Showing papers in "Communications in Mathematical Physics in 1990"

Ribbon graphs and their invariants derived from quantum groups

Abstract: The generalization of Jones polynomial of links to the case of graphs inR 3 is presented. It is constructed as the functor from the category of graphs to the category of representations of the quantum groups.

Topological Gauge Theories and Group Cohomology

Abstract: We show that three dimensional Chern-Simons gauge theories with a compact gauge groupG (not necessarily connected or simply connected) can be classified by the integer cohomology groupH4(BG,Z). In a similar way, possible Wess-Zumino interactions of such a groupG are classified byH3(G,Z). The relation between three dimensional Chern-Simons gauge theory and two dimensional sigma models involves a certain natural map fromH4(BG,Z) toH3(G,Z). We generalize this correspondence to topological “spin” theories, which are defined on three manifolds with spin structure, and are related to what might be calledZ2 graded chiral algebras (or chiral superalgebras) in two dimensions. Finally we discuss in some detail the formulation of these topological gauge theories for the special case of a finite group, establishing links with two dimensional (holomorphic) orbifold models.

Crystalizing the $q$-analogue of universal enveloping algebras

Abstract: For an irreducible representation of theq-analogue of a universal enveloping algebra, one can find a canonical base atq=0, named crystal base (conjectured in a general case and proven forA n, Bn, Cn andD n). The crystal base has a structure of a colored oriented graph, named crystal graph. The crystal base of the tensor product (respectively the direct sum) is the tensor product (respectively the union) of the crystal base. The crystal graph of the tensor product is also explicitly described. This gives a combinatorial description of the decomposition of the tensor product into irreducible components.

Quantum deformation of lorentz group

Abstract: A one parameter quantum deformationSμL(2,ℂ) ofSL(2,ℂ) is introduced and investigated. An analog of the Iwasawa decomposition is proved. The compact part of this decomposition coincides withSμU(2), whereas the solvable part is identified as a Pontryagin dual ofSμU(2). It shows thatSμL(2,ℂ) is the result of the dual version of Drinfeld's double group construction applied toSμU(2). The same construction applied to any compact quantum groupGc is discussed in detail. In particular the explicit formulae for the Haar measures on the Pontryagin dualGd ofGc and on the double groupG are given. We show that there exists remarkable 1-1 correspondence between representations ofG and bicovariant bimodules (“tensor bundles”) overGc. The theory of smooth representations ofSμL(2,ℂ) is the same as that ofSL(2,ℂ) (Clebsh-Gordon coefficients are however modified). The corresponding “tame” bicovariant bimodules onSμU(2) are classified. An application to 4D+ differential calculus is presented. The nonsmooth case is also discussed.

Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory

Abstract: In this paper the nonlinear wave equation $$u_u - u_{xx} + v(x)u(x,t) + \varepsilon u^3 (x,t) = 0$$ is studied. It is shown that for a large class of potentials,v(x), one can use KAM methods to construct periodic and quasi-periodic solutions (in time) for this equation.

Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics

Abstract: Given the local observables in the vacuum sector fulfilling a few basic principles of local quantum theory, we show that the superselection structure, intrinsically determined a priori, can always be described by a unique compact global gauge group acting on a field algebra generated by field operators which commute or anticommute at spacelike separations. The field algebra and the gauge group are constructed simultaneously from the local observables. There will be sectors obeying parastatistics, an intrinsic notion derived from the observables, if and only if the gauge group is non-Abelian. Topological charges would manifest themselves in field operators associated with spacelike cones but not localizable in bounded regions of Minkowski space. No assumption on the particle spectrum or even on the covariance of the theory is made. However the notion of superselection sector is tailored to theories without massless particles. When translation or Poincare covariance of the vacuum sector is assumed, our construction leads to a covariant field algebra describing all covariant sectors.

Q -analogues of Clifford and Weyl algebras-spinor and oscillator representations of quantum enveloping algebras

Abstract: We introduceq-analogues of Clifford and Weyl algebras. Using these, we construct spinor and oscillator representations of quantum enveloping algebras of typeAN−1,BN,CN,DN andAN−1(1). Also we discuss the irreducibility and the unitarity of these representations.

On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential

Abstract: In this paper, we first construct multi-lump (nonlinear) bound states of the nonlinear Schrodinger equation for sufficiently small ℏ>0, in which sense we call them “semiclassical bound states.” We assume that 1≦p<∞ forn=1,2 and 1≦p<1+4/(n−2) forn≧3, and thatV is in the class(V) a in the sense of Kato for somea. For any finite collection {x 1,...,x N} of nondegenerate critical points ofV, we construct a solution of the forme −iEt/ℏv(x) forE0. Indeed, for each such collection of critical points we construct 2 N−1 distinct unstable bound states which may have nodes in general, and the above positive bound state is just one of them.

