Blowup equations for refined topological strings

• Published in: The journal of high energy physics Vol. 2018; no. 10; pp. 1 - 88
• Main Authors: Huang, Min-xin, Sun, Kaiwen, Wang, Xin
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-10-2018; Springer Nature B.V; SpringerOpen
• Subjects: Classical and Quantum Gravitation; Differential and Algebraic Geometry; Elementary Particles; Functional equations; High energy physics; Markov analysis; Mathematical models; Partitions; Partitions (mathematics); Physics; Physics and Astronomy; Polynomials; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; String Theory; Strings; Supersymmetric Gauge Theory; Supersymmetry; Topological Strings; Unity; Topological Strings; Differential and Algebraic Geometry; Supersymmetric Gauge Theory
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP10(2018)196

A
bstract
Göttsche-Nakajima-Yoshioka K-theoretic blowup equations characterize the Nekrasov partition function of five dimensional
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
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
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.