, including AdS and AdS cases, with ABJM, D2 and various quiver theories. A key finding is the index theorem, relating the large- twisted index to the extremal twisted superpotential, and its parallel with the free energy, suggesting deep links between field theory extremization and supergravity attractor mechanisms. The results support a holographic microstate counting program, extend it to massive IIA duals, and reveal universal structures, such as long-range force cancellations and the mapping between matrix-model data and Sasaki-Einstein volumes. Collectively, the work provides exact, large- tools for matching microscopic states with macroscopic black hole entropy in diverse AdS/CFT contexts and highlights a broader thermodynamic/combinatorial dictionary between localization data and gravitational charges.

Abstract

This thesis focuses mainly on understanding the origin of the Bekenstein-Hawking entropy for a class of four- and five-dimensional BPS black holes in string/M-theory. To this aim, important ingredients are holography and supersymmetric localization. Using the method of supersymmetric localization, the Euclidean path integrals for supersymmetric field theories on , with at least four real supercharges, can be reduced to a matrix integral that depends on background magnetic fluxes and chemical potentials for the global symmetries of the theory. This defines the topologically twisted index which, upon extremization with respect to the chemical potentials, is conjectured to reproduce the entropy of magnetically charged static BPS AdS black holes/strings. We solve a number of such matrix models both in three and four dimensions and provide general formulae in the large limit. We then use these results to provide the microscopic realization of the entropy of a class of BPS black holes in gauged supergravity. Finally, inspired by our previous results, we put forward an extremization principle for reproducing the Bekenstein-Hawking entropy of a class of BPS electrically charged rotating black holes in AdS.

Paper Structure

This paper contains 131 sections, 2 theorems, 716 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction and summary
  2. Topological twist
  3. Magnetic black holes/strings in gauged supergravity
  4. Supersymmetric localization
  5. Supersymmetric Chern-Simons-matter theories in three dimensions
  6. Spinor conventions
  7. Background geometry
  8. Supersymmetry transformations
  9. Localizing supercharges
  10. Supersymmetric Lagrangians
  11. ABJM theory
  12. The topologically twisted index
  13. $\ThisStyle{\ooalign{$N=2$\cr$N=2$\cr$N=2$}}$ Chern-Simons-matter theories on $\ThisStyle{\ooalign{$S^2 S^1$\cr$S^2 S^1$\cr$S^2 S^1$}}$
  14. $\ThisStyle{\ooalign{$N=1$\cr$N=1$\cr$N=1$}}$ gauge theories on $\ThisStyle{\ooalign{$S^2 T^2$\cr$S^2 T^2$\cr$S^2 T^2$}}$
  15. The $\ThisStyle{\ooalign{$I$\cr$I$\cr$I$}}$-extremization principle
  16. ...and 116 more sections

Key Result

Theorem 1

The index of any $\mathcal{N} \geq 2$ quiver gauge theory which respects the constraints no long-range-forces Bethe0 and no long-range forces, and satisfies in addition superpotential0, can be written as where $\widetilde{\mathcal{W}} (\Delta_I)$ is the extremal value of the functional bethefunctional and $\mu$ is the Lagrange multiplier appearing in bethefunctional.The second identity in virial

Figures (1)

  • Figure 1: Sequence of consistent truncations from massive type IIA supergravity in ten dimensions, to the dyonic STU model in four dimensions.

Theorems & Definitions (4)

  • Theorem 1
  • proof
  • Theorem 2
  • proof