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21
The witten equation, mirror symmetry and quantum singularity theory
, 2009
"... For any quasihomogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a corresponding cohomological field theory associated to the singularity. This theory is analogous to GromovWitten theory and generalizes the theory of rspin curves, which corresponds ..."
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Cited by 53 (2 self)
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For any quasihomogeneous hypersurface singularity, we describe a family of moduli spaces, a virtual cycle, and a corresponding cohomological field theory associated to the singularity. This theory is analogous to GromovWitten theory and generalizes the theory of rspin curves, which corresponds to the simple singularity Ar−1. The main results are that we resolve two outstanding conjectures of Witten. The first conjecture is that ADEsingularities are selfdual; and the second conjecture is that the total potential functions of ADEsingularities satisfy corresponding ADEintegrable hierarchies. Other cases of integrable hierarchies are also discussed.
Tautological relations and the rspin Witten conjecture
"... In [23, 24], Y.P. Lee introduced a notion of universal relation for formal Gromov–Witten potentials. Universal relations are connected to tautological relations in the cohomology ring of the moduli space Mg,n of stable curves. Y.P. Lee conjectured that the two sets of relations coincide and proved ..."
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Cited by 43 (11 self)
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In [23, 24], Y.P. Lee introduced a notion of universal relation for formal Gromov–Witten potentials. Universal relations are connected to tautological relations in the cohomology ring of the moduli space Mg,n of stable curves. Y.P. Lee conjectured that the two sets of relations coincide and proved the inclusion (tautological relations) ⊂ (universal relations) modulo certain results announced by C. Teleman. He also proposed an algorithm that, conjecturally, computes all universal/tautological relations. Here we give a geometric interpretation of Y.P. Lee’s algorithm. This leads to a much simpler proof of the fact that every tautological relation gives rise to a universal relation. We also show that Y.P. Lee’s algorithm computes the tautological relations correctly if and only if the Gorenstein conjecture on the tautological cohomology ring of Mg,n is true. These results are first steps in the task of establishing an equivalence between formal and geometric Gromov–Witten theories. In particular, it implies that in any semisimple Gromov–Witten theory where arbitrary correlators can be expressed in genus 0 correlators using only tautological relations, the formal and the geometric Gromov–Witten potentials coincide.
The Witten equation and its virtual fundamental cycle
, 2007
"... We study a system of nonlinear elliptic partial differential equations, which we call the Witten Equation, associated to a quasihomogeneous polynomial. We construct a virtual cycle for this equation and prove that it satisfies axioms similar to those of GromovWitten theory and of rspin theory. ..."
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Cited by 31 (5 self)
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We study a system of nonlinear elliptic partial differential equations, which we call the Witten Equation, associated to a quasihomogeneous polynomial. We construct a virtual cycle for this equation and prove that it satisfies axioms similar to those of GromovWitten theory and of rspin theory.
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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Cited by 23 (8 self)
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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.
Identification of the Givental formula with the spectral curve topological recursion procedure
 Comm. Math. Phys
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BCOV theory via Givental group action on . . .
, 2008
"... In a previous paper, Losev, me, and Shneiberg constructed a full descendant potential associated to an arbitrary cyclic Hodge dGBV algebra. This contruction extended the construction of Barannikov and Kontsevich of solution of the WDVV equation, based on the earlier paper of Bershadsky, Cecotti, O ..."
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Cited by 13 (6 self)
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In a previous paper, Losev, me, and Shneiberg constructed a full descendant potential associated to an arbitrary cyclic Hodge dGBV algebra. This contruction extended the construction of Barannikov and Kontsevich of solution of the WDVV equation, based on the earlier paper of Bershadsky, Cecotti, Ooguri, and Vafa. In the present paper, we give an interpretation of this full descendant potential in terms of Givental group action on cohomological field theories. In particular, the fact that it satisfies all
An algebraic scheme associated with the noncommutative KP hierarchy and some of its extensions
, 2005
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Tautological equations in M3,1 via invariance conjectures, preprint 2005
"... This paper is dedicated to Matteo S. Arcara, who was born while the paper was at its last phase of preparation. Abstract. A new tautological equation of M3,1 in codimension 3 is derived and proved, using the invariance condition explained in [1, 9, 10, 11]. 1. ..."
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Cited by 7 (4 self)
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This paper is dedicated to Matteo S. Arcara, who was born while the paper was at its last phase of preparation. Abstract. A new tautological equation of M3,1 in codimension 3 is derived and proved, using the invariance condition explained in [1, 9, 10, 11]. 1.
Invariance of Gromov–Witten theory under simple flops
 J. Reine Angew. Math
"... ABSTRACT. We show that the generating functions of Gromov–Witten invariants with ancestors are invariant under a simple flop, for all genera, after an analytic continuation in the extended Kähler moduli space. This is a sequel to [14]. 0.1. Statement of the main results. Let X be a smooth complex p ..."
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Cited by 7 (5 self)
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ABSTRACT. We show that the generating functions of Gromov–Witten invariants with ancestors are invariant under a simple flop, for all genera, after an analytic continuation in the extended Kähler moduli space. This is a sequel to [14]. 0.1. Statement of the main results. Let X be a smooth complex projective manifold and ψ: X → X ̄ a flopping contraction in the sense of minimal model theory, with ψ ̄ : Z ∼ = Pr → pt the restriction map to the extremal contraction. Assume that NZ/X ∼ = OPr(−1)⊕(r+1). It was shown