1. Matrix Models and Matrix Integrals A.Mironov Lebedev Physical Institute and ITEP

2. 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

3. 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

4. Hermitean 1-matrix integral: is a polynomial 1/N – expansion (saddle point equation): W( )

5. Constraints:

6. Solution to the saddle point equation: 1 2 A B

7. An additional constraint: C i = const in the saddle point equation Therefore, N i (or f n-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

8. Let N i be the parameters! It can be done either by introducing chemical potential or by removing tunneling (G.Bonnet, F.David, B.Eynard) i.e.

9. Virasoro & loop equations A systematic way to construct these expansions (including higher order corrections) is Virasoro (loop) equations Change of variablesin leads to the Ward identities : - Virasoro (Borel sub-) algebra

10. 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?

11. The problem number zero: How is the matrix model integral defined at all? It is a formal series in positive degrees of t k and we are going to solve Virasoro constraints iteratively. t k have dimensions (grade): [t k ]=k (from L n or matrix integral) c k... dimensionful all c k... = 0

12. The Bonnet - David - Eynard matrix representation for the DV construction is obtained by shifting or Then W ( or T k ) can appear in the denominators of the formal series in t k We then solve the Virasoro constraints with the additional requirement

13. Example 1 and The only solution to the Virasoro constraints is the Gaussian model: the integral is treated as the perturbation expansion in t k - Example 2 and One of many solutions is the Bonnet - David - Eynard n-parametric construction N i can be taken non-integer in the perturbative expansion

14. Where. Note that We again shift the couplings and consider Z as a power series in t k ’s but not in T k ’s: i.e. one calculates the moments

15. 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

16. One possible choice: (the standard Airy function) Another choice:

17. Asymptotic expansion of the integral Saddle point equation has two solutions: Generally W‘(x) = 0 has n solutions n-1 solutions have smooth limit T n+1  0

18. Cubic example:

19. Toy matrix model are arbitrary coefficients counterpart of Fourier exponentials counterpart of Fourier coefficients

20. 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 T k enter through n-1 fixed combinations) E.g. In the curve

21. Claim where U w is an (infinite degree) differential operator in T k that does not depend of the choice of arbitrary function Therefore: (T) some proper basis DV construction provides us with a possible basis:

22. DV basis: 1) N i = const, i.e. 2) (More important) adding more times T k does not change analytic structures (e.g. the singularities of should be at the same branching points which, however, begin to depend on T k ) This fixes f n uniquely. 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) Constant monodromies Whitham system In planar limit:

23. Invariant description of the DV basis: - monodromies of minima of W(x) can be diagonalized DV – basis: eigenvectors of (similarly to the condition )

24. 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

25. Conclusion The Hermitean one-matrix integral is well-defined by fixing an arbitrary polynomial W n+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.