Refined stable pair invariants for E-, M- and [p,q]-strings

TL;DR

This paper calculates refined BPS invariants for stable pairs on non-compact Calabi-Yau manifolds using mirror symmetry, modularity, and anomaly equations, linking mathematical invariants to physical theories like M-theory and F-theory.

Contribution

It introduces a novel method combining mirror symmetry, anomaly equations, and modularity to compute refined BPS invariants for complex geometries, with applications to string theory and gauge theories.

Findings

Refined BPS invariants computed for specific Calabi-Yau geometries.

Connections established between mathematical invariants and physical theories.

Enhanced understanding of BPS states in various string theory contexts.

Abstract

We use mirror symmetry, the refined holomorphic anomaly equation and modularity properties of elliptic singularities to calculate the refined BPS invariants of stable pairs on non-compact Calabi-Yau manifolds, based on del Pezzo surfaces and elliptic surfaces, in particular the half K3. The BPS numbers contribute naturally to the five-dimensional N=1 supersymmetric index of M-theory, but they can be also interpreted in terms of the superconformal index in six dimensions and upon dimensional reduction the generating functions count N=2 Seiberg-Witten gauge theory instantons in four dimensions. Using the M/F-theory uplift the additional information encoded in the spin content can be used in an essential way to obtain information about BPS states in physical systems associated to small instantons, tensionless strings, gauge symmetry enhancement in F-theory by [p,q]-strings as well as M-strings.