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MIRROR SYMMETRY FOR ORBIFOLD HURWITZ NUMBERS
"... Abstract. We study mirror symmetry for orbifold Hurwitz numbers. We show that the Laplace transform of orbifold Hurwitz numbers satisfy a differential recursion, which is then proved to be equivalent to the integral recursion of Eynard and Orantin with spectral curve given by the rLambert curve. We ..."
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Abstract. We study mirror symmetry for orbifold Hurwitz numbers. We show that the Laplace transform of orbifold Hurwitz numbers satisfy a differential recursion, which is then proved to be equivalent to the integral recursion of Eynard and Orantin with spectral curve given by the rLambert curve. We argue that the rLambert curve also arises in the infinite framing limit of orbifold GromovWitten theory of [C 3 /(Z/rZ)]. Finally, we prove that the mirror model to orbifold Hurwitz numbers admits a quantum curve.
Spectral curves and the Schrödinger equations for the EynardOrantin recursion
, 2012
"... It is predicted that the principal specialization of the partition function of a Bmodel topological string theory, that is mirror dual to an Amodel enumerative geometry problem, satisfies a Schrödinger equation, and that the characteristic variety of the Schrödinger operator gives the spectral ..."
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Cited by 7 (6 self)
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It is predicted that the principal specialization of the partition function of a Bmodel topological string theory, that is mirror dual to an Amodel enumerative geometry problem, satisfies a Schrödinger equation, and that the characteristic variety of the Schrödinger operator gives the spectral curve of the Bmodel theory, when an algebraic Ktheory obstruction vanishes. In this paper we present two concrete mathematical Amodel examples whose mirror dual partners exhibit these predicted features on the Bmodel side. The Amodel examples we discuss are the generalized Catalan numbers of an arbitrary genus and the single Hurwitz numbers. In each case, we show that the Laplace transform of the counting functions satisfies the EynardOrantin topological recursion, that the Bmodel partition function satisfies the KP equations, and that the principal specialization of the partition function satisfies a Schrödinger equation whose total symbol is exactly the Lagrangian immersion of the spectral curve of the EynardOrantin theory.