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Spectral Theory and Mirror Symmetry
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In this article, it has been shown that the quantization of mirror curves to toric Calabi-Yau threefolds leads to trace class operators whose spectral properties are conjecturally encoded in the enumerative geometry of the Calabi Yau.Abstract:
Recent developments in string theory have revealed a surprising connection between spectral theory and local mirror symmetry: it has been found that the quantization of mirror curves to toric Calabi-Yau threefolds leads to trace class operators, whose spectral properties are conjecturally encoded in the enumerative geometry of the Calabi-Yau. This leads to a new, infinite family of solvable spectral problems: the Fredholm determinants of these operators can be found explicitly in terms of Gromov-Witten invariants and their refinements; their spectrum is encoded in exact quantization conditions, and turns out to be determined by the vanishing of a quantum theta function. Conversely, the spectral theory of these operators provides a non-perturbative definition of topological string theory on toric Calabi-Yau threefolds. In particular, their integral kernels lead to matrix integral representations of the topological string partition function, which explain some number-theoretic properties of the periods. In this paper we give a pedagogical overview of these developments with a focus on their mathematical implicationsread more
Citations
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Seiberg–Witten theory as a Fermi gas
TL;DR: In this paper, the dual partition function of SU(2) Super Yang-Mills theory in a self-dual π-Omega background was shown to be equivalent to the spectral determinant of an ideal Fermi gas.
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Reconstructing WKB from topological recursion
Vincent Bouchard,Bertrand Eynard +1 more
TL;DR: In this article, it was shown that the topological recursion can reconstruct the WKB expansion of a quantum curve for all spectral curves whose Newton polygons have no interior point (and that are smooth as affine curves).
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Resurgence Matches Quantization
TL;DR: In this article, the Borel-Pade-Ecalle resummation of this resurgent transseries, alongside occurrence of Stokes phenomenon, matches the string-theoretic partition function obtained via quantization of the mirror curve.
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Hofstadter’s butterfly in quantum geometry
TL;DR: In this article, it was shown that the quantum A-period, determining the relation between the energy eigenvalue and the Kahler modulus of the Calabi-Yau geometry, can be found explicitly when the quantum parameter $q=e^{i\hbar}$ is a root of unity, and that its branch cuts are given by Hofstadter's butterfly.
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Exact Quantization Conditions, Toric Calabi-Yau and Nonperturbative Topological String
TL;DR: The relation between the Nekrasov-Shatashvili (NS) quantization scheme and Grassi-Hatsuda-Marino conjecture for the mirror curve of arbitrary toric Calabi-Yau was established in this article.
References
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Book
Trace ideals and their applications
TL;DR: In this paper, Calkin's theory of operator ideals and symmetrically normed ideals convergence theorems for trace, determinant, and Lidskii's theorem are discussed.
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Seiberg-Witten Prepotential from Instanton Counting
TL;DR: In this article, a two-parameter generalization of the Seiberg-Witten prepotential is presented, which is rather natural from the M-theory/five dimensional perspective, and conjecture its relation to the tau-functions of KP/Toda hierarchy.
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A Pair of Calabi-Yau manifolds as an exactly soluble superconformal theory
TL;DR: In this paper, the prepotentials and geometry of the moduli spaces for a Calabi-Yau manifold and its mirror were derived and all the sigma model corrections to the Yukawa couplings and moduli space metric were obtained.
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Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes
TL;DR: In this paper, the authors developed techniques to compute higher loop string amplitudes for twisted N = 2 theories with ε = 3 (i.e. the critical case) by exploiting the discovery of an anomaly at every genus in decoupling of BRST trivial states, captured by a master anomaly equation.
Book
Mirror symmetry and algebraic geometry
David A. Cox,Sheldon Katz +1 more
TL;DR: The quintic threefold Toric geometry Mirror symmetry constructions Hodge theory and Yukawa couplings Moduli spaces Gromov-Witten invariants Quantum cohomology Localization Quantum differential equations The mirror theorem Conclusion Singular varieties Physical theories Bibliography Index as mentioned in this paper