Pis'ma v ZhETF, vol.95, iss. 11, pp.664-671
© 2012 June 10
Faces of matrix models
A. Morozov
Institute for Theoretical and Experimental Physics, 117218 Moscow, Russia Submitted 18 April 2012
Partition functions of eigenvalue matrix models possess a number of very different descriptions: as matrix integrals, as solutions to linear and non-linear equations, as r-functions of integrable hierarchies and as special-geometry prepotentials, as result of the action of TV-operators and of various recursions on elementary input data, as gluing of certain elementary building blocks. All this explains the central role of such matrix models in modern mathematical physics: they provide the basic "special functions" to express the answers and relations between them, and they serve as a dream model of what one should try to achieve in any other field.
Matrix model theory [1] studies the integral ZN(g)~ [ dMe-^tlM2 =
JNxN
ЛN N
{[dMu J[d2Mij x
»=i
i<j
N
N
x exp
y \«=1 i<j
(1)
over N x N Hermitian matrices M as a toy-example of quantum field and even string theory. It is spectacular, how much one can learn from this seemingly obvious problem.
What does it mean to study an integral?
First, we can simply take it. In this particular case the answer is simple:
ZN(g) ~ (2*g)N2?2
(2)
and does not look very interesting. However it only seems so. As usual, of interest is not the answer itself, but its decomposition, implied, by internal structure of our "theory". And the more we know about these structures. The more interesting decompositions we can obtain. In this particular case we could notice that M = UDUt, where D = diag{a;i} matrix, made from eigenvalues of M, and U is a unitary matrix. Then the same integral is decomposed into two - over unitary matrix U and over N eigenvalues Factoring away the volume Vn of the unitary group, we obtain:
Zpf —
m S Ö<
M
n e-*V2°dxi =
% i ) J^ 6
i<j t= 1
f dMe-^tlM\
n\Vn JNxN
(3)
This is already a somewhat non-trivial decomposition, because
VN =
(27r)iV(iV+l)/2
(4)
what is a considerably more complicated expression than the original (2).
Second, to study an integral in QFT sense means to treat it as measure, and consider all possible correlators. This means that of interest is not the (1) itself, but the averages
Cilt...tih = (tvMtl...tvMt-) =
f trikP1 ...trikPfee^
ètrM2
dM
fe-iitlM2dM or even their connected counterparts, like
Cconn _
ij — t-'ij
CiCj.
(5)
(6)
This is already a far-less-trivial problem, and looking at the very first examples one immediately observes an emergency of new structure:
C0(N) = N,
C2{N) = gN\ Ca(N) = g2(2N3 + N) ~ 2(gNf ■ Cq(N) = g3(5N4 + 2N2) ~ 5(gN)4 ■
g2(gN), 2 g2(gN)2,
(7)
The fact that each correlator is a polynomial (not a monomial) in N is encoded in the idea of loop expansion. The fact that all coefficients are integers signals about connection to combinatorics and is encoded in the idea of topological theories.
Third, if we move in the direction of string theory, we need not just correlators: we need generating
functions. For the set of C^ options:
there are two obvious
V,N
x /dikfexp ( ^trikf2 + Vifetrikffe | = eF{t} (8)
V 29 ti> J
and
/ m
P(m){z}= ( JJtr Then we have
Cii.....i
u=i
* -M
(9)
dkz
N
c,c
ZN dtij ... dtik 3felog ZN
and
dti 1 • • • d^ih
p{m){z}=^ YLV(zi)ZN,
00 dz 0
(10)
(11)
where V(z) = Tk=o^eTh the connected resolvent
. One can also introduce
{z} = HV(zi)logZN.
(12)
The fact that correlators Cj where polynomials in N is now expressed in the genus expansion of the free energy and connected resolvents:
F{t\g,N} = ^2g">-'Fp{t\gN}
p=0
(13)
and similarly
p= 0
(14)
Already at this stage something highly non-trivial shows up. This becomes clear from a look on the first few resolvents:
pW1Hz) = y(z)dz/2, p№1)(z) = dz/y6(z),
PW2Hzi,Z2) =
dz\dz2
(zi -z2)2 y{z1)y{z2y
(15)
They all are meromorphic (poly)differentials on a Riemann surface
E : y2 = z2 - 4(gN) (16)
which is called the spectral curve.
According to the string-theory approach, from this point we should move far enough in a number of different directions.
