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UDC 530.14
Matrix Integrals and Gluings of Regular 2n-gons A. O. Shishanin
State Technical University n.a. N.E. Bauman, Moscow, Russia
This review is concerned applications of matrix models in combinatorics. We will discuss counting of orientable and nonorientable gluings of regular 2n-gons using gaussian matrix integrals.
Key words and phrases: matrix integrals, generalized Catalan numbers, generating function of gluings, virtual Euler characteristic.
1. Introduction
First time a gaussian matrix model was proposed by E.Wigner to describe the excited states of nuclei. Nowadays matrix models have many applications. In high energy physics the method of 1/^-expansion was presented by G.t' Hooft [1]. He demonstrated that in the limit of a large number of colors N quantum U (N) Yang-Mills theory is described by the diagrams, which can be drawn on a plane or a sphere. Such diagrams are called planar. Contribution (for example, the partition function) of each chart is included with the factor N in the degree of the Euler characteristic of the surface on which you can draw graphs. Therefore, the contributions of other diagrams are suppressed by 1/N.
As is well known, in the main (or planar) approximation of N in the gaussian hermitian matrix model the correlation functions are the Catalan numbers. At the same time, the number for ways of gluings 2n-gon to get the sphere, too, will give the nth Catalan number. This coincidence is not accidental. Moreover, J.Harer and D.Zagier [2-4] was found the generating function for the number for orientable gluings of all sorts. It has a surprisingly simple form. The number of gluings 2n-gon, giving a surface of genus g, can be regarded as generalized Catalan numbers eg (n). These numbers were obtained by the recurrence relations. Correlations in this model are even or odd polynomials of degree N. For odd n, we obtain a polynomial of even degree and vice versa.
By the way, the main achievement of [2] is even more interesting result, namely the computation of virtual Euler characteristic of the moduli space for the two-dimensional surface of genus g with n marked points. The moduli space is the orbifold for which a cell decomposition has the form Rk/Gk, where Gk are finite groups. This implies that its is generally not an integer this is true for the manifold. Euler characteristics are expressed in terms of Bernoulli numbers, and are particularly remarkable form of Q(1 — 2g) for a surface of genus g with one marked point.
In this report also we will discuss gluings regular 2n-gon for nonorientable surfaces. They are described by gaussian integrals on symmetric matrices. We will present examples of calculations for correlators in this model. They already have the form of polynomials in N with non-zero coefficients for all the terms of the lowest degrees.
2. Orientable Case
At first the generating function for orientable gluings of regular 2n-gons was obtained by Harer and Zagier [2]. These results were generalized by many authors [5-7] and so on.
Total number of orientable gluings for regular 2n-gon is (2n — 1)!!.
Received 11th June, 2012.
It is possible to count the number of orientable gluings using gaussian hermitian matrix model. The measure of this model with hermitian matrix H is
1 1 2 N d/J'(H) = n K Nl e~ 1 tTR2n dXii!l dXii '
i=1 i<j
2 t (n) 2
where hij = Xij + iyij. Here correlation functions are given (hijhki) = 5u5jk. Let us define a sequence of polynomials
Tn(N) = Y, (n)Nn+i-2° = Y (™)Nn-i+x(g),
9=0 9=0
where x(s) is Euler characteristic. Then their first few values are
Ti (N) = N2, T2(N) = 2N3 + N, T3(N) = 5N4 + 10N2, T4(N) = 14N5 + 70N3 + 21N ....
First numbers are Catalan numbers 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862 .... They obey the following recursion formula Cn+i = C0Cn + CiCn-i + ... + CnC0 and
r = 2n!
n
n!(n + 1)!'
^ Tn(N)
Consider generating function T(N, s) = 1 + 2Ns + 2s V] -—--— s
n=i (2n - 1)!!
J. Harer and D. Zagier (1986) [2] using matrix integrals
Tn(N) = J tr(H2n)d^(H)Tn(N) = (trH2n) = j tr(H2n)d^(H) obtained that
(f+i)" . (D
For generalized Catalan numbers eg recursion formula takes the form
(n + 1)eg(n) = (4n - 2)eg(n - 1) + (n - 1)(2n - 1)(2n - 3)eg-i(n - 2).
B. Lass [8] obtained pure combinatorial proof of the formula (1). A. Morozov and Sh. Shakirov [7] introduced another generating function for polynomials and calculated that
^ s2i^N ^ 1
S=0= 1-A 1 - A - (1 + A)s2.
They also calculated generating functions for two-and three-point correlators. Surprisingly, these generating functions are elementary.
Moreover authors [2] found virtual Euler characteristics of moduli space M.g,n for curve with genus g and with n punctured points
V(M ) = ( n» (2g - 3 + n)!(2g - 1) R
X(Mg,n) = (-1) -)-B2g,
286 Bulletin of PFUR. Series Mathematics. Information Sciences. Physics. No 1, 2013. Pp. 284-287
where B2g are Bernulli numbers. Here, the virtual Euler characteristic is not an integer, because the moduli space is the orbifold.
Especially simple this formula looks for one punctured point
x(Mg, 1 ) = c (1 — 2g) = — B2g/2g.
Penner [9,10] developed a matrix model that calculate x(Mg>n). By the way famous Kontsevich model [11] computes insertion indices on the moduli spaces.
3. General Case
Total number for nonorientable gluings [12] of regular 2n-gon is 2n(2n — 1)!!. It can be demonstrated that the total number of gluings is described symmetric gaussian matrix model. Matrix models with symmetric matrix were studied many authors [1315].
