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44
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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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.
Yefeng: LandauGinzburg/CalabiYau Correspondence of all Genera for Elliptic Orbifold P1
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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
On deformations of quasiMiura transformations and the DubrovinZhang bracket
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A proof of the Faber intersection number conjecture
"... Abstract. We prove the Faber intersection number conjecture and other more general results by using a recursion formula of npoint functions for intersection numbers on moduli spaces of curves. We also present several conjectural properties of GromovWitten invariants generalizing results on interse ..."
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Cited by 9 (3 self)
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Abstract. We prove the Faber intersection number conjecture and other more general results by using a recursion formula of npoint functions for intersection numbers on moduli spaces of curves. We also present several conjectural properties of GromovWitten invariants generalizing results on intersection numbers. 1.
GromovWitten theory of étale gerbes I: root gerbes
, 2009
"... Let X be a smooth complex projective algebraic variety. Given a line bundle L over X and an integer r> 1 we study the GromovWitten theory of the stack rp L/X of rth root of L. We prove an exact formula expressing genus 0 GromovWitten invariants of rp L/X in terms of those of X. Assuming that ..."
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Cited by 9 (3 self)
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Let X be a smooth complex projective algebraic variety. Given a line bundle L over X and an integer r> 1 we study the GromovWitten theory of the stack rp L/X of rth root of L. We prove an exact formula expressing genus 0 GromovWitten invariants of rp L/X in terms of those of X. Assuming that either rp L/X or X has semisimple quantum cohomology, we prove an exact formula between higher genus invariants. We also present constructions of moduli stacks of twisted stable maps to rp L/X starting from moduli stack of stable maps to X.
On the Convergence of GromovWitten Potentials and Givental’s Formula. Preprint available at 1203.4193
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THE GROMOVWITTEN POTENTIAL ASSOCIATED TO A TCFT
, 2005
"... Abstract. This is the sequel to my preprint“TCFTs and CalabiYau categories”. Here we extend the results of that paper to construct, for certain CalabiYau A∞ categories, something playing the role of the GromovWitten potential. This is a state in the Fock space associated to periodic cyclic homolo ..."
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Cited by 7 (2 self)
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Abstract. This is the sequel to my preprint“TCFTs and CalabiYau categories”. Here we extend the results of that paper to construct, for certain CalabiYau A∞ categories, something playing the role of the GromovWitten potential. This is a state in the Fock space associated to periodic cyclic homology, which is a symplectic vector space. Applying this to a suitable A ∞ version of the derived category of sheaves on a CalabiYau yields the B model potential, at all genera. The construction doesn’t go via the DeligneMumford spaces, but instead uses the BatalinVilkovisky algebra constructed from the uncompactified moduli spaces of curves by Sen and Zwiebach. The fundamental class of DeligneMumford space is replaced here by a certain solution of the quantum master equation, essentially the “string vertices ” of Zwiebach. On the field theory side, the BV operator has an interpretation as the quantised differential on the Fock space for periodic cyclic chains. Passing to homology, something satisfying the master equation yields an element of the Fock space. 1. Notation We work throughout over a ground field K containing Q. Often we will use topological K vector spaces. All tensor products will be completed. All the topological vector spaces we use are inverse limits, so the completed tensor product is also an inverse limit. All the results remain true without any change if we work over a differential graded ground ring R, and use flat R modules. (An R module is flat if the functor of tensor product with it is exact). We could also have only a Z/2 grading on R. 2. Acknowledgements I would like to thank Tom Coates, Ezra Getzler, Alexander Givental and Paul Seidel for very helpful conversations, and Dennis Sullivan for explaining to me his ideas on the BatalinVilkovisky formalism and moduli spaces of curves. 3. Topological conformal field theories Let S be the topological category whose objects are the nonnegative integers, and whose morphism space S(n,m) is the moduli space of Riemann surfaces with n parameterised incoming and m parameterised outgoing boundaries, such that each connected component has at least one incoming boundary. These surfaces are not necessarily connected. Let Sχ(n,m) ⊂ S(n,m) be the space of surfaces of Euler characteristic χ.
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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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