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Blowup Equations for Refined Topological Strings
G\"{o}ttsche-Nakajima-Yoshioka K-theoretic blowup equations characterize the
Nekrasov partition function of five dimensional supersymmetric
gauge theories compactified on a circle, which via geometric engineering
correspond to the refined topological string theory on 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 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 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.Comment: 85 pages. v2: Journal versio
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