Integrable Models and Geometry of Target Spaces from the Partition Function of N=(2,2) theories on S^2

In this thesis we analyze the exact partition function for N=(2,2) supersymmetric theories on the sphere S^2. Especially, its connection to geometry of target spaces of a gauged linear sigma model under consideration is investigated. First of all, such a model has different phases corresponding to different target manifolds as one varies the Fayet-Iliopoulos parameters. It is demonstrated how a single partition function includes information about geometries of all these target manifolds and which operation corresponds to crossing a wall between phases. For a fixed phase we show how one can extract from the partition function the I-function, a central object of Givental's formalism developed to study mirror symmetry. It is in some sense a more fundamental object than the exact Kahler potential, since it is holomorphic in the coordinates of the moduli space (in a very vague sense it is a square root of it), and the main advantage is that one can derive it from the partition function in a more effective way. Both these quantities contain genus zero Gromov--Witten invariants of the target manifold. For manifolds where mirror construction is not known (this happens typically for targets of non-abelian gauged linear sigma models), this method turns out to be the only available one for obtaining these invariants. All discussed features are illustrated on numerous examples throughout the text. Further, we establish a way for obtaining the effective twisted superpotential based on studying the asymptotic behavior of the partition function for large radius of the sphere. Consequently, it allows for connecting the gauged linear sigma model with a quantum integrable system by applying the Gauge/Bethe correspondence of Nekrasov and Shatashvili. The dominant class of examples we study are ''ADHM models``, i.e. gauged linear sigma models with target manifold the moduli space of instantons (on C^2 or C^2/Gamma). For the case of a unitary gauge group we were able to identify the related integrable system, which turned out to be the Intermediate Long Wave system describing hydrodynamics of two layers of liquids in a channel. It has two interesting limits, the Korteweg--deVries integrable system (limit of shallow water with respect to the wavelength) and Benjamin--Ono integrable system (deep water limit). Another integrable model that naturally enters the scene is the (spin) Calogero--Sutherland model. We examine relations among energy eigenvalues of the latter, the spectrum of integrals of motion for Benjamin--Ono and expectation values of chiral correlators in the ADHM model