Vacuum Geometry of the N=2 Wess-Zumino Model

TL;DR

This paper rigorously constructs the vacuum moduli space of N=2 supersymmetric Wess-Zumino models, confirming key assumptions of tt* geometry and providing mathematical foundations for its use in quantum field theory and string theory calculations.

Contribution

It provides a rigorous mathematical construction of the vacuum geometry for N=2 Wess-Zumino models, validating the assumptions of tt* geometry within a constructive quantum field theory framework.

Findings

Confirmed the existence of the moduli space and vacuum geometry mathematically.

Validated the assumptions of tt* geometry for these models.

Provided a simplified proof for holomorphic quantum mechanics.

Abstract

We give a mathematically rigorous construction of the moduli space and vacuum geometry of a class of quantum field theories which are N=2 supersymmetric Wess-Zumino models on a cylinder. These theories have been proven to exist in the sense of constructive quantum field theory, and they also satisfy the assumptions used by Vafa and Cecotti in their study of the geometry of ground states. Since its inception, the Vafa-Cecotti theory of topological-antitopological fusion, or tt* geometry, has proven to be a powerful tool for calculations of exact quantum string amplitudes. However, tt* geometry postulates the existence of certain vector bundles and holomorphic sections built from the ground states. Our purpose in the present article is to give a mathematical proof that this postulate is valid within the context of the two-dimensional N=2 supersymmetric Wess-Zumino models. We also give a simpler proof in the case of holomorphic quantum mechanics.