matrix models, the gelfand-dikii differential polynomials, and (super) string theory

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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

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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. Nathan Seiberg Cambridge, Massachusetts September 1, 2003 Based on: - PowerPoint PPT Presentation

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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

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

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 quantitiesin a power series in the string coupling constant, ~.

Each term in the power series is given by a sum over Riemannsurfaces (the string worldsheet) of a given genus.

Different spacetime backgrounds are described by different two-dimensional quantum field theories on the string worldsheet.

In the superstring we sum over super-Riemann surfaces.

Different types of superstring theories (0A, 0B, etc.) differin the way we sum over the spin structures (e.g. 0A and 0Bdiffer 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.

Matrix Models

• Hermitian or unitary

U(N) symmetry M! U M Uy

• Complex

U(N) x U(N+q) symmetry ! U Vy

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• ?• ?

Interesting Limit N ! 1

Brezin, Itzykson, Parisi and Zuber: Diagonalize M

Look for a dominant configuration (minimum) of n.

The measure leads to repulsion between the eigenvalues.

Transition

Local minimum – unstable

Dyson gas, Wigner distribution

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 V near the maximum

• For complex matrices behavior of V(yt 0)

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

Double Well Potential, Eigenvalues Almost Spilling Out

is determined by solving a differential equation (Brezin and Kazakov, Douglas and Shenker, Gross and Migdal)…

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 x for x > 0 (below the barrier).

Painleve I

Relation to String Theory/Random 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).

Can show that the integral leads to a discrete approximationof Riemann surfaces

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,whichcorrespond to parameters in the potential V

Relation to KdV

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).

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.

More generally (more general potential V), consider the “Hamiltonian”

Its resolvent is given by a matrix of differential polynomials(Gelfand-Dikii)

Hk, Rk, and k are differential polynomials in r(x), and (x).

t0 = x, q is an integration constant.

Relation to mKdV

is determined from the string equation

with parameters tk

Returning to the simplest case

Adding the integration constant q, Painleve II is modified

are polynomials in q2

Focusing on the largest power of q in each term:

Take q ! 1 with finite t

is smooth – no transition at x = t = 0.

This exhibits the same pattern in the asymptotic expansions as before.

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 q is 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.

The system exhibits smooth interpolation between D-branes and fluxes (like geometric transitions).

For large q 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).

Complex Matrix Models

is N x (N+q) complex matrix.

A transition in the eigenvalue distribution (q=0):

For nonzero q, repulsion from the origin

Again, a differential equation for in terms of the Gelfand-Dikii differential polynomials (Morris)

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.

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 ofthe odd spin structures. In this case it is changed by x! – x .)

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.

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) fieldon them, time t. Another interpretation: (super) strings in a two-dimensionaltarget space. Its coordinates are t and another spatial direction, which arises from the conformal factor of the worldsheet metric(Liouville field).

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.