Hirota equation as an example of integrable symplectic map

The hamiltonian formalism is developed for the sine-Gordon model on the space-time light-like lattice, first introduced by Hirota. The evolution operator is explicitely constructed in the quantum variant of the model, the integrability of the corresponding classical finite-dimensional system is established.Comment: 10 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

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

Parafermionic edge zero modes in Z_n-invariant spin chains

A sign of topological order in a gapped one-dimensional quantum chain is the existence of edge zero modes. These occur in the Z_2-invariant Ising/Majorana chain, where they can be understood using free-fermion techniques. Here I discuss their presence in spin chains with Z_n symmetry, and prove that for appropriate coupling they are exact, even in this strongly interacting system. These modes are naturally expressed in terms of parafermions, generalizations of fermions to the Z_n case. I show that parafermionic edge zero modes do not occur in the usual ferromagnetic and antiferromagnetic cases, but rather only when the interactions are chiral, so that spatial-parity and time-reversal symmetries are broken.Comment: 22 pages. v2: small changes, added reference

Geometry of W-algebras from the affine Lie algebra point of view

To classify the classical field theories with W-symmetry one has to classify the symplectic leaves of the corresponding W-algebra, which are the intersection of the defining constraint and the coadjoint orbit of the affine Lie algebra if the W-algebra in question is obtained by reducing a WZNW model. The fields that survive the reduction will obey non-linear Poisson bracket (or commutator) relations in general. For example the Toda models are well-known theories which possess such a non-linear W-symmetry and many features of these models can only be understood if one investigates the reduction procedure. In this paper we analyze the SL(n,R) case from which the so-called W_n-algebras can be obtained. One advantage of the reduction viewpoint is that it gives a constructive way to classify the symplectic leaves of the W-algebra which we had done in the n=2 case which will correspond to the coadjoint orbits of the Virasoro algebra and for n=3 which case gives rise to the Zamolodchikov algebra. Our method in principle is capable of constructing explicit representatives on each leaf. Another attractive feature of this approach is the fact that the global nature of the W-transformations can be explicitly described. The reduction method also enables one to determine the ``classical highest weight (h. w.) states'' which are the stable minima of the energy on a W-leaf. These are important as only to those leaves can a highest weight representation space of the W-algebra be associated which contains a ``classical h. w. state''.Comment: 17 pages, LaTeX, revised 1. and 7. chapter

BPS States in Omega Background and Integrability

We reconsider string and domain wall central charges in N=2 supersymmetric gauge theories in four dimensions in presence of the Omega background in the Nekrasov-Shatashvili (NS) limit. Existence of these charges entails presence of the corresponding topological defects in the theory - vortices and domain walls. In spirit of the 4d/2d duality we discuss the worldsheet low energy effective theory living on the BPS vortex in N=2 Supersymmetric Quantum Chromodynamics (SQCD). We discuss some aspects of the brane realization of the dualities between various quantum integrable models. A chain of such dualities enables us to check the AGT correspondence in the NS limit.Comment: 48 pages, 10 figures, minor changes, references added, typos correcte

Energy loss of pions and electrons of 1 to 6 GeV/c in drift chambers operated with Xe,CO2(15%)

We present measurements of the energy loss of pions and electrons in drift chambers operated with a Xe,CO2(15%) mixture. The measurements are carried out for particle momenta from 1 to 6 GeV/c using prototype drift chambers for the ALICE TRD. Microscopic calculations are performed using input parameters calculated with GEANT3. These calculations reproduce well the measured average and most probable values for pions, but a higher Fermi plateau is required in order to reproduce our electron data. The widths of the measured distributions are smaller for data compared to the calculations. The electron/pion identification performance using the energy loss is also presented.Comment: 15 pages, 10 figures, accepted for publication in Nucl.Instrum.Meth.

Space charge in drift chambers operated with the Xe,CO2(15%) mixture

Using prototype modules of the ALICE Transition Radiation Detector we investigate space charge effects and the dependence of the pion rejection performance on the incident angle of the ionizing particle. The average pulse height distributions in the drift chambers operated with the Xe,CO2(15%) mixture provide quantitative information on the gas gain reduction due to space charge accumulating during the drift of the primary ionization. Our results demonstrate that the pion rejection performance of a TRD is better for tracks which are not at normal incidence to the anode wires. We present detailed simulations of detector signals, which reproduce the measurements and lend strong support to our interpretation of the measurements in terms of space charge effects.Comment: 18 pages, 10 figures, accepted for publication in Nucl.Instrum.Meth. A. Data files available at http://www-alice.gsi.de/tr

Transition Radiation Spectroscopy with Prototypes of the ALICE TRD

We present measurements of the transition radiation (TR) spectrum produced in an irregular radiator at different electron momenta. The data are compared to simulations of TR from a regular radiator.Comment: 4 pages, 5 Figures, Proceedings for "TRDs for the 3rd millennium" (Sept. 4-7, 2003, Bari, Italy

Position Reconstruction in Drift Chambers operated with Xe, CO2 (15%)

We present measurements of position and angular resolution of drift chambers operated with a Xe,CO$_2$(15%) mixture. The results are compared to Monte Carlo simulations and important systematic effects, in particular the dispersive nature of the absorption of transition radiation and non-linearities, are discussed. The measurements were carried out with prototype drift chambers of the ALICE Transition Radiation Detector, but our findings can be generalized to other drift chambers with similar geometry, where the electron drift is perpendicular to the wire planes.Comment: 30 pages, 18 figure