Showing papers in "Communications in Mathematical Physics in 1989"
TL;DR: In this paper, it was shown that 2+1 dimensional quantum Yang-Mills theory with an action consisting purely of the Chern-Simons term is exactly soluble and gave a natural framework for understanding the Jones polynomial of knot theory in three dimensional terms.
Abstract: It is shown that 2+1 dimensional quantum Yang-Mills theory, with an action consisting purely of the Chern-Simons term, is exactly soluble and gives a natural framework for understanding the Jones polynomial of knot theory in three dimensional terms. In this version, the Jones polynomial can be generalized fromS
3 to arbitrary three manifolds, giving invariants of three manifolds that are computable from a surgery presentation. These results shed a surprising new light on conformal field theory in 1+1 dimensions.
5,093 citations
TL;DR: In this article, the authors define the classical limit of a conformal field theory as a limit in which the conformal weights of all primary fields vanish, and define chiral vertex operators and duality matrices and review the fundamental identities they satisfy.
Abstract: We define chiral vertex operators and duality matrices and review the fundamental identities they satisfy. In order to understand the meaning of these equations, and therefore of conformal field theory, we define the classical limit of a conformal field theory as a limit in which the conformal weights of all primary fields vanish. The classical limit of the equations for the duality matrices in rational field theory together with some results of category theory, suggest that (quantum) conformal field theory should be regarded as a generalization of group theory.
1,305 citations
TL;DR: In this paper, a general theory of non-commutative differential geometry on quantum groups is developed, where bicovariant bimodules as objects analogous to tensor bundles over Lie groups are studied.
Abstract: The paper deals with non-commutative differential geometry. The general theory of differential calculus on quantum groups is developed. Bicovariant bimodules as objects analogous to tensor bundles over Lie groups are studied. Tensor algebra and external algebra constructions are described. It is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra, so the external derivative of higher order differential forms is well defined and obeys the usual properties. The proper form of the Cartan Maurer formula is found. The vector space dual to the space of left-invariant differential forms is endowed with a bilinear operation playing the role of the Lie bracket (commutator). Generalized antisymmetry relation and Jacobi identity are proved.
1,248 citations
TL;DR: In this paper, the chiral properties of (orbifold) conformal field theories were analyzed by modding out by a finite symmetry group, and the fusion rules were derived by studying the modular transformation properties of one-loop characters.
Abstract: We analyze the chiral properties of (orbifold) conformal field theories which are obtained from a given conformal field theory by modding out by a finite symmetry group. For a class of orbifolds, we derive the fusion rules by studying the modular transformation properties of the one-loop characters. The results are illustrated with explicit calculations of toroidal andc=1 models.
603 citations
TL;DR: In this paper, the authors consider the variational theory of the symplectic action function perturbed by a Hamiltonian term and associate to each isolated invariant set of its gradient flow an Abelian group with a cyclic grading.
Abstract: LetP be a symplectic manifold whose symplectic form, integrated over the spheres inP, is proportional to its first Chern class. On the loop space ofP, we consider the variational theory of the symplectic action function perturbed by a Hamiltonian term. In particular, we associate to each isolated invariant set of its gradient flow an Abelian group with a cyclic grading. It is shown to have properties similar to the homology of the Conley index in locally compact spaces. As an application, we show that if the fixed point set of an exact diffeomorphism onP is nondegenerate, then it satisfies the Morse inequalities onP. We also discuss fixed point estimates for general exact diffeomorphisms.
552 citations
TL;DR: In this paper, the theory of superselection sectors is generalized to situations in which normal statistics has to be replaced by braid group statistics, and the essential role of the positive Markov trace of algebraic quantum field theory for this analysis is explained.
Abstract: The theory of superselection sectors is generalized to situations in which normal statistics has to be replaced by braid group statistics. The essential role of the positive Markov trace of algebraic quantum field theory for this analysis is explained, and the relation to exchange algebras is established.
495 citations
TL;DR: In this paper, two results on site percolation on the d-dimensional lattice, d≧1 arbitrary, are presented, and they extend to a broad class of finite-dimensional models.
