Equilibrium density matrices for the 2D black hole sigma models from an integrable spin chain

TL;DR

This paper constructs equilibrium density matrices for 2D black hole sigma models, linking them to integrable spin chains and revealing their spectral and symmetry properties.

Contribution

It introduces a novel equilibrium density matrix for Euclidean and Lorentzian black hole sigma models, connecting them to integrable spin chain models and their critical behavior.

Findings

Reproduces the modular invariant partition function for Euclidean models.

Describes the state space structure of Lorentzian models as a gauged WZW model.

Shows the spin chain scaling limit captures key features of black hole sigma models.

Abstract

This work concerns the quantum Lorentzian and Euclidean black hole non-linear sigma models. For the Euclidean black hole sigma model an equilibrium density matrix is proposed, which reproduces the modular invariant partition function from the 2001 paper of Maldacena, Ooguri and Son. For the Lorentzian black hole sigma model, using its formulation as a gauged ${\rm SL}(2,\mathbb{R})$ WZW model, we describe the linear and Hermitian structure of its space of states and also propose an expression for the equilibrium density matrix. Our analysis is guided by the results of the study of a certain critical, integrable spin chain. In the scaling limit, the latter exhibits the key features of the Lorentzian black hole sigma model including the same global symmetries, the same algebra of extended conformal symmetry and a continuous spectrum of conformal dimensions.