892 research outputs found
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
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
Spin-s wavefunctions with algebraic order
We generalize the Gutzwiller wavefunction for s = 1/2 spin chains to
construct a family of wavefunctions for all s > 1/2. Through numerical
simulations, we demonstrate that the spin spin correlation functions for all s
decay as a power law with logarithmic corrections. This is done by mapping the
model to a classical statistical mechanical model, which has coupled Ising spin
chains with long range interactions. The power law exponents are those of the
Wess Zumino Witten models with k = 2s. Thus these simple wavefunctions
reproduce the spin correlations of the family of Hamiltonians obtained by the
Algebraic Bethe Ansatz.Comment: 10 pages, 7 figure
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
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
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
Hyper-elliptic Nambu flow associated with integrable maps
We study hyper-elliptic Nambu flows associated with some dimensional maps
and show that discrete integrable systems can be reproduced as flows of this
class.Comment: 13 page
Berezin quantization, conformal welding and the Bott-Virasoro group
Following Nag-Sullivan, we study the representation of the group 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 .Comment: 26 page
Black Hole Thermodynamics and Riemann Surfaces
We use the analytic continuation procedure proposed in our earlier works to
study the thermodynamics of black holes in 2+1 dimensions. A general black hole
in 2+1 dimensions has g handles hidden behind h horizons. The result of the
analytic continuation is a hyperbolic 3-manifold having the topology of a
handlebody. The boundary of this handlebody is a compact Riemann surface of
genus G=2g+h-1. Conformal moduli of this surface encode in a simple way the
physical characteristics of the black hole. The moduli space of black holes of
a given type (g,h) is then the Schottky space at genus G. The (logarithm of
the) thermodynamic partition function of the hole is the Kaehler potential for
the Weil-Peterson metric on the Schottky space. Bekenstein bound on the black
hole entropy leads us to conjecture a new strong bound on this Kaehler
potential.Comment: 17+1 pages, 9 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
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
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