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93
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 two-dimensional 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 52 (4 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 two-dimensional 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 non-technical. In particular, we supply details of some simple results from the axiomatic theory, including a several-line 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 cobordism-valued 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 extended bigraded Toda hierarchy
- J. Phys. A: Math. Gen
"... Abstract. We generalize the Toda lattice hierarchy by considering N + M dependent variables. We construct roots and logarithms of the Lax operator which are uniquely defined operators with coefficients that are ǫ-series of differential polynomials in the dependent variables, and we use them to provi ..."
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Cited by 32 (4 self)
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Abstract. We generalize the Toda lattice hierarchy by considering N + M dependent variables. We construct roots and logarithms of the Lax operator which are uniquely defined operators with coefficients that are ǫ-series of differential polynomials in the dependent variables, and we use them to provide a Lax pair definition of the extended bigraded Toda hierarchy, generalizing [4]. Using R-matrix theory we give the bihamiltonian formulation of this hierarchy and we prove the existence of a tau function for its solutions. Finally we study the dispersionless limit and its connection with a class of Frobenius manifolds on the orbit space of the extended affine Weyl groups ˜ W (N) (AN+M−1) of the A series, defined in [9].
Higher-genus Gromov-Witten invariants as genus zero invariants of symmetric products
, 2003
"... I prove a formula expressing the descendent genus g Gromov-Witten invariants of a projective variety X in terms of genus 0 invariants of its symmetric product stack S g+1 (X). When X is a point, the latter are structure constants of the symmetric group, and we obtain a new way of calculating the G ..."
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Cited by 22 (0 self)
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I prove a formula expressing the descendent genus g Gromov-Witten invariants of a projective variety X in terms of genus 0 invariants of its symmetric product stack S g+1 (X). When X is a point, the latter are structure constants of the symmetric group, and we obtain a new way of calculating the Gromov-Witten invariants of a point.
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.
Gromov-Witten theory of orbicurves, the space of tri-polynomials and Symplectic Field Theory of
, 808
"... We compute, with Symplectic Field Theory techniques, the Gromov-Witten theory of P1, i.e. the complex projective line with a orbifold points. A natural subclass of α1,...,αa these orbifolds, the ones with polynomial quantum cohomology, gives rise to a family of (polynomial) Frobenius manifolds and i ..."
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Cited by 18 (3 self)
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We compute, with Symplectic Field Theory techniques, the Gromov-Witten theory of P1, i.e. the complex projective line with a orbifold points. A natural subclass of α1,...,αa these orbifolds, the ones with polynomial quantum cohomology, gives rise to a family of (polynomial) Frobenius manifolds and integrable systems of Hamiltonian PDEs, which extend the (dispersionless) bigraded Toda hierarchy ([2]). We then define a Frobenius structure on the spaces of polynomials in three complex variables of the form F(x, y, z) = −xyz + P1(x) + P2(y) + P3(z) which contains as special cases the ones constructed on the space of Laurent polynomials ([5],[13]). We prove a mirror theorem stating that these Frobenius structures are isomorphic to the ones found before for polynomial P1-orbifolds. Finally we link rational Symplectic Field Theory of Seifert fibrations over P1 a,b,c with orbifold Gromov-Witten invariants of the base, extending a known result ([1]) valid in the smooth case.
On Quasitriviality and Integrability of a Class of Scalar Evolutionary PDEs
, 2005
"... For certain class of perturbations of the equation ut = f(u)ux, we prove the existence of change of coordinates, called quasi-Miura transformations, that reduce these perturbed equations to the unperturbed ones. As an application, we propose a criterion for the integrability of these equations. 1 ..."
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Cited by 16 (4 self)
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For certain class of perturbations of the equation ut = f(u)ux, we prove the existence of change of coordinates, called quasi-Miura transformations, that reduce these perturbed equations to the unperturbed ones. As an application, we propose a criterion for the integrability of these equations. 1
An−1 singularities and nKdV hierarchies
"... Abstract. According to a conjecture of E. Witten [18] proved by M. Kontsevich [11], a certain generating function for intersection indices on the Deligne – Mumford moduli spaces of Riemann surfaces coincides with a certain taufunction of the KdV hierarchy. The generating function is naturally genera ..."
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Cited by 16 (1 self)
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Abstract. According to a conjecture of E. Witten [18] proved by M. Kontsevich [11], a certain generating function for intersection indices on the Deligne – Mumford moduli spaces of Riemann surfaces coincides with a certain taufunction of the KdV hierarchy. The generating function is naturally generalized under the name the total descendent potential in the theory of Gromov – Witten invariants of symplectic manifolds. The papers [5, 4] contain two equivalent constructions, motivated by some results in Gromov – Witten theory, which associate a total descendent potential to any semisimple Frobenius structure. In this paper, we prove that in the case of K.Saito’s Frobenius structure [14] on the miniversal deformation of the An−1-singularity, the total descendent potential is a tau-function of the nKdV hierarchy. We derive this result from a more general construction for solutions of the nKdV hierarchy from n − 1 solutions of the KdV hierarchy. 1. Introduction: Singularities and Frobenius
Open topological strings and integrable hierarchies: Remodeling the a-model
, 2011
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