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Matrix Models and Matrix Integrals . A.Mironov Lebedev Physical Institute and ITEP. New structures associated with matrix integrals mostly inspired by studies in low-energy SUSY Gauge theories ( F. Cachazo, K. Intrilligator, C.Vafa; R.Dijkgraaf, C.Vafa )

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Matrix Modelsand Matrix Integrals


Lebedev Physical Institute and ITEP

New structures associated with matrix integrals

mostly inspired by studies in low-energy SUSY

Gauge theories (F. Cachazo, K. Intrilligator, C.Vafa;

R.Dijkgraaf, C.Vafa)

low-energy effective

action in N=2 SUSY

gauge theory




Superpotential in minima

in N=1 SUSY gauge theory

  • Standard dealing with matrix models

  • Dijkgraaf – Vafa (DV)construction (G.Bonnet, F.David, B.Eynard, 2000)

  • Virasoro constraints (=loop equations, =Schwinger-Dyson equations, =Ward identities)

  • Matrix models as solutions to the Virasoro constraints (D-module)

  • What distinguishes the DVconstruction. On Whitham hierarchies and all that

Hermitean 1-matrix integral:


is a polynomial

1/N – expansion (saddle point equation):

DV – construction

An additional constraint:

Ci = constin the saddle point equation

Therefore, Ni (or fn-1)are fixed

Interpretation (F.David,1992):

C1 = C2 = C3 - equal “levels” due to tunneling

= 0 - further minimization in the saddle

point approximation

Let Ni be the parameters!

It can be done either by introducing

chemical potential or by removing tunneling

(G.Bonnet, F.David, B.Eynard)


Virasoro & loop equations

A systematic way to construct these expansions (including

higher order corrections) is Virasoro (loop) equations

Change of variables


leads to the Ward identities:

- Virasoro (Borel sub-) algebra

We define the matrix model as any solution to the Virasoro constraints (i.e. as a D-module). DV construction is a particular case of this general approach, when there exists multi-matrix representation for the solution.


1) How many solutions do the Virasoro constraints have?

2) What is role of the DV - solutions?

3) When do there exist integral (matrix) representations?

The problem number zero:

How is the matrix model integral defined at all?

It is a formal series in positive degrees of tk and we are going to

solve Virasoro constraints iteratively.

tk have dimensions (grade): [tk]=k (from Ln or matrix integral)

ck... dimensionful

all ck... = 0

The Bonnet - David - Eynard matrix representation

for the DV construction is obtained by shifting


ThenW (orTk) can appear in the denominators

of the formal series intk

We then solve the Virasoro constraints

with the additional requirement

Example 1


The only solution to the Virasoro constraints is the Gaussian model:

the integral is treated as the perturbation

expansion intk


Example 2


One of many solutions is the Bonnet - David - Eynard

n-parametric construction

Nican be taken non-integer in the perturbative expansion

Where . Note that

We again shift the couplings

and consider Z as a power series in tk’s but not in Tk’s:

i.e. one calculates the moments

Example: Cubic potential at zero couplings gives the Airy equation


Two solutions = two basic contours.

Contour: the integrand vanishes at its ends

to guarantee Virasoro constraints!

The contour should go to infinity where

One possible choice:

(the standard Airy function)

Another choice:

Asymptotic expansion of the integral

Saddle point equation has two solutions:

Generally W‘(x) = 0 has n solutions

n-1 solutions have smooth limit Tn+1 0

Toy matrix model

are arbitrary coefficients

counterpart of



counterpart of



General solution (A.Alexandrov, A.M., A.Morozov)

At any order in 1/NthesolutionZ of the Virasoro equations

is uniquely defined by an arbitrary function

of n-1 variables (n+2 variablesTkenter through n-1

fixed combinations)


In the curve


whereUwis an (infinite degree) differential operator inTk

that does not depend of the choice of arbitrary function



some proper basis

DV construction provides us with a possible basis:

DV basis:

1) Ni = const, i.e.

This fixesfnuniquely.

2) (More important) adding more timesTkdoes not change analytic

structures (e.g. the singularities of should be at the same

branching points which, however, begin to depend onTk )

Constant monodromies Whitham system

In planar limit:

This concrete Virasoro solution describes Whitham hierarchy

(L.Chekhov, A.M.) andlog Zis itst-function.

It satisfies Witten-Dijkgraaf-Verlinde-Verlinde equations

(L.Chekhov, A.Marshakov, A.M., D.Vasiliev)

Invariant description of the DV basis:

- monodromies of

minima of W(x)

can be diagonalized

DV – basis: eigenvectors of

(similarly to the condition )

Seiberg – Witten – Whitham system

Operator relation (not proved) :

Conditions: blowing up to cuts on the complex plane

Therefore, in the basis of eigenvectors,

can be realized as

Seiberg - Witten -

- Whitham system


  • The Hermitean one-matrix integral is well-defined by fixing an arbitrary polynomial Wn+1(x).

  • The corresponding Virasoro constraints have many solutions parameterized by an arbitrary function of n-1 variables.

  • The DV - Bonnet - David - Eynard solution gives rise to a basis in the space of all solutions to the Virasoro constraints.

  • This basis is distinguished by its property of preserving monodromies, which implies the Whitham hierarchy. The t-function of this hierarchy is associated with logarithm of the matrix model partition function.