A Unified Approach to Holomorphic Anomaly Equations and Quantum Spectral Curves

TL;DR

This paper introduces a unified formalism connecting holomorphic anomaly equations with quantum spectral curves using diagrammatic quantum field theory on moduli spaces, enabling new recursive computations and applications.

Contribution

It develops an abstract quantum field theory framework based on moduli space diagrammatics, unifying approaches to holomorphic anomalies and quantum spectral curves.

Findings

Derived a quadratic recursion relation for free energies.

Unified treatment of holomorphic anomaly equations and quantum spectral curves.

Applicable to various models via different choices of functions and propagators.

Abstract

We present a unified approach to holomorphic anomaly equations and some well-known quantum spectral curves. We develop a formalism of abstract quantum field theory based on the diagrammatics of the Deligne-Mumford moduli spaces $\overline{{\mathcal M}}_{g,n}$ and derive a quadratic recursion relation for the abstract free energies in terms of the edge-cutting operators. This abstract quantum field theory can be realized by various choices of a sequence of holomorphic functions or formal power series and suitable propagators, and the realized quantum field theory can be represented by formal Gaussian integrals. Various applications are given.