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53
Lectures on 2D YangMills Theory, Equivariant Cohomology and Topological Field Theories
, 1996
"... These are expository lectures reviewing (1) recent developments in twodimensional YangMills theory and (2) the construction of topological field theory Lagrangians. Topological field theory is discussed from the point of view of infinitedimensional differential geometry. We emphasize the unifying ..."
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Cited by 99 (7 self)
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These are expository lectures reviewing (1) recent developments in twodimensional YangMills theory and (2) the construction of topological field theory Lagrangians. Topological field theory is discussed from the point of view of infinitedimensional differential geometry. We emphasize the unifying role of equivariant cohomology both as the underlying principle in the formulation of BRST transformation laws and as a central concept in the geometrical interpretation of topological field theory path integrals.
Intersection theory, integrable hierarchies and topological field theory
, 1992
"... In these lecture notes we review the various relations between intersection theory on the moduli space of Riemann surfaces, integrable hierarchies of KdV type, matrix models, and topological field theory. We focus in particular on the question why matrix integrals of the type considered by Kontsevic ..."
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Cited by 93 (5 self)
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In these lecture notes we review the various relations between intersection theory on the moduli space of Riemann surfaces, integrable hierarchies of KdV type, matrix models, and topological field theory. We focus in particular on the question why matrix integrals of the type considered by Kontsevich naturally appear as τfunctions of integrable hierarchies related to topological minimal models.
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.
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.
Loop Equations and Virasoro Constraints in Nonperturbative 2D Quantum Gravity
, 1990
"... We give a derivation of the loop equation for twodimensional gravity from the KdV equations and the string equation of the one matrix model. We find that the loop equation is equivalent to an infinite set of linear constraints on the square root of the partition function satisfying the Virasoro alg ..."
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Cited by 46 (4 self)
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We give a derivation of the loop equation for twodimensional gravity from the KdV equations and the string equation of the one matrix model. We find that the loop equation is equivalent to an infinite set of linear constraints on the square root of the partition function satisfying the Virasoro algebra. We give an interpretation of these equations in topological gravity and discuss their extension to multimatrix models. For the multicritical models the loop equation naturally singles out the operators corresponding to the primary fields of the minimal models. (rhd@pupthy.princeton.edu) y (hlv@pupthy.princeton.edu) z (verlinde@iassns) 1. Introduction Two dimensional quantum gravity can be formulated as a sum over random surfaces. In the matrix model approach to 2d gravity [1, 2] this partition function is defined by an appropriate double scaling limit of the matrix integral Z = Z dOE e \GammaN trV (OE) (1.1) where OE is an N \Theta N hermitean matrix and V (OE) = P n ...
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.
Semisimple Frobenius structures at higher genus
, 2000
"... We describe genus g ≥ 2 potentials of semisimple Frobenius structures. Our formula can be considered as a definition in the axiomatic context of Frobenius manifolds. In GromovWitten theory, it becomes a conjecture expressing higher genus GWinvariants in terms of genus 0 GWinvariants of symplect ..."
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Cited by 42 (4 self)
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We describe genus g ≥ 2 potentials of semisimple Frobenius structures. Our formula can be considered as a definition in the axiomatic context of Frobenius manifolds. In GromovWitten theory, it becomes a conjecture expressing higher genus GWinvariants in terms of genus 0 GWinvariants of symplectic manifolds with generically semisimple quantum cupproduct. The conjecture is supported by the corresponding theorem about equivariant GWinvariants of tori actions with isolated fixed points. The parallel theory of gravitational descendents is also presented.
Moduli spaces of higher spin curves and integrable hierarchies
 Compositio Math
"... Abstract. We prove the genus zero part of the generalized Witten conjecture relating moduli spaces of spin curves to GelfandDickey hierarchies. That is, we show that intersection numbers on the moduli space of stable rspin curves assemble into a generating function which yields a solution of the s ..."
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Cited by 38 (8 self)
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Abstract. We prove the genus zero part of the generalized Witten conjecture relating moduli spaces of spin curves to GelfandDickey hierarchies. That is, we show that intersection numbers on the moduli space of stable rspin curves assemble into a generating function which yields a solution of the semiclassical limit of the KdVr equations. We formulate axioms for a cohomology class on this moduli space which allow one to construct a cohomological field theory of rank r −1 in all genera. In genus zero it produces a Frobenius manifold which is isomorphic to the Frobenius manifold structure on the base of the versal deformation of the singularity Ar−1. We prove analogs of the puncture, dilaton, and topological recursion relations by drawing an analogy with the construction of GromovWitten invariants and quantum cohomology. The moduli space of stable curves of genus g with n marked points Mg,n is a fascinating object. Mumford [37] introduced tautological cohomology classes associated to the universal curve Cg,n
Geometry and analytic theory of Frobenius manifolds
, 1998
"... Main mathematical applications of Frobenius manifolds are in the theory of Gromov Witten invariants, in singularity theory, in differential geometry of the orbit spaces of reflection groups and of their extensions, in the hamiltonian theory of integrable hierarchies. The theory of Frobenius manifol ..."
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Cited by 37 (3 self)
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Main mathematical applications of Frobenius manifolds are in the theory of Gromov Witten invariants, in singularity theory, in differential geometry of the orbit spaces of reflection groups and of their extensions, in the hamiltonian theory of integrable hierarchies. The theory of Frobenius manifolds establishes remarkable relationships between these, sometimes rather distant, mathematical theories.