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48
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.
Moduli of Twisted Spin Curves
"... . In this note we give a new, natural construction of a compactication of the stack of smooth rspin curves, which we call the stack of stable twisted rspin curves. This stack is identied with a special case of a stack of twisted stable maps of Abramovich and Vistoli. Realizations in terms of ad ..."
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Cited by 38 (6 self)
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. In this note we give a new, natural construction of a compactication of the stack of smooth rspin curves, which we call the stack of stable twisted rspin curves. This stack is identied with a special case of a stack of twisted stable maps of Abramovich and Vistoli. Realizations in terms of admissible G m spaces and Qline bundles are given as well. The innitesimal structure of this stack is described in a relatively straightforward manner, similar to that of usual stable curves. We construct representable morphisms from the stacks of stable twisted rspin curves to the stacks of stable rspin curves [4], and show that they are isomorphisms. Many delicate features of rspin curves, including torsion free sheaves with power maps, arise as simple byproducts of twisted spin curves. Various constructions, such as the @operator of Seeley and Singer [9] and Witten's cohomology class [10] go through without complications in the setting of twisted spin curves. The moduli s...
Algebraic construction of Witten’s top Chern class in “Advances in algebraic geometry motivated by physics
"... Abstract. Applying a modification of MacPherson’s graph construction to the case of periodic complexes, we give an algebraic construction of Witten’s “top Chern class ” on the moduli space of algebraic curves with higher spin structures. We show that it satisfies most of the axioms for the spin virt ..."
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Cited by 35 (6 self)
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Abstract. Applying a modification of MacPherson’s graph construction to the case of periodic complexes, we give an algebraic construction of Witten’s “top Chern class ” on the moduli space of algebraic curves with higher spin structures. We show that it satisfies most of the axioms for the spin virtual class and also the socalled descent axiom. 1.
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.
The orbifold quantum cohomology of the classifying space of a finite group
 Orbifolds in Mathematics and Physics, Contemp
, 2002
"... Abstract. We work through, in detail, the quantum cohomology of the orbifold BG, the point with action of a finite group G. We provide a simple description of algebraic structures on the state space of this theory. As a consequence, we find that multiple copies of commuting Virasoro algebras appear ..."
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Cited by 31 (2 self)
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Abstract. We work through, in detail, the quantum cohomology of the orbifold BG, the point with action of a finite group G. We provide a simple description of algebraic structures on the state space of this theory. As a consequence, we find that multiple copies of commuting Virasoro algebras appear in this theory which completely determine the correlators of the theory. For the Proceedings of the Mathematical Aspects of Orbifold String Theory conference in Madison, Wisconsin.
Witten’s top Chern class on the moduli space of higher spin curves, math.AG/0208112, to appear
 in Proceedings of the Workshop on Frobenius manifolds
"... Abstract. We prove that the algebraic Witten’s “top Chern class ” constructed in [9] satisfies the axioms for the spin virtual class formulated in [5]. This paper is a sequel to [9]. Its goal is to verify that the virtual top Chern class c1/r in the Chow group of the moduli space of higher spin curv ..."
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Cited by 24 (1 self)
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Abstract. We prove that the algebraic Witten’s “top Chern class ” constructed in [9] satisfies the axioms for the spin virtual class formulated in [5]. This paper is a sequel to [9]. Its goal is to verify that the virtual top Chern class c1/r in the Chow group of the moduli space of higher spin curves M 1/r g,n, constructed in [9], satisfies all the axioms of spin virtual class formulated in [5]. Hence, according to [5], it gives rise to a cohomological field theory in the sense of KontsevichManin [7]. As was observed in [9], the only nontrivial axioms that have to be checked for the class c1/r are two axioms that we call Vanishing axiom and Ramond factorization axiom. The first of them requires c 1/r to vanish on all the components of the moduli space M 1/r g,n, where one of the markings is equal to r − 1. The second demands vanishing of the pushforward of c 1/r restricted to the components of the moduli space corresponding to the so called Ramond sector, under some natural finite maps. Recall that the virtual top Chern class is a crucial ingredient in the generalized Witten’s conjecture formulated in [10], [11]. The original indextheoretic construction of this
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.
Stable twisted curves and their rspin structures
, 2007
"... The object of this paper is the notion of rspin structure: a line bundle whose rth power is isomorphic to the canonical bundle. Over the moduli functor Mg of smooth genusg curves, rspin structures form a finite torsor under the group of rtorsion line bundles. Over the moduli functor Mg of stable ..."
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Cited by 21 (7 self)
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The object of this paper is the notion of rspin structure: a line bundle whose rth power is isomorphic to the canonical bundle. Over the moduli functor Mg of smooth genusg curves, rspin structures form a finite torsor under the group of rtorsion line bundles. Over the moduli functor Mg of stable curves, rspin structures form an étale stack, but the finiteness and the torsor structure are lost. In the present work, we show how this bad picture can be definitely improved simply by placing the problem in the category of Abramovich and Vistoli’s twisted curves. First, we find that within such category there exist several different compactifications of Mg; each one corresponds to a different multiindex ⃗ l = (l0, l1,...) identifying a notion of stability: ⃗ lstability. Then, we determine the suitable choices of ⃗ l for which rspin structures form a finite torsor over the moduli of ⃗ lstable curves.
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.