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96
Localization of virtual classes
"... We prove a localization formula for the virtual fundamental class in the general context of C∗equivariant perfect obstruction theories. Let X be an algebraic scheme with a C∗action and a C∗equivariant perfect obstruction theory. The virtual fundamental class [X] vir in ..."
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Cited by 174 (26 self)
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We prove a localization formula for the virtual fundamental class in the general context of C∗equivariant perfect obstruction theories. Let X be an algebraic scheme with a C∗action and a C∗equivariant perfect obstruction theory. The virtual fundamental class [X] vir in
Bihamiltonian Hierarchies in 2D Topological Field Theory At OneLoop Approximation
, 1997
"... We compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2D topological field theory. The bihamiltonian structure of the hierarchy is given by a classical Walgebra; we compute the central charge of this algebra. We also express the generating function of ..."
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Cited by 73 (8 self)
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We compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2D topological field theory. The bihamiltonian structure of the hierarchy is given by a classical Walgebra; we compute the central charge of this algebra. We also express the generating function of elliptic Gromov Witten invariants via taufunction of the isomonodromy deformation problem arising in the theory of WDVV equations of associativity.
Exact vs. semiclassical target space of the minimal string
 JHEP
"... We study both the classical and the quantum target space of (p, q) minimal string theory, using the FZZT brane as a probe. By thinking of the target space as the moduli space of FZZT branes, parametrized by the boundary cosmological constant x, we see that classically it consists of a Riemann surfac ..."
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Cited by 47 (4 self)
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We study both the classical and the quantum target space of (p, q) minimal string theory, using the FZZT brane as a probe. By thinking of the target space as the moduli space of FZZT branes, parametrized by the boundary cosmological constant x, we see that classically it consists of a Riemann surface Mp,q which is a psheeted cover of the complex x plane. However, we show using the dual matrix model that the exact quantum FZZT observables exhibit Stokes ’ phenomenon and are entire functions of x. Along the way we clarify some points about the semiclassical limit of Dbrane correlation functions. The upshot is that nonperturbative effects modify the target space drastically, changing it from Mp,q to the complex x plane. To illustrate these ideas, we study in detail the example of (p, q) = (2, 1), which is dual to the Gaussian matrix model. Here we learn that the other sheets of the classical Riemann surface describe instantons in the effective theory on the brane. Finally, we discuss possible applications to black holes and the topological string. August
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.
The MathaiQuillen Formalism and Topological Field Theory
"... These lecture notes give an introductory account of an approach to cohomological field theory due to Atiyah and Jeffrey which is based on the construction of Gaussian shaped Thom forms by Mathai and Quillen. Topics covered are: an explanation of the MathaiQuillen formalism for finite dimensional ve ..."
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Cited by 28 (2 self)
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These lecture notes give an introductory account of an approach to cohomological field theory due to Atiyah and Jeffrey which is based on the construction of Gaussian shaped Thom forms by Mathai and Quillen. Topics covered are: an explanation of the MathaiQuillen formalism for finite dimensional vector bundles; the definition of regularized Euler numbers of infinite dimensional vector bundles; interpretation of supersymmetric quantum mechanics as the regularized Euler number of loop space; the AtiyahJeffrey interpretation of Donaldson theory; the construction of topological gauge theories from infinite dimensional vector bundles over spaces of connections.
The Partition function of 2D string theory
 Nucl. Phys. B
, 1993
"... We derive a compact and explicit expression for the generating functional of all correlation functions of tachyon operators in 2D string theory. This expression makes manifest relations of the c = 1 system to KP flow and W1+ ∞ constraints. Moreover we derive a KontsevichPenner integral representati ..."
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Cited by 27 (1 self)
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We derive a compact and explicit expression for the generating functional of all correlation functions of tachyon operators in 2D string theory. This expression makes manifest relations of the c = 1 system to KP flow and W1+ ∞ constraints. Moreover we derive a KontsevichPenner integral representation of this generating functional.
2D Gravity and Random Matrices
, 1994
"... We review recent progress in 2D gravity coupled to d < 1 conformal matter, based on a representation of discrete gravity in terms of random matrices. We discuss the saddle point approximation for these models, including a class of related O(n) matrix models. For d < 1 matter, the matrix proble ..."
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Cited by 21 (0 self)
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We review recent progress in 2D gravity coupled to d < 1 conformal matter, based on a representation of discrete gravity in terms of random matrices. We discuss the saddle point approximation for these models, including a class of related O(n) matrix models. For d < 1 matter, the matrix problem can be completely solved in many cases by the introduction of suitable orthogonal polynomials. Alternatively, in the continuum limit the orthogonal polynomial method can be shown to be equivalent to the construction of representations of the canonical commutation relations in terms of differential operators. In the case of pure gravity or discrete Ising–like matter, the sum over topologies is reduced to the solution of nonlinear differential equations (the Painlevé equation in the pure gravity case) which can be shown to follow from an action principle. In the case of pure gravity and more generally all unitary models, the perturbation theory is not Borel summable and therefore alone does not define a unique solution. In the nonBorel summable case, the matrix model does not define the sum over topologies beyond perturbation theory. We also review the computation of correlation functions directly in the continuum formulation of matter coupled to 2D gravity, and compare with the matrix model results. Finally, we
A simple proof of Witten conjecture through localization
"... Abstract. We obtain a system of relations between Hodge integrals with one λclass. As an application, we show that its first nontrivial relation implies the Witten’s Conjecture/Kontsevich Theorem [12, 6]. 1. ..."
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Cited by 18 (6 self)
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Abstract. We obtain a system of relations between Hodge integrals with one λclass. As an application, we show that its first nontrivial relation implies the Witten’s Conjecture/Kontsevich Theorem [12, 6]. 1.