Abstract
Göttsche-Nakajima-Yoshioka K-theoretic blowup equations characterize the Nekrasov partition function of five dimensional mathcal{N}=1 supersymmetric gauge theories compactified on a circle, which via geometric engineering correspond to the refined topological string theory on SU(N) 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 r 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 r 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.
Highlights
Blowup formulae originated from the attempt to understand the relation between the Donaldson invariants of a four-manifold X and those of its blowup X = X# P2
We find that both vanishing and unity blowup equations exist for the partition function of refined topological string, and the crucial ingredients are the r fields introduced in our previous paper
We study the modular properties of the blowup equations and provide a procedure to determine all the vanishing and unity r fields from the polynomial part of refined topological string at large radius point
Summary
The blowup formulae satisfied by the K-theoretic Nekrasov partition function can be regarded as the functional equations of the partition function of refined topological string, at least for those local Calabi-Yau which can engineer suitable supersymmetry gauge theories. The main result of this paper is as follows: for an arbitrary local Calabi-Yau threefold X with mirror curve of genus g, suppose there are b = dimH2(X, Z) irreducible curve classes corresponding to Kahler moduli t in which b − g classes correspond to mass parameters m, and denote C as the intersection matrix between the b curve classes and the g irreducible compact divisor classes, there exist infinite constant integral vectors r ∈ Zb such that the following functional equations for the twisted partition function of refined topological string on X hold:.
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