Schwinger and Schwinger-Thirring model on squashed S$^{2}$

The Schwinger model is a model of a two-dimensional $U(1)$ gauge theory coupled to a Dirac fermion. It is an interesting model that exhibits phenomena like confinement and chiral symmetry breaking. In this paper, we study the massless Schwinger and Schwinger-Thirring model on a squashed sphere, $S^2_b$. These models are examples of interacting non-supersymmetric theories where the exact computations in the coupling parameter are possible. Squashing provides a smooth deformation of the metric away from the spherical geometry. We compute the partition function, and the expectation value of the Wilson loop and the fermion condensate exactly in the Schwinger and Schwinger-Thirring model as a function of the squashing parameter and the coupling constant. We then obtain variations in these quantities in response to the squashing deformation. These contain information about correlation functions involving the energy-momentum tensor. We evaluate these variations in the first order in the squashing parameter and exactly in the coupling constant.