Showing papers in "Communications in Mathematical Physics in 1996"

On orthogonal and symplectic matrix ensembles

Abstract: The focus of this paper is on the probability,Eβ(O;J), that a setJ consisting of a finite union of intervals contains no eigenvalues for the finiteN Gaussian Orthogonal (β=1) and Gaussian Symplectic (β=4) Ensembles and their respective scaling limits both in the bulk and at the edge of the spectrum. We show how these probabilities can be expressed in terms of quantities arising in the corresponding unitary (β=2) ensembles. Our most explicit new results concern the distribution of the largest eigenvalue in each of these ensembles. In the edge scaling limit we show that these largest eigenvalue distributions are given in terms of a particular Painleve II function.

Gravity coupled with matter and the foundation of non-commutative geometry

Abstract: We first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length elementds. Its unitary representations correspond to Riemannian metrics and Spin structure whileds is the Dirac propagatords=x−x=D−1, whereD is the Dirac operator. We extend these simple relations to the non-commutative case using Tomita's involutionJ. We then write a spectral action, the trace of a function of the length element, which when applied to the non-commutative geometry of the Standard Model will be shown ([CC]) to give the SM Lagrangian coupled to gravity. The internal fluctuations of the non-commutative geometry are trivial in the commutative case but yield the full bosonic sector of SM with all correct quantum numbers in this slightly non-commutative case. The group of local gauge transformations appears spontaneously as a normal subgroup of the diffeomorphism group.

Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz

Abstract: We construct the quantum versions of the monodromy matrices of KdV theory. The traces of these quantum monodromy matrices, which will be called as “T-operators,” act in highest weight Virasoro modules. TheT-operators depend on the spectral parameter λ and their expansion around λ=∞ generates an infinite set of commuting Hamiltonians of the quantum KdV system. TheT-operators can be viewed as the continuous field theory versions of the commuting transfermatrices of integrable lattice theory. In particular, we show that for the values\(c = 1 - 3\frac{{3(2n + 1)^2 }}{{2n + 3}}\),n=1,2,3 .... of the Virasoro central charge the eigenvalues of theT-operators satisfy a closed system of functional equations sufficient for determining the spectrum. For the ground-state eigenvalue these functional equations are equivalent to those of the massless Thermodynamic Bethe Ansatz for the minimal conformal field theoryM2,2n+3; in general they provide a way to generalize the technique of the Thermodynamic Bethe Ansatz to the excited states. We discuss a generalization of our approach to the cases of massive field theories obtained by perturbing these Conformal Field Theories with the operator Φ1,3. The relation of theseT-operators to the boundary states is also briefly described.

Micro-local approach to the Hadamard condition in quantum field theory on curved space-time

Abstract: For the two-point distribution of a quasi-free Klein-Gordon neutral scalar quantum field on an arbitrary four dimensional globally hyperbolic curved space-time we prove the equivalence of (1) the global Hadamard condition, (2) the property that the Feynman propagator is a distinguished parametrix in the sense of Duistermaat and Hormander, and (3) a new property referred to as the wave front set spectral condition (WFSSC), because it is reminiscent of the spectral condition in axiomatic quantum field theory on Minkowski space. Results in micro-local analysis such as the propagation of singularities theorem and the uniqueness up toC∞ of distinguished parametrices are employed in the proof. We include a review of Kay and Wald's rigorous definition of the global Hadamard condition and the theory of distinguished parametrices, specializing to the case of the Klein-Gordon operator on a globally hyperbolic space-time. As an alternative to a recent computation of the wave front set of a globally Hadamard two-point distribution on a globally hyperbolic curved space-time, given elsewhere by Kohler (to correct an incomplete computation in [32]), we present a version of this computation that does not use a deformation argument such as that used in Fulling, Narcowich and Wald and is independent of the Cauchy evolution argument of Fulling, Sweeny and Wald (both of which are relied upon in Kohler's proof). This leads to a simple micro-local proof of the preservation of Hadamard form under Cauchy evolution (first shown by Fulling, Sweeny and Wald) relying only on the propagation of singularities theorem. In another paper [33], the equivalence theorem is used to prove a conjecture by Kay that a locally Hadamard quasi-free Klein-Gordon state on any globally hyperbolic curved space-time must be globally Hadamard.