Vortices in holomorphic line bundles over closed Kähler manifolds

Abstract: We apply a modified Yang-Mills-Higgs functional to unitary bundles over closed Kahler manifolds and study the equations which govern the global minima. The solutions represent vortices in holomorphic bundles and are direct analogs of the vortices overR 2. We obtain a complete description of the moduli space of these new vortices where the bundle is of rank one. The description is in terms of a class of divisors in the base manifold. There is also a dependence on a real valued parameter which can be attributed to the compactness of the base manifold.

Flat connections and geometric quantization

Abstract: Using the space of holomorphic symmetric tensors on the moduli space of stable bundles over a Riemann surface we construct a projectively flat connection on a vector bundle over Teichmuller space. The fibre of the vector bundle consists of the global sections of a power of the determinant bundle on the moduli space. Both Dolbeault and Cech techniques are used.

Multichannel nonlinear scattering for nonintegrable equations

Abstract: We consider a class of nonlinear Schr6dinger equations (conservative and dispersive systems) with localized and dispersive solutions. We obtain a class of initial conditions, for which the asymptotic behavior (t ~ + oo) of solutions is given by a linear combination of nonlinear bound state (time periodic and spatially localized solution) of the equation and a purely dispersive part (decaying to zero with time at the free dispersion rate). We also obtain a result of asymptotic stability type: given data near a nonlinear bound state of the system, there is a nonlinear bound state of nearby energy and phase, such that the difference between the solution (adjusted by a phase) and the latter disperses to zero. It turns out that in general, the time-period (and energy) of the localized part is different for t ~ + ov from that+for t --. - or. Moreover the solution acquires an extra constant asymptotic phase e '~-.

Relativistic Toda Systems

Abstract: We present and study Poincare-invariant generalizations of the Galilei-invariant Toda systems. The classical nonperiodic systems are solved by means of an explicit action-angle transformation.

Einstein metrics on S 3 , R 3 and R 4 bundles

Abstract: Starting from a 4n-dimensional quaternionic Kahler base space, we construct metrics of cohomogeneity one in (4n+3) dimensions whose level surfaces are theS 2 bundle space of almost complex structures on the base manifold We derive the conditions on the metric functions that follow from imposing the Einstein equation, and obtain solutions both for compact and non-compact (4n+3)-dimensional spaces Included in the non-compact solutions are two Ricci-flat 7-dimensional metrics withG 2 holonomy We also discuss two other Ricci-flat solutions, one on theR 4 bundle overS 3 and the other on anR 4 bundle overS 4 These haveG 2 and Spin(7) holonomy respectively

Affine Kac-Moody algebras and semi-infinite flag manifolds

Abstract: We study representations of affine Kac-Moody algebras from a geometric point of view. It is shown that Wakimoto modules introduced in [18], which are important in conformal field theory, correspond to certain sheaves on a semi-infinite flag manifold with support on its Schhubert cells. This manifold is equipped with a remarkable semi-infinite structure, which is discussed; in particular, the semi-infinite homology of this manifold is computed. The Cousin-Grothendieck resolution of an invertible sheaf on a semi-infinite flag manifold gives a two-sided resolution of an irreducible representation of an affine algebras, consisting of Wakimoto modules. This is just the BRST complex. As a byproduct we compute the homology of an algebra of currents on the real line with values in a nilpotent Lie algebra.

Fock representations and BRST cohomology in SL (2) current algebra

Abstract: We investigate the structure of the Fock modules overA1(1) introduced by Wakimoto. We show that irreducible highest weight modules arise as degree zero cohomology groups in a BRST-like complex of Fock modules. Chiral primary fields are constructed as BRST invariant operators acting on Fock modules. As a result, we obtain a free field representation of correlation functions of theSU(2) WZW model on the plane and on the torus. We also consider representations of fractional level arising in Polyakov's 2D quantum gravity. Finally, we give a geometrical, Borel-Weil-like interpretation of the Wakimoto construction.

$q$-Weyl group and a multiplicative formula for universal $R$-matrices

Abstract: We define theq-version of the Weyl group for quantized universal enveloping algebras of simple Lie group and we find explicit multiplicative formulas for the universalR-matrix.