Abstract: Two results on site percolation on thed-dimensional lattice,d≧1 arbitrary, are presented. In the first theorem, we show that for stationary underlying probability measures, each infinite cluster has a well-defined density with probability one. The second theorem states that if in addition, the probability measure satisfies the finite energy condition of Newman and Schulman, then there can be at most one infinite cluster with probability one. The simple arguments extend to a broad class of finite-dimensional models, including bond percolation and regular lattices.
468 citations
TL;DR: In this article, a class of diffeomorphism invariant theories for which the Hilbert space of quantum states can be explicitly constructed is described, including Witten's solution to 2+1 dimensional gravity as a special case.
Abstract: A class of diffeomorphism invariant theories is described for which the Hilbert space of quantum states can be explicitly constructed. These theories can be formulated in any dimension and include Witten's solution to 2+1 dimensional gravity as a special case. Higher dimensional generalizations exist which start with an action similar to the Einstein action inn dimensions. Many of these theories do not involve a spacetime metric and provide examples of topological quantum field theories. One is a version of Yang-Mills theory in which the only quantum states onS3×R are the θ vacua. Finally it is shown that the three dimensional Chern-Simons theory (which Witten has shown is intimately connected with knot theory) arises naturally from a four dimensional topological gauge theory.
454 citations
TL;DR: In this article, the Weil-Petersson metric is defined on the moduli space of Calabi-Yau manifolds and its potential is explicitely defined on these manifolds.
Abstract: The Weil-Petersson metric is defined on the moduli space of Calabi-Yau manifolds. The curvature of this Weil-Petersson metrics is computed and its potential is explicitely defined. It is proved that the moduli space of Calabi-Yau manifolds is unobstructed (see Tian).
438 citations
TL;DR: In this article, the authors identify the statistical dimension of a superselection sector in a local quantum field theory with the square root of the index of a localized endomorphism of the quasi-local C*-algebra that represents the sector.
Abstract: We identify the statistical dimension of a superselection sector in a local quantum field theory with the square root of the index of a localized endomorphism of the quasi-local C*-algebra that represents the sector. As a consequence in a two-dimensional theory the possible values of the statistical dimension below 2 are restricted to a given discrete set.
408 citations
TL;DR: In this paper, the authors define and investigate a multiparametric deformation of the general linear supergroup, which is the simplest example of some general constructions described in [5, 6].
Abstract: In the work L. D. Faddeev and his collaborators, and subsequently V. G. Drinfeld, M. Jimbo, S. L. Woronowicz, a new class of Hopf algebras was constructed. They can be considered as one-parametric deformations of either group ring or the universal enveloping algebra of a simple algebraic group. In this paper we define and investigate a multiparametric deformation of the general linear supergroup. This is the simplest example of some general constructions described in [5, 6].
TL;DR: In this article, the authors give a new proof of exponential localization in the Anderson tight binding model which uses many ideas of the Frohlich, Martinelli, Scoppola and Spencer proof.
Abstract: We give a new proof of exponential localization in the Anderson tight binding model which uses many ideas of the Frohlich, Martinelli, Scoppola and Spencer proof, but is technically simpler-particularly the probabilistic estimates.
TL;DR: In this article, for a smooth manifold equipped with a Poisson bracket, the authors formulate a C*-algebra framework for deformation quantization, including the possibility of invariance under a Lie group of diffeomorphisms preserving the bracket.
Abstract: ForM a smooth manifold equipped with a Poisson bracket, we formulate aC*-algebra framework for deformation quantization, including the possibility of invariance under a Lie group of diffeomorphisms preserving the Poisson bracket. We then show that the much-studied non-commutative tori give examples of such deformation quantizations, invariant under the usual action of ordinary tori. Going beyond this, the main results of the paper provide a construction of invariant deformation quantizations for those Poisson brackets on Heisenberg manifolds which are invariant under the action of the Heisenberg Lie group, and for various generalizations suggested by this class of examples. Interesting examples are obtained of simpleC*-algebras on which the Heisenberg group acts ergodically.
