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139
Convergence of Quantum Cohomology by Quantum Lefschetz
, 2005
"... Quantum Lefschetz theorem by Coates and Givental [3] gives a relationship between the genus 0 GromovWitten theory of X and the twisted theory by a line bundle L on X. We prove the convergence of the twisted theory under the assumption that the genus 0 theory for original X converges. As a corolla ..."
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Cited by 27 (5 self)
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Quantum Lefschetz theorem by Coates and Givental [3] gives a relationship between the genus 0 GromovWitten theory of X and the twisted theory by a line bundle L on X. We prove the convergence of the twisted theory under the assumption that the genus 0 theory for original X converges. As a corollary, we prove the semisimplicity and the Virasoro conjecture for the GromovWitten theories of (not necessarily Fano) projective toric manifolds.
Computation of open GromovWitten invariants for toric CalabiYau 3folds by topological recursion, a proof of the BKMP conjecture
, 2013
"... ..."
Equivariant mirrors and the Virasoro conjecture for flag manifolds
 Int. Math. Res. Not
"... Abstract. We found an explicit description of all GL(n, R)Whittaker functions as oscillatory integrals and thus constructed equivariant mirrors of flag manifolds. As a consequence we proved the Virasoro conjecture for flag manifolds. 1. ..."
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Cited by 26 (1 self)
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Abstract. We found an explicit description of all GL(n, R)Whittaker functions as oscillatory integrals and thus constructed equivariant mirrors of flag manifolds. As a consequence we proved the Virasoro conjecture for flag manifolds. 1.
The moduli space of curves and GromovWitten theory
, 2006
"... The goal of this article is to motivate and describe how GromovWitten theory can and has provided tools to understand the moduli space of curves. For example, ideas and methods from GromovWitten theory have led to both conjectures and theorems showing that the tautological part of the cohomology r ..."
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Cited by 26 (4 self)
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The goal of this article is to motivate and describe how GromovWitten theory can and has provided tools to understand the moduli space of curves. For example, ideas and methods from GromovWitten theory have led to both conjectures and theorems showing that the tautological part of the cohomology ring has a remarkable and profound structure. As an illustration, we describe a new approach to Faber’s intersection number conjecture via branched covers of the projective line (work with I.P. Goulden and D.M. Jackson, based on work with T. Graber). En route we review the work of a large number of mathematicians.
Partition Functions of Matrix Models as the First Special Functions of String Theory II. Kontsevich Model
, 811
"... In arXiv:hepth/0310113 we started a program of creating a referencebook on matrixmodel τfunctions, the new generation of special functions, which are going to play an important role in string theory calculations. The main focus of that paper was on the onematrix Hermitian model τfunctions. The ..."
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In arXiv:hepth/0310113 we started a program of creating a referencebook on matrixmodel τfunctions, the new generation of special functions, which are going to play an important role in string theory calculations. The main focus of that paper was on the onematrix Hermitian model τfunctions. The present paper is devoted to a direct counterpart for the Kontsevich and Generalized Kontsevich Model (GKM) τfunctions. We mostly focus on calculating resolvents (=loop operator averages) in the Kontsevich model, with a special emphasis on its simplest (Gaussian) phase, where exists a surprising integral formula, and the expressions for the resolvents in the genus zero and one are especially simple (in particular, we generalize the known genus zero result to genus one). We also discuss various features of generic phases of the Kontsevich model, in particular, a counterpart of the unambiguous Gaussian solution in the generic case, the solution called DijkgraafVafa (DV) solution. Further, we extend the results to the GKM and, in particular, discuss the pq duality in terms of resolvents and corresponding Riemann surfaces in the example of dualities between (2,3) and (3,2) models.
Quantum RiemannRoch, Lefschetz and Serre
 OF MATH
"... Given a holomorphic vector bundle E: EX → X over a compact Kähler manifold, one introduces twisted GWinvariants of X replacing virtual fundamental cycles of moduli spaces of stable maps f: Σ → X by their capproduct with a chosen multiplicative characteristic class of H 0 (Σ, f ∗ E) − H 1 (Σ, f ∗ E ..."
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Cited by 25 (3 self)
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Given a holomorphic vector bundle E: EX → X over a compact Kähler manifold, one introduces twisted GWinvariants of X replacing virtual fundamental cycles of moduli spaces of stable maps f: Σ → X by their capproduct with a chosen multiplicative characteristic class of H 0 (Σ, f ∗ E) − H 1 (Σ, f ∗ E). Using the formalism [17] of quantized quadratic hamiltonians, we express the descendent potential for the twisted theory in terms of that for X. The result (Theorem 1) is a consequence of Mumford’s Riemann – Roch – Grothendieck formula [31, 13] applied to the universal stable map. When E is concave, and the inverse C ×equivariant Euler class is chosen, the twisted theory yields GWinvariants of EX. The “nonlinear Serre duality principle ” [19, 20] expresses GWinvariants of EX via those of the supermanifold ΠE ∗ X, where the Euler class and E ∗ replace the inverse Euler class and E. We derive from Theorem 1 the nonlinear Serre duality in a very general form (Corollary 2). When the bundle E is convex, and a submanifold Y ⊂ X is defined by
Melting crystal, quantum torus and Toda hierarchy, arXiv:0701.5339 [hepth
"... Searching for the integrable structures of supersymmetric gauge theories and topological strings, we study melting crystal, which is known as random plane partition, from the viewpoint of integrable systems. We show that a series of partition functions of melting crystals gives rise to a tau functio ..."
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Searching for the integrable structures of supersymmetric gauge theories and topological strings, we study melting crystal, which is known as random plane partition, from the viewpoint of integrable systems. We show that a series of partition functions of melting crystals gives rise to a tau function of the onedimensional Toda hierarchy, where the models are defined by adding suitable potentials, endowed with a series of coupling constants, to the standard statistical weight. These potentials can be converted to a commutative subalgebra of quantum torus Lie algebra. This perspective reveals a remarkable connection between random plane partition and quantum torus Lie algebra, and substantially enables to prove the statement. Based on the result, we briefly argue the integrable structures of fivedimensional N = 1 supersymmetric gauge theories and Amodel topological strings. The aforementioned potentials correspond to gauge theory observables analogous to the Wilson loops, and thereby the partition functions are translated in the gauge theory to generating functions of their correlators. In topological strings, we particularly comment on a possibility of topology change caused by condensation of these observables, giving a simple example.
Invariance of tautological equations I: conjectures and applications
"... Abstract. The main goal of this paper is to introduce a set of conjectures on the relations in the tautological rings. In particular, the conjectures gives an efficient algorithm to calculate, conjecturally, all tautological equations using only finite dimensional linear algebra. Other applications ..."
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Abstract. The main goal of this paper is to introduce a set of conjectures on the relations in the tautological rings. In particular, the conjectures gives an efficient algorithm to calculate, conjecturally, all tautological equations using only finite dimensional linear algebra. Other applications include the proofs of Witten’s conjecture on the relations between higher spin curves and Gelfand– Dickey hierarchy and Virasoro conjecture for target manifolds with conformal semisimple quantum cohomology, both for genus up to two. 1.
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.