the harder narasimhan trace and unitarity of the kz hitchin connection genus 0 let a reductive group g act on a projective variety x and suppose given a lift of the action to an ample line bundle by definition all g invariant sections of vanish on the nonsemistable locus x nss taking an appropriate normal derivative defines a map h0 x g h0 s v g where v is a g vector bundle on a g variety s we call this the harder narasimhan trace applying this to the geometric invariant theory construction of the moduli space of parabolic bundles on a curve we discover generalisations of coulomb gas representations which map conformal blocks to hypergeometric local systems in this paper we prove the unitarity of the kz hitchin connection in the genus zero rank two case by proving that the above map lands in a unitary factor of the hypergeometric system an ingredient in the proof is a lower bound on the degree of polynomials vanishing on partial diagonals this elucidates the work of k gawedzki

self duality for the two component asymmetric simple exclusion process we study a two component asymmetric simple exclusion process asep that is equivalent to the asep with second class particles we prove self duality with respect to a family of duality functions which are shown to arise from the reversible measures of the process and the symmetry of the generator under the quantum algebra uq x1d524 x1d529 3 we construct all invariant measures in explicit form and discuss some of their properties we also prove a sum rule for the duality functions

phase transitions in an exactly soluble one dimensional exclusion process we consider an exclusion process with particles injected with rate at the origin and removed with rate at the right boundary of a one dimensional chain of sites the particles are allowed to hop onto unoccupied sites to the right only for the special case of 1 the model was solved previously by derridaet al here we extend the solution to general the phase diagram obtained from our exact solution differs from the one predicted by the mean field approximation

q deformations of the o 3 symmetric spin 1 heisenberg chain the authors present the general expression for the spin 1 heisenberg chain invariant under the uq so 3 quantum algebra several physical and mathematical implications are discussed

unitarity of the knizhnik zamolodchikov bernard connection and the bethe ansatz for the elliptic hitchin systems we work out finite dimensional integral formulae for the scalar product of genus one states of the groupg chern simons theory with insertions of wilson lines assuming convergence of the integrals we show that unitarity of the elliptic knizhnik zamolodchikov bernard connection with respect to the scalar product of cs states is closely related to the bethe ansatz for the commuting hamiltonians building up the connection and quantizing the quadratic hamiltonians of the elliptic hitchin system

critical discontinuous phase transition in the threshold contact process we analyze a threshold contact process on a square lattice in which particles are created on empty sites with at least two neighboring particles and are annihilated spontaneously we show by means of monte carlo simulations that the process undergoes a discontinuous phase transition at a definite value of the annihilation parameter in accordance with the gibbs phase rule and that the discontinuous transition exhibits critical behavior the simulations were performed by using boundary conditions in which the sites of the border of the lattice are permanently occupied by particles

exact solution of the master equation for the asymmetric exclusion process using the bethe ansatz we obtain the exact solution of the master equation for the totally asymmetric exclusion process on an infinite one dimensional lattice we derive explicit expressions for the conditional probabilitiesp x1 xn t y1 yn 0 of findingn particles on lattices sitesx1 xn at timet with initial occupationy1 yn at timet 0

duality relations for asymmetric exclusion processes we derive duality relations for a class ofu q su 2 symmetric stochastic processes including among others the asymmetric exclusion process in one dimension like the known duality relations for symmetric hopping processes these relations express certainm point correlation functions inn particle systems n m in terms of sums of correlation functions of the same system but with onlym particles for the totally asymmetric case we obtain exact expressions for some boundary density correlation functions the dynamical exponent for these correlators isz 2 which is different from the dynamical exponent for bulk density correlations which is known to bez 3 2

reaction diffusion processes described by three state quantum chains and integrability the master equation of one dimensional three species reaction diffusion processes is mapped onto an imaginary time schrodinger equation in many cases the hamiltonian obtained is that of an integrable quantum chain with known properties within this approach we search for three state integrable quantum chains with known spectra and which are related to diffusive reactive systems two integrable models are found to appear naturally in this context the uqsu 2 invariant model with external fields and the three state uqsu p m invariant perk schultz models with external fields a non local similarity transformation which brings the hamiltonian governing the chemical processes to the known standard forms is described leading in the case of periodic boundary conditions to a generalization of the dzialoshinsky moriya interaction for n state hamiltonians n 2

factorisation of generalised theta functions i we prove a version of factorisation relating the space of sections of theta bundles on the moduli spaces of parabolic rank 2 vector bundles on curves of genusg andg 1

