Showing papers in "Communications in Mathematical Physics in 1991"

On quantum gauge theories in two dimensions

Abstract: Two dimensional quantum Yang-Mills theory is studied from three points of view: (i) by standard physical methods; (ii) by relating it to the largek limit of three dimensional Chern-Simons theory and two dimensional conformal field theory; (iii) by relating its weak coupling limit to the theory of Reidemeister-Ray-Singer torsion. The results obtained from the three points of view agree and give formulas for the volumes of the moduli spaces of representations of fundamental groups of two dimensional surfaces.

Discrete versions of some classical integrable systems and factorization of matrix polynomials

Abstract: Discrete versions of several classical integrable systems are investigated, such as a discrete analogue of the higher dimensional force-free spinning top (Euler-Arnold equations), the Heisenberg chain with classical spins and a new discrete system on the Stiefel manifold. The integrability is shown with the help of a Lax-pair representation which is found via a factorization of certain matrix polynomials. The complete description of the dynamics is given in terms of Abelian functions; the flow becomes linear on a Prym variety corresponding to a spectral curve. The approach is also applied to the billiard problem in the interior of anN-dimensional ellipsoid.

The Proper Formula for Relative Entropy and its Asymptotics in Quantum Probability

Abstract: Umegaki's relative entropyS(ω,ϕ)=TrDω(logDω−logDϕ) (of states ω and ϕ with density operatorsDω andDϕ, respectively) is shown to be an asymptotic exponent considered from the quantum hypothesis testing viewpoint. It is also proved that some other versions of the relative entropy give rise to the same asymptotics as Umegaki's one. As a byproduct, the inequality TrA logAB ≧TrA(logA+logB) is obtained for positive definite matricesA andB.

Universal R-matrix for quantized (super)algebras

Abstract: For quantum deformations of finite-dimensional contragredient Lie (super)algebras we give an explicit formula for the universalR-matrix. This formula generalizes the analogous formulae for quantized semisimple Lie algebras obtained by M. Rosso, A. N. Kirillov, and N. Reshetikhin, Ya. S. Soibelman, and S. Z. Levendorskii. Our approach is based on careful analysis of quantized rank 1 and 2 (super)algebras, a combinatorial structure of the root systems and algebraic properties ofq-exponential functions. We don't use quantum Weyl group.

An example of a generalized Brownian motion

Abstract: We present an example of a generalized Brownian motion. It is given by creation and annihilation operators on a “twisted” Fock space ofL2(ℝ). These operators fulfill (for a fixed −1≦μ≦1) the relationsc(f)c*(g)−μc*(g)c(f)=〈f,g〉1 (f, g ∈L2(ℝ)). We show that the distribution of these operators with respect to the vacuum expectation is a generalized Gaussian distribution, in the sense that all moments can be calculated from the second moments with the help of a combinatorial formula. We also indicate that our Brownian motion is one component of ann-dimensional Brownian motion which is invariant under the quantum groupS ν U(n) of Woronowicz (withμ =v2).

Quantum affine algebras

Abstract: We classify the finite-dimensional irreducible representations of the quantum affine algebra $$U_q (\hat sl_2 )$$ in terms of highest weights (this result has a straightforward generalization for arbitrary quantum affine algebras). We also give an explicit construction of all such representations by means of an evaluation homomorphism $$U_q (\hat sl_2 ) \to U_q (sl_2 )$$ , first introduced by M. Jimbo. This is used to compute the trigonometricR-matrices associated to finite-dimensional representations of $$U_q (\hat sl_2 )$$ .

Discrete Painlevé equations and their appearance in quantum gravity

Abstract: We discuss an algorithmic approach for both deriving discrete analogues of Painleve equations as well as using such equations to characterize “similarity” reductions of spatially discrete integrable evolution equations. As a concrete example we show that a discrete analogue of Painleve I can be used to characterize “similarity” solutions of the Kac-Moerbeke equation. It turns out that these similarity solutions also satisfy a special case of Painleve IV equation. In addition we discuss a methodology for obtaining the relevant continuous limits not only at the level of equations but also at the level of solutions. As an example we use the WKB method in the presence of two turning points of the third order to parametrize (at the continuous limit) the solution of Painleve I in terms of the solution of discrete Painleve I. Finally we show that these results are useful for investigating the partition function of the matrix model in 2D quantum gravity associated with the measure exp [−t1z2−t2z4−t3z6].

