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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
Topological Strings and Quantum Spectral Problems
We consider certain quantum spectral problems appearing in the study of local
Calabi-Yau geometries. The quantum spectrum can be computed by the
Bohr-Sommerfeld quantization condition for a period integral. For the case of
small Planck constant, the periods are computed perturbatively by deformation
of the Omega background parameters in the Nekrasov-Shatashvili limit. We
compare the calculations with the results from the standard perturbation theory
for the quantum Hamiltonian. There have been proposals in the literature for
the non-perturbative contributions based on singularity cancellation with the
perturbative contributions. We compute the quantum spectrum numerically with
some high precisions for many cases of Planck constant. We find that there are
also some higher order non-singular non-perturbative contributions, which are
not captured by the singularity cancellation mechanism. We fix the first few
orders formulas of such corrections for some well known local Calabi-Yau
models.Comment: 47 pages, 3 figures. v2: journal version, typos correcte
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