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The spectral curve and the Schrödinger equation of double Hurwitz numbers and higher spin structures
"... Abstract. We derive the spectral curves for qpart double Hurwitz numbers, rspin simple Hurwitz numbers, and arbitrary combinations of these cases, from the analysis of the unstable (0, 1)geometry. We quantize this family of spectral curves and obtain the Schrödinger equations for the partition fun ..."
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Abstract. We derive the spectral curves for qpart double Hurwitz numbers, rspin simple Hurwitz numbers, and arbitrary combinations of these cases, from the analysis of the unstable (0, 1)geometry. We quantize this family of spectral curves and obtain the Schrödinger equations for the partition function of the corresponding Hurwitz problems. We thus confirm the conjecture for the existence of quantum curves in these generalized Hurwitz number cases.
QUANTUM CURVES FOR HITCHIN FIBRATIONS AND THE EYNARDORANTIN THEORY
, 2014
"... We generalize the topological recursion of Eynard–Orantin (JHEP 0612:053, 2006; Commun Number Theory Phys 1:347–452, 2007) to the family of spectral curves of Hitchin fibrations. A spectral curve in the topological recursion, which is defined to be a complex plane curve, is replaced with a generic ..."
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We generalize the topological recursion of Eynard–Orantin (JHEP 0612:053, 2006; Commun Number Theory Phys 1:347–452, 2007) to the family of spectral curves of Hitchin fibrations. A spectral curve in the topological recursion, which is defined to be a complex plane curve, is replaced with a generic curve in the cotangent bundle T ∗C of an arbitrary smooth base curve C. We then prove that these spectral curves are quantizable, using the new formalism. More precisely, we construct the canonical generators of the formal deformation family of D modules over an arbitrary projective algebraic curve C of genus greater than 1, from the geometry of a prescribed family of smooth Hitchin spectral curves associated with the SL(2,C)character variety of the fundamental group π1(C). We show that the semiclassical limit through the WKB approximation of these deformed D
QUANTUM CURVES FOR SIMPLE HURWITZ NUMBERS OF AN ARBITRARY BASE CURVE
"... Abstract. The generating functions of simple Hurwitz numbers of the projective line are known to satisfy many properties. They include a heat equation, the EynardOrantin topological recursion, an infiniteorder differential equation called a quantum curve equation, and a Schrödinger like partial ..."
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Abstract. The generating functions of simple Hurwitz numbers of the projective line are known to satisfy many properties. They include a heat equation, the EynardOrantin topological recursion, an infiniteorder differential equation called a quantum curve equation, and a Schrödinger like partial differential equation. In this paper we generalize these properties to simple Hurwitz numbers with an arbitrary base curve. Contents
THE LAPLACE TRANSFORM, MIRROR SYMMETRY, AND THE TOPOLOGICAL RECURSION OF EYNARDORANTIN
"... Abstract. This paper is based on the author’s talk at the 2012 Workshop on Geometric Methods in Physics held in Bia lowieża, Poland. The aim of the talk is to introduce the audience to the EynardOrantin topological recursion. The formalism is originated in random matrix theory. It has been predict ..."
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Abstract. This paper is based on the author’s talk at the 2012 Workshop on Geometric Methods in Physics held in Bia lowieża, Poland. The aim of the talk is to introduce the audience to the EynardOrantin topological recursion. The formalism is originated in random matrix theory. It has been predicted, and in some cases it has been proven, that the theory provides an effective mechanism to calculate certain quantum invariants and a solution to enumerative geometry problems, such as open GromovWitten invariants of toric CalabiYau threefolds, single and double Hurwitz numbers, the number of lattice points on the moduli space of smooth algebraic curves, and quantum knot invariants. In this paper we use the Laplace transform of generalized Catalan numbers of an arbitrary genus as an example, and present the EynardOrantin recursion. We examine various aspects of the theory, such as its relations to mirror symmetry, GromovWitten invariants, integrable hierarchies such as the KP equations, and the Schrödinger equations. Contents
QUANTIZATION OF SPECTRAL CURVES FOR MEROMORPHIC HIGGS BUNDLES THROUGH TOPOLOGICAL RECURSION
"... Abstract. 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 quantu ..."
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Abstract. 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 Dmodule on the base. The topological recursion then gives an allorder 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 semiclassical 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
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"... Quantum curves for the enumeration of ribbon graphs and hypermaps ..."
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