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49
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.
Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and GromovWitten invariants
, 2001
"... We present a project of classification of a certain class of bihamiltonian 1+1 PDEs depending on a small parameter. Our aim is to embed the theory of Gromov Witten invariants of all genera into the theory of integrable systems. The project is focused at describing normal forms of the PDEs and their ..."
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Cited by 45 (2 self)
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We present a project of classification of a certain class of bihamiltonian 1+1 PDEs depending on a small parameter. Our aim is to embed the theory of Gromov Witten invariants of all genera into the theory of integrable systems. The project is focused at describing normal forms of the PDEs and their local bihamiltonian structures satisfying certain simple axioms. A Frobenius manifold or its degeneration is associated to every bihamiltonian structure of our type. The main result is a universal loop equation on the jet space of a semisimple Frobenius manifold that can be used for perturbative reconstruction of the integrable hierarchy. We show that first few terms of the perturbative expansion correctly reproduce the universal identities between intersection numbers of Gromov Witten classes and their descendents.
New moduli spaces of pointed curves and pencils of flat connections
 Fulton’s Festschrift), 2000, 443–472. Preprint math.AG/0001003
"... Abstract. It is well known that formal solutions to the Associativity Equations are the same as cyclic algebras over the homology operad (H∗(M0,n+1)) of the moduli spaces of n–pointed stable curves of genus zero. In this paper we establish a similar relationship between the pencils of formal flat co ..."
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Cited by 27 (3 self)
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Abstract. It is well known that formal solutions to the Associativity Equations are the same as cyclic algebras over the homology operad (H∗(M0,n+1)) of the moduli spaces of n–pointed stable curves of genus zero. In this paper we establish a similar relationship between the pencils of formal flat connections (or solutions to the Commutativity Equations) and homology of a new series Ln of pointed stable curves of genus zero. Whereas M0,n+1 parametrizes trees of P 1 ’s with pairwise distinct nonsingular marked points, Ln parametrizes strings of P 1 ’s stabilized by marked points of two types. The union of all Ln’s forms a semigroup rather than operad, and the role of operadic algebras is taken over by the representations of the appropriately twisted homology algebra of this union. 0. Introduction and plan of the paper One of the remarkable basic results in the theory of the Associativity Equations (or Frobenius manifolds) is the fact that their formal solutions are the same as cyclic algebras over the homology operad (H∗(M0,n+1)) of the moduli spaces of n– pointed stable curves of genus zero. This connection was discovered by physicists,
Stability conditions on a noncompact CalabiYau threefold
"... Abstract. We study the space of stability conditions on the noncompact CalabiYau threefold X which is the total space of the canonical bundle of P 2. We give a combinatorial description of an open subset of Stab(X) and state a conjecture relating Stab(X) to the Frobenius manifold obtained from the ..."
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Cited by 26 (1 self)
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Abstract. We study the space of stability conditions on the noncompact CalabiYau threefold X which is the total space of the canonical bundle of P 2. We give a combinatorial description of an open subset of Stab(X) and state a conjecture relating Stab(X) to the Frobenius manifold obtained from the quantum cohomology of P 2. We give some evidence from mirror symmetry for this conjecture. 1.
The discrete and continuous Painlevé VI hierarchy and the Garnier systems
 Glasgow Math. J. 43A
"... We present a general scheme to derive higherorder members of the Painlevé VI (PVI) hierarchy of ODE’s as well as their difference analogues. The derivation is based on a discrete structure that sits on the background of the PVI equation and that consists of a system of partial difference equations ..."
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Cited by 22 (6 self)
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We present a general scheme to derive higherorder members of the Painlevé VI (PVI) hierarchy of ODE’s as well as their difference analogues. The derivation is based on a discrete structure that sits on the background of the PVI equation and that consists of a system of partial difference equations on a multidimensional lattice. The connection with the isomonodromic Garnier systems is discussed.
AN INTEGRAL STRUCTURE IN QUANTUM COHOMOLOGY AND MIRROR SYMMETRY FOR TORIC ORBIFOLDS
, 2009
"... We introduce an integral structure in orbifold quantum cohomology associated to the Kgroup and the b Γclass. In the case of compact toric orbifolds, we show that this integral structure matches with the natural integral structure for the LandauGinzburg model under mirror symmetry. By assuming the ..."
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Cited by 18 (1 self)
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We introduce an integral structure in orbifold quantum cohomology associated to the Kgroup and the b Γclass. In the case of compact toric orbifolds, we show that this integral structure matches with the natural integral structure for the LandauGinzburg model under mirror symmetry. By assuming the existence of an integral structure, we give a natural explanation for the specialization to a root of unity in Y. Ruan’s crepant resolution conjecture [66].
Spaces of stability conditions
"... Abstract. Stability conditions are a mathematical way to understand Πstability for Dbranes in string theory. Spaces of stability conditions seem to be related to moduli spaces of conformal field theories. This is a survey article describing what is currently known about spaces of stability conditi ..."
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Cited by 17 (3 self)
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Abstract. Stability conditions are a mathematical way to understand Πstability for Dbranes in string theory. Spaces of stability conditions seem to be related to moduli spaces of conformal field theories. This is a survey article describing what is currently known about spaces of stability conditions, and giving some pointers for future research. 1.
On a Poisson structure on the space of Stokes matrices
 Internat. Math. Res. Notices 1999
"... Abstract: In this paper we study the map associating to a linear differential operator with rational coefficients its monodromy data. The operator is of the form Λ(z) = d V dz − U − z, with one regular and one irregular singularity of Poincaré rank 1, where U is a diagonal and V is a skewsymmetric ..."
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Cited by 8 (0 self)
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Abstract: In this paper we study the map associating to a linear differential operator with rational coefficients its monodromy data. The operator is of the form Λ(z) = d V dz − U − z, with one regular and one irregular singularity of Poincaré rank 1, where U is a diagonal and V is a skewsymmetric n × n matrix. We compute the Poisson structure of the corresponding Monodromy Preserving Deformation Equations (MPDE) on the space of the monodromy data. Preprint SISSA 120/98/FM Monodromy preserving deformation equations (MPDE) of linear differential operators with rational coefficients are known since the beginning of the century [Fu, Schl, G]. Particularly, the famous six Painlevé equations are known [G] to be of this type. MPDE were included in the framework of the general theory of integrable systems much later, at the end of 70s [ARS, FN1, JMU]; see also [IN]). Many authors were
Symmetries and solutions of Getzler’s equation for Coxeter and extended affine Weyl Frobenius manifolds
"... Abstract. The Gfunction associated to the semisimple Frobenius manifold C n /W (where W is a Coxeter group or an extended affine Weyl group) is studied. The general form of the G function is given in terms of a logarithmic singularity over caustics in the manifold. The main result in this paper is ..."
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Cited by 7 (1 self)
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Abstract. The Gfunction associated to the semisimple Frobenius manifold C n /W (where W is a Coxeter group or an extended affine Weyl group) is studied. The general form of the G function is given in terms of a logarithmic singularity over caustics in the manifold. The main result in this paper is a universal formula for the Gfunction corresponding to the Frobenius manifold C n / ˜ W (k) (An−1) , where ˜ W (k) (An−1) is a certain extended affine Weyl group (or, equivalently, corresponding to the Hurwitz space ˆ M0;k−1,n−k−1), together with the general form of the Gfunction in terms of data on caustics. Symmetries of the G function are also studied.