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;
action in N=2 SUSY
Superpotential in minima
in N=1 SUSY gauge theory
Hermitean 1-matrix integral:
is a polynomial
1/N – expansion (saddle point equation):
Solution to the saddle point equation:
DV – construction
An additional constraint:
Ci = constin the saddle point equation
Therefore, Ni (or fn-1)are fixed
C1 = C2 = C3 - equal “levels” due to tunneling
= 0 - further minimization in the saddle
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)
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
The only solution to the Virasoro constraints is the Gaussian model:
the integral is treated as the perturbation
One of many solutions is the Bonnet - David - Eynard
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)
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
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
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:
1) Ni = const, i.e.
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