Showing papers in "Communications in Mathematical Physics in 1987"

Compact matrix pseudogroups

Abstract: The compact matrix pseudogroup is a non-commutative compact space endowed with a group structure. The precise definition is given and a number of examples is presented. Among them we have compact group of matrices, duals of discrete groups and twisted (deformed)SU(N) groups. The representation theory is developed. It turns out that the tensor product of representations depends essentially on their order. The existence and the uniqueness of the Haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived. The form of these relations differs from that in the group case. This is due to the fact that the Haar measure on pseudogroups is not central in general. The corresponding modular properties are discussed. The Haar measures on the twistedSU(2) group and on the finite matrix pseudogroup are found.

Hyperkahler Metrics and Supersymmetry

Abstract: We describe two constructions of hyperkahler manifolds, one based on a Legendre transform, and one on a sympletic quotient. These constructions arose in the context of supersymmetric nonlinear σ-models, but can be described entirely geometrically. In this general setting, we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry, along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience.

Complete integrability of relativistic Calogero-Moser systems and elliptic function identities

Abstract: Poincare-invariant generalizations of the Galilei-invariant Calogero-Moser AΓ-particle systems are studied. A quantization of the classical integrals S1?...,SN is presented such that the operators Si9...,SN mutually commute. As a corollary it follows that Si9...9SN Poisson commute. These results hinge on functional equations satisfied by the Weierstrass σ- and 0*- functions. A generalized Cauchy identity involving the σ-function leads to an N x N matrix L whose symmetric functions are proportional to Sl5 ...9SN.

The decorated Teichmüller space of punctured surfaces

Abstract: A principal ℝ + 5 -bundle over the usual Teichmuller space of ans times punctured surface is introduced. The bundle is mapping class group equivariant and admits an invariant foliation. Several coordinatizations of the total space of the bundle are developed. There is furthermore a natural cell-decomposition of the bundle. Finally, we compute the coordinate action of the mapping class group on the total space; the total space is found to have a rich (equivariant) geometric structure. We sketch some connections with arithmetic groups, diophantine approximations, and certain problems in plane euclidean geometry. Furthermore, these investigations lead to an explicit scheme of integration over the moduli spaces, and to the construction of a “universal Teichmuller space,” which we hope will provide a formalism for understanding some connections between the Teichmuller theory, the KP hierarchy and the Virasoro algebra. These latter applications are pursued elsewhere.

Solutions of Hartree-Fock equations for Coulomb systems

Abstract: This paper deals with the existence of multiple solutions of Hartree-Fock equations for Coulomb systems and related equations such as the Thomas-Fermi-Dirac-Von Weizacker equation.

The A-D-E classification of minimal andA 1 (1) conformal invariant theories

Abstract: We present a detailed and complete proof of our earlier conjecture on the classification of minimal conformal invariant theories. This is based on an exhaustive construction of all modular invariant sesquilinear forms, with positive integral coefficients, in the characters of the Virasoro or of theA1(1) Kac-Moody algebras, which describe the corresponding partition functions on a torus. A remarkable correspondence emerges with simply laced Lie algebras.

Hyperbolic conservation laws with relaxation

Abstract: The effect of relaxation is important in many physical situations. It is present in the kinetic theory of gases, elasticity with memory, gas flow with thermo-non-equilibrium, water waves, etc. The governing equations often take the form of hyperbolic conservation laws with lower-order terms. In this article, we present and analyze a simple model of hyperbolic conservation laws with relaxation effects. Dynamic subcharacteristics governing the propagation of disturbances over strong wave forms are identified. Stability criteria for diffusion waves, expansion waves and traveling waves are found and justified nonlinearly. Time-asymptotic expansion and the energy method are used in the analysis. For dissipative waves, the expansion is similar in spirit to the Chapman-Enskog expansion in the kinetic theory. For shock waves, however, a different approach is needed.

