FZZT branes in JT gravity and topological gravity

TL;DR

This work develops a comprehensive framework for FZZT branes in JT gravity and 2d topological gravity, modeling branes as determinant insertions that induce a simple boundary factor and a shift of the infinite coupling set t_k. The genus expansion is constructed by gluing trumpets to generalized Weil-Petersson volumes, and the nonperturbative content is captured via Baker-Akhiezer functions and the Christoffel-Darboux kernel, enabling closed expressions for correlators of multiple FZZT branes and macroscopic loops. In the Airy (JT) case, the analysis reveals eigenbrane-induced voids in the eigenvalue density and oscillatory behavior in the spectral form factor, aligning with the eigenbrane picture and suggesting paths toward factorization and Page-curve interpretations. The results unify perturbative and nonperturbative aspects of FZZT branes, establishing a bridge between matrix-model determinants, Liouville wavefunctions, and topological gravity, with practical implications for understanding holographic ensembles and black-hole information in two dimensions.

Abstract

We study Fateev-Zamolodchikov-Zamolodchikov-Teschner (FZZT) branes in Witten-Kontsevich topological gravity, which includes Jackiw-Teitelboim (JT) gravity as a special case. Adding FZZT branes to topological gravity corresponds to inserting determinant operators in the dual matrix integral and amounts to a certain shift of the infinitely many couplings of topological gravity. We clarify the perturbative interpretation of adding FZZT branes in the genus expansion of topological gravity in terms of a simple boundary factor and the generalized Weil-Petersson volumes. As a concrete illustration we study JT gravity in the presence of FZZT branes and discuss its relation to the deformations of the dilaton potential that give rise to conical defects. We then construct a non-perturbative formulation of FZZT branes and derive a closed expression for the general correlation function of multiple FZZT branes and multiple macroscopic loops. As an application we study the FZZT-macroscopic loop correlators in the Airy case. We observe numerically a void in the eigenvalue density due to the eigenvalue repulsion induced by FZZT-branes and also the oscillatory behavior of the spectral form factor which is expected from the picture of eigenbranes.

Figures (6)

• Figure 1: Trumpet can end on a FZZT brane. We attach the factor $\mathcal{M}(b)=-e^{-zb}$ to the geodesic boundary and integrate over $b$.
• Figure 2: Contributions of \ref{['sfig:wormhole']} wormhole and \ref{['sfig:half-wormhole']} half-wormholes.
• Figure 3: Plot of the threshold energy $E_0$ as a function of $t=g_{\rm s} K$. We set $z=1$ in this figure.
• Figure 4: Plot of the eigenvalue density $\rho_0(E)$ for $t=3,z=1,g_s=1$ (blue solid curve). The orange dashed curve represents the pure JT gravity density of states in \ref{['eq:JT-rho']}.
• Figure 5: Plot of $\widetilde{\rho}(E,E')$ in \ref{['eq:tilrho']} for \ref{['sfig:Em3-2']}$E'=-3$ and \ref{['sfig:E6-2']}$E'=6$, as a function of $E$. The solid curves are the deformed eigenvalue density $\widetilde{\rho}(E,E')$ while the orange dashed curve represents the original eigenvalue density $\widetilde{\rho}(E)=K(-E,-E)$ without FZZT branes.