Localizing non-linear ${\cal N}=(2,2)$ sigma model on $S^2$

TL;DR

This paper systematically studies ${ m N}=(2,2)$ supersymmetric non-linear sigma models on $S^2$, reformulating them as cohomological theories, analyzing localization, and exploring their relation to models over holomorphic disk moduli spaces.

Contribution

It introduces a novel cohomological formulation of these sigma models using 2D self-duality and explores their localization and reduction to models on moduli spaces.

Findings

Localization locus identified for the models

One-loop calculations performed around constant maps

Connection established to models over holomorphic disk moduli spaces

Abstract

We present a systematic study of ${\cal N}=(2,2)$ supersymmetric non-linear sigma models on $S^2$ with the target being a K\"ahler manifold. We discuss their reformulation in terms of cohomological field theory. In the cohomological formulation we use a novel version of 2D self-duality which involves a $U(1)$ action on $S^2$. In addition to the generic model we discuss the theory with target space equivariance corresponding to a supersymmetric sigma model coupled to a non-dynamical supersymmetric background gauge multiplet. We discuss the localization locus and perform a one-loop calculation around the constant maps. We argue that the theory can be reduced to some exotic model over the moduli space of holomorphic disks.