The geometry of gauged linear sigma model correlation functions

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 Kähler moduli and fourfolds with a single Kähler 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.