Spiders for rank 2 Lie algebras

Abstract: A spider is an axiomatization of the representation theory of a group, quantum group, Lie algebra, or other group or group-like object. It is also known as a spherical category, or a strict, monoidal category with a few extra properties, or by several other names. A recently useful point of view, developed by other authors, of the representation theory of sl(2) has been to present it as a spider by generators and relations. That is, one has an algebraic spider, defined by invariants of linear representations, and one identifies it as isomorphic to a combinatorial spider, given by generators and relations. We generalize this approach to the rank 2 simple Lie algebras, namelyA 2,B 2, andG 2. Our combinatorial rank 2 spiders yield bases for invariant spaces which are probably related to Lusztig's canonical bases, and they are useful for computing quantities such as generalized 6j-symbols and quantum link invariants. Their definition originates in definitions of the rank 2 quantum link invariants that were discovered independently by the author and Francois Jaeger.

Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics

Abstract: We study systems of conservation laws arising in two models of adhesion particle dynamics. The first is the system of free particles which stick under collision. The second is a system of gravitationally interacting particles which also stick under collision. In both cases, mass and momentum are conserved at the collisions, so the dynamics is described by 2×2 systems of conservations laws. We show that for these systems, global weak solutions can be constructed explicitly using the initial data by a procedure analogous to the Lax-Oleinik variational principle for scalar conservation laws. However, this weak solution is not unique among weak solutions satisfying the standard entropy condition. We also study a modified gravitational model in which, instead of momentum, some other weighted velocity is conserved at collisions. For this model, we prove both existence and uniqueness of global weak solutions. We then study the qualitative behavior of the solutions with random initial data. We show that for continuous but nowhere differentiable random initial velocities, all masses immediately concentrate on points even though they were continuously distributed initially, and the set of shock locations is dense.

Hecke algebras at roots of unity and crystal bases of quantum affine algebras

Abstract: We present a fast algorithm for computing the global crystal basis of the basic $$U_q (\widehat{\mathfrak{s}\mathfrak{l}}_n )$$ -module. This algorithm is based on combinatorial techniques which have been developed for dealing with modular representations of symmetric groups, and more generally with representations of Hecke algebras of typeA at roots of unity. We conjecture that, upon specializationq→1, our algorithm computes the decomposition matrices of all Hecke algebras at an th root of 1.

The microlocal spectrum condition and Wick polynomials of free fields on curved spacetimes

Abstract: Quantum fields propagating on a curved spacetime are investigated in terms of microlocal analysis. We discuss a condition on the wave front set for the correspondingn-point distributions, called “microlocal spectrum condition” (μSC). On Minkowski space, this condition is satisfied as a consequence of the usual spectrum condition. Based on Radzikowski's determination of the wave front set of the two-point function of a free scalar field, satisfying the Hadamard condition in the Kay and Wald sense, we construct in the second part of this paper all Wick polynomials including the energy-momentum tensor for this field as operator valued distributions on the manifold and prove that they satisfy our “microlocal spectrum condition”.

Invariant measures for the 2D -defocusing nonlinear Schrödinger equation

Abstract: Consider the2D defocusing cubic NLSiu t+Δu−u|u|2=0 with Hamiltonian $$\smallint \left( {\left| { abla \phi } \right|^2 + \tfrac{1}{2}\left| \phi \right|^4 } \right)$$ . It is shown that the Gibbs measure constructed from the Wick ordered Hamiltonian, i.e. replacing |φ|4 by |φ|4 :, is an invariant measure for the appropriately modified equationiu t + Δu‒ [u|u 2−2(∫|u|2 dx)u]=0. There is a well defined flow on thesupport of the measure. In fact, it is shown that for almost all data ϕ the solutionu, u(0)=ϕ, satisfiesu(t)−e itΔφ ∈C Hs (ℝ), for somes>0. First a result local in time is established and next measure invariance considerations are used to extend the local result to a global one (cf. [B2]).

