A Note on Fractional KdV Hierarchies

We introduce a hierarchy of mutually commuting dynamical systems on a finite number of Laurent series. This hierarchy can be seen as a prolongation of the KP hierarchy, or a ``reduction'' in which the space coordinate is identified with an arbitrarily chosen time of a bigger dynamical system. Fractional KdV hierarchies are gotten by means of further reductions, obtained by constraining the Laurent series. The case of sl(3)^2 and its bihamiltonian structure are discussed in detail.Comment: Final version to appear in J. Math. Phys. Some changes in the order of presentation, with more emphasis on the geometrical picture. One figure added (using epsf.sty). 30 pages, Late

Hungry Volterra equation, multi boson KP hierarchy and Two Matrix Models

We consider the hungry Volterra hierarchy from the view point of the multi boson KP hierarchy. We construct the hungry Volterra equation as the B\"{a}cklund transformations (BT) which are not the ordinary ones. We call them ``fractional '' BT. We also study the relations between the (discrete time) hungry Volterra equation and two matrix models. From this point of view we study the reduction from (discrete time) 2d Toda lattice to the (discrete time) hungry Volterra equation.Comment: 13 pages, LaTe

Lattice WW algebras and quantum groups

We represent Feigin's construction [22] of lattice W algebras and give some simple results: lattice Virasoro and $W_3$ algebras. For simplest case $g=sl(2)$ we introduce whole $U_q(sl(2))$ quantum group on this lattice. We find simplest two-dimensional module as well as exchange relations and define lattice Virasoro algebra as algebra of invariants of $U_q(sl(2))$. Another generalization is connected with lattice integrals of motion as the invariants of quantum affine group $U_q(\hat{n}_{+})$. We show that Volkov's scheme leads to the system of difference equations for the function from non-commutative variables.Comment: 13 pages, misprints have been correcte

Exceptional structure of the dilute A3_3 model: E8_8 and E7_7 Rogers--Ramanujan identities

The dilute A$_3$ lattice model in regime 2 is in the universality class of the Ising model in a magnetic field. Here we establish directly the existence of an E$_8$ structure in the dilute A$_3$ model in this regime by expressing the 1-dimensional configuration sums in terms of fermionic sums which explicitly involve the E$_8$ root system. In the thermodynamic limit, these polynomial identities yield a proof of the E$_8$ Rogers--Ramanujan identity recently conjectured by Kedem {\em et al}. The polynomial identities also apply to regime 3, which is obtained by transforming the modular parameter by $q\to 1/q$. In this case we find an A_1\times\mbox{E}_7 structure and prove a Rogers--Ramanujan identity of A_1\times\mbox{E}_7 type. Finally, in the critical $q\to 1$ limit, we give some intriguing expressions for the number of $L$-step paths on the A$_3$ Dynkin diagram with tadpoles in terms of the E$_8$ Cartan matrix. All our findings confirm the E$_8$ and E$_7$ structure of the dilute A$_3$ model found recently by means of the thermodynamic Bethe Ansatz.Comment: 9 pages, 1 postscript figur

Exotic resonant level models in non-Abelian quantum Hall states coupled to quantum dots

In this paper we study the coupling between a quantum dot and the edge of a non-Abelian fractional quantum Hall state. We assume the dot is small enough that its level spacing is large compared to both the temperature and the coupling to the spatially proximate bulk non-Abelian fractional quantum Hall state. We focus on the physics of level degeneracy with electron number on the dot. The physics of such a resonant level is governed by a $k$-channel Kondo model when the quantum Hall state is a Read-Rezayi state at filling fraction $\nu=2+k/(k+2)$ or its particle-hole conjugate at $\nu=2+2/(k+2)$. The $k$-channel Kondo model is channel symmetric even without fine tuning any couplings in the former state; in the latter, it is generically channel asymmetric. The two limits exhibit non-Fermi liquid and Fermi liquid properties, respectively, and therefore may be distinguished. By exploiting the mapping between the resonant level model and the multichannel Kondo model, we discuss the thermodynamic and transport properties of the system. In the special case of $k=2$, our results provide a novel venue to distinguish between the Pfaffian and anti-Pfaffian states at filling fraction $\nu=5/2$. We present numerical estimates for realizing this scenario in experiment.Comment: 18 pages, 2 figures. Clarified final discussio

Exact and semiclassical approach to a class of singular integral operators arising in fluid mechanics and quantum field theory

A class of singular integral operators, encompassing two physically relevant cases arising in perturbative QCD and in classical fluid dynamics, is presented and analyzed. It is shown that three special values of the parameters allow for an exact eigenfunction expansion; these can be associated to Riemannian symmetric spaces of rank one with positive, negative or vanishing curvature. For all other cases an accurate semiclassical approximation is derived, based on the identification of the operators with a peculiar Schroedinger-like operator.Comment: 12 pages, 1 figure, amslatex, bibtex (added missing label eq.11

Worldsheet correlators in AdS(3)/CFT(2)

The AdS_3/CFT_2 correspondence is checked beyond the supergravity approximation by comparing correlation functions. To this end we calculate 2- and 3-point functions on the sphere of certain chiral primary operators for strings on AdS_3 x S^3 x T^4. These results are then compared with the corresponding amplitudes in the dual 2-dimensional conformal field theory. In the limit of small string coupling, where the sphere diagrams dominate the string perturbation series, beautiful agreement is found.Comment: 23 page

D-branes and SQCD in Non-Critical Superstring Theory

Using exact boundary conformal field theory methods we analyze the D-brane physics of a specific four-dimensional non-critical superstring theory which involves the N=2 SL(2)/U(1) Kazama-Suzuki model at level 1. Via the holographic duality of hep-th/9907178 our results are relevant for D-brane dynamics in the background of NS5-branes and D-brane dynamics near a conifold singularity. We pay special attention to a configuration of D3- and D5-branes that realizes N=1 supersymmetric QCD and discuss the massless spectrum and classical moduli of this setup in detail. We also comment briefly on the implications of this construction for the recently proposed generalization of the AdS/CFT correspondence by Klebanov and Maldacena within the setting of non-critical superstrings.Comment: harvmac, 47 pages, 6 figures; v4 same as v3 due to submission erro

Non-Critical String Duals of N=1 Quiver Theories

We construct N=1 non-critical strings in four dimensions dual to strongly coupled N=1 quiver gauge theories in the Coulomb phase, generalizing the string duals of Argyres-Douglas points in N=2 gauge theories. They are the first examples of superstrings vacua with an exact worldsheet description dual to chiral N=1 theories. We identify the dual of the non-critical superstring using a brane setup describing the field theory in the classical limit. We analyze the spectrum of chiral operators in the strongly coupled regime and show how worldsheet instanton effects give non-perturbative information about the gauge theory. We also consider aspects of D-branes relevant for the holographic duality.Comment: JHEP style; 40 pages, 3 figures; v2: minor corrections, refs added, version to appear in JHE