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132
Feynman diagrams and lowdimensional topology
, 2006
"... We shall describe a program here relating Feynman diagrams, topology of manifolds, homotopical algebra, noncommutative geometry and several kinds of “topological physics”. The text below consists of 3 parts. The first two parts (topological sigma model and ChernSimons theory) are formally independ ..."
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Cited by 237 (3 self)
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We shall describe a program here relating Feynman diagrams, topology of manifolds, homotopical algebra, noncommutative geometry and several kinds of “topological physics”. The text below consists of 3 parts. The first two parts (topological sigma model and ChernSimons theory) are formally independent and could be read separately. The third part describes the common algebraic background of both theories.
Euler characteristics of moduli spaces of curves
, 2008
"... Let M n g be the moduli space of npointed Riemann surfaces of genus g. Denote by M n g the DeligneMumford compactification of M n g. In the present paper, we calculate the orbifold and the ordinary Euler characteristic of M n g for any g and n such that n> 2 − 2g. ..."
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Cited by 207 (2 self)
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Let M n g be the moduli space of npointed Riemann surfaces of genus g. Denote by M n g the DeligneMumford compactification of M n g. In the present paper, we calculate the orbifold and the ordinary Euler characteristic of M n g for any g and n such that n> 2 − 2g.
Semiclassical asymptotics of orthogonal polynomials, RiemannHilbert problem, and universality in the matrix model
, 1999
"... We derive semiclassical asymptotics for the orthogonal polynomials Pn(z) on the line with respect to the exponential weight exp(−NV (z)), where V (z) is a doublewell quartic polynomial, in the limit when n,N → ∞. We assume that ε ≤ (n/N) ≤ λcr − ε for some ε> 0, where λcr is the critical value ..."
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Cited by 174 (12 self)
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We derive semiclassical asymptotics for the orthogonal polynomials Pn(z) on the line with respect to the exponential weight exp(−NV (z)), where V (z) is a doublewell quartic polynomial, in the limit when n,N → ∞. We assume that ε ≤ (n/N) ≤ λcr − ε for some ε> 0, where λcr is the critical value which separates orthogonal polynomials with two cuts from the ones with one cut. Simultaneously we derive semiclassical asymptotics for the recursive coefficients of the orthogonal polynomials, and we show that these coefficients form a cycle of period two which drifts slowly with the change of the ratio n/N. The proof of the semiclassical asymptotics is based on the methods of the theory of integrable systems and on the analysis of the appropriate matrix RiemannHilbert problem. As an application of the semiclassical asymptotics of the orthogonal polynomials, we prove the universality of the local distribution of eigenvalues in the matrix model with the doublewell quartic interaction in the presence of two cuts.
Noncommutative geometry, quantum fields and motives
 Colloquium Publications, Vol.55, American Mathematical Society
, 2008
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Matrix Model as a Mirror of ChernSimons Theory
, 2002
"... Using mirror symmetry, we show that ChernSimons theory on certain manifolds such as lens spaces reduces to a novel class of Hermitian matrix models, where the measure is that of unitary matrix models. We show that this agrees with the more conventional canonical quantization of ChernSimons theory. ..."
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Cited by 134 (17 self)
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Using mirror symmetry, we show that ChernSimons theory on certain manifolds such as lens spaces reduces to a novel class of Hermitian matrix models, where the measure is that of unitary matrix models. We show that this agrees with the more conventional canonical quantization of ChernSimons theory. Moreover, large N dualities in this context lead to computation of all genus Amodel topological amplitudes on toric CalabiYau manifolds in terms of matrix integrals. In the context of type IIA superstring compactifications on these CalabiYau manifolds with wrapped D6 branes (which are dual to Mtheory on G2 manifolds) this leads to engineering and solving Fterms for N = 1 supersymmetric gauge theories with superpotentials involving certain multitrace operators
Modular Operads
 COMPOSITIO MATH
, 1994
"... We develop a "higher genus" analogue of operads, which we call modular operads, in which graphs replace trees in the definition. We study a functor F on the category of modular operads, the Feynman transform, which generalizes Kontsevich's graph complexes and also the bar constructi ..."
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Cited by 103 (5 self)
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We develop a "higher genus" analogue of operads, which we call modular operads, in which graphs replace trees in the definition. We study a functor F on the category of modular operads, the Feynman transform, which generalizes Kontsevich's graph complexes and also the bar construction for operads. We calculate the Euler characteristic of the Feynman transform, using the theory of symmetric functions: our formula is modelled on Wick's theorem. We give applications to the theory of moduli spaces of pointed algebraic curves.
Double scaling limit in the random matrix model: the RiemannHilbert approach
"... Abstract. We derive the double scaling limit of eigenvalue correlations in the random matrix model at critical points and we relate it to a nonlinear hierarchy of ordinary differential equations. 1. ..."
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Cited by 86 (11 self)
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Abstract. We derive the double scaling limit of eigenvalue correlations in the random matrix model at critical points and we relate it to a nonlinear hierarchy of ordinary differential equations. 1.
Asymptotics of the partition function for random matrices via RiemannHilbert techniques, and applications to graphical enumeration
 Internat. Math. Research Notices
, 2003
"... Abstract. We study the partition function from random matrix theory using a well known connection to orthogonal polynomials, and a recently developed RiemannHilbert approach to the computation of detailed asymptotics for these orthogonal polynomials. We obtain the first proof of a complete large N ..."
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Cited by 72 (9 self)
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Abstract. We study the partition function from random matrix theory using a well known connection to orthogonal polynomials, and a recently developed RiemannHilbert approach to the computation of detailed asymptotics for these orthogonal polynomials. We obtain the first proof of a complete large N expansion for the partition function, for a general class of probability measures on matrices, originally conjectured by Bessis, Itzykson, and Zuber. We prove that the coefficients in the asymptotic expansion are analytic functions of parameters in the original probability measure, and that they are generating functions for the enumeration of labelled maps according to genus and valence. Central to the analysis is a large N expansion for the mean density of eigenvalues, uniformly valid on the entire real axis.
The Spectrum of Coupled Random Matrices
, 1999
"... this paper, we will use the following operators e ..."
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Cited by 58 (12 self)
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this paper, we will use the following operators e