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Integrability in Modern Theoretical and Mathematical Physics

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Organized by Nikita Nekrasov and Samson Shatasvili
Fall 2012

Exactly solvable quantum many body systems, lattice models of statistical physics and integrable 1+1 dimensional quantum field theories have very rich and long history which substantially influenced the development of the mathematical physics in the 20th century. During the recent years it has become clear that supersymmetric quantum field theories in various dimensions contain sectors that are equivalent to these quantum integrable models (and their classical counterparts). This observation connected two communities of mathematicians and physicists and thus is expected to have profound consequences.

Just to list a few appearances of quantum integrable systems in modern quantum field theory and string theory:

a) supersymmetric vacua of supersymmetric gauge theories with four supercharges in various dimensions are connected to Bethe eigenstates of lattice models of statistical physics as well as quantized Hitchin systems and its limits corresponding to many celebrated quantum many body systems; b) thermodynamic Bethe (TBA) ansatz type of equations, originally developed for quantum integrable systems, play the central role in above-mentioned correspondence; c) these TBA type equations appear in the study of wall-crossing phenomena in counting of BPS states in N=2 theories; d) the same equations recently appeared in computing the amplitudes and the expectations values of Wilson and ‘t Hooft loops in the maximally supersymmetric gauge theories; e) quantum integrability has been a central topic of study in maximally supersymmetric gauge theories in four dimensions when computing the anomalous dimensions and in AdS/CFT correspondence; f) the spectrum of the equivariant Donaldson theory and its generalizations coincides with the spectrum of the quantized Seiberg-Witten theory; g) recently discovered correspondence between four dimensional instanton calculus and two dimensional conformal field theory has important consequences, both for conformal field theory and gauge theory; h)  partition functions of closed topological strings are the tau-functions of classical integrable hierarchies, and the inclusion of open strings connects to quantum integrability; i) The dimer models and their applications to the topological strings on the toric Calabi-Yau manifolds provide yet another link to the quantum integrability; j) geometric Langlands correspondence, its quantum field theory realization, and the possibility to reach out to number theory; k) the connections to SLE, random growth models, emergent geometry and matrix models; and, of course l) the integrable quantum field theories in 1+1 dimensions like sine-Gordon theory which contain rich structure still under very active development.

The Simons Center Program will focus on most of the topics mentioned in previous paragraph, as well as on some unexpected developments which clearly will take place within next year, and bring together experts in all these fields.