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 SL(2, 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.
Highlights
U V = 1 just as the four dimensional Schwarzschild black hole in terms of Kruskal coordinates
We found the following relation between the partition function of the Euclidean black hole Non-Linear Sigma Model (NLSM) and that which occurs in the scaling limit of the Z2 invariant spin chain: 2 ZEBH = Z(cont) + Z(disc)
In this work we apply the results obtained for the Z2 invariant integrable spin chain to the study of two NLSMs
Summary
The Lorentzian black hole NLSM can be obtained by gauging a non-compact one dimensional subgroup of the classical SL(2, R) WZW model. The action is invariant w.r.t. the infinitesimal gauge transformation of the form δX = δω X , δY = −δω Y , δU = δV = 0 ; δaμ = ∂μ(δω) This can be seen by rewriting the Lagrangian density corresponding to the action (2.1) as. As was pointed out in [1], if we take the SL(2, R) picture literally the full target space of the Lorentzian black hole NLSM would contain two copies of the regions III and IV in figure 1 corresponding to the cases X, Y > 0 and X, Y < 0
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