Local fluctuation of the spectrum of a multidimensional Anderson tight binding model

Abstract: We consider the Anderson tight binding modelH=−Δ+V acting inl 2(Z d ) and its restrictionH Λ to finite hypercubes Λ⊂Z d . HereV={V x ;x∈Z d } is a random potential consisting of independent identically distributed random variables. Let {E j (Λ)} j be the eigenvalues ofH Λ, and let ξ j (Λ,E)=|Λ|(E j (Λ)−E),j≧1, be its rescaled eigenvalues. Then assuming that the exponential decay of the fractional moment of the Green function holds for complex energies nearE and that the density of statesn(E) exists atE, we shall prove that the random sequence {ξ j (Λ,E)} j , considered as a point process onR 1, converges weakly to the stationary Poisson point process with intensity measuren(E)dx as Λ gets large, thus extending the result of Molchanov proved for a one-dimensional continuum random Schrodinger operator. On the other hand, the exponential decay of the fractional moment of the Green function was established recently by Aizenman, Molchanov and Graf as a technical lemma for proving Anderson localization at large disorder or at extreme energy. Thus our result in this paper can be summarized as follows: near the energyE where Anderson localization is expected, there is no correlation between eigenvalues ofH Λ if Λ is large.

Exact operator solution of the Calogero-Sutherland model

Abstract: The wave functions of the Calogero-Sutherland model are known to be expressible in terms of Jack polynomials. A formula which allows to obtain the wave functions of the excited states by acting with a string of creation operators on the wave function of the ground state is presented and derived. The creation operators that enter in this formula of Rodrigues-type for the Jack polynomials involve Dunkl operators.

The Euler-Poincaré equations and double bracket dissipation

Abstract: This paper studies the perturbation of a Lie-Poisson (or, equivalently an Euler-Poincare) system by a special dissipation term that has Brockett's double bracket form. We show that a formally unstable equilibrium of the unperturbed system becomes a spectrally and hence nonlinearly unstable equilibrium after the perturbation is added. We also investigate the geometry of this dissipation mechanism and its relation to Rayleigh dissipation functions. This work complements our earlier work (Bloch, Krishnaprasad, Marsden and Ratiu [1991, 1994]) in which we studied the corresponding problem for systems with symmetry with the dissipation added to the internal variables; here it is added directly to the group or Lie algebra variables. The mechanisms discussed here include a number of interesting examples of physical interest such as the Landau-Lifschitz equations for ferromagnetism, certain models for dissipative rigid body dynamics and geophysical fluids, and certain relative equilibria in plasma physics and stellar dynamics.

Holomorphic bundles and many-body systems

Abstract: We show that spin generalization of elliptic Calogero-Moser system, elliptic extension of Gaudin model and their cousins are the degenerations of Hitchin systems. Applications to the constructions of integrals of motion, angle-action variables and quantum systems are discussed. The constructions of classical systems are motivated by Conformal Field Theory, and their quantum counterparts can be thought of as being the degenerations of the critical level Knizhnik-Zamolodchikov-Bernard equations.

The conformal spin and statistics theorem

Abstract: We prove the equality between the statistics phase and the conformal univalence for a superselection sector with finite index in Conformal Quantum Field Theory onS1. A relevant point is the description of the PCT symmetry and the construction of the global conjugate charge.

Quantum $\scr W$-algebras and elliptic algebras

Abstract: We define a quantum Open image in new window -algebra associated to\(\mathfrak{s}\mathfrak{l}_N \) as an associative algebra depending on two parameters. For special values of the parameters, this algebra becomes the ordinary Open image in new window -algebra of\(\mathfrak{s}\mathfrak{l}_N \), or theq-deformed classical Open image in new window -algebra algebra of\(\mathfrak{s}\mathfrak{l}_N \). We construct free field realizations of the quantum Open image in new window -algebra and the screening currents. We also point out some interesting elliptic structures arising in these algebras. In particular, we show that the screening currents satisfy elliptic analogues of the Drinfeld relations in Open image in new window .

