Results 1  10
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24
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.
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.
The Witten top Chern class via Ktheory
 J. Algebraic Geom
"... Abstract. The Witten top Chern class is the crucial cohomology class needed to state a conjecture by Witten relating the Gelfand–Dikiĭ hierarchies to higher spin curves. In [PV01], Polishchuk and Vaintrob provide an algebraic construction of such a class. We present a more straightforward constructi ..."
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Cited by 17 (2 self)
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Abstract. The Witten top Chern class is the crucial cohomology class needed to state a conjecture by Witten relating the Gelfand–Dikiĭ hierarchies to higher spin curves. In [PV01], Polishchuk and Vaintrob provide an algebraic construction of such a class. We present a more straightforward construction via Ktheory. In this way we shortcircuit the passage through bivariant intersection theory and the use of MacPherson’s graph construction. Furthermore, we show that the Witten top Chern class admits a natural lifting to the Ktheory ring. 1.
GEOMETRY AND ANALYSIS OF SPIN EQUATIONS
, 2004
"... Abstract. We introduce Wspin structures on a Riemann surface and give a precise definition to the corresponding Wspin equations for any quasihomogeneous polynomial W. Then, we construct examples of nonzero solutions of spin equations in the presence of Ramond marked points. The main result of the ..."
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Cited by 15 (5 self)
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Abstract. We introduce Wspin structures on a Riemann surface and give a precise definition to the corresponding Wspin equations for any quasihomogeneous polynomial W. Then, we construct examples of nonzero solutions of spin equations in the presence of Ramond marked points. The main result of the paper is a compactness theorem for the moduli space of the solutions of Wspin equations when W = W(x1,..., xt) is a nondegenerate quasihomogeneous polynomial with fractional degrees (or weights) wt(xi) = qi < 1/2 for all i. In particular, the compactness theorem holds for the superpotentials E6, E7, E8, and An−1, Dn+1 for n ≥ 3. 1.
Towards an enumerative geometry of the moduli space of twisted curves and rth roots
, 2008
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Intersection numbers with Witten’s top Chern class
 Geom. Topol
"... Witten’s top Chern class is a particular cohomology class on the moduli space of Riemann surfaces endowed with rspin structures. It plays a key role in Witten’s conjecture relating to the intersection theory on these moduli spaces. Our first goal is to compute the integral of Witten’s class over th ..."
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Cited by 7 (4 self)
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Witten’s top Chern class is a particular cohomology class on the moduli space of Riemann surfaces endowed with rspin structures. It plays a key role in Witten’s conjecture relating to the intersection theory on these moduli spaces. Our first goal is to compute the integral of Witten’s class over the socalled double ramification cycles in genus 1. We obtain a simple closed formula for these integrals. This allows us, using the methods of [15], to find an algorithm for computing the intersection numbers of the Witten class with powers of the ψclasses (or tautological classes) over any moduli space of rspin structures, in short, all numbers involved in Witten’s conjecture. 1
Open/closed string duality for topological gravity with matter
"... The exact FZZT brane partition function for topological gravity with matter is computed using the dual twomatrix model. We show how the effective theory of open strings on a stack of FZZT branes is described by the generalized Kontsevich matrix integral, extending the earlier result for pure topolo ..."
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Cited by 5 (1 self)
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The exact FZZT brane partition function for topological gravity with matter is computed using the dual twomatrix model. We show how the effective theory of open strings on a stack of FZZT branes is described by the generalized Kontsevich matrix integral, extending the earlier result for pure topological gravity. Using the wellknown relation between the Kontsevich integral and a certain shift in the closedstring background, we conclude that these models exhibit open/closed string duality explicitly. Just as in pure topological gravity, the unphysical sheets of the classical FZZT moduli space are eliminated in the exact answer. Instead, they contribute small, nonperturbative corrections to the exact answer through Stokes ’ phenomenon. January
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.