quantum kz equation with q 1 and correlation functions of the xxz model in the gapless regime an integral solution to the quantum knizhnik zamolodchikov qkz equation with q 1 is presented upon specialization it leads to a conjectural formula for correlation functions of the xxz model in the gapless regime the validity of this conjecture is verified in special cases including the nearest neighbour correlator with an arbitrary coupling constant and general correlators in the xxx and xy limits

new formulae for solutions of quantum knizhnik zamolodchikov equations on level 4 we present a new form of solution to the quantum knizhnik zamolodchikov equation qkz on level 4 in a special case corresponding to the heisenberg xxx spin chain our form is equivalent to the integral representation obtained by jimbo and miwa in 1996 7 an advantage of our form is that it is reduced to the product of single integrals this fact is deeply related to a cohomological nature of our formulae our approach is also based on the deformation of hyper elliptic integrals and their main property deformed riemann bilinear relation jimbo and miwa also suggested a nice conjecture which relates solution of the qkz on level 4 to any correlation function of the xxx model this conjecture together with our form of solution to the qkz makes it possible to prove a conjecture that any correlation function of the xxx model can be expressed in terms of the riemann function at odd arguments and rational coefficients suggested in 8 9 this issue will be discussed in our next publication

curvature and rank of teichmuller space let 5 be a surface with genus g and n boundary components and let d s 3g 3 n denote the number of curves in any pants decomposition of s we employ metric properties of the graph of pants decompositions p s to prove that the weil petersson metric on teichmiiller space teich s is gromov hyperbolic if and only if d s 3 the weil petersson metric has higher rank in the sense of gromov it admits a quasi isometric embedding of rk k 2 when d s 3 and that no complete riemannian metric of pinched negative curvature exists on moduli space m s when d s 2 1 introduction the weil petersson metric on teichmiiller space teich 5 has many curious properties it is a riemannian metric with negative sectional curvature but its curvatures are not bounded away from zero or negative infinity it is geodesically convex but it is not complete in this paper we show that in spite of exhibiting negative curvature behavior the weil petersson metric is not coarsely negatively curved except for topologically simple surfaces s our main theorem answers a question of bowditch be question 11 4

the weil petersson hessian of length on teichmuller space we present a brief but nearly self contained proof of a formula for the weil petersson hessian of the geodesic length of a closed curve either simple or not simple on a hyperbolic surface the formula is the sum of the integrals of two naturally defined positive functions over the geodesic proving convexity of this functional over teichmuller space due to wolpert 1987 we then estimate this hessian from below in terms of local quantities and distance along the geodesic the formula extends to proper arcs on punctured hyperbolic surfaces and the estimate to laminations wolpert s result that the thurston metric is a multiple of the weil petersson metric directly follows on taking a limit of the formula over an appropriate sequence of curves we give further applications to upper bounds of the hessian especially near pinching loci recover through a geometric argument wolpert s result on the convexity of length to the half power and give a lower bound for growth of length in terms of twist

cohomologies of affine hyperelliptic jacobi varieties and integrable systems we study the affine ring of the affine jacobi variety of a hyperelliptic curve the matrix construction of the affine hyperelliptic jacobi varieties due to mumford is used to calculate the character of the affine ring by decomposing the character we make several conjectures on the cohomology groups of the affine hyperelliptic jacobi varieties in the integrable system described by the familly of these affine hyperelliptic jacobi varieties the affine ring is closely related to the algebra of functions on the phase space classical observables we show that the affine ring is generated by the highest cohomology group over the action of the invariant vector fields on the jacobi variety

the geometry of the universal teichmuller space and the euler weil petersson equation on the identity component of the universal teichmuller space endowed with the takhtajan teo topology the geodesics of the weil petersson metric are shown to exist for all time this component is naturally a subgroup of the quasisymmetric homeomorphisms of the circle viewed this way the regularity of its elements is shown to be h3 2 epsilon for all epsilon 0 the evolutionary pde associated to the spatial representation of the geodesics of the weil petersson metric is derived using multiplication and composition below the critical sobolev index 3 2 geodesic completeness is used to introduce special classes of solutions of this pde analogous to peakons our setting is used to prove that there exists a unique geodesic between each two shapes in the plane in the context of the application of the weil petersson metric in imaging c 2015 elsevier inc all rights reserved

reduced qkz equation and correlation functions of the xxz model correlation functions of the xxz model in the massive and massless regimes are known to satisfy a system of linear equations the main relations among them are the difference equations obtained from the qkz equation by specializing the variables 1 2n as 1 n n 1 1 1 we call it the reduced qkz equation in this article we construct a special family of solutions to this system they can be written as linear combinations of products of two transcendental functions open image in new window with coefficients being rational functions we show that correlation functions of the xxz model in the massive regime are given by these formulas with an appropriate choice of open image in new window we also present a conjectural formula in the massless regime

