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

TL;DR

This work analyzes the massless Schwinger and Schwinger-Thirring models on the squashed sphere $S^2_b$, deriving exact results for the partition function and gauge-invariant observables as functions of the squashing parameter $b$ and couplings. By perturbing in $b^2$ and treating the coupling exactly, the authors compute first-order deformations of the free-energy for free scalars and Dirac fermions, study fermion determinants in monopole backgrounds, and determine condensates and Wilson-loop expectations, with the Schwinger-Thirring case yielding an effective scalar mass $\mu_{\text{eff}}=\frac{e^2}{\pi(1+2\lambda/\pi)}$. The results reveal how geometric deformation probes energy-momentum tensor correlations and topological sectors in these 2D gauge theories, and they establish a framework for extending to massive, non-abelian, or more refined observables. Overall, the paper provides exact, analytically tractable benchmarks for non-supersymmetric gauge dynamics under curved-space deformations and paves the way for exploring richer interactions on curved manifolds.

Abstract

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.

Figures (1)

• Figure 1: X($\mu$) vs $\mu$. The above plot is obtained after numerically evaluating the sum in \ref{['WilsonLoopon S2b.2']} for $\ell_{\text{max}}=10000$.