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11
Ribbon graphs, quadratic differentials on Riemann surfaces, and algebraic curves defined over Q, Asian
 Journal of Mathematics
, 1998
"... Abstract. It is well known that there is a bijective correspondence between metric ribbon graphs and compact Riemann surfaces with meromorphic Strebel differentials. In this article, we prove that Grothendieck’s correspondence between dessins d’enfants and Belyi morphisms is a special case of this c ..."
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Cited by 34 (9 self)
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Abstract. It is well known that there is a bijective correspondence between metric ribbon graphs and compact Riemann surfaces with meromorphic Strebel differentials. In this article, we prove that Grothendieck’s correspondence between dessins d’enfants and Belyi morphisms is a special case of this correspondence through an explicit construction of Strebel differentials. For a metric ribbon graph with edge length 1, an algebraic curve over Q and a Strebel differential on it is constructed. It is also shown that the critical trajectories of the measured foliation that is determined by the Strebel differential recover the original metric ribbon graph. Conversely, for every Belyi morphism, a unique Strebel differential is constructed such that the critical leaves of the measured foliation it determines form a metric ribbon graph of edge length 1,
Lectures on the asymptotic expansion of a hermitian matrix integral
 in Supersymmetry and Integrable Models, Henrik Aratin et al., Editors, Springer Lecture Notes in Physics 502
, 1998
"... Abstract. In these lectures three different methods of computing the asymptotic expansion of a Hermitian matrix integral is presented. The first one is a combinatorial method using Feynman diagrams. This leads us to the generating function of the reciprocal of the order of the automorphism group of ..."
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Cited by 10 (7 self)
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Abstract. In these lectures three different methods of computing the asymptotic expansion of a Hermitian matrix integral is presented. The first one is a combinatorial method using Feynman diagrams. This leads us to the generating function of the reciprocal of the order of the automorphism group of a tiling of a Riemann surface. The second method is based on the classical analysis of orthogonal polynomials. A rigorous asymptotic method is established, and a special case of the matrix integral is computed in terms of the Riemann ζfunction. The third method is derived from a formula for the τfunction solution to the KP equations. This method leads us to a new class of solutions of the KP equations that are transcendental, in the sense that theycannot be obtained bythe celebrated Krichever construction and its generalizations based on algebraic geometryof vector bundles on Riemann surfaces. In each case a mathematicallyrigorous wayof dealing with asymptotic series in an infinite number of variables is established. Contents
Duality of orthogonal and symplectic matrix integrals and quaternionic feynman graphs
 Commun. Math. Phys
"... ABSTRACT. We present an asymptotic expansion for quaternionic selfadjoint matrix integrals. The Feynman diagrams appearing in the expansion are ordinary ribbon graphs and their nonorientable counterparts. We show that the 2N × 2N Gaussian Orthogonal Ensemble (GOE) and N × N Gaussian Symplectic Ense ..."
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Cited by 8 (0 self)
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ABSTRACT. We present an asymptotic expansion for quaternionic selfadjoint matrix integrals. The Feynman diagrams appearing in the expansion are ordinary ribbon graphs and their nonorientable counterparts. We show that the 2N × 2N Gaussian Orthogonal Ensemble (GOE) and N × N Gaussian Symplectic Ensemble (GSE) have exactly the same expansion term by term, except that the contributions from graphs on a nonorientable surface with odd Euler characteristic carry the opposite sign. As an application, we give a new topological proof of the known duality for correlations of characteristic polynomials, demonstrating that this duality is equivalent to Poincaré duality of graphs drawn on a compact surface. Another consequence of our graphical expansion formula is a simple and simultaneous (re)derivation of the Central Limit Theorem for GOE, GUE (Gaussian Unitary Ensemble) and GSE: The three cases have exactly the same graphical limiting formula except for an overall
Noncommutative matrix integrals and representation varieties of surface groups in a finite group, Annales de l’Institut Fourier 55
, 2005
"... 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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Cited by 6 (2 self)
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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. The purpose of this paper is to establish Feynman diagram expansion formulas for noncommutative matrix integrals over a finitedimensional real or complex von Neumann algebra. An interesting case is the real or complex group algebra of a finite group. Using the graphical expansion formulas, we give a new proof of the classical formulas for the number
A generating function of the number of homomorphisms from a surface group into a finite group
, 209
"... Abstract. A generating function of the number of homomorphisms from the fundamental group of a compact oriented or nonorientable surface without boundary into a finite group is obtained in terms of an integral over a real group algebra. We calculate the number of homomorphisms using the decompositi ..."
