Aspects of classical and quantum Nambu mechanics

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

On a class of n-Leibniz deformations of the simple Filippov algebras

We study the problem of the infinitesimal deformations of all real, simple, finite-dimensional Filippov (or n-Lie) algebras, considered as a class of n-Leibniz algebras characterized by having an n-bracket skewsymmetric in its n-1 first arguments. We prove that all n>3 simple finite-dimensional Filippov algebras are rigid as n-Leibniz algebras of this class. This rigidity also holds for the Leibniz deformations of the semisimple n=2 Filippov (i.e., Lie) algebras. The n=3 simple FAs, however, admit a non-trivial one-parameter infinitesimal 3-Leibniz algebra deformation. We also show that the $n\geq 3$ simple Filippov algebras do not admit non-trivial central extensions as n-Leibniz algebras of the above class.Comment: 19 pages, 30 refs., no figures. Some text rearrangements for better clarity, misprints corrected. To appear in J. Math. Phy

Semiclassical and quantum Liouville theory

We develop a functional integral approach to quantum Liouville field theory completely independent of the hamiltonian approach. To this end on the sphere topology we solve the Riemann-Hilbert problem for three singularities of finite strength and a fourth one infinitesimal, by determining perturbatively the Poincare' accessory parameters. This provides the semiclassical four point vertex function with three finite charges and a fourth infinitesimal. Some of the results are extended to the case of n finite charges and m infinitesimal. With the same technique we compute the exact Green function on the sphere on the background of three finite singularities. Turning to the full quantum problem we address the calculation of the quantum determinant on the background of three finite charges and of the further perturbative corrections. The zeta function regularization provides a theory which is not invariant under local conformal transformations. Instead by employing a regularization suggested in the case of the pseudosphere by Zamolodchikov and Zamolodchikov we obtain the correct quantum conformal dimensions from the one loop calculation and we show explicitly that the two loop corrections do not change such dimensions. We then apply the method to the case of the pseudosphere with one finite singularity and compute the exact value for the quantum determinant. Such results are compared to those of the conformal bootstrap approach finding complete agreement.Comment: 12 pages, 1 figure, Contributed to 5th Meeting on Constrained Dynamics and Quantum Gravity (QG05), Cala Gonone, Sardinia, Italy, 12-16 Sep 200

Berezin quantization, conformal welding and the Bott-Virasoro group

Following Nag-Sullivan, we study the representation of the group ${\rm Diff}^+(S^1)$ of diffeomorphisms of the circle on the Hilbert space of holomorphic functions. Conformal welding provides a triangular decompositions for the corresponding symplectic transformations. We apply Berezin formalism and lift this decomposition to operators acting on the Fock space. This lift provides quantization of conformal welding, gives a new representative of the Bott-Virasoso cocylce class, and leads to a surprising identity for the Takhtajan-Teo energy functional on ${\rm Diff}^+(S^1)$.Comment: 26 page

Non-regular eigenstate of the XXX model as some limit of the Bethe state

For the one-dimensional XXX model under the periodic boundary conditions, we discuss two types of eigenvectors, regular eigenvectors which have finite-valued rapidities satisfying the Bethe ansatz equations, and non-regular eigenvectors which are descendants of some regular eigenvectors under the action of the SU(2) spin-lowering operator. It was pointed out by many authors that the non-regular eigenvectors should correspond to the Bethe ansatz wavefunctions which have multiple infinite rapidities. However, it has not been explicitly shown whether such a delicate limiting procedure should be possible. In this paper, we discuss it explicitly in the level of wavefunctions: we prove that any non-regular eigenvector of the XXX model is derived from the Bethe ansatz wavefunctions through some limit of infinite rapidities. We formulate the regularization also in terms of the algebraic Bethe ansatz method. As an application of infinite rapidity, we discuss the period of the spectral flow under the twisted periodic boundary conditions.Comment: 53 pages, no figur

On the Absence of Continuous Symmetries for Noncommutative 3-Spheres

A large class of noncommutative spherical manifolds was obtained recently from cohomology considerations. A one-parameter family of twisted 3-spheres was discovered by Connes and Landi, and later generalized to a three-parameter family by Connes and Dubois-Violette. The spheres of Connes and Landi were shown to be homogeneous spaces for certain compact quantum groups. Here we investigate whether or not this property can be extended to the noncommutative three-spheres of Connes and Dubois-Violette. Upon restricting to quantum groups which are continuous deformations of Spin(4) and SO(4) with standard co-actions, our results suggest that this is not the case.Comment: 15 pages, no figure

Quantum Liouville theory and BTZ black hole entropy

In this paper I give an explicit conformal field theory description of (2+1)-dimensional BTZ black hole entropy. In the boundary Liouville field theory I investigate the reducible Verma modules in the elliptic sector, which correspond to certain irreducible representations of the quantum algebra U_q(sl_2) \odot U_{\hat{q}}(sl_2). I show that there are states that decouple from these reducible Verma modules in a similar fashion to the decoupling of null states in minimal models. Because ofthe nonstandard form of the Ward identity for the two-point correlation functions in quantum Liouville field theory, these decoupling states have positive-definite norms. The explicit counting from these states gives the desired Bekenstein-Hawking entropy in the semi-classical limit when q is a root of unity of odd order.Comment: LaTeX, 33 pages, 4 eps figure

Algebraic Structures of Quantum Projective Field Theory Related to Fusion and Braiding. Hidden Additive Weight

The interaction of various algebraic structures describing fusion, braiding and group symmetries in quantum projective field theory is an object of an investigation in the paper. Structures of projective Zamolodchikov al- gebras, their represntations, spherical correlation functions, correlation characters and envelopping QPFT-operator algebras, projective \"W-algebras, shift algebras, braiding admissible QPFT-operator algebras and projective G-hypermultiplets are explored. It is proved (in the formalism of shift algebras) that sl(2,C)-primary fields are characterized by their projective weights and by the hidden additive weight, a hidden quantum number discovered in the paper (some discussions on this fact and its possible relation to a hidden 4-dimensional QFT maybe found in the note by S.Bychkov, S.Plotnikov and D.Juriev, Uspekhi Matem. Nauk 47(3) (1992)[in Russian]). The special attention is paid to various constructions of projective G-hyper- multiplets (QPFT-operator algebras with G-symmetries).Comment: AMS-TEX, amsppt style, 16 pages, accepted for a publication in J.MATH.PHYS. (Typographical errors are excluded

Quantum State Transfer in Spin-1 Chains

We study the transfer of quantum information through a Heisenberg spin-1 chain prepared in its ground state. We measure the efficiency of such a quantum channel {\em via} the fidelity of retrieving an arbitrarily prepared state and {\em via} the transfer of quantum entanglement. The Heisenberg spin-1 chain has a very rich quantum phase diagram. We show that the phase boundaries are reflected in sharp variations of the transfer efficiency. In the vicinity of the border between the dimer and the ferromagnetic phase (in the conjectured spin-nematic region), we find strong indications for a qualitative change of the excitation spectrum. Moreover, we identify two regions of the phase diagram which give rise to particularly high transfer efficiency; the channel might be non-classical even for chains of arbitrary length, in contrast to spin-1/2 chains.Comment: 4 pages, 4 figures, published versio