2,777 research outputs found
Quantized Nambu-Poisson Manifolds and n-Lie Algebras
We investigate the geometric interpretation of quantized Nambu-Poisson
structures in terms of noncommutative geometries. We describe an extension of
the usual axioms of quantization in which classical Nambu-Poisson structures
are translated to n-Lie algebras at quantum level. We demonstrate that this
generalized procedure matches an extension of Berezin-Toeplitz quantization
yielding quantized spheres, hyperboloids, and superspheres. The extended
Berezin quantization of spheres is closely related to a deformation
quantization of n-Lie algebras, as well as the approach based on harmonic
analysis. We find an interpretation of Nambu-Heisenberg n-Lie algebras in terms
of foliations of R^n by fuzzy spheres, fuzzy hyperboloids, and noncommutative
hyperplanes. Some applications to the quantum geometry of branes in M-theory
are also briefly discussed.Comment: 43 pages, minor corrections, presentation improved, references adde
n-ary algebras: a review with applications
This paper reviews the properties and applications of certain n-ary
generalizations of Lie algebras in a self-contained and unified way. These
generalizations are algebraic structures in which the two entries Lie bracket
has been replaced by a bracket with n entries. Each type of n-ary bracket
satisfies a specific characteristic identity which plays the r\^ole of the
Jacobi identity for Lie algebras. Particular attention will be paid to
generalized Lie algebras, which are defined by even multibrackets obtained by
antisymmetrizing the associative products of its n components and that satisfy
the generalized Jacobi identity (GJI), and to Filippov (or n-Lie) algebras,
which are defined by fully antisymmetric n-brackets that satisfy the Filippov
identity (FI). Three-Lie algebras have surfaced recently in multi-brane theory
in the context of the Bagger-Lambert-Gustavsson model. Because of this,
Filippov algebras will be discussed at length, including the cohomology
complexes that govern their central extensions and their deformations
(Whitehead's lemma extends to all semisimple n-Lie algebras). When the
skewsymmetry of the n-Lie algebra is relaxed, one is led the n-Leibniz
algebras. These will be discussed as well, since they underlie the
cohomological properties of n-Lie algebras.
The standard Poisson structure may also be extended to the n-ary case. We
shall review here the even generalized Poisson structures, whose GJI reproduces
the pattern of the generalized Lie algebras, and the Nambu-Poisson structures,
which satisfy the FI and determine Filippov algebras. Finally, the recent work
of Bagger-Lambert and Gustavsson on superconformal Chern-Simons theory will be
briefly discussed. Emphasis will be made on the appearance of the 3-Lie algebra
structure and on why the A_4 model may be formulated in terms of an ordinary
Lie algebra, and on its Nambu bracket generalization.Comment: Invited topical review for JPA Math.Theor. v2: minor changes,
references added. 120 pages, 318 reference
Selfdual strings and loop space Nahm equations
We give two independent arguments why the classical membrane fields should be
loops. The first argument comes from how we may construct selfdual strings in
the M5 brane from a loop space version of the Nahm equations. The second
argument is that there appears to be no infinite set of finite-dimensional Lie
algebras (such as for any ) that satisfies the algebraic structure
of the membrane theory.Comment: 28 pages, various additional comment
Topological Many-Body States in Quantum Antiferromagnets via Fuzzy Super-Geometry
Recent vigorous investigations of topological order have not only discovered
new topological states of matter but also shed new light to "already known"
topological states. One established example with topological order is the
valence bond solid (VBS) states in quantum antiferromagnets. The VBS states are
disordered spin liquids with no spontaneous symmetry breaking but most
typically manifest topological order known as hidden string order on 1D chain.
Interestingly, the VBS models are based on mathematics analogous to fuzzy
geometry. We review applications of the mathematics of fuzzy super-geometry in
the construction of supersymmetric versions of VBS (SVBS) states, and give a
pedagogical introduction of SVBS models and their properties [arXiv:0809.4885,
1105.3529, 1210.0299]. As concrete examples, we present detail analysis of
supersymmetric versions of SU(2) and SO(5) VBS states, i.e. UOSp(N|2) and
UOSp(N|4) SVBS states whose mathematics are closely related to fuzzy two- and
four-superspheres. The SVBS states are physically interpreted as hole-doped VBS
states with superconducting property that interpolate various VBS states
depending on value of a hole-doping parameter. The parent Hamiltonians for SVBS
states are explicitly constructed, and their gapped excitations are derived
within the single-mode approximation on 1D SVBS chains. Prominent features of
the SVBS chains are discussed in detail, such as a generalized string order
parameter and entanglement spectra. It is realized that the entanglement
spectra are at least doubly degenerate regardless of the parity of bulk
(super)spins. Stability of topological phase with supersymmetry is discussed
with emphasis on its relation to particular edge (super)spin states.Comment: Review article, 1+104 pages, 37 figures, published versio
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