Holomorphic Anomaly in Gauge Theories and Matrix Models
Min-xin Huang, Albrecht Klemm
TL;DR
The paper develops a holomorphic-anomaly based framework to compute gravitational corrections $F^{(g)}(\tau,\bar{\tau})$ for 4d $N=2$ gauge theories and a two-cut matrix model related by the Dijkgraaf-Vafa conjecture to a local Calabi-Yau B-model. By introducing propagators, they build a genus-$g$ recursion for $F^{(g)}$ and determine boundary data to fix holomorphic ambiguities; SW theory yields $F^{(g)}$ as quasimodular functions of $\Gamma(2)$, while the DV-related matrix model fixes the ambiguity up to genus two and provides closed-form, generalized hypergeometric expressions. Contributions include (i) closed-form quasimodular expressions for SW gravitational corrections, (ii) fixation of the holomorphic ambiguity up to genus $2$ in the matrix-model case, establishing DV at that genus, and (iii) a practical genus-by-genus solution method via generalized hypergeometric functions. Significance lies in unifying gauge theory, matrix models, and topological strings under the holomorphic anomaly with modular structure, enabling systematic computation of higher-genus gravitational corrections.
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.