Quantization of Chern-Simons gauge theory with complex gauge group

Abstract: The canonical quantization of Chern-Simons gauge theory in 2+1 dimensions is generalized from the case in which the gauge group is a compact Lie groupG to the case in which the gauge group is a complex Lie groupG ℂ. Though the physical Hilbert spaces become infinite dimensional in the latter case, the quantization can be described as precisely as for compact gauge groups and using similar methods. The special case in which the gauge group isSL(2,ℂ) gives a description of 2+1 dimensional quantum gravity with Lorentz signature and positive cosmological constant or with Euclidean signature and negative cosmological constant. While it is not clear whether there is a 1+1 dimensional conformal field theory related to these 2+1 dimensional models, there are natural, computable candidates for the central charge and the conformal blocks of such a hypothetical theory.

Quantum group symmetries and non-local currents in 2D QFT

Abstract: We construct and study the implications of some new non-local conserved currents that exist is a wide variety of massive integrable quantum field theories in 2 dimensions, including the sine-Gordon theory and its generalization to affine Toda theory. These non-local currents provide a non-perturbative formulation of the theories. The symmetry algebras correspond to the quantum affine Kac-Moody algebras. TheS-matrices are completely characterized by these symmetries. FormalS-matrices for the imaginary-coupling affine Toda theories are thereby derived. The application of theseS-matrices to perturbed coset conformal field theory is studied. Non-local charges generating the finite dimensional Quantum Group in the Liouville theory are briefly presented. The formalism based on non-local charges we describe provides an algernative to the quantum inverse scattering method for solving integrable quantum field theories in 2d.

Unbounded elements affiliated with C *-algebras and non-compact quantum groups

Abstract: The affiliation relation that allows to include unbounded elements (operators) into theC *-algebra framework is introduced, investigated and applied to the quantum group theory. The quantum deformation of (the two-fold covering of) the group of motions of Euclidean plane is constructed. A remarkable radius quantization is discovered. It is also shown that the quantumSU(1, 1) group does not exist on theC *-algebra level for real value of the deformation parameter.

Long range scattering for nonlinear Schrödinger equations in one space dimension

Abstract: We consider the scattering problem for the nonlinear Schrodinger equation in 1+1 dimensions: Open image in new window where ∂ = ∂/∂x,λ∈R∖{0},μ∈R,p>3. We show that modified wave operators for (*) exist on a dense set of a neighborhood of zero in the Lebesgue spaceL2(R) or in the Sobolev spaceH1(R)., The modified wave operators are introduced in order to control the long range nonlinearity λ|u|2u.

Hydrodynamic limit for attractive particle systems on 417-1417-1417-1

Abstract: We study the hydrodynamic behavior of asymmetric simple exclusions and zero range processes in several dimensions. Under Euler scaling, a nonlinear conservation law is derived for the time evolution of the macroscopic particle density.

Fusion rings and geometry

Abstract: The algebraic structure of fusion rings in rational conformal field theories is analyzed in detail in this paper. A formalism which closely parallels classical tools in the study of the cohomology of homogeneous spaces is developed for fusion rings, in general, and for current algebra theories, in particular. It is shown that fusion rings lead to a natural orthogonal polynomial structure. The rings are expressed through generators and relations. The relations are then derived from some potentials leading to an identification of the fusion rings with deformations of affine varieties. In general, the fusion algebras are mapped to affine varieties which are the locus of the relations. The connection with modular transformations is investigated in this picture. It is explained how chiral algebras, arising inN=2 superconformal field theory, can be derived from fusion rings. In particular, it is argued that theories of the typeSU(N) k /SU(n−1) are theN=2 counterparts of Grassmann manifolds and that there is a natural identification of the chiral fields with Schubert varieties, which is a graded algebra isomorphism.

Computer calculation of Witten's $3$-manifold invariant

Abstract: Witten's 2+1 dimensional Chern-Simons theory is exactly solvable. We compute the partition function, a topological invariant of 3-manifolds, on generalized Seifert spaces. Thus we test the path integral using the theory of 3-manifolds. In particular, we compare the exact solution with the asymptotic formula predicted by perturbation theory. We conclude that this path integral works as advertised and gives an effective topological invariant.

Every gauge orbit passes inside the Gribov horizon

Abstract: TheL 2 topology is introduced on the space of gauge connectionsA and a natural topology is introduced on the group of local gauge transformationsGT. It is shown that the mappingGT×A→A defined byA→A g=g*Ag+g*dg is continuous and that each gauge orbit is closed. The Hilbert norm of the gauge connection achieves its absolute minimum on each gauge orbit, at which point the orbit intersects the region bounded by the Gribov horizon.

Conformal field theories via Hamiltonian reduction

Abstract: Constraining theSL(3) WZW-model we construct a reduced theory which is invariant with respect to the new chiral algebraW 3 2 . This symmetry is generated by the stress-energy tensor, two bosonic currents with spins 3/2 and theU(1) current. We conjecture a Kac formula that describes the highly reducible representation for this algebra. We also discuss the quantum Hamiltonian reduction for the general type of constraints that leads to the new extended conformal algebras.

