no. 6
Non-commutative matrix integrals and representation varieties of surface groups in a finite group
[Intégrales matricielles non-commutatives et variétés de représentations du groupe d'une surface dans un groupe fini]
Mulase, Motohico1 ; T. Yu, Josephine
1 University of California, department of mathematics, One Shields Avenue Davis CA 95616 (USA), University of California, department of mathematics, Evans Hall 3840 Berkeley CA 94720-3840 (USA)
Annales de l'Institut Fourier, Tome 55 (2005) no. 6, pp. 2161-2196.

Une nouvelle formule est établie pour l'expansion asymptotique d'une intégrale matricielle avec des valeurs dans une algèbre de von Neumann de dimension finie en terme de graphes sur les surfaces orientables ou non-orientables.

A new formula is established for the asymptotic expansion of a matrix integral with values in a finite-dimensional von Neumann algebra in terms of graphs on surfaces which are orientable or non-orientable.

DOI : 10.5802/aif.2157
Classification : 15A52, 20C05, 32G13, 81Q30
Keywords: Random matrices, non-commutative matrix integral, Feynman diagram expansion, ribbon graph, Moebius graph, von Neumann algebra, representation variety
Mot clés : matrices aléatoires, intégrale non commutative de matrice, expansion de diagramme de Feynman, graphe de ruban, graphe de Moebius, algèbre de von Neumann, variété de représentations

Mulase, Motohico 1 ; T. Yu, Josephine 

1 University of California, department of mathematics, One Shields Avenue Davis CA 95616 (USA), University of California, department of mathematics, Evans Hall 3840 Berkeley CA 94720-3840 (USA) 
@article{AIF_2005__55_6_2161_0,
     author = {Mulase, Motohico and T. Yu, Josephine},
     title = {Non-commutative matrix integrals and representation varieties of surface groups in a finite group},
     journal = {Annales de l'Institut Fourier},
     pages = {2161--2196},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {55},
     number = {6},
     year = {2005},
     doi = {10.5802/aif.2157},
     zbl = {1092.15020},
     mrnumber = {2187951},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2157/}
}
TY  - JOUR
AU  - Mulase, Motohico
AU  - T. Yu, Josephine
TI  - Non-commutative matrix integrals and representation varieties of surface groups in a finite group
JO  - Annales de l'Institut Fourier
PY  - 2005
SP  - 2161
EP  - 2196
VL  - 55
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2157/
DO  - 10.5802/aif.2157
LA  - en
ID  - AIF_2005__55_6_2161_0
ER  - 
%0 Journal Article
%A Mulase, Motohico
%A T. Yu, Josephine
%T Non-commutative matrix integrals and representation varieties of surface groups in a finite group
%J Annales de l'Institut Fourier
%D 2005
%P 2161-2196
%V 55
%N 6
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2157/
%R 10.5802/aif.2157
%G en
%F AIF_2005__55_6_2161_0
Mulase, Motohico; T. Yu, Josephine. Non-commutative matrix integrals and representation varieties of surface groups in a finite group. Annales de l'Institut Fourier, Tome 55 (2005) no. 6, pp. 2161-2196. doi : 10.5802/aif.2157. https://aif.centre-mersenne.org/articles/10.5802/aif.2157/

[1] M. Adler; P. van Moerbeke Hermitian, symmetric and symplectic random ensembles: PDEs for the distribution of the spectrum, Ann. of Math. (2), Volume 153 (2001) no. 1, pp. 149-189 | DOI | MR | Zbl

[2] J. Baik; P. Deift; K. Johansson On the distribution of the length of the longest increasing subsequence of random permutations, J. Amer. Math. Soc., Volume 12 (1999), pp. 1119-1178 | DOI | MR | Zbl

[3] J. Baik; P. Deift; K. Johansson On the distribution of the length of the second row of a Young diagram under Plancherel measure (1999) (math.CO/9901118, http://arxiv.org/abs/math.CO/9901118) | Zbl

[4] J. Baik; E. Rains Symmetrized random permutations (1999) (math.CO/9910019, http://arxiv.org/abs/math.CO/9910019) | Zbl

[5] G.V. Belyi On galois extensions of a maximal cyclotomic fields, Math. USSR Izvestija, Volume 14 (1980), pp. 247-256 | DOI | MR | Zbl

