# Extended Eigenstate Thermalization and the role of FZZT branes in the Schwarzian theory

## TL;DR

This paper extends the eigenstate thermalization hypothesis in Schwarzian theories, linking operator behavior in pure states to monodromy classes, and explores the role of FZZT branes in low-dimensional holography.

## Contribution

It introduces an extended ETH framework for Schwarzian theories and clarifies the role of FZZT branes in the holographic microstate structure.

## Key findings

- Expectation values are thermal in energy eigenstates.
- Operator behavior varies with state coherence and monodromy class.
- FZZT branes influence the microstate interpretation in holography.

## Abstract

In this paper we provide a universal description of the behavior of the basic operators of the Schwarzian theory in pure states. When the pure states are energy eigenstates, expectation values of non-extensive operators are thermal. On the other hand, in coherent pure states, these same operators can exhibit ergodic or non-ergodic behavior, which is characterized by elliptic, parabolic or hyperbolic monodromy of an auxiliary equation; or equivalently, which coadjoint Virasoro orbit the state lies on. These results allow us to establish an extended version of the eigenstate thermalization hypothesis (ETH) in theories with a Schwarzian sector. We also elucidate the role of FZZT-type boundary conditions in the Schwarzian theory, shedding light on the physics of microstates associated with ZZ branes and FZZT branes in low dimensional holography.