Random matrices and integrable systems
March 4 - 9, 2012
Les Houches
Vladimir Kazakov, Ecole Normale Supérieure, Paris:
Quantum integrability made classical
Integrability of a quantum system often manifests itself in the form of classical integrablility for a related, usually discrete system. Among well known examples of such relation are the fusion procedure in quantum spin chains or the discrete Y-systems for quantum sigma models in the finite volume. In all known cases, this classical integrability takes the form of an integrable discrete Hirota dynamics, with many parallels to other applications of classical integrability, such as matrix models, group characters, integrable PDEs etc. Moreover, with a certain analyticity input, the Hirota dynamics allows to efficiently study various quantum integrable systems. This point of view recently allowed to formulate an alternative approach to the solution of rational quantum spin chains and of 2D relativistic sigma-models, and culminated in the exact solution for the full spectrum of anomalous dimensions of the four-dimensional N=4 syper-Yang-Mills theory at any 't Hooft coupling, thus giving the most convincing demonstration of the AdS/CFT correspondence. I will try to explain the basics of this approach on the example of the Heisenberg spin chain, employing a rather elementary language of GL(N) characters, matrices and matrix co-derivatives. I will construct in this way Baxter's Q-operators and derive the Bethe ansatz equations. If the time permits, I will sketch out the generalisation of the formalism to the quantum 2D sigma-models.
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