the weil petersson curvature operator on the universal teichmuller space the universal teichmuller space is an infinitely dimensional generalization of the classical teichmuller space of riemann surfaces it carries a natural hilbert structure on which one can define a natural riemannian metric the weil petersson metric in this paper we investigate the weil petersson riemannian curvature operator tilde q of the universal teichmuller space with the hilbert structure and prove the following i tilde q is non positive definite ii tilde q is a bounded operator iii tilde q is not compact the set of the spectra of tilde q is not discrete as an application we show that neither the quaternionic hyperbolic space nor the cayley plane can be totally geodesically immersed in the universal teichmuller space endowed with the weil petersson metric

quantum spin chains and riemann zeta function with odd arguments the riemann zeta function is an important object of number theory we argue that it is related to the heisenberg spin 1 2 anti ferromagnet in the xxx spin chain we study the probability of formation of a ferromagnetic string in the anti ferromagnetic ground state in the thermodynamics limit we prove that for short strings the probability can be expressed in terms of the riemann zeta function with odd arguments

canonical metrics on the moduli space of riemann surfaces i in this paper we continue our study of the canonical metrics on the moduli space of curves we first prove the bounded geometry of the ricci and perturbed ricci metrics by carefully choosing the pertubation constant and by studying the asymptotics we show that the ricci and holomorphic sectional curvatures of the perturbed ricci metric are bounded from above and below by negative constants based on our understanding of the kahler einstein metric we show that the logarithmic cotangent bundle over the deligne mumford moduli space is stable with respect to the canonical polarization finally in the last section we prove the strongly bounded geometry of the kahler einstein metric by using the kahler ricci flow and a priori estimates of the complex monge ampere equation

the riemannian sectional curvature operator of the weil petersson metric and its application fix a number g gt 1 let s be a close surface of genus g and let mathrm teich s be the teichmuller space of s endowed with the weil petersson metric in this paper we show that the riemannian sectional curvature operator of mathrm teich s is non positive definite as an application we show that any twist harmonic map from rank one hyperbolic spaces h q m sp m 1 sp m cdot sp 1 or h o 2 f 4 20 so 9 into mathrm teich s is a constant

the weak ekeland variational principle and fixed points abstract we use the weak ekeland variational principle to obtain a new proof of an improved version of the nadler fixed point theorem for set valued mappings given by dontchev and hager 1994 separately we apply the fixed point theorem to obtain an unconventional existence result for differential inclusions

a q operator identity for the correlation functions of the infinite xxz spin chain an explicit construction for q operators of the finite xxz spin chain with twisted boundary conditions is presented the massless and the massive regimes are considered as well as the root of unity case it is explained how these results yield an alternative expression for the trace function employed in the description of the correlation functions of the inhomogeneous xxz model on the infinite lattice by boos jimbo miwa smirnov and takeyama

traces on the sklyanin algebra and correlation functions of the eight vertex model we propose a conjectural formula for correlation functions of the z invariant inhomogeneous eight vertex model we refer to this conjecture as ansatz it states that correlation functions are linear combinations of products of three transcendental functions with theta functions and derivatives as coefficients the transcendental functions are essentially logarithmic derivatives of the partition function per site the coefficients are given in terms of a linear functional tr on the sklyanin algebra which interpolates the usual trace on finite dimensional representations we establish the existence of tr and discuss the connection to the geometry of the classical limit we also conjecture that the ansatz satisfies the reduced qkz equation as a non trivial example of the ansatz we present a new formula for the next nearest neighbour correlation functions

a general formula for soliton form factors in the quantum sine gordon model a general formula for soliton form factors of several local fields in the quantum sine gordon model is presented

finite temperature correlation functions from discrete functional equations we present a new approach to the static finite temperature correlation functions of the heisenberg chain based on functional equations an inhomogeneous generalization of the n site density operator is considered the lattice path integral formulation with a finite but arbitrary trotter number allows us to derive a set of discrete functional equations with respect to the spectral parameters we show that these equations yield a unique characterization of the density operator our functional equations are a discrete version of the reduced q knizhnik zamolodchikov equations which played a central role in the study of the zero temperature case as a natural result and independent of the arguments given by jimbo et al 2009 j phys a math theor 42 304018 we prove that the inhomogeneous finite temperature correlation functions have the same remarkable structure as for zero temperature they are a sum of products of nearest neighbour correlators