Holomorphic Anomaly in Gauge Theories and Matrix Models

TL;DR

This paper applies the holomorphic anomaly equation to compute gravitational corrections in Seiberg-Witten theory and a related matrix model, providing recursive solutions and confirming the Dijkgraaf-Vafa conjecture up to genus two.

Contribution

It introduces a recursive method using propagators for solving gravitational corrections in gauge theories and matrix models, confirming the Dijkgraaf-Vafa conjecture at genus two.

Findings

Gravitational corrections expressed as quasimodular functions of Gamma(2).

Holomorphic ambiguity fixed up to genus two in the matrix model.

New closed-form solutions for matrix models using hypergeometric functions.

Abstract

We use the holomorphic anomaly equation to solve the gravitational corrections to Seiberg-Witten theory and a two-cut matrix model, which is related by the Dijkgraaf-Vafa conjecture to the topological B-model on a local Calabi-Yau manifold. In both cases we construct propagators that give a recursive solution in the genus modulo a holomorphic ambiguity. In the case of Seiberg-Witten theory the gravitational corrections can be expressed in closed form as quasimodular functions of Gamma(2). In the matrix model we fix the holomorphic ambiguity up to genus two. The latter result establishes the Dijkgraaf-Vafa conjecture at that genus and yields a new method for solving the matrix model at fixed genus in closed form in terms of generalized hypergeometric functions.