Blowup Equations for Topological Strings and Supersymmetric Gauge Theories

Blowup equations and their K-theoretic version were proposed by Nakajima-Yoshioka and G"{o}ttsche as functional equations for Nekrasov partition functions of supersymmetric gauge theories in 4d and 5d. We generalize the blowup equations in two directions: one is for the refined topological string theory on arbitrary local Calabi-Yau threefolds, the other is an elliptic version for arbitrary 6d $(1,0)$ superconformal field theories (SCFTs) in the "atomic classification" of Heckman-Morrison-Rudelius-Vafa. In general, blowup equations fall into two types: the unity and the vanishing. We find the unity part of generalized blowup equations can be used to efficiently solve all refined BPS invariants of local Calabi-Yau geometries and the elliptic genera of 6d $(1,0)$ SCFTs, while the vanishing part can derive the compatibility formulas between two quantization schemes of algebraic curves, which are the exact Nekrasov-Shatashivili quantization conditions and the Grassi-Hatsuda-Mari~no conjecture. Blowup equations also give many interesting identities among modular forms and Jacobi forms. Furthermore, we study the relation between the elliptic genera of pure gauge 6d $(1,0)$ SCFTs and the superconformal indices of certain 4d $mathcal{N}=2$ SCFTs. At last, we study the K-theoretic blowup equations on $mathbb{Z}_2$ orbifold space and their connection with the bilinear relations of $q$-deformed periodic Toda systems.