Algebras of functions on compact quantum groups, Schubert cells and quantum tori

Abstract: The structures of Poisson Lie groups on a simple compact group are parametrized by pairs (a, u), wherea∈R,\(u \in \Lambda ^2 \mathfrak{h}_R\), and\(\mathfrak{h}_R\) is a real Cartan subalgebra of complexification of Lie algebra of the group in question. In the present article the description of the symplectic leaves for all pairs (a,u) is given. Also, the corresponding quantized algebras of functions are constructed and their irreducible representations are described. In the course of investigation Schubert cells and quantum tori appear. At the end of the article the quantum analog of the Weyl group is constructed and some of its applications, among them the formula for the universalR-matrix, are given.

Hidden Yangians in 2D massive current algebras

Abstract: We define non-local conserved currents in massive current algebras in two dimensions. Our approach is algebraic and non-perturbative. The non-local currents give a quantum field realization of the Yangians. We show how the noncocommutativity of the Yangians is related to the non-locality of the currents. We discuss the implications of the existence of non-local conserved charges on theS-matrices.

Critical droplets and metastability for a Glauber dynamics at very low temperatures

Abstract: We consider the metastable behavior in the so-called pathwise approach of a ferromagnetic spin system with a Glauber dynamics in a finite two dimensional torus under a positive magnetic field in the limit as the temperature goes to zero. First we consider the evolution starting from a single rectangular droplet of spins +1 in a sea of spins −1. We show that small droplets are likely to disappear while large droplets are likely to grow; the threshold between the two cases being sharply defined and depending only on the external field. This result is used to prove that starting from the configuration with all spins down (−1) the pattern of evolution leading to the more stable configuration with all spins up (+1) approaches, as the temperature vanishes, a metastable behavior: the system stays close to −1 for an unpredictable time until a critical square droplet of a precise size is eventually formed and nucleates the decay to +1 in a relatively short time. The asymptotic magnitude of the total decay time is shown to be related to the height of an energy barrier, as expected from heuristic and mean field studies of metastability.

Quasiperiodic motions in superquadratic time-periodic potentials

Abstract: It is shown that for a large class of potentials on the line with superquadratic growth at infinity and with the additional time-periodic dependence all possible motions under the influence of such potentials are bounded for all time and that most (in a precise sense) motions are in fact quasiperiodic. The class of potentials includes, as very particular examples, the exponential, polynomial and much more. This extends earlier results and gives an answer to a problem posed by Littlewood in the mid 1960's. Along the way machinery is developed for estimating the action-angle transformation directly in terms of the potential and also some apparently new identities involving singular integrals are derived.

The bi-Hamiltonian structure of fully supersymmetric Korteweg-de Vries systems

Abstract: The bi-Hamiltonian structure of integrable supersymmetric extensions of the Korteweg-de Vries (KdV) equation related to theN=1 and theN=2 superconformal algebras is found. It turns out that some of these extensions admit inverse Hamiltonian formulations in terms of presymplectic operators rather than in terms of Poisson tensors. For one extension related to theN=2 case additional symmtries are found with bosonic parts that cannot be reduced to symmetries of the classical KdV. They can be explained by a factorization of the corresponding Lax operator. All the bi-Hamiltonian formulations are derived in a systematic way from the Lax operators.

Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes

Abstract: Spacelike hypersurfaces of prescribed mean curvature in cosmological spacetimes are constructed as asymptotic limits of a geometric evolution equation. In particular, an alternative, constructive proof is given for the existence of maximal and constant mean curvature slices.

Semi-global existence and convergence of solutions of the Robinson-Trautman (2-dimensional Calabi) equation

Abstract: It is shown that for smooth initial data solutions of the Robinson-Trautman equation (also known as the two-dimensional Calabi equation) exist for all positive “times,” and asymptotically converge to a constant curvature metric.

An integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows

Abstract: Precise necessary and sufficient conditions on the velocity statistics for mean field behavior in advection-diffusion by a steady incompressible velocity field are developed here. Under these conditions, a rigorous Stieltjes integral representation for effective diffusivity in turbulent transport is derived. This representation is valid for all Peclet numbers and provides a rigorous resummation of the divergent perturbation expansion in powers of the Peclet number. One consequence of this representation is that convergent upper and lower bounds on effective diffusivity for all Peclet numbers can be obtained utilizing a prescribed finite number of terms in the perturbation series. Explicit rigorous examples of steady incompressible velocity fields are constructed which have effective diffusivities realizing the simplest upper or lower bounds for all Peclet numbers. A nonlocal variational principle for effective diffusivity is developed along with applications to advection-diffusion by random arrays of vortices. A new class of rigorous examples is introduced. These examples have an explicit Stieltjes measure for the effective diffusivity; furthermore, the effective diffusivity behaves likek0(Pe)1/2 in the limit of large Peclet numbers wherek0 is the molecular diffusivity. Formal analogies with the theory of composite materials are exploited systematically.