Rounding effects of quenched randomness on first-order phase transitions

Abstract: Frozen-in disorder in an otherwise homogeneous system, is modeled by interaction terms with random coefficients, given by independent random variables with a translation-invariant distribution. For such systems, it is proven that ind=2 dimensions there can be no first-order phase transition associated with discontinuities in the thermal average of a quantity coupled to the randomized parameter. Discontinuities which would amount to a continuous symmetry breaking, in systems which are (stochastically) invariant under the action of a continuous subgroup ofO(N), are suppressed by the randomness in dimensionsd≦4. Specific implications are found in the Random-Field Ising Model, for which we conclude that ind=2 dimensions at all (β,h) the Gibbs state is unique for almost all field configurations, and in the Random-Bond Potts Model where the general phenomenon is manifested in the vanishing of the latent heat at the transition point. The results are explained by the argument of Imry and Ma [1]. The proofs involve the analysis of fluctuations of free energy differences, which are shown (using martingale techniques) to be Gaussian on the suitable scale.

Area-preserving diffeomorphisms and higher-spin algebras

Abstract: We show that there exists a one-parameter family of infinite-dimensional algebras that includes the bosonicd=3 Fradkin-Vasiliev higher-spin algebra and the non-Euclidean version of the algebra of area-preserving diffeomorphisms of the two-sphereS2 as two distinct members. The non-Euclidean version of the area preserving algebra corresponds to the algebra of area-preserving diffeomorphisms of the hyperbolic spaceS1,1, and can be rewritten as\(\mathop {\lim }\limits_{N \to \infty } su(N,N)\). As an application of our results, we formulate a newd=2+1 massless higher-spin field theory as the gauge theory of the area-preserving diffeomorphisms ofS1,1.

Construction of solutions with exactly $k$ blow-up points for the Schrödinger equation with critical nonlinearity

Abstract: We consider the nonlinear Schrodinger equation: (1) $${{i\partial u} \mathord{\left/ {\vphantom {{i\partial u} {\partial t}}} \right. \kern- ulldelimiterspace} {\partial t}} = - \Delta u - \left| u \right|^{{4 \mathord{\left/ {\vphantom {4 N}} \right. \kern- ulldelimiterspace} N}} uandu\left( {0,.} \right) = \varphi \left( . \right),$$ whereu:[0,T)×ℝ N →ℂ. For any given pointsx 1,x 2,...,x k in ℝ N , we construct a solution of Eq. (1),u(t), which blows up in a finite timeT at exactlyx 1,x 2,...,x k . In addition, we describe the precise behavior of the solutionu(t) whent→T, at the blow-up points {x 1,x 2,...,x k } and in ℝ N −{x 1,x 2,...,x k }.

Quantum groups and subfactors of type $B$, $C$, and $D$

Abstract: The main object of this paper is the study of a sequence of finite dimensional algebras, depending on 2 parameters, which appear in connection with the Kauffman link invariant and with Drinfeld's and Jimbo'sq deformation of Lie algebras of typesB, C andD We determine for which parameters these algebras are semisimple Moreover, we classify all unitary representations of the infinite braid groupB∞ factoring through the inductive limit of these algebras This yields new examples of irreducible subfactors of finite depth, whose indices are squares ofq dimensions of irreducible representations of sympletic and orthogonal groups In the combinatorial description of these subfactors one naturally obtains truncated Weyl chambers (as for loop groups for a given level) and multiplicity coefficients of fusion rules for Wess-Zumino-Witten models

Mean-Field Critical Behaviour for Percolation in High Dimensions

Abstract: The triangle condition for percolation states that\(\sum\limits_{x,y} {\tau (0,x)\tau (0,y) \cdot \tau (y,0)} \) is finite at the critical point, where τ(x, y) is the probability that the sitesx andy are connected. We use an expansion related to the lace expansion for a self-avoiding walk to prove that the triangle condition is satisfied in two situations: (i) for nearest-neighbour independent bond percolation on thed-dimensional hypercubic lattice, ifd is sufficiently large, and (ii) in more than six dimensions for a class of “spread-out” models of independent bond percolation which are believed to be in the same universality class as the nearest-neighbour model. The class of models in (ii) includes the case where the bond occupation probability is constant for bonds of length less than some large number, and is zero otherwise. In the course of the proof an infrared bound is obtained. The triangle condition is known to imply that various critical exponents take their mean-field (Bethe lattice) values\((\gamma = \beta = 1,\delta = \Delta _t = 2, t\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } 2)\) and that the percolation density is continuous at the critical point. We also prove thatv2 in (i) and (ii), wherev2 is the critical exponent for the correlation length.

Geometry of the string equations

Abstract: The string equations of hermitian and unitary matrix models of 2D gravity are flatness conditions. These flatness conditions may be interpreted as the consistency conditions for isomonodromic deformation of an equation with an irregular singularity. In particular, the partition function of the matrix model is shown to be the tau function for isomonodromic deformation. The physical parameters defining the string equation are interpreted as moduli of meromorphic gauge fields, and the compatibility conditions can be interpreted as defining a “quantum” analog of a Riemann surface. In the latter interpretation, the equations may be viewed as compatibility conditions for transport on “quantum moduli space” of correlation functions in a theory of free fermions. We discuss how the free fermion field theory may be deduced directly from the matrix model integral. As an application of our formalism we discuss some properties of the BMP solutions of the string equations. We also mention briefly a possible connection to twistor theory.

