Transcript Document

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)
low-energy effective
action in N=2 SUSY
gauge theory
Prepotential
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 (Dmodule)
• What distinguishes the DV construction. On Whitham
hierarchies and all that
Hermitean 1-matrix integral:
W(l) is a polynomial
1/N – expansion (saddle point equation):
Constraints:
Solution to the saddle point equation:
1
2
A
B
DV – construction
An additional constraint:
Ci = const in 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)
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 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
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
coefficients
counterpart of
Fourier
exponentials
General solution (A.Alexandrov, A.M., A.Morozov)
At any order in 1/N the solution Z of the Virasoro equations
is 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
Claim
where Uw is an (infinite degree) differential operator in Tk
(T)
that does not depend of the choice of arbitrary function
Therefore:
some proper basis
DV construction provides us with a possible basis:
DV basis:
1) Ni = const, i.e.
This fixes fn uniquely.
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 )
Constant monodromies
Whitham system
In planar limit:
This concrete Virasoro solution describes Whitham hierarchy
(L.Chekhov, A.M.) and log Z is its t -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
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 t function of this hierarchy is associated with logarithm of the
matrix model partition function.