Oscillations and concentrations in weak solutions of the incompressible fluid equations

Abstract: The authors introduce a new concept of measure-valued solution for the 3-D incompressible Euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations. One application of the concepts developed here is the following important result: a sequence of Leray-Hopf weak solutions of the Navier-Stokes equations converges in the high Reynolds number limit to a measure-valued solution of 3-D Euler defined for all positive times. The authors present several explicit examples of solution sequences for 3-D incompressible Euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations. An extensions of the concept of Young measure is developed to incorporate these complex phenomena in the measure-valued solutions constructed here.

Elliptic genera and quantum field theory

Abstract: It is shown that in elliptic cohomology — as recently formulated in the mathematical literature — the supercharge of the supersymmetric nonlinear signa model plays a role similar to the role of the Dirac operator inK-theory. This leads to several insights concerning both elliptic cohomology and string theory. Some of the relevant calculations have been done previously by Schellekens and Warner in a different context.

Sharpness of the phase transition in percolation models

Abstract: The equality of two critical points - the percolation threshold pH and the point pτ where the cluster size distribution ceases to decay exponentially - is proven for all translation invariant independent percolation models on homogeneous d-dimensional lattices (d^ 1). The analysis is based on a pair of new nonlinear partial differential inequalities for an order parameter M(β, h\ which for h = Q reduces to the percolation density P^ - at the bond density p = l—e~β in the single parameter case. These are: (1) M^hdM/dh + M2 + βMdM/dβ, and (2) dM/dβ^\J\MdM/dh. Inequality (1) is intriguing in that its derivation provides yet another hint of a "φ3 structure" in percolation models. Moreover, through the elimination of one of its derivatives, (1) yields a pair of ordinary differential inequalities which provide information on the critical exponents β and δ. One of these resembles an Ising model inequality of Frόhlich and Sokal and yields the mean field bound (5^2, and the other implies the result of Chayes and Chayes that β^ί. An inequality identical to (2) is known for Ising models, where it provides the basis for Newman's universal relation /?((5 —1)^1 and for certain extrapolation principles, which are now made applicable also to independent percolation. These results apply to both finite and long range models, with or without orientation, and extend to periodic and weakly inhomogeneous systems.

Spectral functions, special functions and the Selberg zeta function

Abstract: The functional determinant of an eigenvalue sequence, as defined by zeta regularization, can be simply evaluated by quadratures. We apply this procedure to the Selberg trace formula for a compact Riemann surface to find a factorization of the Selberg zeta function into two functional determinants, respectively related to the Laplacian on the compact surface itself, and on the sphere. We also apply our formalism to various explicit eigenvalue sequences, reproducing in a simpler way classical results about the gamma function and the BarnesG-function. Concerning the latter, our method explains its connection to the Selberg zeta function and evaluates the related Glaisher-Kinkelin constantA.

The Chandrasekhar Theory of Stellar Collapse as the Limit of Quantum Mechanics

Abstract: Starting with a “relativistic” Schrodinger Hamiltonian for neutral gravitating particles, we prove that as the particle number N → ∞ and the gravitation constant G → 0 we obtain the well known semiclassical theory for the ground state of stars. For fermions; the correct limit is to fix GN 2/3 and the Chandrasekhar formula is obtained. For bosons the correct limit is to fix GN and a Hartree type equation is obtained. In the fermion case we also prove that the semiclassical equation has a unique solution — a fact which had not been established previously.

Existence of solutions for Schrödinger evolution equations

Abstract: We study the existence, uniqueness and regularity of the solution of the initial value problem for the time dependent Schrodinger equationi∂u/∂t=(−1/2)Δu+V(t,x)u,u(0)=u 0. We provide sufficient conditions onV(t,x) such that the equation generates a unique unitary propagatorU(t,s) and such thatU(t,s)u 0eC 1(ℝ,L 2) ∩C 0(ℝH 2(ℝ n )) foru 0eH 2(ℝ n ). The conditions are general enough to accommodate moving singularities of type ∣x∣−2+ɛ(n≧4) or ∣x∣−n/2+ɛ(n≧3).