For correlators of elements in the case of symmetric matrix S we can obtain
(sij Ski) = Sji + 5n 5jk. Here the correlators or polynomials take the form [16,17]
J tr(S2) d^(S) = N2 + N, J tr(S4) d»(S) = 2N3 + 5N2 + 5N,
J tr(S6) d»(S) = 5N4 + 22N3 + 52N2 + 41N,
J tr(S8) d»(S) = 14N5 + 93N4 + 374N3 + 690N2 + 509N.
Here as before first numbers are Catalan numbers. Surfaces with the same Euler characteristic are taken into account in a suitable coefficient of the polynomial. To illustrate this let us take gluings of an 4-gon with surfaces of zero Euler characteristics. As a result we have one torus and four Klein bottle.
Similarly, the oriented case let us define the coefficients of these polynomials as
Un(N) = Y, ^(n)Nn-1+^.
9=0
Then these numbers /j,g(n) have the following recurrent formula [16,17] for every n > 4
(n +1) Vg (n) = (8n — 2)pg-i(n — 1) — (4n — 1) Vg (n — 1) + n(2n — 3)(10n — 9)^g (n — 2) — — 8(2n — 3)^g-2 (n — 2) + 8(2n — 3)pg-i(n — 2) — 10(2n — 3)(2n — 4)(2n — 5)pg-i(n — 3)+ +5(2n—3)(2n—4)(2n—5) (n—3)—2(2n—3)(2n—4)(2n—5)(2n—6)(2n—7)^g (n—4).
4. Conclusion and Outlook
We have considered the current state of the Harer-Zagier problem for general case with nonorientable gluings of a regular 2n-gon. Virtual Euler characteristics for nonorientable two-dimensional surfaces via symmetric Penner model were counted in [15]. Symmetric Kontsevich model is not investigated.
Author would like to thank V.O. Manturov for discussions, L.O. Chekhov for providing the article [15] and A.V. Borisov for reading manuscript and remarks.
References
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2. Harer J., Zagier D. The Euler Characteristics of the Moduli Space of Curves // Invent. Math. — 1986. — Vol. 85. — Pp. 457-485.
3. Zvonkin A. K., Lando S. K. Graphs on Surfaces and Their Applications. — Springer, 2003. — Vol. 459.
4. Alexandrov A., Mironov A., Morozov A. Partition Functions of Matrix Models as the First Special Functions of String Theory I. Finite Size Hermitean 1-Matrix Model // Int.J.Mod.Phys. A. — 2004. — Vol. 19. — Pp. 4127-4165.
5. Adrianov N. M. An Analog of the Harer-Zagier Formula for Unicellular Bicolored Maps // Funk. Analys and its Appl. — 1997. — Vol. 31. — Pp. 1-9.
6. Akhmedov E. T., Shakirov S. Gluings of Surfaces with Polygonal Boundaries // Funk. Analys and its Appl. — 2009. — Vol. 43, issue 4. — Pp. 3-13.
7. Morozov A.and Shakirov S. Exact 2-Point Function in Hermitian Matrix Model // J. of High Energy Phys. — 2009. — Vol. 0912. — Pp. 002-033.
8. Lass B. Demonstration Combinatoire de la Formule de Harer-Zagier // C. R. Acad. Sci. Paris. — 2009. — Vol. 0912. — Pp. 002-033.
9. Penner R. C. The Moduli Space of a Punctured Surface and Perturbative Series // Bull. Amer. Math. Soc. — 1986. — Vol. 15. — Pp. 73-77.
10. Distler J., Vafa C. A Critical Matrix Model at c = 1 // Mod. Phys. Lett. A. — 1991. — Vol. 6. — Pp. 259-270.
11. Kontsevich M. Intersection Theory on the Moduli Space of Curves and the Matrix Airy Function // Comm. Math. Phys. — 1992. — Vol. 147. — Pp. 1-23.
12. Golden I., Jackson D. Maps in Locally Orientable Surfaces and Integrals over Real Symmetric Surfaces // Can. J. Math. — 1997. — Vol. 49. — Pp. 865-882.
13. Harris G., Martinec E. Unoriented Strings and Matrix Ensembles // Phys. Lett. B. — 1990. — Vol. 245. — Pp. 384-392.
14. Brezin E., Neuberger H. Multicritical Points of Unoriented Random Surfaces // Nucl. Phys. B. — 1991. — Vol. 350. — Pp. 513-553.
15. Chekhov L., Zabrodin A. A Critical Matrix Model for Nonoriented String // Mod. Phys. Lett. — 1991. — Vol. 6. — Pp. 3143-3152.
16. Ledoux M. A Recursion Formula for the Moments of the Gaussian Orthogonal Ensemble // Ann. Inst. H. Poincare Probab. Statist. — 2009. — Vol. 45. — Pp. 754769.
17. Bernardi O. An Analogue of the Harer-Zagier Formula for Unicellular Maps on General Surfaces // Adv. in Appl. Math. — 2012. — Vol. 48. — Pp. 164-180.
УДК 530.14
Матричные интегралы и склейки правильных 2п-угольников
А. О. Шишанин
Кафедра физики
Московский государственный технический университет им. Н.Э. Баумана Россия, 105005, Москва, 2-я Бауманская ул., д. 5
Рассматриваются приложения матричных моделей в комбинаторике. Обсуждается подсчёт ориентируемых и неориентируемых склеек правильных 2п-угольников с помощью гауссовых интегралов по ортогональным матрицам.
Ключевые слова: матричные интегралы, обобщённые числа Каталана, производящая функция склеек, виртуальная эйлерова характеристика.
