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125
Symplectic surgery and GromovWitten invariants of CalabiYau 3folds
 I, Invent. Math
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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 139 (5 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
 I. ASIAN J. MATH
, 1997
"... 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 obstruc ..."
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Cited by 126 (13 self)
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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
Stationary phase integrals, quantum Toda lattices, flag manifolds and the mirror conjecture
, 1996
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Semisimple Frobenius structures at higher genus
, 2000
"... We describe genus g ≥ 2 potentials of semisimple Frobenius structures. Our formula can be considered as a definition in the axiomatic context of Frobenius manifolds. In GromovWitten theory, it becomes a conjecture expressing higher genus GWinvariants in terms of genus 0 GWinvariants of symplect ..."
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Cited by 74 (4 self)
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We describe genus g ≥ 2 potentials of semisimple Frobenius structures. Our formula can be considered as a definition in the axiomatic context of Frobenius manifolds. In GromovWitten theory, it becomes a conjecture expressing higher genus GWinvariants in terms of genus 0 GWinvariants of symplectic manifolds with generically semisimple quantum cupproduct. The conjecture is supported by the corresponding theorem about equivariant GWinvariants of tori actions with isolated fixed points. The parallel theory of gravitational descendents is also presented.
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 66 (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
Instanton counting via affine Lie algebras I: Equivariant . . .
, 2004
"... Let g be a simple complex Lie algebra, G the corresponding simply connected group; let also gaff be the corresponding untwisted affine Lie algebra. For a parabolic subgroup P ⊂ G we introduce a generating function Z aff G,P which roughly speaking counts framed Gbundles on P 2 endowed with a Pst ..."
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Cited by 63 (6 self)
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Let g be a simple complex Lie algebra, G the corresponding simply connected group; let also gaff be the corresponding untwisted affine Lie algebra. For a parabolic subgroup P ⊂ G we introduce a generating function Z aff G,P which roughly speaking counts framed Gbundles on P 2 endowed with a Pstructure on the horizontal line (the formal definition uses the corresponding Uhlenbeck type compactifications studied in [3]). In the case P = G the function Z aff G,P coincides with Nekrasov’s partition function introduced in [23] and studied thoroughly in [24] and [22] for G = SL(n). In the ”opposite case ” when P is a Borel subgroup of G we show that Z aff G,P is equal (roughly speaking) to the Whittaker matrix coefficient in the universal Verma module for the Lie algebra ˇgaff – the Langlands dual Lie algebra of ˇg. This clarifies somewhat the connection between certain asymptotic of Z aff G,P (studied in loc. cit. for P = G) and the classical affine Toda system. We also explain why the above result gives rise to a calculation of (not yet rigorously defined) equivariant quantum cohomology ring of the affine flag manifold associated with G. In particular, we reprove the results of [13] and [18] about quantum cohomology of ordinary flag manifolds using methods which are totally different from loc. cit. We shall show in a subsequent publication how this allows one to connect certain asymptotic of the function Z aff G,P with the SeibergWitten prepotential (cf. [2], thus proving the main conjecture of [23] for an arbitrary gauge group G (for G = SL(n) it has been proved in [24] and [22] by other methods.