[6] D. Bessis; C. Itzykson; J.B. Zuber Quantum field theory techniques in graphical enumeration, Advances Applied Math., Volume 1 (1980), pp. 109-157 | DOI | MR | Zbl

[7] P.M. Bleher; A.R. Its Random matrix models and their applications, Math. Sci. Research Institute Publications, 40, Cambridge University Press, 2001 | MR | Zbl

[8] A. Borodin; A. Okounkov; G. Olshanski On asymptotics of Plancherel measures for symmetric groups (1999) (math.CO/990532, http://arxiv.org/abs/math.CO/9905032) | Zbl

[9] C. Brézin; C. Itzykson; G. Parisi; J.-B. Zuber Planar diagrams, Comm. Math. Physics, Volume 59 (1978), pp. 35-51 | DOI | MR | Zbl

[10] W. Burnside Theory of groups of finite order, 2nd ed., Cambridge University Press, 1991 | JFM

[11] P. Deift Integrable systems and combinatorial theory, Notices AMS, Volume 47 (2000), pp. 631-640 | MR | Zbl

[12] R.P. Feynman Space-time approach to quantum electrodynamics, Phys. Review, Volume 76 (1949), pp. 769-789 | DOI | MR | Zbl

[13] D.S. Freed; Frank Quinn Chern-Simons theory with finite gauge group, Communications in Mathematical Physics, Volume 156 (1993), pp. 435-472 | DOI | MR | Zbl

[14] G. Frobenius Über Gruppencharaktere, Sitzungsberichte der königlich preussischen Akademie der Wissenschaften (1896), pp. 985-1021 | JFM

[15] G. Frobenius; I. Schur Über die reellen 11arstellungen der endlichen Gruppen, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften (1906), pp. 186-208 | JFM

[16] F.P. Gardiner Teichmüller theory and quadratic differentials, John Wiley \& Sons, 1987 | MR | Zbl

[17] I.P. Goulden; J.L. Harer; J.L.; D.M. Jackson A geometric parametrization for the virtual Euler characteristics of the moduli spaces of real and complex algebraic curves, Trans. Amer. Math. Soc., Volume 353 (2001), pp. 4405-4427 | DOI | MR | Zbl

[18] D.J. Gross; W. Taylor Two dimensional QCD is a string theory, Nucl. Phys., Volume B 400 (1993), pp. 181-210 | MR | Zbl

[19] J.L. Gross; T.W. Tucker Topological graph theory, John Wiley \& Sons, 1987 | MR | Zbl

[20] A. Grothendieck Esquisse d'un programme (1984) (reprinted in [46], 7–48)

[21] J.L. Harer The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math., Volume 84 (1986), pp. 157-176 | DOI | MR | Zbl

[22] J.L. Harer; D. Zagier The Euler characteristic of the moduli space of curves, Invent. Math., Volume 85 (1986), pp. 457-485 | DOI | MR | Zbl

[23] A. Hatcher On triangulations of surfaces, Topology and Appl., Volume 40 (1991), pp. 189-194 | DOI | MR | Zbl

[24] I.M. Isaacs Character theory of finite groups, Academic Press, 1976 | MR | Zbl

[25] K. Johansson Discrete orthogonal polynomial ensembles and the Plancherel measure (1999) (math.CO/9906120, http://arxiv.org/abs/math.CO/9906120) | Zbl

[26] G.A. Jones Characters and surfaces: a survey, Lecture Note Series, 249, London Math. Soc., 1998 | MR | Zbl

[27] M. Kontsevich Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Physics, Volume 147 (1992), pp. 1-23 | DOI | MR | Zbl

[28] K. Liu Heat kernel and moduli space, Math. Research Letters, Volume 3 (1996), pp. 743-762 | MR | Zbl

[29] K. Liu Heat kernel and moduli space II, Math. Research Letters, Volume 4 (1996), pp. 569-588 | MR | Zbl

[30] K. Liu Heat kernels, symplectic geometry, moduli spaces and finite groups, Surveys Diff. Geom., Volume 5 (1999), pp. 527-542 | MR | Zbl

