All Genus Open-Closed Mirror Symmetry for Affine Toric Calabi-Yau 3-Orbifolds

Fang, Bohan; Liu, Chiu-Chu Melissa; Zong, Zhengyu

doi:10.48550/arxiv.1310.4818

Cited by 26 publications

(33 citation statements)

References 78 publications

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“…The coefficients of certain series expansions of correlation differentials are often solutions to problems in enumerative geometry and mathematical physics. In this way, topological recursion governs intersection theory on moduli spaces of curves [16], Weil-Petersson volumes of moduli spaces of hyperbolic surfaces [17], enumeration of ribbon graphs [29,10], stationary Gromov-Witten theory of P 1 [30,12], simple Hurwitz numbers and their generalisations [6,15,8,4], and Gromov-Witten theory of toric Calabi-Yau threefolds [5,18,19]. There are also conjectural relations to spin Hurwitz numbers [27] and quantum invariants of knots [7,1].…”

Section: Introductionmentioning

confidence: 99%

Topological recursion and a quantum curve for monotone Hurwitz numbers

Dyer

Mathews

2017

Journal of Geometry and Physics

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Abstract. Classical Hurwitz numbers count branched covers of the Riemann sphere with prescribed ramification data, or equivalently, factorisations in the symmetric group with prescribed cycle structure data. Monotone Hurwitz numbers restrict the enumeration by imposing a further monotonicity condition on such factorisations. In this paper, we prove that monotone Hurwitz numbers arise from the topological recursion of Eynard and Orantin applied to a particular spectral curve. We furthermore derive a quantum curve for monotone Hurwitz numbers. These results extend the collection of enumerative problems known to be governed by the paradigm of topological recursion and quantum curves, as well as the list of analogues between monotone Hurwitz numbers and their classical counterparts.

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Section: Introductionmentioning

confidence: 99%

Topological recursion and a quantum curve for monotone Hurwitz numbers

Dyer

Mathews

2017

Journal of Geometry and Physics

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“…It has been proposed that the topological recursion is an effective tool for defining a genus g B-model topological string theory on a holomorphic curve (known as an Eynard-Orantin spectral curve), that should be the mirror symmetric dual to the genus g Gromov-Witten theory on the A-model side [9,10,43]. This correspondence has been rigorously established for several examples, most notably for an arbitrary toric Calabi-Yau orbifold of 3 dimensions [26], and many other enumerative geometry problems [8,16,19,23,25,47].…”

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confidence: 99%

Quantization of spectral curves for meromorphic Higgs bundles through topological recursion

Dumitrescu¹,

Mulase

2018

Proceedings of Symposia in Pure Mathematics

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A geometric quantization using the topological recursion is established for the compactified cotangent bundle of a smooth projective curve of an arbitrary genus. In this quantization, the Hitchin spectral curve of a rank 2 meromorphic Higgs bundle on the base curve corresponds to a quantum curve, which is a Rees D-module on the base. The topological recursion then gives an all-order asymptotic expansion of its solution, thus determining a state vector corresponding to the spectral curve as a meromorphic Lagrangian. We establish a generalization of the topological recursion for a singular spectral curve. We show that the partial differential equation version of the topological recursion automatically selects the normal ordering of the canonical coordinates, and determines the unique quantization of the spectral curve. The quantum curve thus constructed has the semi-classical limit that agrees with the original spectral curve. Typical examples of our construction includes classical differential equations, such as Airy, Hermite, and Gauß hypergeometric equations. The topological recursion gives an asymptotic expansion of solutions to these equations at their singular points, relating Higgs bundles and various quantum invariants.

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“…This formalism turns out to be very successful in reformulating local mirror symmetry of toric Calabi-Yau 3-folds [6]. Both the proofs in the case of toric Calabi-Yau 3-manifold [14] and 3-orbifolds [16] use comparison with recursions in the Givental formalism. By these results, one may expect to establish some connections between the global EO topological recursions and the Frobenius manifold theory by associating spectral curves to Froebnius manifolds.…”

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confidence: 99%

“…By combining Conjecture 1 with the proof of the BKMP Remodelling Conjecture [14,16], we make the following: One can also compare Conjecture 3 and Conjecture 4 and make a connection between them. It was conjectured in [1] that the partition functions in Conjecture 4 are tau-functions of n-component KP hierarchy and this leads to a fermionic reformulation of the theory.…”

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confidence: 99%