BPS relations from spectral problems and blowup equations

TL;DR

This paper proves constraints on refined BPS invariants of toric Calabi-Yau threefolds by connecting topological string dualities, spectral theory, and blowup equations, advancing understanding of their mathematical structure.

Contribution

It establishes the validity of BPS relation constraints for $Y^{N,m}$ geometries using $K$-theoretic blowup equations, linking different theoretical frameworks.

Findings

Proved constraints for $Y^{N,m}$ geometries.

Connected topological string dualities with spectral theory.

Utilized $K$-theoretic blowup equations for $SU(N)$ SYM.

Abstract

Recently an exact duality between topological string and the spectral theory of operators constructed from mirror curves to toric Calabi-Yau threefolds has been proposed. At the same time an exact quantization condition for the cluster integrable systems associated to these geometries has been conjectured. The consistency between the two approaches leads to an infinite set of constraints for the refined BPS invariants of the toric Calabi-Yau threefolds. We prove these constraints for the $Y^{N,m}$ geometries using the $K$-theoretic blowup equations for $SU(N)$ SYM with generic Chern-Simons invariant $m$.