Quantum 401-1401-1401-1algebras and Macdonald polynomials

Abstract: We derive a quantum deformation of theW N algebra and its quantum Miura transformation, whose singular vectors realize the Macdonald polynomials.

Quantum affine algebras and deformations of the Virasoro and 237-1237-1237-1

Abstract: Using the Wakimoto realization of quantum affine algebras we define new Poisson algebras, which areq-deformations of the classicalW. We also define their free field realizations, i.e. homomorphisms into some Heisenberg-Poisson algebras. The formulas for these homomorphisms coincide with formulas for spectra of transfer-matrices in the corresponding quantum integrable models derived by the Bethe-Ansatz method.

Ergodicity of eigenfunctions for ergodic billiards

Abstract: We give a simple proof of ergodicity of eigenfunctions of the Laplacian with Dirichlet boundary conditions on compact Riemannian manifolds with piecewise smooth boundaries and ergodic billiards. Examples include the “Bunimovich stadium”, the “Sinai billiard” and the generic polygonal billiard tables of Kerckhoff, Masur and Smillie.

Heat-kernels and functional determinants on the generalized cone

Abstract: We consider zeta functions and heat-kernel expansions on the bounded, generalized cone in arbitrary dimensions using an improved calculational technique. The specific case of a global monopole is analysed in detail and some restrictions thereby placed on theA 5/2 coefficient. The computation of functional determinants is also addressed. General formulas are given and known results are incidentally, and rapidly, reproduced.

Simple currents and extensions of vertex operator algebras

Abstract: We consider how a vertex operator algebra can be extended to an abelian interwining algebra by a family of weak twisted modules which aresimple currents associated with semisimple weight one primary vectors. In the case that the extension is again a vertex operator algebra, the rationality of the extended algebra is discussed. These results are applied to affine Kac-Moody algebras in order to construct all the simple currents explicitly (except forE 8) and to get various extensions of the vertex operator algebras associated with integrable representations.

Equipartition of the eigenfunctions of quantized ergodic maps on the torus

Abstract: We give a simple proof of the equipartition of the eigenfunctions of a class of quantized ergodic area-preserving maps on the torus. Examples are the irrational translations, the skew translations, the hyperbolic automorphisms and some of their perturbations.

Topologically nontrivial field configurations in noncommutative geometry

Abstract: In the framework of noncommutative geometry we describe spinor fields with nonvanishing winding number on a truncated (fuzzy) sphere. The corresponding field theory actions conserve all basic symmetries of the standard commutative version (space isometries and global chiral symmetry), but due to the noncommutativity of the space the fields are regularized and they contain only a finite number of modes.

Deformation quantizations with separation of variables on a Kähler manifold

Abstract: We give a simple geometric description of all formal differentiable deformation quantizations on a Kahler manifoldM such that for each open subsetU⊂M ⋆-multiplication from the left by a holomorphic function and from the right by an antiholomorphic function onU coincides with the pointwise multiplication by these functions. We show that these quantizations are in 1-1 correspondence with the formal deformations of the original Kahler metrics onM.

On finite 4d quantum field theory in non-commutative geometry

Abstract: The truncated 4-dimensional sphereS4 and the action of the self-interacting scalar field on it are constructed. The path integral quantization is performed while simultaneously keeping theSO(5) symmetry and the finite number of degrees of freedom. The usual field theory UV-divergences are manifestly absent.

Strong connections on quantum principal bundles

Abstract: A gauge invariant notion of a strong connection is presented and characterized. It is then used to justify the way in which a global curvature form is defined. Strong connections are interpreted as those that are induced from the base space of a quantum bundle. Examples of both strong and non-strong connections are provided. In particular, such connections are constructed on a quantum deformation of the two-sphere fibrationS 2→RP 2. A certain class of strongU q (2)-connections on a trivial quantum principal bundle is shown to be equivalent to the class of connections on a free module that are compatible with theq-dependent hermitian metric. A particular form of the Yang-Mills action on a trivialU q (2)-bundle is investigated. It is proved to coincide with the Yang-Mills action constructed by A. Connes and M. Rieffel. Furthermore, it is shown that the moduli space of critical points of this action functional is independent ofq.