Parabolic problems for the Anderson model. I. Intermittency and related topics

Abstract: The objective of this paper is a mathematically rigorous investigation of intermittency and related questions intensively studied in different areas of physics, in particular in hydrodynamics. On a qualitative level, intermittent random fields are distinguished by the appearance of sparsely distributed sharp “peaks” which give the main contribution to the formation of the statistical moments. The paper deals with the Cauchy problem (∂/∂t)u(t,x)=Hu(t, x), u(0,x)=t 0(x) ≥ 0, (t, x) ∈ ℝ+ × ℤ d , for the Anderson HamiltonianH = κΔ + ξ(·), ξ(x),x ∈ ℤd where is a (generally unbounded) spatially homogeneous random potential. This first part is devoted to some basic problems. Using percolation arguments, a complete answer to the question of existence and uniqueness for the Cauchy problem in the class of all nonnegative solutions is given in the case of i.i.d. random variables. Necessary and sufficient conditions for intermittency of the fieldsu(t,·) ast→∞ are found in spectral terms ofH. Rough asymptotic formulas for the statistical moments and the almost sure behavior ofu(t,x) ast→∞ are also derived.

Hidden quantum group symmetry and integrable perturbations of conformal field theories

Abstract: The hidden quantum group symmetry in the quantum Sine-Gordon model is found. This symmetry provides the possibility to restrict the operator algebra of the model to subalgebras. It is shown that these subalgebras are massive deformations of minimal conformal field theories.

Localization for a class of one-dimensional quasi-periodic Schrödinger operators

Abstract: We prove for small ɛ and α satisfying a certain Diophantine condition the operator $$H = - \varepsilon ^2 \Delta + \frac{1}{{2\pi }}\cos 2\pi (j\alpha + \theta ) j \in \mathbb{Z}$$ has pure point spectrum for almost all θ. A similar result is established at low energy for\(H = - \frac{{d^2 }}{{dx^2 }} - K^2 (\cos 2\pi x + \cos 2\pi (\alpha x + \theta ))\) providedK is sufficiently large.

Crystal base for the basic representation of\(U_q (\widehat{\mathfrak{s}\mathfrak{l}}(n))\)

Abstract: We show the existence of the crystal base for the basic representation of\(U_q (\widehat{\mathfrak{s}\mathfrak{l}}(n))\) by giving an explicit description in terms of Young diagrams.

On Listing's law

Abstract: Listing's law states that visual directions of sight are related to rotations of the eye so that all rotation axes lie in a plane. The geometry ofSO(3) indicates several plausible algorithms how the human brain could relate vision to eye movements satisfying Listing's law, and suggests crucial experiments which we have carried out.

Isoholonomic problems and some applications

Abstract: We study the problem of finding the shortest loops with a given holonomy. We show that the solutions are the trajectories of particles in Yang-Mills potentials (Theorem 4), or, equivalently, the projections of Kaluza-Klein geodesics (Theorem 2). Applications to quantum mechanics (Berry's phase, Sect. 3) and the optimal control of deformable bodies (Sect. 6) are touched upon.

Homological representations of the Hecke algebra

Abstract: In this paper a topological construction of representations of theAn(1)-series of Hecke algebras, associated with 2-row Young diagrams will be given. This construction gives the representations in terms of the monodromy representation obtained from a vector bundle on which there is a natural flat connection. The fibres of the vector bundle are homology spaces of configuration spaces of points in C, with a suitable twisted local coefficient system. It is also shown that there is a close correspondence between this construction and the work of Tsuchiya and Kanie, who constructed Hecke algebra representations from the monodromy ofn-point functions in a conformal field theory onP1. This work has significance in relation to the one-variable Jones polynomial, which can be expressed in terms of characters of the Iwahori-Hecke algebras associated with 2-row Young diagrams; it gives rise to a topological description of the Jones polynomial, which will be discussed elsewhere [L2].

On the derivation of Hawking radiation associated with the formation of a black hole

Abstract: We show how in gravitational collapse the Hawking radiation at large times is precisely related to a scaling limit on the sphere where the star radius crosses the Schwarzschild radius (as long as the back reaction of the radiation on the metric is neglected). For a free quantum field it can be exactly evaluated and the result agrees with Hawking's prediction. For a realistic quantum field theory no evaluation based on general principles seems possible. The outcoming radiation depends on the field theoretical model.