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Subleading Weingartens

Douglas Stanford, Zhenbin Yang, Shunyu Yao

TL;DR

The work connects subleading unitarity-correcting terms in Haar unitary averages to gravity and chaos-based quantum models. By analyzing JT gravity with bulk matter and Brownian SYK, it shows that the first subleading terms in JT gravity arise from bulk dynamics and topology (notably the handle-disk) and that Brownian SYK configurations reproduce the complete series using a small set of off-shell chaotic modes. The results reveal how slightly off-shell, chaos-amplified modes govern unitarity-preserving corrections and demonstrate a close correspondence between gravitational topologies and random-unitary corrections. This suggests a universal skeleton behind unitarity in chaotic quantum systems, with potential broad applicability beyond the specific models studied.

Abstract

Haar integrals over the unitary group contain subleading terms that are needed for unitarity. We study analogous effects in the time evolution operators of JT gravity and Brownian SYK. In JT gravity with bulk matter we find an explanation for the first subleading terms, and in Brownian SYK we find configurations that can explain the full series. An important role is played by slightly off-shell modes that are exponentially amplified by chaos.

Subleading Weingartens

TL;DR

The work connects subleading unitarity-correcting terms in Haar unitary averages to gravity and chaos-based quantum models. By analyzing JT gravity with bulk matter and Brownian SYK, it shows that the first subleading terms in JT gravity arise from bulk dynamics and topology (notably the handle-disk) and that Brownian SYK configurations reproduce the complete series using a small set of off-shell chaotic modes. The results reveal how slightly off-shell, chaos-amplified modes govern unitarity-preserving corrections and demonstrate a close correspondence between gravitational topologies and random-unitary corrections. This suggests a universal skeleton behind unitarity in chaotic quantum systems, with potential broad applicability beyond the specific models studied.

Abstract

Haar integrals over the unitary group contain subleading terms that are needed for unitarity. We study analogous effects in the time evolution operators of JT gravity and Brownian SYK. In JT gravity with bulk matter we find an explanation for the first subleading terms, and in Brownian SYK we find configurations that can explain the full series. An important role is played by slightly off-shell modes that are exponentially amplified by chaos.

Paper Structure

This paper contains 23 sections, 167 equations, 4 figures.

Table of Contents

  1. Introduction
  2. A first look at the minus sign
  3. JT gravity
  4. Working at disk order
  5. Handle-disk without dynamical matter fields
  6. Handle-disk with dynamical bulk matter
  7. Brownian SYK
  8. Fermions on an OTOC contour
  9. Disorder averaged theory and collective fields
  10. Large $N$ equations of motion
  11. Reducing to a smaller set of variables
  12. Conserved quantities and qualitative behavior of solutions
  13. Boundary conditions
  14. Solutions
  15. The classical action of the solutions
  16. ...and 8 more sections

Figures (4)

  • Figure 1: Cutting the handle-disk geometry on the red and blue geodesics leads to the shaded portion of the disk geometry shown at right. The renormalized proper distance along the boundary between the operators is $u_1,u_2,u_3,u_4$, which will eventually be continued to Lorentzian values.
  • Figure 2: The cylinder geometry that gives a nonzero value for $\langle \mathcal{O}\rangle^2$.
  • Figure 3: Integration parameters on the disk.
  • Figure 4: Useful variables for discussing the handle disk