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arXiv:1306.2646 · Journal of Physics A Mathematical and Theoretical

Nonlinear integral equations for the $\mathrm{SL}(2,\mathbb {R})/\mathrm{U}(1)$ black hole sigma model

Constantin Candu, Yacine Ikhlef

Published Sep 27, 2013• 40 citations
arXivHTMLPDF

Abstract

It was previously established that the critical staggered XXZ spin chain provides a lattice regularization of the black hole conformal field theory (CFT). We reconsider the continuum limit of this spin chain with the exact method of nonlinear integral equations (NLIEs), paying particular attention to the effects of a singular integration kernel. With the help of the NLIEs, we rederive the continuous black hole spectrum, but also numerically match the density of states of the spin chain with that of the CFT, which is a new result. Finally, we briefly discuss the integrable structure of the black hole CFT and the identification of its massive integrable perturbation on the lattice.

DOI
10.1088/1751-8113/46/41/415401

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Nonlinear integral equations for the SL(2,R)/U(1)\mathrm{SL}(2,\mathbb {R})/\mathrm{U}(1)SL(2,R)/U(1) black hole sigma model
It was previously established that the critical staggered XXZ spin chain provides a lattice regularization of the black hole conformal field theory (CFT). We reconsider the continuum limit of this spin chain with the exact method of nonlinear integral equations (NLIEs), paying particular attention to the effects of a singular integration kernel. With the help of the NLIEs, we rederive the continuous black hole spectrum, but also numerically match the density of states of the spin chain with that of the CFT, which is a new result. Finally, we briefly discuss the integrable structure of the black hole CFT and the identification of its massive integrable perturbation on the lattice.
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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.
In an earlier work [1] J. Suzuki proposed a set of nonlinear integral equations (NLIE) to describe the excited state spectrum of the integrable spin-1 XXZ chain in its repulsive regime. In this paper we extend his equations for the attractive regime of the model, and calculate analytically the conformal spectrum of the spin chain. We also discuss the typical root configurations of the thermodynamic limit as well as the 2-string deviations of certain excited states of the model. Special objects appearing in the NLIE are also treated with special care.
We show that the radial differential equation for a scalar field in an anti-de Sitter black brane background can be encoded in an exactly solvable semi-infinite spin chain independent of dimensionality. The radial coordinate emerges as lattice quasimomentum. Spin coherence encodes the geometry, while the spectrum and response recover gravitational observables and yield exact screening-mass sum rules. We implement the finite chain on a superconducting processor. At finite filling and low energies, the model yields a conformal field theory in a coordinate proportional to proper radial distance.