BPS relations from spectral problems and blowup equations

Understand

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 $m$.