Blowup Equations and Holomorphic Anomaly Equations

TL;DR

This paper introduces a new recursive consistency equation linking blowup and holomorphic anomaly equations in refined topological string theory, revealing their relation and suggesting a non-holomorphic extension.

Contribution

It proposes a novel recursive consistency equation that governs the relation between blowup and holomorphic anomaly equations in topological string theory.

Findings

Introduces a recursive consistency equation for the two approaches.

Computes the modular anomaly order by order.

Suggests a non-holomorphic extension of blowup equations.

Abstract

Blowup equations and holomorphic anomaly equations are two universal yet completely different approaches to solve refined topological string theory on local Calabi-Yau threefolds corresponding to A- and B-model respectively. The former originated from comparing Nekrasov partition functions of 4d $\mathcal{N}=2$ gauge theories on $\Omega$ defomed spacetime $\mathbb{C}^2_{\epsilon_1,\epsilon_2}$ and its one-point blown-up, while the latter takes root in the degeneration of wordsheet Riemann surfaces. The relation between the two approaches is an open question. In this short note, we find a novel recursive equation governing their consistency, which we call the consistency equation. This new equation computes the modular anomaly of blowup equations order by order. The consistency equation also suggests a non-holomorphic extension of blowup equations.