Other phases. As soon as we introduced the generating function Z{t}, we can start treating it non-perturbatively. This means that tu are considered not just as infinitesimal expansion parameters, defining a germe, but as the coupling constants, and study what happens when they take finite (or even infinite) values. Then Z{t} defines a partition function of a family of theories, called non-perturbative partition function. This partition function can be re-expanded not only around the Gaussian point, but around any background potential V (IkI) = Partition function (particular branch of it) then becomes also a function of parameters J*., which parameterize the moduli of the spectral curve. Phase transitions take place when the genus of the curve changes - it is controlled by the number of extrema of the background potential. The study of these dependencies is the subject of Seiberg-Witten theory [2], in matrixmodel context the corresponding field is sometime called the theory multi-cut solutions or of the Dijkgraaf-Vafa phases [3]. The particular branch of partition function is also known as CIV prepotential [4]. The most interesting feature of this prepotential are Seiberg-Witten special-geometry equations, describing dependence on the moduli by introducing very special "flat" coordinates fflfe instead of J*.:
afe = Ф П 'Ah
OF
дак
П
(17)
Bk
and the role of the Seiberg-Witten differential on the spectral curve is presumably played by the 1-point resolvent 0(z) = p^(z) [5]. The system of interrelated mul-tidensities pjcan in fact be built in a universal way for arbitrary Seiberg-Witten family of spectral curves £ - this procedure is now known as AMM/EO topological recursion [6] and has surprisingly many applications. Whenever partition function can be reconstructed in this way, this signals about the matrix-model hidden behind the scene - and there are already numerous examples, when recursion works, but the matrix model is not yet found.
Various limits. Non-perturbative partition function has a huge variety of different limits and critical behaviors in the vicinities of all its numerous singularities. The standard large-iV, genus-zero and multiscal-ing limits are just the examples. Related problem is the study of convergency properties of various pertur-
m
conn
bative series. All this is very important in applications and constitutes, perhaps, the biggest parts of traditional matrix-model theory.
Other observables. In string-theory paradigm there is no special preference for any obvious choice of observables. Instead of the correlators C¡ of the monomials trikfj one could study those, say, of the "Wilson loops" tr (esM), and form many other generating functions, different from (8) and (9), like the celebrated Harer-Zagier exact 1-point function [7]
z2h{tvM2h) _
N=0
k=0 A
(2fc-l)!!
(1-A)[(1-A)-(1 + A)**]
and Brezin-Hikami integrals [8]
(18)
,SiM\
n
p«?/2
¡duí
i+^i Ui
N
n
(Ui - Uj)(Ui - Uj + Si - Sj)
i<j
-. (Ui
Ui
Si)(Ui
Ui
Sj)'
(19)
The number of integrals here is k, not N, as in (3). In fact, these two subjects are unexpectedly closely related [9]. Harer-Zagier functions capture contributions from all genera - they differ from (8) by a kind of Pade transform and allow to put under the control the divergence of perturbative genus expansion. Instead they hide all the information related to spectral curves and Seiberg-Witten equations - but are capable to provide a closed expression for the Seiberg-Witten differential 0(z) = pW(z). Unfortunately, they are much more difficult to study than the resolvents.
Alternative formulations. For non-perturbative partition functions integrals (be they matrix or functional) provide only a description of particular phases: or, in worst case just the perturbative germes at particular points. More adequate are formulations in terms of D-modules or r-functions, characterizing partition functions as solutions to linear or quadratic equations respectively. It is still unclear, how general is the existence of quadratic (integrability-theory) structures and if higher non-linearities can also be relevant. At this moment, the "matrix-model r-functions" - usually, KP/Toda-functions, satisfying also a linear string equation, and, as a corollary, a whole infinite set of linear "Virasoro constraints" [10] are the most profound special functions, encountered in modern mathematical physics. They are natural for presentation of quantitative results in various fields of string theory, and their investigation is one of the primary purposes of modern science.
Integrability and II- r e p r e s e nt a t i o n. Emergency of non-linear (integrable) relations, like [11]
d2logZpf _ Z\ . iZ\ i
at2
(20)
for (8), is so non-trivial and so universal in string theory, that it can be considered as one of the main features of non-perturbative physics - still very mysterious. One should look for adequate ways to characterize these structures. Non-trivial r-functions can be made from the "trivial" ones by integrability-preserving transforms, described in terms of the IF-operators, which move the points in the Universal Grassmannian, parameterizing the space of the KP/Toda (free-fermion) r-functions. In other words, a matrix-model r-function can be considered as a result of the "evolution", driven by cut-and-join (IF) operators from some simple "initial conditions" [12]:
Z{t} = eWT0{t}. For (8) this ¡^-representation looks as follows:
(21)
ZN{t} = exp \ £
a, b
atabtb
dt,
+ (a + b+2)ta+b+2
dtadtb
a+6-2
oNt0
(22)
Generalizations. According to string-theory paradigm, one should not just embed original model in a set the similar ones by exponentiating all naive observables, one should also deform everything else, including the discrete parameters. In application to matrix models this means that sta
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