## Lectures on the asymptotic expansion of a hermitian matrix integral (1998)

Venue: | in Supersymmetry and Integrable Models, Henrik Aratin et al., Editors, Springer Lecture Notes in Physics 502 |

Citations: | 10 - 7 self |

### BibTeX

@INPROCEEDINGS{Mulase98lectureson,

author = {Motohico Mulase},

title = {Lectures on the asymptotic expansion of a hermitian matrix integral},

booktitle = {in Supersymmetry and Integrable Models, Henrik Aratin et al., Editors, Springer Lecture Notes in Physics 502},

year = {1998},

pages = {91--134}

}

### OpenURL

### Abstract

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 ζ-function. The third method is derived from a formula for the τ-function 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

### Citations

372 |
Intersection theory on the moduli space of curves and the matrix Airy function
- Kontsevich
- 1992
(Show Context)
Citation Context ...combinatorial one using the technique of Feynman diagram expansion. This method leads us directly to the connection between the matrix integrals and the moduli spaces of pointed Riemann surfaces [3], =-=[4]-=-, [11], [15]. The second method is the classical asymptotic analysis of orthogonal polynomials. It allows us to compute the integral explicitly in the special case known as the Penner Model, which is ... |

362 |
Two-dimensional gravity and intersection theory on moduli space
- Witten
- 1990
(Show Context)
Citation Context ...al one using the technique of Feynman diagram expansion. This method leads us directly to the connection between the matrix integrals and the moduli spaces of pointed Riemann surfaces [3], [4], [11], =-=[15]-=-. The second method is the classical asymptotic analysis of orthogonal polynomials. It allows us to compute the integral explicitly in the special case known as the Penner Model, which is related to t... |

191 |
Soliton Equations as Dynamical Systems on Infinite
- Sato, Sato
- 1982
(Show Context)
Citation Context ...th in this article are transcendental solutions of the KP equations. This is closely related to the unexpected sl(2) stability condition of the points of the infinite-dimensional Grassmannian of Sato =-=[12]-=- that correspond to the matrix integrals. Again we do not have any satisfactory explanation why the KP equations, the sl(2) stability condition, and the Euler characteristic of the moduli spaces of po... |

138 | The Euler characteristic of the moduli space of curves, Invent
- Harer, Zagier
- 1986
(Show Context)
Citation Context ...is a combinatorial one using the technique of Feynman diagram expansion. This method leads us directly to the connection between the matrix integrals and the moduli spaces of pointed Riemann surfaces =-=[3]-=-, [4], [11], [15]. The second method is the classical asymptotic analysis of orthogonal polynomials. It allows us to compute the integral explicitly in the special case known as the Penner Model, whic... |

98 |
Quantum field theory techniques in graphical enumeration
- Bessis, Itzykson, et al.
- 1980
(Show Context)
Citation Context ... n! Then we obtain � � f(X)dµ(X)=c(n) ◦ Hn � � U(n) T n ◦ R n i<j � Re(dωij) ∧ℑ(dωij). i<j ∆(k) 2 f(k0, ···,kn−1)dk0 ···dkn−1. For computation of c(n), we refer to, for example, Bessis-Itzykson-Zuber =-=[2]-=-. Using formula (3.6), we can reduce our integral to Pn(z,m)= c(n) � ∆(k) N 2 ⎛ ⎛ n−1 � 2m� ⎝exp ⎝− ( √ z) j−2 k j j ⎞ ⎞ ⎠ i dki⎠ . R n i=0 j=2 R n . LetsASYMPTOTIC EXPANSION OF MATRIX INTEGRAL 25 At ... |

82 |
Perturbative series and the moduli space of Riemann surfaces
- Penner
- 1988
(Show Context)
Citation Context ...natorial one using the technique of Feynman diagram expansion. This method leads us directly to the connection between the matrix integrals and the moduli spaces of pointed Riemann surfaces [3], [4], =-=[11]-=-, [15]. The second method is the classical asymptotic analysis of orthogonal polynomials. It allows us to compute the integral explicitly in the special case known as the Penner Model, which is relate... |

59 |
Characterization of Jacobian varieties in terms of soliton equations
- Shiota
- 1986
(Show Context)
Citation Context ...n of the Riemann theta functions. We know that Riemann theta functions associated with Riemann surfaces are characterized as finite-dimensional solutions to the system of KP equations [1], [5], [10], =-=[13]-=-. The matrix integrals that we will investigate in this article satisfy again the same KP equations, though this time they are truly infinite-dimensional solutions [7]. Using a combinatorial and numbe... |

51 |
An algebra-geometric construction of commuting operators
- Mumford
- 1977
(Show Context)
Citation Context ...ization of the Riemann theta functions. We know that Riemann theta functions associated with Riemann surfaces are characterized as finite-dimensional solutions to the system of KP equations [1], [5], =-=[10]-=-, [13]. The matrix integrals that we will investigate in this article satisfy again the same KP equations, though this time they are truly infinite-dimensional solutions [7]. Using a combinatorial and... |

38 |
Cohomological structure in soliton equations and Jacobian varieties
- Mulase
- 1984
(Show Context)
Citation Context ...neralization of the Riemann theta functions. We know that Riemann theta functions associated with Riemann surfaces are characterized as finite-dimensional solutions to the system of KP equations [1], =-=[5]-=-, [10], [13]. The matrix integrals that we will investigate in this article satisfy again the same KP equations, though this time they are truly infinite-dimensional solutions [7]. Using a combinatori... |

