The Geometry of Gauged Linear Sigma Model Correlation Functions

TL;DR

This paper explores the structure of correlation functions in 2D supersymmetric gauge theories, revealing universal relations, differential equations, and geometric interpretations, especially for theories with Calabi-Yau target spaces.

Contribution

It introduces a unified framework connecting correlation functions, differential equations, and geometric data in 2D N=(2,2) gauge theories, including Calabi-Yau cases.

Findings

Universal relations among correlation functions

Differential equations governing ground state dependence

Geometric interpretation via Givental I-function

Abstract

Applying advances in exact computations of supersymmetric gauge theories, we study the structure of correlation functions in two-dimensional N=(2,2) Abelian and non-Abelian gauge theories. We determine universal relations among correlation functions, which yield differential equations governing the dependence of the gauge theory ground state on the Fayet-Iliopoulos parameters of the gauge theory. For gauge theories with a non-trivial infrared N=(2,2) superconformal fixed point, these differential equations become the Picard-Fuchs operators governing the moduli-dependent vacuum ground state in a Hilbert space interpretation. For gauge theories with geometric target spaces, a quadratic expression in the Givental I-function generates the analyzed correlators. This gives a geometric interpretation for the correlators, their relations, and the differential equations. For classes of Calabi-Yau target spaces, such as threefolds with up to two Kahler moduli and fourfolds with a single Kahler modulus, we give general and universally applicable expressions for Picard-Fuchs operators in terms of correlators. We illustrate our results with representative examples of two-dimensional N=(2,2) gauge theories.