From dynkin diagram symmetries to fixed point structures

Abstract: Any automorphism of the Dynkin diagram of a symmetrizable Kac-Moody algebra g induces an automorphism of g and a mappingτ ω between highest weight modules of g. For a large class of such Dynkin diagram automorphisms, we can describe various aspects of these maps in terms of another Kac-Moody algebra, the “orbit Lie algebra” g. In particular, the generating function for the trace ofτ ω over weight spaces, which we call the “twining character” of g (with respect to the automorphism), is equal to a character of g. The orbit Lie algebras of untwisted affine Lie algebras turn out to be closely related to the fixed point theories that have been introduced in conformal field theory. Orbit Lie algebras and twining characters constitute a crucial step towards solving the fixed point resolution problem in conformal field theory.

Gkz-generalized hypergeometric systems in mirror symmetry of calabi-yau hypersurfaces

Abstract: We present a detailed study of the generalized hypergeometric system introduced by Gel'fand, Kapranov and Zelevinski (GKZ-hypergeometric system) in the context of toric geometry. GKZ systems arise naturally in the moduli theory of Calabi-Yau toric varieties, and play an important role in applications of the mirror symmetry. We find that the Grobner basis for the so-called toric ideal determines a finite set of differential operators for the local solutions of the GKZ system. At the special point called the large radius limit, we find a close relationship between the principal parts of the operators in the GKZ system and the intersection ring of a toric variety. As applications, we analyze general three dimensional hypersurfaces of Fermat and non-Fermat types with Hodge numbers up toh 1,1=3. We also find and analyze several non-Landau-Ginzburg models which are related to singular models.

Arithmetic properties of mirror map and quantum coupling

Abstract: We study some arithmetic properties of the mirror maps and the quantum Yukawa couplings for some 1-parameter deformations of Calabi-Yau manifolds. First we use the Schwarzian differential equation, which we derived previously, to characterize the mirror map in each case. For algebraic K3 surfaces, we solve the equation in terms of theJ-function. By deriving explicit modular relations we prove that some K3 mirror maps are algebraic over the genus zero function fieldQ(J). This leads to a uniform proof that those mirror maps have integral Fourier coefficients. Regarding the maps as Riemann mappings, we prove that they are genus zero functions. By virtue of the Conway-Norton conjecture (proved by Borcherds using Frenkel-Lepowsky-Meurman's Moonshine module), we find that these maps are actually the reciprocals of the Thompson series for certain conjugacy classes in the Griess-Fischer group. This also gives, as an immediate consequence, a second proof that those mirror maps are integral. We thus conjecture a surprising connection between K3 mirror maps and the Thompson series. For threefolds, we construct a formal nonlinear ODE for the quantum coupling reduced modp. Under the mirror hypothesis and an integrality assumption, we derive modp congurences for the Fourier coefficients. For the quintics, we deduce, (at least for 5×d) that the degreed instanton numbersnd are divisible by 53 — a fact first conjectured by Clemens.

Scaling exponents and multifractal dimensions for independent random cascades

Abstract: This paper is concerned with Mandelbrot's stochastic cascade measures. The problems of (i) scaling exponents of structure functions of the measure, τ(q), and (ii) multifractal dimensions are considered for cascades with a generator vector (w1...wc) of the general type. These problems were previously studied for independent strongly bounded variableswi: 0

Meyer's concept of quasicrystal and quasiregular sets

Abstract: This paper relates two mathematical concepts of long-range order of a set of atoms Λ, each of which is based on restrictions on the set of interatomic distances Λ−Λ. A set Λ in ℝn is aMeyer set if Λ is a Delone set and there is a finite setF such that $$\Lambda - \Lambda \subseteq \Lambda + F.{\text{ Y}}$$ . Meyer proposed that such sets include the possible structures of quasicrystals. He obtained a structure theory for such sets, which reformulates results that he obtained in harmonic analysis around 1970, and which relates these sets to cut-and-project sets. In geometric crystallography V.I. Galiulin introduced the concept ofquasiregular set, which is a set Λ such that both Λ and Λ−Λ are Delone sets. This paper shows that these two concepts are identical.