A kinetic equation with kinetic entropy functions for scalar conservation laws

Abstract: We construct a nonlinear kinetic equation and prove that it is welladapted to describe general multidimensional scalar conservation laws. In particular we prove that it is well-posed uniformly in e — the microscopic scale. We also show that the proposed kinetic equation is equipped with a family of kinetic entropy functions — analogous to Boltzmann's microscopicH-function, such that they recover Krushkov-type entropy inequality on the macroscopic scale. Finally, we prove by both — BV compactness arguments in the multidimensional case and by compensated compactness arguments in the one-dimensional case, that the local density of kinetic particles admits a “continuum” limit, as it converges strongly with e↓0 to the unique entropy solution of the corresponding conservation law.

Interfaces in the Potts model I: Pirogov-Sinai theory of the Fortuin-Kasteleyn representation

Abstract: We develop a new analysis of the order-disorder transition in ferromagnetic Potts models for large numberq of spin states We use the Pirogov-Sinai theory which we adapt to the Fortuin-Kasteleyn representation of the models This theory applies in a rather direct way in our approach and leads to a system of non-interacting contours with small activities As a consequence, simpler and more natural techniques are found, allowing us to recover previous results on the bulk properties of the model (which then extend to non-integer values ofq) and to deal with non-translation invariant boundary conditions This will be applied in a second part of this work to study the behaviour of the interfaces at the transition point

Free fermions and the Alexander-Conway polynomial

Abstract: We show how the Conway Alexander polynomial arises from theq deformation of (Z 2 graded)sl(n, n) algebras. In the simplestsl(1, 1) case we then establish connection between classical knot theory and its modern versions based on quantum groups. We first shown how the crystal and the fundamental group of the complement of a knot give rise naturally to the Burau representation of the braid group. The Burau matrix is then transformed into theU q sl(1, 1) R matrix by going to the exterior power algebra. Using a det=str identity, this allows us to recover the state model of [K2, 89] as well. We also show how theU q> sl(1, 1) algebra describes free fermions “propagating” on the knot diagram. We rewrite the Conway Alexander polynomial as a Berezin integral, and thus as an apparently new determinant.

Positive Lyapunov exponents for Schrödinger operators with quasi-periodic potentials

Abstract: We present a new, simple way to estimate the rate of exponential growth (Lyapunov exponent) of solutions of the finite-difference Schrodinger equation: $$((H - E)\psi )(n)\mathop = \limits^{def} - [\psi (n + 1) + \psi (n - 1)] + [\lambda f(\alpha n + \theta )]\psi (n).$$ Heref is a non-constant real-analytic function of period 1 and α is irrational. For λ large we prove that the Lyapunov exponent is positive for every energyE in the spectrum ofH and a.e. θ. In particular, the absolutely continuous spectrum ofH is empty. In the continuum we study the quasi-periodic operator onL2(R) $$H = - \frac{{d^2 }}{{dx^2 }} - K^2 [\cos x + \cos (\alpha x + \theta )]$$ for largeK and show that for wide intervals of low energies the Lyapunov exponent is positive. The main idea, which originated from M. Herman's subharmonic argument [11], is to deform the phase θ to the complex plane. This enables us to avoid small denominator problems by moving them off the axis, making estimates much easier to perform. We recover the information for real θ using an elementary extension of Jensen's formula (subharmonicity).

Spectral Properties of a Tight Binding Hamiltonian with Period Doubling Potential

Abstract: We study a one dimensional tight binding hamiltonian with a potential given by the period doubling sequence. We prove that its spectrum is purely singular continuous and supported on a Cantor set of zero Lebesgue measure, for all nonzero values of the potential strength. Moreover, we obtain the exact labelling of all spectral gaps and compute their widths asymptotically for small potential strength.

Smooth static solutions of the Einstein/Yang-Mills equations

Abstract: We consider the Einstein/Yang-Mills equations in 3+1 space time dimensions withSU(2) gauge group and prove rigorously the existence of a globally defined smooth static solution We show that the associated Einstein metric is asymptotically flat and the total mass is finite Thus, for non-abelian gauge fields the Yang-Mills repulsive force can balance the gravitational attractive force and prevent the formation of singularities in spacetime