A block spin construction of Ondelettes. Part i: Lemarié Functions

Abstract: Using block spin assignments, we construct anL2-orthonormal basis consisting of dyadic scalings and translates of just a finite number of functions. These functions have exponential localization, and for even values of a construction parameterM one can make them classCM−1 with vanishing moments up to orderM inclusive. Such a basis has an important application to phase cell cluster expansions in quantum field theory.

Anderson localization for Bernoulli and other singular potentials

Abstract: We prove exponential localization in the Anderson model under very weak assumptions on the potential distribution. In one dimension we allow any measure which is not concentrated on a single point and possesses some finite moment. In particular this solves the longstanding problem of localization for Bernoulli potentials (i.e., potentials that take only two values). In dimensions greater than one we prove localization at high disorder for potentials with Holder continuous distributions and for bounded potentials whose distribution is a convex combination of a Holder continuous distribution with high disorder and an arbitrary distribution. These include potentials with singular distributions. We also show that for certain Bernoulli potentials in one dimension the integrated density of states has a nontrivial singular component.

Kahler-einstein Metrics on Complex Surfaces With C(1) > 0

Abstract: Various estimates of the lower bound of the holomorphic invariant α(M), defined in [T], are given here by using branched coverings, potential estimates and Lelong numbers of positive,d-closed (1, 1) currents of certain type, etc. These estimates are then applied to produce Kahler-Einstein metrics on complex surfaces withC1>0, in particular, we prove that there are Kahler-Einstein structures withC1>0 on any manifold of differential type\(CP^2 \# \overline {nCP^2 } (3 \leqq n \leqq 8)\).

Scaling relations for $2$D-percolation

Abstract: We prove that the relations 2D-percolation hold for the usual critical exponents for 2D-percolation, provided the exponents δ andv exist Even without the last assumption various relations (inequalities) are obtained for the singular behavior near the critical point of the correlation length, the percolation probability, and the average cluster size We show that in our models the above critical exponents have the same value for approach ofp to the critical probability from above and from below

Dynamical entropy of $C^*$ algebras and von Neumann algebras

Abstract: The definition of the dynamical entropy is extended for automorphism groups ofC* algebras. As an example, the dynamical entropy of the shift of a lattice algebra is studied, and it is shown that in some cases it coincides with the entropy density.

Bosonization on Higher Genus Riemann Surfaces

Abstract: We prove the equivalence between certain fermionic and bosonic theories in two spacetime dimensions. The theories have fields of arbitrary spin on compact surfaces with any number of handles. Global considerations require that we add new topological terms to the bosonic action. The proof that our prescription is correct relies on methods of complex algebraic geometry.

Determinants of Laplacians

Abstract: The determinant of the Laplacian on spinor fields on a Riemann surface is evaluated in terms of the value of the Selberg zeta function at the middle of the critical strip. A key role in deriving this relation is played by the Barnes double gamma function.

A mathematical reformulation of Derrida's REM and GREM

Abstract: The large system limit of the Random Energy Model (REM) and generalized Random Energy Model (GREM) of Derrida is investigated, and found to be universal. This permits systematic calculations of relevance in particular to Parisi's solution of the Sherrington-Kirkpatrick spin-glass model.

Analytic fields on Riemann surfaces. II

Abstract: The properties of analytic fields on a Riemann surface represented by a branch covering of ℂℙ1 are investigated in detail. Branch points are shown to correspond to the vertex operators with simple conformal properties. As applications we compute determinants of\(\bar \partial _j\) operators forZn-symmetric surfaces and obtain various representations for the two-loop measure in the bosonic string theory together with various identities for theta-functions of hyperelliptic surfaces. We also present an integral representation for the quantum part of the twist field correlation functions, which describe propagation of the string on the orbifold background. We also calculate the quantum part of the structure constants of the twist-field operator algebra, generalizing the results of Dixon, Friedan, Martinec, and Shenker.

Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation

Abstract: For independent translation-invariant irreducible percolation models, it is proved that the infinite cluster, when it exists, must be unique. The proof is based on the convexity (or almost convexity) and differentiability of the mean number of clusters per site, which is the percolation analogue of the free energy. The analysis applies to both site and bond models in arbitrary dimension, including long range bond percolation. In particular, uniqueness is valid at the critical point of one-dimensional 1/∣x−y∣2 models in spite of the discontinuity of the percolation density there. Corollaries of uniqueness and its proof are continuity of the connectivity functions and (except possibly at the critical point) of the percolation density. Related to differentiability of the free energy are inequalities which bound the “specific heat” critical exponent α in terms of the mean cluster size exponent γ and the critical cluster size distribution exponent δ; e.g., 1+α≦γ (δ/2−1)/(δ−1).

The spectrum of a quasiperiodic Schrödinger operator

Abstract: The spectrum σ(H) of the tight binding Fibonacci Hamiltonian (Hmn=δm,n+1+δm+1,n+δm,nμv(n),v(n)=\(\chi _{[ - \omega ^3 ,\omega ^2 [} \)((n−1)ω), 1/ω is the golden number) is shown to coincide with the dynamical spectrum, the set on which an infinite subsequence of traces of transfer matrices is bounded. The point spectrum is absent for any μ, and σ(H) is a Cantor set for ∣μ∣≧4. Combining this with Casdagli's earlier result, one finds that the spectrum is singular continuous for ∣μ∣≧16.

Ergodicité et limite semi-classique

Abstract: Consider a self adjoint quantic hamiltonian:P(h)=p(x, hDx) whereh>0 is the Planck's constant andp some smooth classical observable on the phase space R2n. When the classical flow on a compact energy shell {p=λ} is ergodic we prove that in the limith ↓ 0 almost all the eigenfunctions ofP(h) whose energy is near of λ are distributed according to the Liouville measure on {p=λ}.

The universal structure of local algebras

Abstract: It is shown that a few physically significant conditions fix the global structure of the local algebras appearing in quantum field theory: it is isomorphic to that of ℜ where ℜ is the unique hyperfinite factor of typeIII1 and the center of the respective algebra. The argument is based on results in [1, 2] relating to the type of the local algebras and an improvement of an argument in [3] concerning the “split property.”

A mathematical theory of gravitational collapse

Abstract: We study the asymptotic behaviour, as the retarded timeu tends to infinity, of the solutions of Einstein's equations in the spherically symmetric case with a massless scalar field as the material model. We prove that when the final Bondi massM1 is different from zero, asu → ∞, a black hole forms of massM1 surrounded by vacuum. We find the rate of decay of the metric functions and the behaviour of the scalar field on the horizon.

Some rigorous results on the Sherrington-Kirkpatrick spin glass model

Abstract: We prove that in the high temperature regime (T/J>1) the deviation of the total free energy of the Sherrington-Kirkpatrick (S-K) spin glass model from the easily computed log Av(Z N ({βJ})) converges in distribution, asN → ∞, to a (shifted) Gaussian variable. Some weak results about the low temperature regime are also obtained.

A Generalization of the Momentum Mapping Construction for Quaternionic Kahler Manifolds

Abstract: We present a method of reduction of any quaternionic Kahler manifold with isometries to another quaternionic Kahler manifold in which the isometries are divided out. Our method is a generalization of the Marsden-Weinstein construction for symplectic manifolds to the non-symplectic geometry of the quaternionic Kahler case. We compare our results with the known construction for Kahler and hyperKahler manifolds. We also discuss the relevance of our results to the physics of supersymmetric non-linear σ-models and some applications of the method. In particular, we show that the Wolf spaces can be obtained as theU(1) andSU(2) quotients of quaternionic projective spaceHP(n). We also construct an interesting example of compact riemannianV-manifolds(orbifolds) whose metrics are quaternionic Kahler and not symmetric.

Geometry of skyrmions

Abstract: A Skyrmion may be regarded as a topologically non-trivial map from one Riemannian manifold to another, minimizing a particular energy functional. We discuss the geometrical interpretation of this energy functional and give examples of Skyrmions on various manifolds. We show how the existence of conformal transformations can cause a Skyrmion on a 3-sphere to become unstable, and how this may be related to chiral symmetry breaking.