FZZT branes in JT gravity and topological gravity

Kazuhiro Sakai,Kazumi Okuyama

doi:10.1007/jhep09(2021)191

Kazuhiro Sakai, Kazumi Okuyama

Open Access

https://doi.org/10.1007/jhep09(2021)191

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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.

Highlights

Two-dimensional Jackiw-Teitelboim (JT) gravity [1, 2] is a useful toy model to study various aspects of quantum gravity and holography
In this paper we have studied FZZT branes in JT gravity and topological gravity
We found that FZZT branes can be introduced in the matrix model of JT gravity [3] by attaching M(b) = −e−zb to the geodesic boundary of length b and integrating over b

Summary

Introduction

Two-dimensional Jackiw-Teitelboim (JT) gravity [1, 2] is a useful toy model to study various aspects of quantum gravity and holography. In a remarkable paper [3] it was shown that JT gravity is holographically dual to a certain double-scaled matrix model. This is an example of the holography involving the ensemble average, where the Hamiltonian H of the boundary theory becomes the random matrix. The trumpet can end on a FZZT brane as shown in figure 1a This implies that the two-boundary correlator Z(β1)Z(β2) in the presence of FZZT brane receives a contribution depicted, which reminds us of the “halfwormhole” introduced in [10]. Can use the same “trumpet” and M(b) as in JT gravity, but the WP volume should be replaced by the generalized WP volume defined in (2.36) This construction defines an FZZT brane amplitude only perturbatively in genus expansion. In appendix B we provide an alternative derivation of (5.11) using the correlator of inverse determinant

FZZT branes in the genus expansion

FZZT branes in general background of topological gravity

Example of generalized WP volume

Half-wormholes and factorization

Relation to Liouville wavefunction

Genus-zero density of states

E dv t

String equation via the Lagrange reversion

Comment on dilaton gravity

Multi-FZZT amplitude from matrix integral

Genus expansion of BA function

Annulus amplitude between two FZZT branes