Blowup Equations for Refined Topological Strings

TL;DR

This paper develops blowup equations for refined topological strings on local Calabi-Yau threefolds, providing a new functional approach to compute refined BPS invariants and analyze modular properties, complementing existing methods.

Contribution

It introduces both vanishing and unity blowup equations for refined topological string partition functions on general local Calabi-Yau threefolds, expanding the toolkit for calculating BPS invariants.

Findings

Existence of vanishing and unity blowup equations for refined topological strings.

Blowup equations lead to infinite identities among refined free energies.

Method to determine all $f r$ fields from polynomial parts at large radius.

Abstract

G\"{o}ttsche-Nakajima-Yoshioka K-theoretic blowup equations characterize the Nekrasov partition function of five dimensional $\mathcal{N}=1$ supersymmetric gauge theories compactified on a circle, which via geometric engineering correspond to the refined topological string theory on $SU(N)$ geometries. In this paper, we study the K-theoretic blowup equations for general local Calabi-Yau threefolds. We find that both vanishing and unity blowup equations exist for the partition function of refined topological string, and the crucial ingredients are the $\bf r$ fields introduced in our previous paper. These blowup equations are in fact the functional equations for the partition function and each of them results in infinite identities among the refined free energies. Evidences show that they can be used to determine the full refined BPS invariants of local Calabi-Yau threefolds. This serves an independent and sometimes more powerful way to compute the partition function other than the refined topological vertex in the A-model and the refined holomorphic anomaly equations in the B-model. We study the modular properties of the blowup equations and provide a procedure to determine all the vanishing and unity $\bf r$ fields from the polynomial part of refined topological string at large radius point. We also find that certain form of blowup equations exist at generic loci of the moduli space.