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60
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.
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
An algebro–geometric proof of Witten’s conjecture
, 2006
"... We present a new proof of Witten’s conjecture. The proof is based on the analysis of the relationship between intersection indices on moduli spaces of complex curves and Hurwitz numbers enumerating ramified coverings of the 2sphere. ..."
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Cited by 28 (0 self)
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We present a new proof of Witten’s conjecture. The proof is based on the analysis of the relationship between intersection indices on moduli spaces of complex curves and Hurwitz numbers enumerating ramified coverings of the 2sphere.
Towards the geometry of double Hurwitz numbers
 Advances Math
"... ABSTRACT. Double Hurwitz numbers count branched covers of CP 1 with fixed branch points, with simple branching required over all but two points 0 and ∞, and the branching over 0 and ∞ points specified by partitions of the degree (with m and n parts respectively). Single Hurwitz numbers (or more usua ..."
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Cited by 25 (5 self)
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ABSTRACT. Double Hurwitz numbers count branched covers of CP 1 with fixed branch points, with simple branching required over all but two points 0 and ∞, and the branching over 0 and ∞ points specified by partitions of the degree (with m and n parts respectively). Single Hurwitz numbers (or more usually, Hurwitz numbers) have a rich structure, explored by many authors in fields as diverse as algebraic geometry, symplectic geometry, combinatorics, representation theory, and mathematical physics. A remarkable formula of Ekedahl, Lando, M. Shapiro, and Vainshtein (the ELSV formula) relates single Hurwitz numbers to intersection theory on the moduli space of curves. This connection has led to many consequences, including Okounkov and Pandharipande’s proof of Witten’s conjecture (Kontsevich’s theorem) connecting intersection theory on the moduli space of curves to integrable systems. In this paper, we determine the structure of double Hurwitz numbers using techniques from geometry, algebra, and representation theory. Our motivation is geometric: we give strong evidence that double Hurwitz numbers are top intersections on a moduli space of curves with a line bundle (a universal Picard variety). In particular, we prove a piecewisepolynomiality
Virasoro constraints for target curves
, 2003
"... We prove generalized Virasoro constraints for the relative GromovWitten theories of all nonsingular target curves. Descendents of the even cohomology classes are studied first by localization, degeneration, and completed cycle methods. Descendents of the odd cohomology are then controlled by monodr ..."
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Cited by 20 (6 self)
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We prove generalized Virasoro constraints for the relative GromovWitten theories of all nonsingular target curves. Descendents of the even cohomology classes are studied first by localization, degeneration, and completed cycle methods. Descendents of the odd cohomology are then controlled by monodromy and geometric vanishing relations. As an outcome of our results, the relative theories of target curves are
Generating functions for intersection numbers on moduli spaces of curves
, 2000
"... Using the connection between intersection theory on the DeligneMumford spaces Mg,n and the edge scaling of the GUE matrix model (see [12, 14]), we express the npoint functions for the intersection numbers as ndimensional errorfunctiontype integrals and also give a derivation of Witten’s KdV equ ..."
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Cited by 18 (4 self)
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Using the connection between intersection theory on the DeligneMumford spaces Mg,n and the edge scaling of the GUE matrix model (see [12, 14]), we express the npoint functions for the intersection numbers as ndimensional errorfunctiontype integrals and also give a derivation of Witten’s KdV equations using the higher Fay identities of Adler, Shiota, and van Moerbeke.
Phase transitions, double–scaling limit, and topological strings
, 2007
"... Topological strings on Calabi–Yau manifolds are known to undergo phase transitions at small distances. We study this issue in the case of perturbative topological strings on local Calabi–Yau threefolds given by a bundle over a twosphere. This theory can be regarded as a q–deformation of Hurwitz the ..."
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Cited by 17 (5 self)
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Topological strings on Calabi–Yau manifolds are known to undergo phase transitions at small distances. We study this issue in the case of perturbative topological strings on local Calabi–Yau threefolds given by a bundle over a twosphere. This theory can be regarded as a q–deformation of Hurwitz theory, and it has a conjectural nonperturbative description in terms of q–deformed 2d Yang–Mills theory. We solve the planar model and find a phase transition at small radius in the universality class of 2d gravity. We give strong evidence that there is a double–scaled theory at the critical point whose all genus free energy is governed by the Painlevé I equation. We compare the critical behavior of the perturbative theory to the critical behavior of its nonperturbative description, which belongs to the universality class of 2d supergravity, and we comment on possible implications for nonperturbative 2d gravity. We also give evidence for a new open/closed duality relating these Calabi–Yau backgrounds to open strings with framing.