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 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.
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.
Matrix Models
• Hermitian or unitary
U(N) symmetry
M! U M Uy
• Complex
U(N) x U(N+q) symmetry F ! U F Vy
Many Applications in Mathematics and Physics
In physics:
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Nuclear physics
Models of quantum field theory
Condensed matter physics
String theory/random surfaces
Supersymmetric field theories
Superstrings
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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.
Dyson gas, Wigner distribution
Local minimum – unstable
Transition
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(FyF t 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)…
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 x for x > 0 (below the
barrier).
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).
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
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 Qk are differential polynomials in r(x), and w(x).
is determined from the string equation
with parameters tk
t0 = x, q is an integration constant.
Relation to mKdV
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 Dbranes 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
F 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 of
the 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) 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).
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.