25 | Algebraic theory of the KP equations
- Mulase
- 1994
(Show Context)
Citation Context ...of KP equations [1], [5], [10], [13]. The matrix integrals that we will investigate in this article satisfy again the same KP equations, though this time they are truly infinite-dimensional solutions =-=[7]-=-. Using a combinatorial and number-theoretic method, Harer and Zagier [3] obtained a formula for the Euler number of the moduli space of pointed Riemann surfaces (defined as an algebraic stack or an o... |

22 |
On a set of equations characterizing Riemann matrices
- Arbarello, Concini
- 1984
(Show Context)
Citation Context ...of generalization of the Riemann theta functions. We know that Riemann theta functions associated with Riemann surfaces are characterized as finite-dimensional solutions to the system of KP equations =-=[1]-=-, [5], [10], [13]. The matrix integrals that we will investigate in this article satisfy again the same KP equations, though this time they are truly infinite-dimensional solutions [7]. Using a combin... |

22 | Category of vector bundles on algebraic curves and infinite-dimensional Grassmannians
- Mulase
- 1990
(Show Context)
Citation Context ...ertain class of solutions of the KP equations and a set of geometric data consisting of an arbitrary irreducible algebraic curve, which can be singular as well, and a torsion-free sheaf defined on it =-=[6]-=-. Let us call a solution to the KP equations transcendental if it does not correspond to any algebraic curve. How can we construct a transcendental solution, then? An answer has been obtained by an ac... |

18 |
A Planar Diagram Theory for Strong Interactions
- ’tHooft
- 1974
(Show Context)
Citation Context ... δikδjjδkiδℓℓ ij jk kl li ij jk kl li = δiℓδjkδjiδkℓ ij jk kl = δiiδjℓδjℓδkk li Figure 2.7. Pairing Contribution An interpretation of Figure 2.7 in terms of Feynman Diagrams was introduced by ’tHooft =-=[14]-=-. The set of four indexed dots •ij •jk •kℓ•ℓi is replaced by a crossroad (Figure 2.8). Since •ij = ∂ ∂yij is different from •ji = ∂ , the different roles of the indices are ∂yji represented by an arro... |

14 | Asymptotic analysis of a hermitian matrix integral
- Mulase
- 1995
(Show Context)
Citation Context ...f Harer and Zagier, except for the subtle point of giving an ordering to the set of marked points or not. The calculation of the asymptotic expansion of the Penner model has been rigorously performed =-=[9]-=-. The theorem of Harer and Zagier gives an amazing relation between the Riemann zeta function and the Riemann theta functions, if we think the latter to be essentially related to the moduli spaces of ... |

2 | Matrix integrals and integrable systems
- Mulase
- 1994
(Show Context)
Citation Context ...n and the Riemann theta functions, if we think the latter to be essentially related to the moduli spaces of Riemann surfaces. We add to this link yet another player: the KP equations. The observation =-=[8]-=- that the Hermitian matrix integral is a continuum soliton solution to the KP equation is suggestive from the geometric point of view. Soliton solutions represent singular Riemann surfaces with ration... |

2 |
Two-dimensional gravityand intersection theory on moduli space”, Surveys in Diff
- Witten
- 1991
(Show Context)
Citation Context ...al one using the technique of Feynman diagram expansion. This method leads us directly to the connection between the matrix integrals and the moduli spaces of pointed Riemann surfaces [3], [4], [11], =-=[15]-=-. The second method is the classical asymptotic analysis of orthogonal polynomials. It allows us to compute the integral explicitly in the special case known as the Penner Model, which is related to t... |

1 |
Intersection theoryon the moduli space of curves and the matrix Airy function
- Kontsevich
- 1992
(Show Context)
Citation Context ...combinatorial one using the technique of Feynman diagram expansion. This method leads us directly to the connection between the matrix integrals and the moduli spaces of pointed Riemann surfaces [3], =-=[4]-=-, [11], [15]. The second method is the classical asymptotic analysis of orthogonal polynomials. It allows us to compute the integral explicitly in the special case known as the Penner Model, which is ... |

1 |
Algebraic theoryof the KP equations
- Mulase
- 1994
(Show Context)
Citation Context ...of KP equations [1], [5], [10], [13]. The matrix integrals that we will investigate in this article satisfy again the same KP equations, though this time they are truly infinite-dimensional solutions =-=[7]-=-. Using a combinatorial and number-theoretic method, Harer and Zagier [3] obtained a formula for the Euler number of the moduli space of pointed Riemann surfaces (defined as an algebraic stack or an o... |

1 |
A planer diagram theoryfor strong interactions
- ’tHooft
- 1974
(Show Context)
Citation Context ...kiδℓℓ ij jk kl li ij jk kl li = δiℓδjkδjiδkℓ ij jk kl Figure 2.7. Pairing Contribution ⎞ ⎟ ⎠ . = δiiδjℓδjℓδkk li An interpretation of Figure 2.7 in terms of Feynman Diagrams was introduced by ’tHooft =-=[14]-=-. The set of four indexed dots •ij •jk •kℓ•ℓi is replaced by a crossroad (Figure 2.8). i l i j l Figure 2.8. Indexed Crossroad Since •ij = ∂ ∂yij is different from •ji = ∂ ∂yji , the different roles o... |