TL;DR: In this article, an irreducible representation of the Virasoro algebra can be extracted from the current algebra by putting a constraint on the latter using the Becchi-Rouet-Stora-Tyutin formalism.
Abstract: We prove that an irreducible representation of the Virasoro algebra can be extracted from an irreducible representation space of theSL(2, ℛ) current algebra by putting a constraint on the latter using the Becchi-Rouet-Stora-Tyutin formalism. Thus there is aSL(2, ℛ) symmetry in the Virasoro algebra, but it is gauged and hidden. This construction of the Virasoro algebra is the quantum analogue of the Hamiltonian reduction. We then are naturally lead to consider a constrainedSL(2, ℛ) Wess-Zumino-Witten model. This system is also related to quantum field theory of coadjoint orbit of the Virasoro group. Based on this result, we present a canonical derivation of theSL(2, ℛ) current algebra in Polyakov's theory of two-dimensional gravity; it is a manifestation of theSL(2, ℛ) symmetry in conformal field theory hidden by the quantum Hamiltonian reduction. We also discuss the quantum Hamiltonian reduction of theSL(2, ℛ) current algebra and its relation to theW
n
-algebra of Zamolodchikov. This makes it possible to define a natural generalization of the geometric action for theW
n
-algebra despite its non-Lie-algebraic nature.
TL;DR: In this article, the spectral properties of transfer operators and corresponding analytic properties of the generating function are discussed, and new results are proved and some natural conjectures are proposed, and applications to Julia sets are also discussed.
Abstract: Letf:X↦X be an expanding map of a compact space (small distances are increased by a factor >1). A generating functionζ(z) is defined which countsf-periodic points with a weight. One can expressζ in terms of nonstandard “Fredholm determinants” of certain “transfer operators”, which can be studied by methods borrowed from statistical mechanics. In this paper we review the spectral properties of the transfer operators and the corresponding analytic properties ofζ(z). Gibbs distributions and applications to Julia sets are also discussed. Some new results are proved, and some natural conjectures are proposed.
TL;DR: In this paper, it was shown that the one dimensional Schrodinger operator has a Cantor spectrum of zero Lebesgue measure for any irrational α and any λ>0.
Abstract: In this paper we prove that the one dimensional Schrodinger operator onl
2(ℤ) with potential given by:
$$\upsilon (n) = \lambda \chi _{[1 - \alpha , 1[} (x + n\alpha )\alpha
otin \mathbb{Q}$$
has a Cantor spectrum of zero Lebesgue measure for any irrationalα and any λ>0. We can thus extend the Kotani result on the absence of absolutely continuous spectrum for this model, to all
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.
TL;DR: An extension of the formalism of quantum mechanics to the case where the canonical variables are valued in a field of p-adic numbers is considered in this article, where the free particle and the harmonic oscillator are considered.
Abstract: An extension of the formalism of quantum mechanics to the case where the canonical variables are valued in a field ofp-adic numbers is considered. In particular the free particle and the harmonic oscillator are considered. In classicalp-adic mechanics we consider time as ap-adic variable and coordinates and momentum orp-adic or real. For the case ofp-adic coordinates and momentum quantum mechanics with complex amplitudes is constructed. It is shown that the Weyl representation is an adequate formulation in this case. For harmonic oscillator the evolution operator is constructed in an explicit form. For primesp of the form 4l+1 generalized vacuum states are constructed. The spectra of the evolution operator have been investigated. Thep-adic quantum mechanics is also formulated by means of probability measures over the space of generalized functions. This theory obeys an unusual property: the propagator of a massive particle has power decay at infinity, but no exponential one.
TL;DR: In this paper, a solution for the Klein-Gordon and Dirac equations in the 2+1 dimensional space-time created by a massive point particle with arbitrary angular momentum is presented.
