Matrix Models, The Gelfand-Dikii Differential Polynomials, And (Super) String Theory

1. Matrix Models, The Gelfand-Dikii Differential Polynomials, And (Super) String Theory The Unity of Mathematics In honor of the ninetieth birthday of I.M. Gelfand

2. Nathan Seiberg Cambridge, Massachusetts September 1, 2003 Based on: Douglas, Klebanov, Kutasov, Maldacena, Martinec, and NS, hep-th/0307195 Klebanov, Maldacena and NS, to appear

3. Crash Course In String Theory Our understanding of string theory is still in its infancy. In most cases, we only know how to expand physical quantities in a power series in the string coupling constant, ~. Each term in the power series is given by a sum over Riemann surfaces (the string worldsheet) of a given genus. Different spacetime backgrounds are described by different two-dimensional quantum field theories on the string worldsheet.

4. In the superstring we sum over super-Riemann surfaces. Different types of superstring theories (0A, 0B, etc.) differ in the way we sum over the spin structures (e.g. 0A and 0B differ in the sign of the odd spin structures). Boundaries in the worldsheet correspond to objects in spacetime – D-branes. Big challenge: find a complete definition of the theory, which reproduces the power series expansion. This is known only in a few cases; no unified principle yet.

5. Matrix Models • Hermitian or unitary U(N) symmetry M! U M Uy • Complex U(N) x U(N+q) symmetry F! U F Vy

6. Many Applications in Mathematics and Physics In physics: • Nuclear physics • Models of quantum field theory • Condensed matter physics • String theory/random surfaces • Supersymmetric field theories • Superstrings • ? • ?

7. Interesting Limit N !1 Brezin, Itzykson, Parisi and Zuber: Diagonalize M Look for a dominant configuration (minimum) of ln. The measure leads to repulsion between the eigenvalues.

8. Dyson gas, Wigner distribution Local minimum – unstable Transition

9. Different critical behaviors: • Eigenvalues are on the verge of spilling out • Transition from one group of eigenvalues to two groups (same in hermitian with two groups and in unitary) • Different shapes of the potential Vnear the maximum • For complex matrices behavior of V(FyF t 0)

10. It is of interest to examine the vicinity of the critical point as a function of the distance x from it.

11. Double Well Potential, Eigenvalues Almost Spilling Out is determined by solving a differential equation (Brezin and Kazakov, Douglas and Shenker, Gross and Migdal)…

12. Painleve I This is not an expansion around the global minimum of the potential V. Correspondingly, the differential equation does not have real and smooth solutions. The solution only has real expansion in inverse powers of xfor x > 0(below the barrier).

13. Relation to String Theory/Random Surfaces Can show that the integral leads to a discrete approximation of Riemann surfaces x is like a two-dimensional cosmological constant Genus g surfaces contribute terms of order The sum does not converge to a real smooth function F(x).

14. More generally (for generic potential V), consider the Hamiltonian Its resolvent is given by the Gelfand-Dikii differential polynomials u(x) is determined by the string equation with parameters tk,which correspond to parameters in the potential V Relation to KdV

15. Double Well Potential With The Transition (or Unitary Matrix Model) r(x) satisfies Painleve II (Periwal and Shevitz) Below the barrier (x>0), r(x) has a nontrivial expansion in negative powers of x. Above the barrier (x<0), r(x) is exponential in x. This is a global minimum of the potential, and correspondingly there exists a smooth real solution for r(x).

16. Conjecture: This is type 0B superstrings (NS and Witten, Crnkovic, Douglas and Moore). The expansions for large |x|are the sums of super-surfaces. Unlike the previous case, here the exact answer is a real and smooth function for all x. Leading order expressions (Gross-Witten transition) Exact F(x) is smooth.

17. More generally (more general potential V), consider the “Hamiltonian” Its resolvent is given by a matrix of differential polynomials (Gelfand-Dikii) Hk, Rk, andQkare differential polynomials inr(x),and w(x).

18. is determined from the string equation with parameterstk t0 = x, q is an integration constant. Relation to mKdV

19. Returning to the simplest case Adding the integration constant q, Painleve II is modified are polynomials in q2

20. Focusing on the largest power of q in each term: Take q !1 with finitet is smooth – no transition at x = t = 0. This exhibits the same pattern in the asymptotic expansions as before.

21. Interpret In Type 0B Superstring Theory: For x>0, the parameter q represents a certain flux (Ramond-Ramond) in the system. The power series has only even powers of q. A power of qis associated with adding a puncture to the surface. For x<0, q represents the number of D-branes. There are even and odd powers of q. Each power of q represents a boundary in the sum over surfaces. Without boundaries the power series vanishes.

22. The system exhibits smooth interpolation between D-branes and fluxes (like geometric transitions). For largeq this can be seen in the leading order of large |x| – only spherical worldsheets (with boundaries). The behavior at |x| !1 leads to a transition. It is smoothed out either by the finite x corrections (adding handles) or by nonzero q(adding boundaries).

23. Complex Matrix Models F is N x (N+q) complex matrix. A transition in the eigenvalue distribution (q=0):

24. For nonzero q, repulsion from the origin Again, a differential equation for in terms of the Gelfand-Dikii differential polynomials (Morris)

25. Interpretation: This is 0A superstring theory in various backgrounds. Here it is natural to identify q with the number of D-branes. As in the previous example (0B superstring theory), but unlike the nonsupersymmetric example, here we study the global minimum of the integrand. Therefore, there is a smooth and real solution for all x.

26. The simplest case is described by Substituting it is the same as the 0B theory (Painleve II) up to: Conclude: in this simple case 0A is essentially the same as 0B. (In the sum over surfaces, 0A differs from 0B in the sign of the odd spin structures. In this case it is changed by x! – x .)

27. More complicated potentials correspond to other superstring backgrounds: The expansion coefficients arise from two dimensional supersymmetric field theories on random supersurfaces, or even more complicated non-field-theoretic constructions… In general 0A is not the same as 0B. There is a rich structure as a function of tk and q.

28. Generalization To A One-Parameter (Time) Family Of Large Matrices Study quantum mechanics of large matrices. It corresponds to a sum over surfaces with a free (super) field on them, time t. Another interpretation: (super) strings in a two-dimensional target space. Its coordinates are t and another spatial direction, which arises from the conformal factor of the worldsheet metric (Liouville field).

29. Here 0A is not the same as 0B – the unitary matrix model is not the same as the complex matrix model. Instead, we have a certain duality symmetry: unlike the previous examples, the theory with x is the same as with –x. The 0A theory has a parameter like q. It represents background flux or D-brane charge.