[31] A.D. Mednykh Determination of the number of nonequivalent coverings over a compact Riemann surface, Soviet Math. Doklady, Volume 19 (1978), pp. 318-320 | Zbl

[32] M. Lal Mehta Random matrices, 2nd ed., Academic Press, 1991 | MR | Zbl

[33] M. Mulase; R. Penner and S.T. Yau Algebraic theory of the KP equations (Perspectives in Mathematical Physics) (1994), pp. 157-223 | Zbl

[34] M. Mulase; K. Fukaya et al. Matrix integrals and integrable systems (Topology, geometry and field theory) (1994), pp. 111–127 | Zbl

[35] M. Mulase Asymptotic analysis of a hermitian matrix integral, Int. J. Math., Volume 6 (1995), pp. 881-892 | DOI | MR | Zbl

[36] M. Mulase; Henrik Aratin et al. Lectures on the asymptotic expansion of a hermitian matrix integral, Supersymmetry and Integrable Models (Springer Lecture Notes in Physics), Volume 502 (1998), pp. 91–134 | Zbl

[37] M. Mulase; M. Penkava Ribbon graphs, quadratic differentials on Riemann surfaces, and algebraic curves defined over ℚ ¯, Asian J. Math., Volume 2 (1998), pp. 875-920 | MR | Zbl

[38] M. Mulase; A. Waldron Duality of orthogonal and symplectic matrix integrals and quaternionic Feynman graphs (2002) (math-ph/0206011, http://arxiv.org/abs/math-ph/0206011) | MR | Zbl

[39] M. Mulase; J.T. Yu A generating function of the number of homomorphisms from a surface groupin to a ?nite group (2002) (math.QA/0209008, http://arxiv.org/abs/math.QA/0209008)

[40] A. Okounkov Random matrices and random permutations (1999) (math.CO/9903176, http://arxiv.org/abs/math.CO/9903176) | MR | Zbl

[41] A. Okounkov; R. Pandharipande Mapcolor theorem, Springer-Verlag, 1974

[42] A. Okounkov; R. Pandharipande The equivariant Gromov-Witten theory of P 1 (2002) (math.AG/0207233, http://arxiv.org/abs/math.AG/0207233) | Zbl

[43] R.C. Penner Perturbation series and the moduli space of Riemann surfaces, J. Diff. Geom., Volume 27 (1988), pp. 35-53 | MR | Zbl

[44] G. Ringel Map color theorem, Springer-Verlag, 1974 | MR | Zbl

[45] L. Schneps The Grothendieck theory of dessins d'enfants, Lecture Notes Series, 200, London Math. Soc., 1994 | Zbl

[46] L. Schneps; P. Lochak; eds. Geometric Galois actions: around Grothendieck's esquisse d'un programme, Lecture Notes Series, 242, London Math. Soc., 1997 | Zbl

[47] J.-P. Serre Linear representations of finite groups, Springer-Verlag, 1987 | MR | Zbl

[48] R.P. Stanley Enumerative combinatorics, 2, Cambridge University Press, 2001 | Zbl

[49] K. Strebel Quadratic differentials, Springer-Verlag, 1984 | MR | Zbl

[50] G. 't Hooft A planer diagram theory for strong interactions, Nuclear Physics B, Volume 72 (1974), pp. 461-473 | DOI

[51] C.A. Tracy; H. Widom Fredholm Determinants, Differential Equations and Matrix Models, hep-th/9306042, Comm. Math. Physics, Volume 163 (1994), pp. 33-72 | DOI | MR | Zbl

[52] P. van Moerbeke; Bleher and Its Integrable lattices: random matrics and random permutations, Random Matrix Models and Their Applications (MSRI Publications), Volume 40 (2001), pp. 321-406 | Zbl

[53] E. Witten On quantum gauge theories in two dimensions, Comm. Math. Physics, Volume 141 (1991), pp. 153-209 | DOI | MR | Zbl

[54] E. Witten Two dimensional gravity and intersection theory on moduli space, Surveys Diff. Geom., Volume 1 (1991), pp. 243-310 | MR | Zbl

[55] J. Yu Graphical expansion of matrix integrals with values in a Clifford algebra (Explorations: A Journal of Undergraduate Research), Volume 6 (2003)

Cité par Sources :