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Noncommutative matrix integrals and representation varieties of surface groups in a finite group
, 2005
"... A graphical expansion formula for noncommutative matrix integrals with values in a finitedimensional real or complex von Neumann algebra is obtained in terms of ribbon graphs and their nonorientable counterpart called Möbius graphs. The contribution of each graph is an invariant of the topologica ..."
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Cited by 9 (2 self)
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A graphical expansion formula for noncommutative matrix integrals with values in a finitedimensional real or complex von Neumann algebra is obtained in terms of ribbon graphs and their nonorientable counterpart called Möbius graphs. The contribution of each graph is an invariant of the topological type of the surface on which the graph is drawn. As an example, we calculate the integral on the group algebra of a finite group. We show that the integral is a generating function of the number of homomorphisms from the fundamental group of an arbitrary closed surface into the finite group. The graphical expansion formula yields a new proof of the classical theorems of Frobenius, Schur and Mednykh on these numbers.
MEDNYKH’S FORMULA VIA LATTICE TOPOLOGICAL QUANTUM FIELD THEORIES
, 2008
"... Abstract. Mednykh [Me78] proved that for any finite group G and any orientable surface S, there is a formula for #Hom(π1(S), G) in terms of the Euler characteristic of S and the dimensions of the irreducible representations of G. A similar formula in the nonorientable case was proved by Frobenius an ..."
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Abstract. Mednykh [Me78] proved that for any finite group G and any orientable surface S, there is a formula for #Hom(π1(S), G) in terms of the Euler characteristic of S and the dimensions of the irreducible representations of G. A similar formula in the nonorientable case was proved by Frobenius and Schur [FS06]. Both of these proofs use character theory and an explicit presentation for π1. These results have been reproven using quantum field theory ([FQ93], [MY05], and others). Here we present a greatly simplified proof of these results which uses only elementary topology and combinatorics. The main tool is an elementary invariant of surfaces attached to a semisimple algebra called a lattice topological quantum field theory. 1.
QUANTUM CURVES FOR SIMPLE HURWITZ NUMBERS OF AN ARBITRARY BASE CURVE
"... Abstract. The generating functions of simple Hurwitz numbers of the projective line are known to satisfy many properties. They include a heat equation, the EynardOrantin topological recursion, an infiniteorder differential equation called a quantum curve equation, and a Schrödinger like partial ..."
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Abstract. The generating functions of simple Hurwitz numbers of the projective line are known to satisfy many properties. They include a heat equation, the EynardOrantin topological recursion, an infiniteorder differential equation called a quantum curve equation, and a Schrödinger like partial differential equation. In this paper we generalize these properties to simple Hurwitz numbers with an arbitrary base curve. Contents
VOLUME OF REPRESENTATION VARIETIES
, 2002
"... Abstract. We introduce the notion of volume of the representation variety of a finitely presented discrete group in a compact Lie group using the pushforward measure associated to a map defined by a presentation of the discrete group. We show that the volume thus defined is invariant under the Andr ..."
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Abstract. We introduce the notion of volume of the representation variety of a finitely presented discrete group in a compact Lie group using the pushforward measure associated to a map defined by a presentation of the discrete group. We show that the volume thus defined is invariant under the AndrewsCurtis moves of the generators and relators of the discrete group, and moreover, that it is actually independent of the choice of presentation if the difference of the number of generators and the number of relators remains the same. We then calculate the volume of the representation variety of a surface group in an arbitrary compact Lie group using the classical technique of Frobenius and Schur on finite groups. Our formulas recover the results of Witten and Liu on the symplectic volume and the Reidemeister torsion of the moduli space of flat Gconnections on a
GEOMETRY OF CHARACTER VARIETIES OF SURFACE GROUPS
"... Abstract. This article is based on a talk delivered at the RIMS–OCAMI Joint International ..."
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Abstract. This article is based on a talk delivered at the RIMS–OCAMI Joint International
GEOMETRY OF CHARACTER VARIETIES OF SURFACE GROUPS
, 2008
"... This article is based on a talk delivered at the RIMS–OCAMI Joint International ..."
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This article is based on a talk delivered at the RIMS–OCAMI Joint International
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"... Abstract. A graphical expansion formula for noncommutative matrix integrals with values in a finitedimensional real or complex von Neumann algebra is obtained in terms of ribbon graphs and their nonorientable counterpart called Möbius graphs. The contribution of each graph is an invariant of the ..."
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Abstract. A graphical expansion formula for noncommutative matrix integrals with values in a finitedimensional real or complex von Neumann algebra is obtained in terms of ribbon graphs and their nonorientable counterpart called Möbius graphs. The contribution of each graph is an invariant of the topological type of the surface on which the graph is drawn. As an example, we calculate the integral on the group algebra of a finite group. We show that the integral is a generating function of the number of homomorphisms from the fundamental group of an arbitrary closed surface into the finite group. The graphical expansion formula yields a new proof of the classical theorems of Frobenius, Schur and Mednykh on these numbers.