Background:This project is about the relation between supersymmetric gauge theories and integrable models. We know from past studies that SUSY gauge theory actually possess an integrable structure. An example is the relation between the four dimensional Seiberg-Witten theory and certain classical integrable models such as Toda and elliptic Calogero-Moser system, discovered in 1990s. Recently a relation between gauge theory and quantum integrable system is outlined in the work of Nekrasov and Shatashvili(NS), and in the work of Alday, Gaiotto, Tachikawa(AGT). The NS correspondence states that the vacua of supersymmetric gauge theories are in one to one correspondence with the spectrum of certain quantum integrable systems, both are determined by the Bethe ansatz equation. The gauge theories include two dimensional N=(2,2) supersymmetric gauge theories, their relative theories in three and four dimensions; Seiberg-Witten theory in the Omega background. The integrable models include the delta potential system, Heisenberg spin chain, the Toda, the Calogero-Moser system, etc. The Bethe/gauge correspondence establishes a precise relation between various quantities on the gauge theory side and integrable models side. The AGT correspondence is a statement about four dimensional Seiberg-Witten gauge theory and two dimensional Liouville conformal field theory(CFT). The gauge theory partition function is identified with the conformal blocks of CFT. The AGT is related to several topics such as matrix model, topological string, M5 brane, etc. The NS and AGT correspondence are closely related for the models they both involve: when part of the Omega background field vanish, the AGT reduces to the NS. Planned Research:Recent works explore the NS and AGT correspondence and give strong support for their validity. However, at the moment not every aspect of the two sides has been correctly identified and carefully checked. We only know a part of the picture. For example, there are many known integrable models, however only few have been identified with gauge theories. For many interesting integrable models it is no known gauge theory counterpart. There is also a rich story about quantum groups and correlation functions on the integrable side, however only little is known about the corresponding story on the gauge theory side. On the other hand, the gauge theories demonstrate the electro-magnetic duality; the corresponding duality for integrable models has been studied only for special cases, but not in general. The project continues our previous work in this direction, aims to fill some details of the correspondence, extend existing results, use ideas and techniques developed in gauge theory to attack problems in integrable theory and vice versa. Our goals include:1) Prove the factorization of the 4D gauge theory partition function. The S-matrix of integrable system is factorizable, the gauge theory partition function, in its present form, is not obviously consistent with this fact. 2) Further study the exact WKB method in the complex domain, its relation to analysis on Riemann surface, special functions, etc. Apply this method to other quantum mechanics problems, especially the non-Hermitian quantum mechanics. 3) Elaborate the implication of the novel phase of gauge theory for the corresponding integrable model. Study the duality of integrable models that corresponds to the electro-magnetic duality, identify the degrees of freedom in the dual phases. 4) Study explicit correspondence for other gauge theories/integrable models. Find gauge theory dual for models such as Hubbard model, Ising model, etc. Study the quantum integrable models corresponding to 4D gauge theory with fundamental matters, study their spectrum. 5) Study the quantization of Hitchin system in the context of NS and AGT correspondence. The main difficulty is to identify the correct gauge theory for an Hitchin system.
News published in Agência FAPESP Newsletter about the scholarship: