Showing papers in "Letters in Mathematical Physics in 1996"

The hyperbolic volume of knots from quantum dilogarithm

Abstract: The invariant of a link in three-sphere, associated with the cyclic quantum dilogarithm, depends on a natural number N. By the analysis of particularexamples, it is argued that, for a hyperbolic knot (link), the absolute valueof this invariant grows exponentially at large N, the hyperbolic volume of the knot (link) complement being the growth rate.

A Quantum deformation of the Virasoro algebra and the Macdonald symmetric functions

Abstract: A quantum deformation of the Virasoro algebra is defined. The Kac determinants at arbitrary levels are conjectured. We construct a bosonic realization of the quantum deformed Virasoro algebra. Singular vectors are expressed by the Macdonald symmetric functions. This is proved by constructing screening currents acting on the bosonic Fock space.

Some remarks concerning Nambu mechanics

Abstract: The structure of Nambu-Poisson brackets is studied and we establish that any Nambu tensor is decomposable. We show that every Nambu-Poisson manifold admits a local foliation by canonical Nambu-Poisson manifolds. Finally, a cohomology for Nambu (Lie) algebras which is adapted to the study of formal deformations of Nambu structures is introduced.

A q-analog of the sixth Painlevé equation

Abstract: A q-difference analog of the sixth Painleve equation is presented. It arises as the condition for preserving the connection matrix of linear q-difference equations, in close analogy with the monodromy-preserving deformation of linear differential equations. The continuous limit and special solutions in terms of q-hypergeometric functions are also discussed.

A Coassociative C-Quantum Group with Nonintegral Dimensions

Abstract: By weakening the counit and antipode axioms of aC*-Hopf algebra and allowing for the coassociative coproduct to be nonunital, we obtain a quantum group, that we call aweak C*-Hopf algebra, which is sufficiently general to describe the symmetries of essentially arbitrary fusion rules. This amounts to generalizing the Baaj-Skandalis multiplicative unitaries to multipicative partial isometries. Every finite-dimensional weakC*-Hopf algebra has a dual which is again a weakC*-Hopf algebra. An explicit example is presented with Lee-Yang fusion rules. We briefly discuss applications to amalgamated crossed products, doubles, and quantum chains.

On the extremality of the disordered state for the Ising model on the Bethe lattice

Abstract: We give a simple proof that the limit Ising Gibbs measure with free boundary conditions on the Bethe lattice with the forward branching ratio k≥2 is extremal if and only if β is less or equal to the spin glass transition value, given by tanh(β c SG = 1/√k.

Continued fractions and fermionic representations for characters of M(p,p′) minimal models

Abstract: We present fermionic sum representations of the characters χτ, s(p, p′) of the minimal M(p,p′) models for all relatively prime integers p′>p for some allowed values of r and s. Our starting point is biomial (q-binomial) identities derived from a truncation of the state counting equations of the XXZ spin 1/2 chain of anisotropy −Δ=−cos(π(p/p′)). We use the Takahashi-Suzuki method to express the allowed values of r (and s) in terms of the continued fraction decomposition of {p/p′} (and p/p′), where {x} stands for the fractional part of x. These values are, in fact, the dimensions of the Hermitian irreducible representations of SUq- (2) (and SUq+ (2)) with q−=exp(iπ{p/p}) (and q+=exp(iπ(p/p′))). We also establish the duality relation M(p,p′) ↔ M(p′−p,p′) and discuss the action of the Andrews-Bailey transformation in the space of minimal models. Many new identities of the Rogers-Ramanujan type are presented.

On the first-order operators in bimodules

Abstract: We analyse the structure of the first-order operators in bimodules introduced by A. Connes. We apply this analysis to the theory of connections on bimodules, thereby generalizing several proposals.

On discrete three-dimensional equations associated with the local Yang-Baxter relation

Abstract: The local Yang-Baxter equation (YBE), introduced by Maillet and Nijhoff, is a proper generalization to three dimensions of the zero curvature relation. Recently, Korepanov has constructed an infinite set of integrable three-dimensional lattice models, and has related them to solutions to the local YBE. The simplest Korepanov model is related to the star-triangle relation in the Ising model. In this Letter the corresponding discrete equation is derived. In the continuous limit it leads to a differential three-dimensional equation, which is symmetric with respect to all permutations of the three coordinates. A similar analysis of the star-triangle transformation in electric networks leads to the discrete bilinear equation of Miwa, associated with the BKP hierarchy.

The lie bialgebroid of a Poisson-Nijenhuis manifold

Abstract: We describe a new class of Lie bialgebroids associated with Poisson-Nijenhuis structures. Resume. Nous etudions une nouvelle classe de bigebroides de Lie, associes aux structures de Poisson-Nijenhuis. Introduction. Nijenhuis operators have been introduced in the theory of integrable systems in the work of Magri, Gelfand and Dorfman (see the book [4]), and, under the name of hereditary operators, in that of Fuchssteiner and Fokas. Poisson-Nijenhuis structures were defined by Magri and Morosi in 1984 [15] in their study of completely integrable systems. There is a compatibility condition between the Poisson structure and the Nijenhuis structure that is expressed by the vanishing of a rather complicated tensorial expression. In this letter, we shall prove that this condition can be expressed in a very simple way, using the notion of a Lie bialgebroid [14] [7] [12]. A Lie bialgebroid is a pair of vector bundles in duality, each of which is a Lie algebroid, such that the differential defined by one of them on the exterior algebra of its dual is a derivation of the Schouten bracket. Here we show that a Poisson structure and a Nijenhuis structure constitute a Poisson-Nijenhuis structure if and only if the following condition is satisfied: the cotangent and tangent bundles are a Lie bialgebroid when equipped respectively with the bracket of 1-forms defined by the Poisson structure, and with the deformed bracket of vector fields defined by the Nijenhuis structure. Let me add three ”historical” remarks. This result was first conjectured by Magri during a conversation that we held at the time of the Semestre Symplectique at the Centre Emile Borel. Secondly, the Lie bracket of differential 1-forms on a Poisson manifold, defining the Lie-algebroid structure of its cotangent bundle, was defined by Fuchssteiner in an article of 1982 [6] which is not often cited, though it is certainly one of the first papers to mention this important definition. Thirdly, as A. Weinstein has shown [18] [19], Sophus Lie’s book [11] contains a comprehensive theory of Poisson manifolds under the name of function groups, including, among many results, a proof of the contravariant form of the Jacobi identity, a proof of the duality between Lie algebra structures and linear Poisson structures on vector spaces, the notions of distinguished functions (Casimir functions) and polar groups (dual pairs), and the existence of canonical coordinates. Moreover, Caratheodory, in his book [2], proves explicitly the tensorial character of the Poisson bivector and gives a rather complete account of this theory, based on a short article by Lie [10] that appeared even earlier than the famous “Theorie der Transformationsgruppen”, Part II, of 1890.

On the Riemannian metric of α-entropies of density matrices

Abstract: The Riemannian metric induced by quantum α-entropies is proven to be monotone under stochastic mappings on the set of density matrices. The length of tangent vectors is essentially the Wigner-Yanase-Dyson skew information in this setting.

Four-dimensional BF theory as a topological quantum field theory

Abstract: Starting from a Lie group G whose Lie algebra is equipped with an invariant nondegenerate symmetric bilinear form, we show that four-dimensional BF theory with cosmological term gives rise to a TQFT satisfying a generalization of Atiyah's axioms to manifolds equipped with principal G-bundle. The case G=GL(4,ℝ) is especially interesting because every 4-manifold is then naturally equipped with a principal G-bundle, namely its frame bundle. In this case, the partition function of a compact oriented 4-manifold is the exponential of its signature, and the resulting TQFT is isomorphic to that constructed by Crane and Yetter using a state sum model, or by Broda using a surgery presentation of four-manifolds.

Phase space reduction for star-products: An explicit construction for ℂP n

Abstract: We derive a closed formula for a star-product on complex projective space and on the domain SU(n+1)/S(U(1)×U(n)) using a completely elementary construction: Starting from the standard starproduct of Wick type on ℂn+1\{0} and performing a quantum analogue of Marsden-Weinstein reduction, we can give an easy algebraic description of this star-product. Moreover, going over to a modified star-product on ℂn+1\{0}, obtained by an equivalence transformation, this description can be even further simplified, allowing the explicit computation of a closed formula for the star-product on ℂP n which can easily be transferred to the domain SU(n+1)/S(U(1)×(n)).

Aspects of classical and quantum Nambu mechanics

Abstract: We present recent developments in the theory of Nambu mechanics, which include new examples of Nambu-Poisson manifolds with linear Nambu brackets and new representations of Nambu-Heisenberg commutation relations.

Dynamical symmetries and Nambu mechanics

Abstract: It is shown that several Hamiltonian systems possessing dynamical or hidden symmetries can be realized within the framework of Nambu's generalized mechanics. Among such systems are the SU(n)-isotropic harmonic oscillator and the SO(4) Kepler problem. As required by the formulation of Nambu dynamics, the integrals of motion for these systems necessarily become the so-called generalized Hamiltonians. Furthermore, in most of these problems, the definition of these generalized Hamiltonians is not unique.

Sigma models having supermanifolds as target spaces

Abstract: We study a topological sigma-model (A-model) in the case when the target space is an (m 0|m 1)-dimensional supermanifold. We prove under certain conditions that such a model is equivalent to an A-model having an (m 0−m 1)-dimensional manifold as a target space. We use this result to prove that in the case when the target space of A-model is a complete intersection in a toric manifold, this A-model is equivalent to an A-model having a toric supermanifold as a target space.

The gl ( M|N ) super Yangian and its finite-dimensional representations

Abstract: Methods are developed for systematically constructing the finite-dimensional irreducible representations of the super Yangian Y (gl(M|N)) associated with the Lie superalgebra gl(M|N). It is also shown that every finite-dimensional irreducible representation of Y (gl(M|N)) is of highest weight type, and is uniquely characterized by a highest weight. The necessary and sufficient conditions for an irrep to be finite-dimensional are given.

Weakly coupled states on branching graphs

Abstract: We consider a Schrodinger particle on a graph consisting of N links joined at a single point. Each link supports a real locally integrable potential V j ; the self-adjointness is ensured by the δ type boundary condition at the vertex. If all the links are semi-infinite and ideally coupled, the potential decays as x −1−∈ along each of them, is nonrepulsive in the mean and weak enough, the corresponding Schrodinger operator has a single negative eigenvalue; we find its asymptotic behavior. We also derive a bound on the number of bound states and explain how the δ coupling constant may be interpreted in terms of a family of squeezed potentials.

Symmetries of the heat equation on the lattice

Abstract: Discrete versions of the heat equation on two-dimensional uniform lattices are shown to possess the same symmetry algebra as their continuum limits. Solutions with definite symmetry properties are presented.

On the euler equation: Bi-Hamiltonian structure and integrals in involution

Abstract: We propose a bi-Hamiltonian formulation of the Euler equation for the free n-dimensional rigid body moving about a fixed point. This formulation lives on the ‘physical’ phase space so(n), and is different from the bi-Hamiltonian formulation on the extended phase space sl(n), considered previously in the literature. Using the bi-Hamiltonian structure on so(n), we construct new recursion schemes for the Mishchenko and Manakov integrals of motion.

Entropy functionals of Kolmogorov-Sinai type and their limit theorems

Abstract: Two functionals\(\tilde S\) and\(\tilde I\) are introduced forC*-dynamical systems with invariant states and stationary channels. It is shown that the Kolmogorov-Sinai-type theorems hold for these functionals\(\tilde S\) and\(\tilde I\). Our functionals\(\tilde S\) and\(\tilde I\) are set within the framework of quantum information theory and generalize a quantum KS entropy by CNT and the mutual entropy by Ohya.

Modular inclusion, the Hawking temperature, and quantum field theory in curved spacetime

Abstract: A recent result by Borchers connecting geometric modular action, modular inclusion and spectrum condition, is applied in quantum field theory on spacetimes with a bifurcate Killing horizon (these are generalizations of black-hole spacetimes, comprising the familiar black-hole spacetime models). Within this framework, we give sufficient, model-independent conditions ensuring that the temperature of thermal equilibrium quantum states is the Hawking temperature.

Explicit Integration of the Full Symmetric Toda Hierarchy and the Sorting Property

Abstract: We give an explicit formula for the solution to the initial-value problem of the full symmetric Toda hierarchy. The formula is obtained by the orthogonalization procedure of Szego, and is also interpreted as a consequence of the QR factorization method of Symes. The sorting property of the dynamics is also proved for the case of a generic symmetric matrix in the sense described in the text, and generalizations of tridagonal formulae are given for the case of matrices with 2M+1 nonzero diagonals.

Decay of the metastable state in a chemical system: Different predictions between discrete and continuous models

Abstract: We show, using a specific example of a chemical reaction, that the rate constants predicted from the discrete master equation and its continuum Fokker-Planck approximations differ in exponential order with respect to the size of the system.

On a q-analogue of the multiple gamma functions

Abstract: A q-analogue of the multiple gamma functions is introduced and is shown to satisfy the generalized Bohr-Morellup theorem. Furthermore, we give some expressions of these functions.

A central extension of DY h (gl 2 ) and its vertex representations

Abstract: A central extension of DYh(gl2) is proposed. The bosonization of level 1 module and vertex operators are also given.

Z 3-Graded exterior differential calculus and gauge theories of higher order

Abstract: We present a possible generalization of the exterior differential calculus, based on the operator d such that d3=0, but d2≠0. The entities dx i and d2x k generate an associative algebra; we shall suppose that the products dx i dx k are independent of dx k dx i , while theternary products will satisfy the relation: dx i dx k dx m =jdx k dx m dx i =j2dx m dx m dx i dx k , complemented by the relation dx i d2x k =jd2x k dx i , withj:=e2πi/3.

Minimal affinizations of representations of quantum groups: the irregular case

Abstract: Let\(\mathfrak{g}\) be a finite-dimensional complex simple Lie algebra and Uq(\(\mathfrak{g}\)) the associated quantum group (q is a nonzero complex number which we assume is transcendental). IfV is a finitedimensional irreducible representation of Uq(\(\mathfrak{g}\)), an affinization ofV is an irreducible representationVV of the quantum affine algebra Uq(\(\hat {\mathfrak{g}}\)) which containsV with multiplicity one and is such that all other irreducible Uq(\(\mathfrak{g}\))-components ofV have highest weight strictly smaller than the highest weight λ ofV. There is a natural partial order on the set of Uq(\(\mathfrak{g}\)) classes of affinizations, and we look for the minimal one(s). In earlier papers, we showed that (i) if\(\mathfrak{g}\) is of typeA, B, C, F orG, the minimal affinization is unique up to Uq(\(\mathfrak{g}\))-isomorphism; (ii) if\(\mathfrak{g}\) is of typeD orE and λ is not orthogonal to the triple node of the Dynkin diagram of\(\mathfrak{g}\), there are either one or three minimal affinizations (depending on λ). In this paper, we show, in contrast to the regular case, that if Uq(\(\mathfrak{g}\)) is of typeD4 and λ is orthogonal to the triple node, the number of minimal affinizations has no upper bound independent of λ.

On ground states of the one-dimensional ferromagnetic XXZ model

Abstract: We obtain the complete list of pure infinite volume ground states for the one-dimensional ferromagnetic XXZ model.

A counterexample on idempotent states on a compact quantum group

Abstract: A simple example is given to illustrate that an idempotent state may not be the Haar state of any subgroup in the case of compact quantum groups.