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Cited by 5 (3 self)
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Abstract. A generating function of the number of homomorphisms from the fundamental group of a compact oriented or nonorientable surface without boundary into a finite group is obtained in terms of an integral over a real group algebra. We calculate the number of homomorphisms using the decomposition of the group algebra into irreducible factors. This gives a new proof of the classical formulas of Frobenius, Schur, and Mednykh. Let S be a compact oriented or nonorientable surface without boundary, and χ(S) its Euler characteristic. The subject of our study is a generating function of the number Hom(π1(S), G)  of homomorphisms from the fundamental group of S into a finite group G. We give a generating function in terms of a noncommutative integral Eqn.(2.7) or Eqn.(3.2),
PERIODS OF STREBEL DIFFERENTIALS AND ALGEBRAIC CURVES DEFINED OVER THE FIELD OF ALGEBRAIC NUMBERS
"... Abstract. In [8] we have shown that if a compact Riemann surface admits a Strebel differential with rational periods, then the Riemann surface is the complex model of an algebraic curve defined over the field of algebraic numbers. We will show in this article that even if all geometric data are defi ..."
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Cited by 1 (0 self)
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Abstract. In [8] we have shown that if a compact Riemann surface admits a Strebel differential with rational periods, then the Riemann surface is the complex model of an algebraic curve defined over the field of algebraic numbers. We will show in this article that even if all geometric data are defined over Q, the Strebel differential can still have a transcendental period. We construct a Strebel differential q on an arbitrary complete nonsingular algebraic curve C defined over Q such that (i) all poles of q are Qrational points of C; (ii) the residue of √ q at each pole is a positive integer; and (iii) q has a transcendental period. 1.
Communications in Mathematical Physics Duality of Orthogonal and Symplectic Matrix Integrals
, 2003
"... Abstract: We present an asymptotic expansion for quaternionic selfadjoint matrix integrals. The Feynman diagrams appearing in the expansion are ordinary ribbon graphs and their nonorientable counterparts. We show that the 2N × 2N Gaussian Orthogonal Ensemble (GOE) and N × N Gaussian Symplectic Ens ..."
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Abstract: We present an asymptotic expansion for quaternionic selfadjoint matrix integrals. The Feynman diagrams appearing in the expansion are ordinary ribbon graphs and their nonorientable counterparts. We show that the 2N × 2N Gaussian Orthogonal Ensemble (GOE) and N × N Gaussian Symplectic Ensemble (GSE) have exactly the same expansion term by term, except that the contributions from graphs on a nonorientable surface with odd Euler characteristic carry the opposite sign. As an application, we give a new topological proof of the known duality for correlations of characteristic polynomials, demonstrating that this duality is equivalent to Poincaré duality of graphs drawn on a compact surface. Another consequence of our graphical expansion formula is a simple and simultaneous (re)derivation of the Central Limit Theorem for GOE, GUE (Gaussian Unitary Ensemble) and GSE: The three cases have exactly the same graphical limiting formula except for an overall constant that represents the type of the ensemble.
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
H(E) =
, 2009
"... In this paper we prove that the integral of the Wishart ensemble is, in a certain sense, a KP τ function, and generalize the result to other random matrix models, especially the Hermitian matrix model with external source. Besides potential application in multivariate statistics, we obtain some inte ..."
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In this paper we prove that the integral of the Wishart ensemble is, in a certain sense, a KP τ function, and generalize the result to other random matrix models, especially the Hermitian matrix model with external source. Besides potential application in multivariate statistics, we obtain some interesting combinatorial results. 1 Motivition and the main result In random matrix theory, the simplest and most studied model is the Gaussian Unitary Ensemble (GUE) [12], for which we consider the integral