an abelianization of su 2 wzw model the purpose of this paper is to carry out the abelianization program pro posed by atiyah 1 and hitchin 9 for the geometric quantization of su 2 wess zumino witten model let c be a riemann surface of genus g let mg be the moduli space of semi stable rank 2 holomorphic vector bundles on c with trivial determinant for a positive integer k let t mg ck be the space of holomorphic sections of the fc th tensor product of the determinant line bundle c on mg an element of t mg ck is called a rank 2 theta function of level k the main result of our abelianization is to give an explicit representation of a base of t mg ck as well as its transformation formula in terms of classical riemann theta functions with automorphic form coefficients defined on the prym variety p associated with a two fold branched covering surface c of c r mg ck can be identified with the conformai block of level k of the su 2 wzw model 5 15 the abelianization procedure enables us to de duce the various known results about the conformai block in a uniform way firstly we construct a projectively flat connection on the vector bundle over the teichm ller space with fibre t mg ck secondly making use of our ex plicit representation of rank 2 theta functions we construct a hermitian product on the vector bundle preserved by the connection also our explicit represen tation enables us to prove that y mg ck has the predicted dimension from the quantum clebsh gordan conditions a natural connection on the said vector bundle for the su iv wzw model was first constructed by hitchin 11 it will turn out that the connection constructed in this paper coincides with the hitchin connection laszlo 16 showed that the hitchin connection coincides with the con nection constructed by tsuchiya ueno and yamada 21 through the above identification on the other hand kirillov 13 14 constructed a hermitian product on the conformai block compatible with the tsuchiya ueno yamada connection using the representation theory of affine lie algebras together with the theory of hermitian modular tensor categories cf 22 laszlo s result

time dependent correlation functions in a one dimensional asymmetric exclusion process we study a one dimensional anisotropic exclusion process describing particles injected at the origin moving to the right on a chain of l sites and being removed at the right boundary we construct the steady state and compute the density profile exact expressions for all equal time n point density correlation functions and the time dependent two point function in the steady state as functions of the injection and absorption rates we determine the phase diagram of the model and compare our results with predictions from dynamical scaling and discuss some conjectures for other exclusion models

cellular automation models and traffic flow a cellular automaton model for the description of traffic flow is investigated it generalizes asymmetric exclusion models which have attracted a lot of interest in the past the authors calculate the so called fundamental diagram flow versus density for parallel dynamics using an improved mean field approximation which takes into account short range correlations for maximum velocity they find that the simplest of these non trivial approximations gives the exact result for higher velocities their results are in excellent agreement with numerical data

a six vertex model as a diffusion problem derivation of correlation functions a cellular automaton which describes diffusion of particles with exclusion in one dimension is shown to be equivalent to a six vertex model on a critical line the arrow arrow correlation function of the six vertex model is calculated exactly on this line using a transfer matrix method

generalized bethe ansatz solution of a one dimensional asymmetric exclusion process on a ring with blockage we present a model for a one dimensional anisotropic exclusion process describing particles moving deterministically on a ring of lengthl with a single defect across which they move with probability 0 p 1 this model is equivalent to a two dimensional six vertex model in an extreme anisotropic limit with a defect line interpolating between open and periodic boundary conditions we solve this model with a bethe ansatz generalized to this kind of boundary condition we discuss in detail the steady state and derive exact expressions for the currentj the density profilen x and the two point density correlation function in the thermodynamic limitl the phase diagram shows three phases a low density phase a coexistence phase and a high density phase related to the low density phase by a particle hole symmetry in the low density phase the density profile decays exponentially with the distance from the boundary to its bulk value on a length scale on the phase transition line diverges and the currentj approaches its critical valuejc p as a power law jc j 1 2 in the coexistence phase the width of the interface between the high density region and the low density region is proportional tol1 2 if the density f 1 2 and 0 independent ofl if 1 2 the connected two point correlation function turns out to be of a scaling form with a space dependent amplitude n x1 x2 a x2 ake r withr x2 x1 and a critical exponent 0

matrix product ground states for exclusion processes with parallel dynamics we show in the example of a one dimensional asymmetric exclusion process that stationary states of models with parallel dynamics may be written in a matrix product form the corresponding algebra is quadratic and involves three different matrices using this formalism we prove previous conjectures for the equal time correlation functions of the model

non abelian symmetries of stochastic processes derivation of correlation functions for random vertex models and disordered interacting particle systems we consider systems of particles hopping stochastically on d dimensional lattices with space dependent probabilities we map the master equation onto an evolution equation in a fock space where the dynamics are given by a quantum hamiltonian continuous time or a transfer matrix discrete time using non abelian symmetries of these operators we derive duality relations expressing the time evolution of a given initial configuration in terms of correlation functions of simpler dual processes particularly simple results are obtained for the time evolution of the density profile as a special case we show that for any su 2 symmetric system the two point and three point density correlation functions in the n particle steady state can be computed from the probability distribution of a single particle moving in the same environment we apply our results to various models among them partial exclusion a simple diffusion reaction system and the two dimensional six vertex model with space dependent vertex weights for a random distribution of the vertex weights one obtains a version of the random barrier model describing diffusion of particles in disordered media we derive exact expressions for the averaged two point density correlation functions in the presence of weak correlated disorder