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107
Three questions in GromovWitten theory
 Proceedings of the ICM (Beijing 2002), Vol II
"... Three conjectural directions in GromovWitten theory are discussed: Gorenstein properties, BPS states, and Virasoro constraints. Each points to basic structures in the subject which are not yet understood. ..."
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Cited by 51 (12 self)
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Three conjectural directions in GromovWitten theory are discussed: Gorenstein properties, BPS states, and Virasoro constraints. Each points to basic structures in the subject which are not yet understood.
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 44 (6 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 38 (9 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
The local GromovWitten theory of curves
, 2008
"... We study the equivariant GromovWitten theory of a rank 2 vector bundle N over a nonsingular curve X of genus g: (i) We define a TQFT using the GromovWitten partition functions. The full theory is determined in the TQFT formalism from a few exact calculations. We use a reconstruction result proven ..."
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Cited by 38 (10 self)
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We study the equivariant GromovWitten theory of a rank 2 vector bundle N over a nonsingular curve X of genus g: (i) We define a TQFT using the GromovWitten partition functions. The full theory is determined in the TQFT formalism from a few exact calculations. We use a reconstruction result proven jointly with C. Faber and A. Okounkov in the appendix.
Hodge integrals and invariants of the unknots
"... We prove the GopakumarMariñoVafa formula for special cubic Hodge integrals. The GMV formula arises from ChernSimons/string duality applied to the unknot in the three sphere. The GMV formula is a qanalog of the ELSV formula for linear Hodge integrals. We find a system of bilinear localization equ ..."
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Cited by 31 (4 self)
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We prove the GopakumarMariñoVafa formula for special cubic Hodge integrals. The GMV formula arises from ChernSimons/string duality applied to the unknot in the three sphere. The GMV formula is a qanalog of the ELSV formula for linear Hodge integrals. We find a system of bilinear localization equations relating linear and special cubic Hodge integrals. The GMV formula then follows easily from the ELSV formula. An operator form of the GMV formula is presented in the last section of the paper.
All orders asymptotic expansion of large partitions
, 2008
"... The generating function which counts partitions with the Plancherel measure (and its qdeformed version), can be rewritten as a matrix integral, which allows to compute its asymptotic expansion to all orders. There are applications in statistical physics of growing/melting crystals, T.A.S.E.P., and ..."
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Cited by 28 (6 self)
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The generating function which counts partitions with the Plancherel measure (and its qdeformed version), can be rewritten as a matrix integral, which allows to compute its asymptotic expansion to all orders. There are applications in statistical physics of growing/melting crystals, T.A.S.E.P., and also in algebraic geometry. In particular we compute the GromovWitten invariants of the Xp = O(p − 2) ⊕ O(−p) → P1 CalabiYau 3fold, and we prove a conjecture of M. Mariño, that the generating functions Fg of Gromov–Witten invariants of Xp, come from a matrix model, and are the symplectic invariants of the mirror spectral curve.
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 27 (7 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.
Melting crystal, quantum torus and Toda hierarchy, arXiv:0701.5339 [hepth
"... Searching for the integrable structures of supersymmetric gauge theories and topological strings, we study melting crystal, which is known as random plane partition, from the viewpoint of integrable systems. We show that a series of partition functions of melting crystals gives rise to a tau functio ..."
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Cited by 24 (15 self)
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Searching for the integrable structures of supersymmetric gauge theories and topological strings, we study melting crystal, which is known as random plane partition, from the viewpoint of integrable systems. We show that a series of partition functions of melting crystals gives rise to a tau function of the onedimensional Toda hierarchy, where the models are defined by adding suitable potentials, endowed with a series of coupling constants, to the standard statistical weight. These potentials can be converted to a commutative subalgebra of quantum torus Lie algebra. This perspective reveals a remarkable connection between random plane partition and quantum torus Lie algebra, and substantially enables to prove the statement. Based on the result, we briefly argue the integrable structures of fivedimensional N = 1 supersymmetric gauge theories and Amodel topological strings. The aforementioned potentials correspond to gauge theory observables analogous to the Wilson loops, and thereby the partition functions are translated in the gauge theory to generating functions of their correlators. In topological strings, we particularly comment on a possibility of topology change caused by condensation of these observables, giving a simple example.
Noncommutative twodimensional topological field theories and Hurwitz numbers for real algebraic curves
 math.GT/0202164. 67 [BK1] [BK2] [C1] [C2] [C3] [C4] [DLM
, 2006
"... It is wellknown that classical twodimensional topological field theories are in onetoone correspondence with commutative Frobenius algebras. An important extension of classical twodimensional topological field theories is provided by openclosed twodimensional topological field theories. In th ..."
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Cited by 21 (3 self)
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It is wellknown that classical twodimensional topological field theories are in onetoone correspondence with commutative Frobenius algebras. An important extension of classical twodimensional topological field theories is provided by openclosed twodimensional topological field theories. In this paper we extend openclosed twodimensional topological field theories to nonorientable surfaces. We call them Klein topological field theories (KTFT). We prove that KTFTs bijectively correspond to algebras with certain additional structures, called structure algebras. Semisimple structure algebras are classified. Starting from an arbitrary finite group, we construct a structure algebra and prove that it is semisimple. We define an analog of Hurwitz numbers for real algebraic curves and prove that they are correlators of a KTFT. The structure algebra of this KTFT is the structure algebra of the symmetric group.