Abstract: Solutions are presented for the Klein-Gordon and Dirac equations in the 2+1 dimensional space-time created by a massive point particle, with arbitrary angular momentum. A universal formula for the scattering amplitude holds when a required self-adjoint extension of the Dirac operator is specified uniquely. Various obstacles to a consistent quantum mechanical interpretation of these results are noted.
TL;DR: In this article, an explicit formula for the quantum R-matrix is given for the Poincare-Birkhoff-Witt algorithm. But this formula is based on a quantum double construction.
Abstract: One uses Drinfeld's quantum double construction and a basis a la Poincare-Birkhoff-Witt inU
h
n
+ to compute an explicit formula for the quantumR-matrix.
TL;DR: In this article, a simple martingale argument is presented which proves that directed polymers in random environments satisfy a central limit theorem ford ≥ 3 and if the disorder is small enough.
Abstract: A simple martingale argument is presented which proves that directed polymers in random environments satisfy a central limit theorem ford≧3 and if the disorder is small enough. This simplifies and extends an approach by J. Imbrie and T. Spencer.
TL;DR: In this paper, the authors developed the quantization of topological solitons (vortices) in three-dimensional quantum field theory, in terms of the Euclidean region functional integral.
Abstract: We develop the quantization of topological solitons (vortices) in three-dimensional quantum field theory, in terms of the Euclidean region functional integral. We analyze in some detail the vortices of the abelian Higgs model. If a Chern-Simons term is added to the action, the vortices turn out to be “anyons,” i.e. particles with arbitrary real spin and intermediate (Θ) statistics. Localization properties of the interpolating field, scattering theory and spin-statistics connection of anyons are discussed. Such analysis might be relevant in connection with the fractional quantum Hall effect and two-dimensional models of HighTcsuperconductors.
TL;DR: In this article, a new class of spectral problems called quasi-exactly-solvable two-dimensional hamiltonian problems is introduced, which occupies an intermediate position between the exactly solvable problems (e.g., harmonic oscillator) and all others.
Abstract: We discuss a new class of spectral problems discovered recently which occupies an intermediate position between the exactly-solvable problems (e.g., harmonic oscillator) and all others. The problems belonging to this class are distinguished by the fact that a part of the eigenvalues, and eigenfunctions can be found algebraically, but not the whole spectrum. The reason explaining the existence of the quasi-exactly-solvable problems is a hidden dynamical symmetry present in the hamiltonian. For one-dimensional motion this hidden symmetry isSL(2,R). It is shown that other groups lead to a partial algebraization in multidimensional quantal problems. In particular,SL(2,R)×SL(2,R),SO(3) andSL(3,R) are relevant to two-dimensional motion inducing a class of quasi-exactly-solvable two-dimensional hamiltonians. Typically they correspond to systems in a curved space, but sometimes the curvature turns out to be zero. Graded algebras open the possibility of constructing quasi-exactlysolvable hamiltonians acting on multicomponent wave functions. For example, with a (non-minimal) superextension ofSL(2,R) we get a hamiltonian describing the motion of a spinor particle.
TL;DR: In this article, a complete canonical formulation of the BRST theory of systems with redundant gauge symmetries is presented, including the superparticle, the superstring, and superstring generators.
Abstract: A complete canonical formulation of the BRST theory of systems with redundant gauge symmetries is presented. These systems includep-form gauge fields, the superparticle, and the superstring. We first define the Koszul-Tate differential and explicitly show how the introduction of the momenta conjugate to the ghosts of ghosts makes it acyclic. The global existence of the BRST generator is then demonstrated, and the BRST charge is proved to be unique up to canonical transformations in the extended phase space, which includes the ghosts. Finally, the BRST cohomology in classical mechanics is investigated and shown to be equal to the cohomology of the exterior derivative along the gauge orbits, as in the irreducible case. This is done by re-expressing the exterior algebra along the gauge orbits as a free differential algebra containing generators of higher degree, which are identified with the ghosts of ghosts. The quantum cohomology is not dealt with.
TL;DR: In this article, a convergence theorem of the fractional step Lax-Friedrichs scheme and the Godunov scheme for an inhomogeneous system of isentropic gas dynamics was established by using the framework of compensated compactness.
