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93
GromovWitten invariants and quantization of quadratic Hamiltonians
, 2001
"... We describe a formalism based on quantization of quadratic hamiltonians and symplectic actions of loop groups which provides a convenient home for most of known general results and conjectures about GromovWitten invariants of compact symplectic manifolds and, more generally, Frobenius structures at ..."
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Cited by 86 (3 self)
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We describe a formalism based on quantization of quadratic hamiltonians and symplectic actions of loop groups which provides a convenient home for most of known general results and conjectures about GromovWitten invariants of compact symplectic manifolds and, more generally, Frobenius structures at higher genus. We state several results illustrating the formalism and its use. In particular, we establish Virasoro constraints for semisimple Frobenius structures and outline a proof of the Virasoro conjecture for Gromov – Witten invariants of complex projective spaces and other Fano toric manifolds. Details will be published elsewhere.
Mirror principle
 I. Asian J. Math
, 1997
"... Abstract. We propose and study the following Mirror Principle: certain sequences of multiplicative equivariant characteristic classes on Kontsevich’s stable map moduli spaces can be computed in terms of certain hypergeometric type classes. As applications, we compute the equivariant Euler classes of ..."
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Cited by 77 (9 self)
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Abstract. We propose and study the following Mirror Principle: certain sequences of multiplicative equivariant characteristic classes on Kontsevich’s stable map moduli spaces can be computed in terms of certain hypergeometric type classes. As applications, we compute the equivariant Euler classes of obstruction bundles induced by any concavex bundles – including any direct sum of line bundles – on Pn. This includes proving the formula of Candelasde la OssaGreenParkes hence completing the program of Candelas et al, Kontesevich, Manin, and Givental, to compute rigorously the instanton prepotential function for the quintic in P4. We derive, among many other examples, the multiple cover formula for GromovWitten invariants of P1, computed earlier by MorrisonAspinwall and by Manin in different approaches. We also prove a formula for enumerating Euler classes which arise in the socalled local mirror symmetry for some noncompact CalabiYau manifolds. At the end we interprete an infinite dimensional transformation group, called the mirror group, acting on Euler data, as a certain duality group of the linear sigma
Mirror symmetry for weighted projective planes and their noncommutative deformations
, 2004
"... ..."
Dirichlet branes, homological mirror symmetry, and stability
 Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), 395–408, Higher Ed
, 2002
"... We discuss some mathematical conjectures which have come out of the study of Dirichlet branes in superstring theory, focusing on the case of supersymmetric branes in CalabiYau compactification. This has led to the formulation of a notion of stability for objects in a derived category, contact with ..."
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Cited by 50 (1 self)
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We discuss some mathematical conjectures which have come out of the study of Dirichlet branes in superstring theory, focusing on the case of supersymmetric branes in CalabiYau compactification. This has led to the formulation of a notion of stability for objects in a derived category, contact with Kontsevich’s homological mirror symmetry conjecture, and “physics proofs” for many of the subsequent conjectures based on it, such as the representation of CalabiYau monodromy by autoequivalences of the derived category.
Elliptic GromovWitten invariants and the generalized mirror conjecture
"... A conjecture expressing genus 1 GromovWitten invariants in mirrortheoretic terms of semisimple Frobenius structures and complex oscillating integrals is formulated. The proof of the conjecture is given for torusequivariant Gromov Witten invariants of compact Kähler manifolds with isolated fixed ..."
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Cited by 48 (5 self)
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A conjecture expressing genus 1 GromovWitten invariants in mirrortheoretic terms of semisimple Frobenius structures and complex oscillating integrals is formulated. The proof of the conjecture is given for torusequivariant Gromov Witten invariants of compact Kähler manifolds with isolated fixed points and for concave bundle spaces over such manifolds. Several results on genus 0 Gromov Witten theory include: a nonlinear Serre duality theorem, its application to the genus 0 mirror conjecture, a mirror theorem for concave bundle spaces over toric manifolds generalizing a recent result of B. Lian, K. Liu and S.T. Yau. We also establish a correspondence (see the extensive footnote in section 4) between their new proof of the genus 0 mirror conjecture for quintic 3folds and our proof of the same conjecture given two years ago. Research supported by NSF grants DMS9321915 and DMS9704774
Conifold transitions and mirror symmetry for complete intersections in Grassmannians
 IN GRASSMANNIANS, PREPRINT
, 1997
"... In this paper we show that conifold transitions between CalabiYau 3folds can be used for the construction of mirror manifolds and for the computation of the instanton numbers of rational curves on complete intersection CalabiYau 3folds in Grassmannians. Using a natural degeneration of Grassmanni ..."
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Cited by 35 (8 self)
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In this paper we show that conifold transitions between CalabiYau 3folds can be used for the construction of mirror manifolds and for the computation of the instanton numbers of rational curves on complete intersection CalabiYau 3folds in Grassmannians. Using a natural degeneration of Grassmannians G(k,n) to some Gorenstein toric Fano varieties P(k,n) with conifolds singularities which was recently described by Sturmfels, we suggest an explicit mirror construction for CalabiYau complete intersections X ⊂ G(k,n) of arbitrary dimension. Our mirror construction is consistent with the formula for the Lax operator conjectured by Eguchi, Hori and Xiong for gravitational
Symplectic geometry of Frobenius structures
"... The concept of a Frobenius manifold was introduced by B. Dubrovin [9] to capture in an axiomatic form the properties of correlators found by physicists (see [8]) in twodimensional topological field theories “coupled to gravity at the tree level”. The purpose of these notes is to reiterate and expan ..."
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Cited by 30 (1 self)
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The concept of a Frobenius manifold was introduced by B. Dubrovin [9] to capture in an axiomatic form the properties of correlators found by physicists (see [8]) in twodimensional topological field theories “coupled to gravity at the tree level”. The purpose of these notes is to reiterate and expand the viewpoint, outlined in the paper [7] of T. Coates and the author, which recasts this concept in terms of linear symplectic geometry and exposes the role of the twisted loop group L (2) GLN of hidden symmetries. We try to keep the text introductory and nontechnical. In particular, we supply details of some simple results from the axiomatic theory, including a severalline proof of the genus 0 Virasoro constraints not mentioned elsewhere, but merely quote and refer to the literature for a number of less trivial applications, such as the quantum Hirzebruch–Riemann–Roch theorem in the theory of cobordismvalued Gromov–Witten invariants. The latter is our joint work in progress with Tom Coates, and we would like to thank him for numerous discussions of the subject.
The Quantum Orbifold Cohomology of Weighted Projective Spaces
, 2007
"... We calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces. We generalize Givental’s heuristic argument, which relates small quantum cohomology to S 1equivariant Floer cohomology of loop space, to weighted projective spaces and use this to conjecture an explicit for ..."
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Cited by 29 (11 self)
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We calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces. We generalize Givental’s heuristic argument, which relates small quantum cohomology to S 1equivariant Floer cohomology of loop space, to weighted projective spaces and use this to conjecture an explicit formula for the small Jfunction, a generating function for certain genuszero Gromov–Witten invariants. We prove this conjecture using a method due to Bertram. This provides the first nontrivial example of a family of orbifolds of arbitrary dimension for which the small quantum orbifold cohomology is known. In addition we obtain formulas for the small Jfunctions of weighted projective complete intersections satisfying a combinatorial condition; this condition