Matrix Modelsand

Matrix Integrals

A.Mironov Lebedev Physical Institute and ITEP

New structures associated with matrix integrals mostly inspired by studies in low-energy SUSYGauge theories (F. Cachazo, K. Intrilligator, C.Vafa;R.Dijkgraaf, C.Vafa)

low-energy effective action in N=2 SUSY gauge theoryPrepotential massless BPS-states

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 DV construction. On Whitham hierarchies and all that

Hermitean 1-matrix integral:

is a polynomial

1/N – expansion (saddle point equation):

W()

Constraints:

Solution to the saddle point equation:

1

2 A

B

An additional constraint:

Ci = const in the saddle point equation

Therefore, Ni (or fn-1) are fixed

Interpretation (F.David,1992):

DV – construction

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 introducingchemical potential or by removing tunneling (G.Bonnet, F.David, B.Eynard)

i.e.

Virasoro & loop equations

A systematic way to construct these expansions (including higher order corrections) is Virasoro (loop) equations

Change of variables in

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.

PROBLEMS: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

or

Then W (or Tk) can appear in the denominators

of the formal series in tk

We then solve the Virasoro constraints with the additional requirement

Example 1 and

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

the integral is treated as the perturbation

expansion in tk

-

Example 2 and

One of many solutions is the Bonnet - David - Eynard n-parametric construction

Ni can be taken non-integer in the perturbative expansion

Where . Note that

We again shift the couplingsand 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

Solution:

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

Cubic example:

Toy matrix model

are arbitrary coefficients

counterpart of Fourier exponentials

counterpart ofFouriercoefficients

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

At any order in 1/N the solution Z of the Virasoro equationsis uniquely defined by an arbitrary function

of n-1 variables (n+2 variables Tk enter through n-1

fixed combinations)

E.g.

In the curve

Claimwhere Uw is an (infinite degree) differential operator in Tk that does not depend of the choice of arbitrary function

Therefore:

(T)

some proper basis

DV construction provides us with a possible basis:

DV basis:

1) Ni = const, i.e.

2) (More important) adding more times Tk does not change analytic

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

branching points which, however, begin to depend on Tk )

This fixes fn uniquely.

This concrete Virasoro solution describes Whitham hierarchy

(L.Chekhov, A.M.) and log Z is its -function.It satisfies Witten-Dijkgraaf-Verlinde-Verlinde equations (L.Chekhov, A.Marshakov, A.M., D.Vasiliev)

Constant monodromies Whitham system

In planar limit:

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

Conclusion

• 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 -function of this hierarchy is associated with logarithm of the matrix model partition function.