Abstract: A convergence theorem of the fractional step Lax-Friedrichs scheme and Godunov scheme for an inhomogeneous system of isentropic gas dynamics (1<γ≦5/3) is established by using the framework of compensated compactness. Meanwhile, a corresponding existence theorem of global solutions with large data containing the vacuum is obtained.
TL;DR: In this article, a natural Lax pair for the Kowalewski top is derived by using a general group-theoretic approach, which gives a new insight into the algebraic geometry of the top.
Abstract: A “natural” Lax pair for the Kowalewski top is derived by using a general group-theoretic approach. This gives a new insight into the algebraic geometry of the top and leads to its complete solution via finite-band integration theory.
TL;DR: In this paper, the energy-dependent Schrodinger operator was derived by asingle construction from the Lax approach, and the time evolutions of the eigenfunctions of the Schroffinger operator, the associated Hamiltonian operators and the Hamiltonian functionals were derived.
Abstract: We consider the energy dependent Schrodinger operator\(\mathbb{L} = \sum\limits_{i = 0}^N {\lambda ^i (\varepsilon _i \partial ^2 + u_i )} \), which we have previously shown to be associated with multi-Hamiltonian structures [2]. In this paper we use an unusual form of the Lax approach to derive by asingle construction the time evolutions of the eigenfunctions of\(\mathbb{L}\), the associated Hamiltonian operators and the Hamiltonian functionals. We then generalise the well known factorisation of standard Lax operators to the case of energy-dependent operators. The simple product of linear factors is replaced by a λ-dependent quadratic form. We thus generalise the resulting construction of Miura maps and modified equations. We show that for some of our systems there exists a sequence ofN such modifications, therth modification possessing (N−r+1) Hamiltonian structures.
TL;DR: In this paper, a connection between one-dimensional Toda lattices, constructed on the basis of the systems of simple roots of classical and affine Lie algebras, and other integrable systems of interacting particles is established.
Abstract: Connection is established between one-dimensional Toda lattices, constructed on the basis of the systems of simple roots of classical and affine Lie algebras, and other integrable systems of interacting particles. That connection allows us to find new lattices differing from the known ones by the interaction of particles near the ends. Some of the new lattices admit non-Abelian generalizations.
TL;DR: The Nahm transformation as mentioned in this paper maps anti-self dual connections on the four-torus (S1)4 onto the dual torus (2)4 and shows that this map is a hyper-Kahler isometry.
Abstract: We describe in mathematical detail the Nahm transformation which maps anti-self dual connections on the four-torus (S1)4 onto anti-self-dual connections on the dual torus. This transformation induces a map between the relevant instanton moduli spaces and we show that this map is a (hyperKahler) isometry.
TL;DR: In this article, it was shown that the finite temperature KMS states of the spin-boson hamiltonian have a limit as the temperature drops to zero, and that the number of bosons is finite below and infinite at and above the critical coupling strength.
Abstract: We establish that the finite temperature KMS states of the spin-boson hamiltonian have a limit as the temperature drops to zero. Using recent advances on the one-dimensional Ising model with long range, 1/r2, interactions, we prove qualitative properties of the ground state(s) in the ohmic case. We show (i) the asymptotics of the critical coupling as the bare energy gap goes to zero and to infinity, (ii) a jump in the order parameter, and (iii) that the number of bosons is finite below and infinite at and above the critical coupling strength.
TL;DR: In this article, the exact temperature of phase transition for the Ising model on an arbitrary infinite tree with arbitrary interaction strengths and no external field was calculated, and the same problem was solved for the diluted models and more general random interaction strengths.
Abstract: We calculate the exact temperature of phase transition for the Ising model on an arbitrary infinite tree with arbitrary interaction strengths and no external field. In the same setting, we calculate the critical temperature for spin percolation. The same problems are solved for the diluted models and for more general random interaction strengths. In the case of no interaction, we generalize to percolation on certain tree-like graphs. This last calculation supports a general conjecture on the coincidence of two critical probabilities in percolation theory.