Matrix Models and Matrix Integrals
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Transcript of Matrix Models and Matrix Integrals
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 tosolve 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.