Exact Results for Supersymmetric Sigma Models

TL;DR

This paper derives exact formulas for the metric and Berry curvature in N=2 supersymmetric sigma models, revealing non-singular behavior at zero size and implications for string theory target spaces.

Contribution

It provides exact computations of ground state metrics and Berry curvature for supersymmetric sigma models, including instanton corrections, and explores their behavior as the target space shrinks.

Findings

Ground state metric remains non-singular at zero size

Exact instanton corrections are computed for specific models

Zero-size manifolds can serve as non-singular string theory targets

Abstract

We show that the metric and Berry's curvature for the ground states of $N=2$ supersymmetric sigma models can be computed exactly as one varies the Kahler structure. For the case of $CP^n$ these are related to special solutions of affine toda equations. This allows us to extract exact results (including exact instanton corrections). We find that the ground state metric is non-singular as the size of the manifold shrinks to zero thus suggesting that 2d QFT makes sense even beyond zero radius. In other words it seems that manifolds with zero size are non-singular as target spaces for string theory (even when they are not conformal). The cases of $CP^1$ and $CP^2$ are discussed in more detail.