  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.
