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Algebraic methods in random matrices and enumerative geometry
, 2008
"... We review the method of symplectic invariants recently introduced to solve matrix models loop equations, and further extended beyond the context of matrix models. For any given spectral curve, one defines a sequence of differential forms, and a sequence of complex numbers Fg. We recall the definitio ..."
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Cited by 38 (9 self)
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We review the method of symplectic invariants recently introduced to solve matrix models loop equations, and further extended beyond the context of matrix models. For any given spectral curve, one defines a sequence of differential forms, and a sequence of complex numbers Fg. We recall the definition of the invariants Fg, and we explain their main properties, in particular symplectic invariance, integrability, modularity,... Then, we give several examples of applications, in particular matrix models, enumeration of discrete surfaces (maps), algebraic geometry and topological strings, nonintersecting brownian motions,...
Computation of open GromovWitten invariants for toric CalabiYau 3folds by topological recursion, a proof of the BKMP conjecture
, 2013
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Witten’s conjecture, Virasoro conjecture, and semisimple Frobenius manifolds
, 2002
"... Abstract. The main goal of this paper is to prove the following two conjectures for genus up to two: (1) Witten’s conjecture on the relations between higher spin curves and Gelfand–Dickey hierarchy. (2) Virasoro conjecture for target manifolds with conformal semisimple Frobenius manifolds. The main ..."
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Cited by 21 (7 self)
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Abstract. The main goal of this paper is to prove the following two conjectures for genus up to two: (1) Witten’s conjecture on the relations between higher spin curves and Gelfand–Dickey hierarchy. (2) Virasoro conjecture for target manifolds with conformal semisimple Frobenius manifolds. The main technique used in the proof is the invariance of tautological equations under loop group action. 1.
Wconstraints for the total descendant potential of a simple singularity
 Compositio Math
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Milanov T., Simple singularities and integrable hierarchies, in The breadth of symplectic and Poisson geometry
 2005, 173–201, math.AG/0307176. (n, 1)Reduced DKP Hierarchy 19
"... Abstract. The paper [11] gives a construction of the total descendent potential corresponding to a semisimple Frobeniusmanifold. In [12], it is proved that the total descendent potential corresponding to K. Saito’s Frobenius structure on the parameter space of the miniversal deformation of the An−1 ..."
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Cited by 8 (0 self)
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Abstract. The paper [11] gives a construction of the total descendent potential corresponding to a semisimple Frobeniusmanifold. In [12], it is proved that the total descendent potential corresponding to K. Saito’s Frobenius structure on the parameter space of the miniversal deformation of the An−1singularity satisfies the modulon reduction of the KPhierarchy. In this paper, we identify the hierarchy satisfied by the total descendent potential of a simple singularity of the A,D,Etype. Our description of the hierarchy is parallel to the vertex operator construction of Kac – Wakimoto [17] except that we give both some general integral formulas and explicit numerical values for certain coefficients which in the Kac – Wakimoto theory are studied on a casebycase basis and remain, generally speaking, unknown. 1. The ADEhierarchies. According to Date–Jimbo–Kashiwara–Miwa [6] and I. Frenkel [10], the KdVhierarchy of integrable systems can be placed under the name A1 into the list of more general integrable hierarchies corresponding to the ADE Dynkin diagrams. These hierarchies are usually constructed (see [16]) using representation theory of the corresponding loop groups. V. Kac and M. Wakimoto [17] describe the hierarchies even more explicitly in the form of the so called Hirota quadratic equations expressed in terms of suitable vertex operators. One of the goals of the present paper is to show how the vertex operator description of the Hirota quadratic equations (certainly the same ones, even though we don’t quite prove this) emerges from the theory of vanishing cycles associated with the ADE singularities. Let f be a weightedhomogeneous polynomial inC3 with a simple critical point at the origin. According to V. Arnold [1] simple singularities of holomorphic functions are classified by simplylaced Dynkin diagrams: AN, N ≥ 1: f = x
WN+1CONSTRAINTS FOR SINGULARITIES OF TYPE AN
, 2008
"... Using Picard–Lefschetz periods for the singularity of type AN, we construct a projective representation of the Lie algebra of differential operators on the circle with central charge h: = N + 1. We prove that the total descendant potential DAN of ANsingularity is a highest weight vector. It is kno ..."
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Cited by 2 (1 self)
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Using Picard–Lefschetz periods for the singularity of type AN, we construct a projective representation of the Lie algebra of differential operators on the circle with central charge h: = N + 1. We prove that the total descendant potential DAN of ANsingularity is a highest weight vector. It is known that DAN can be interpreted as a generating function of a certain class of intersection numbers on the moduli space of hspin curves. In this settings our constraints provide a complete set of recursion relations between the intersection numbers. Our methods are based entirely on the symplectic loop space formalism of A. Givental and therefore they can be applied to the mirror models of symplectic manifolds.
GromovWitten theory of Fano orbifold curves, Gamma integral structures and ADEToda Hierarchies
"... Abstract. We construct an integrable hierarchy in the form of Hirota quadratic equations (HQE) that governs the Gromov–Witten (GW) invariants of the Fano orbifold projective curve P1a1,a2,a3 with positive orbifold Euler characteristic. We also identify our HQEs with an appropriate Kac–Wakimoto hiera ..."
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Abstract. We construct an integrable hierarchy in the form of Hirota quadratic equations (HQE) that governs the Gromov–Witten (GW) invariants of the Fano orbifold projective curve P1a1,a2,a3 with positive orbifold Euler characteristic. We also identify our HQEs with an appropriate Kac–Wakimoto hierarchy of ADE type. In particular, we obtain a generalization of the famous Toda conjecture about the GW invariants of P1. Contents