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Finite $N$ black hole cohomologies

Jaehyeok Choi, Sunjin Choi, Seok Kim, Jehyun Lee, Siyul Lee

TL;DR

The work tackles finite-$N$ black hole microstates in $\mathcal{N}=4$ SYM by formulating the problem in terms of $Q$-cohomologies of $1/16$-BPS operators and exploiting a BMN truncation to render calculations tractable. It develops strategies to separate graviton from non-graviton cohomologies, computes the graviton index via eigenvalue and Gröbner-basis methods, and identifies non-graviton cohomologies by analyzing the index, including an explicit SU(3) threshold cohomology at $j=24$. The findings reveal structured towers, partial no-hair phenomena, and black-hole-like entropy growth in the BMN sector at large $N$, suggesting rich finite-$N$ microstate structure and guiding future exact index computations at larger $N$. These insights offer a path toward understanding black hole microstates in AdS/CFT through controlled, finite-$N$ $Q$-cohomology constructions and their spectral organization.

Abstract

We study new cohomologies for the BPS operators of the $\mathcal{N}=4$ Yang-Mills theory with $SU(3)$ and $SU(4)$ gauge groups, to better understand the black hole microstates. We first study the index of these black hole operators and identify their apparent threshold levels. For $SU(3)$, we find many towers of states and partial no-hair behaviors. We explicitly construct the threshold cohomology in the $SU(3)$ theory. We study throughout this paper a subsector of the field theory corresponding to the BMN matrix theory. We also argue that the BMN sector exhibits a black hole like entropy growth at large $N$.

Finite $N$ black hole cohomologies

TL;DR

The work tackles finite- black hole microstates in SYM by formulating the problem in terms of -cohomologies of -BPS operators and exploiting a BMN truncation to render calculations tractable. It develops strategies to separate graviton from non-graviton cohomologies, computes the graviton index via eigenvalue and Gröbner-basis methods, and identifies non-graviton cohomologies by analyzing the index, including an explicit SU(3) threshold cohomology at . The findings reveal structured towers, partial no-hair phenomena, and black-hole-like entropy growth in the BMN sector at large , suggesting rich finite- microstate structure and guiding future exact index computations at larger . These insights offer a path toward understanding black hole microstates in AdS/CFT through controlled, finite- -cohomology constructions and their spectral organization.

Abstract

We study new cohomologies for the BPS operators of the Yang-Mills theory with and gauge groups, to better understand the black hole microstates. We first study the index of these black hole operators and identify their apparent threshold levels. For , we find many towers of states and partial no-hair behaviors. We explicitly construct the threshold cohomology in the theory. We study throughout this paper a subsector of the field theory corresponding to the BMN matrix theory. We also argue that the BMN sector exhibits a black hole like entropy growth at large .
Paper Structure (22 sections, 74 equations, 1 figure, 2 tables)

This paper contains 22 sections, 74 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction and summary
  2. The cohomology problem
  3. The BMN sector
  4. The graviton cohomologies
  5. The index
  6. $SU(3)$
  7. $SU(4)$
  8. Constructing cohomologies
  9. $SU(3)$ threshold cohomology from ansätze
  10. i) $(R,J) = (\frac{3}{2}, \frac{5}{2})$
  11. ii) $(R,J) = (2, 2)$
  12. iii) $(R,J) = (\frac{5}{2}, \frac{3}{2})$
  13. iv) $(R,J) = (3, 1)$
  14. v) $(R,J) = (\frac{7}{2}, \frac{1}{2})$
  15. $Q$-exactness checks and ansatz-independent studies
  16. ...and 7 more sections

Figures (1)

  • Figure 1: An example distribution of 1.5$\times 10^6$ evaluated values of $Q$-action of basis polynomials in paragraph iii). Width of each bin is $10^{27}$.