Top Menu

Exactly Solvable Models of Quantum Field Theory and Statistical Mechanics: September 4 – November 30, 2018

Apply to a Program Now

Organized by: Vladimir E. Korepin, Sergei L. Lukyanov, Nikita A. Nekrasov, Samson L. Shatashvili and Alexander B. Zamolodchikov

Integrability is a traditional area of mathematical physics. For 1+1 dimensional field theory the inverse scattering method is an appropriate method. It is based on the zero-curvature representation. In quantum theory it leads to the Yang-Baxter algebras and quantum groups. These are useful for description of the connections between the supersymmetric gauge theories and quantum integrable systems. Bethe/gauge-correspondence [based on BPS/CFT-correspondence] relates Bethe ansatz solvable spin chains to the twisted chiral rings of the gauge theories with two dimensional Poincare supersymmetry. The planar maximal super-Yang-Mills theory in four dimensions is related to quantum and classical integrable systems [at the level of the anomalous dimensions of local operators and scattering amplitudes]. An important phenomena is the ODE/CFT correspondence. Many deep properties of representations of Yang-Baxter algebras in integrable Conformal Field Theories can be encoded in the monodromies of certain linear Ordinary Differential Equations. This can be extended to massive Integrable Quantum Field Theories: the ODE/IQFT correspondence. A related problem is the application of the ODE/IQFT method to non-linear sigma models, including the supersymmetric ones. The sigma model associated with the AdS side of the correspondence for the N = 4 theory was argued to be integrable. It is natural to start with simpler models, like principal chiral models, O(n) models, and such. Integrable structures of such symmetric models correspond to the Yangian reductions of the Yang-Baxter algebras. Previous experience with the ODE/IQFT approach shows that this reduction leads to a subtle limiting case on the ODE side of the correspondence. An intriguing generalization lies in study of a two-parameter deformation of the general principal chiral. In the SU(2) case, coincides with the Fateev sigma model. We believe it is the best testing ground for the ODE/IQFT approach in the sigma model context. The ultimate goal, is to extend the approach to sigma models of direct interest AdS/CFT duality and in condensed matter physics.

Spin chains are in the center of high energy physics, statistical mechanic, condensed matter, quantum optics and quantum information. Bethe Ansatz and Yang-Baxter equations helped to construct multiple examples. Some spin chains are solvable in a weaker cense: only the ground state can be described analytically. For example Fredkin model has high level of quantum fluctuations.

Another important development in statistical mechanics is the failure of van Hove theorem. The most notable case is six vertex model.

The goal of the program is to connect these traditional problems and methods to the recent developments, notably complex saddle points, resurgence, and the Bethe/gauge correspondence.