Reductions of Multicomponent mKdV Equations on Symmetric Spaces of DIII-Type

New reductions for the multicomponent modified Korteveg-de Vries (MMKdV) equations on the symmetric spaces of {\bf DIII}-type are derived using the approach based on the reduction group introduced by A.V. Mikhailov. The relevant inverse scattering problem is studied and reduced to a Riemann-Hilbert problem. The minimal sets of scattering data $\mathcal{T}_i$, $i=1,2$ which allow one to reconstruct uniquely both the scattering matrix and the potential of the Lax operator are defined. The effect of the new reductions on the hierarchy of Hamiltonian structures of MMKdV and on $\mathcal{T}_i$ are studied. We illustrate our results by the MMKdV equations related to the algebra $\mathfrak{g}\simeq so(8)$ and derive several new MMKdV-type equations using group of reductions isomorphic to ${\mathbb Z}_{2}$, ${\mathbb Z}_{3}$, ${\mathbb Z}_{4}$.Comment: This is a contribution to the Proc. of the Seventh International Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007, Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Classical Limit of the Three-Point Function from Integrability

We give analytic expression for the three-point function of three large classical non-BPS operators N=4 Super-Yang-Mills theory at weak coupling. We restrict ourselves to operators belonging to an su(2) sector of the theory. In order to carry out the calculation we derive, by unveiling a hidden factorization property, the thermodynamical limit of Slavnov's determinant.Comment: 4 pages, 2 figure

On the timelike Liouville three-point function

In a recent paper, D. Harlow, J. Maltz, and E. Witten showed that a particular proposal for the timelike Liouville three-point function, originally due to Al. Zamolodchikov and to I. Kostov and V. Petkova, can actually be computed by the original Liouville path integral evaluated on a new integration cycle. Here, we discuss a Coulomb gas computation of the timelike three-point function and show that an analytic extension of the Selberg type integral formulas involved reproduces the same expression, including the adequate normalization. A notable difference with the spacelike calculation is pointed out.Comment: 11 pages. v2 comments and references added. Appropriate credit is given to Ref. arXiv:hep-th/0512346, where the Coulomb gas computation of the c<1 theory has already been discusse

D-particles, Matrix Integrals and KP hierachy

We study the regularized correlation functions of the light-like coordinate operators in the reduction to zero dimensions of the matrix model describing $D$-particles in four dimensions. We investigate in great detail the related matrix model originally proposed and solved in the planar limit by J. Hoppe. It also gives the solution of the problem of 3-coloring of planar graphs. We find interesting strong/weak 't Hooft coupling dependence. The partition function of the grand canonical ensemble turns out to be a tau-function of KP hierarchy. As an illustration of the method we present a new derivation of the large-N and double-scaling limits of the one-matrix model with cubic potential.Comment: harvmac, 35 pp. v2. typos correcte

Two-dimensional gauge theories of the symmetric group S(n) and branched n-coverings of Riemann surfaces in the large-n limit

Branched n-coverings of Riemann surfaces are described by a 2d lattice gauge theory of the symmetric group S(n) defined on a cell discretization of the surface. We study the theory in the large-n limit, and we find a rich phase diagram with first and second order transition lines. The various phases are characterized by different connectivity properties of the covering surface. We point out some interesting connections with the theory of random walks on group manifolds and with random graph theory.Comment: Talk presented at the "Light-cone physics: particles and strings", Trento, Italy, September 200