708 research outputs found
N-complexes as functors, amplitude cohomology and fusion rules
We consider N-complexes as functors over an appropriate linear category in
order to show first that the Krull-Schmidt Theorem holds, then to prove that
amplitude cohomology only vanishes on injective functors providing a well
defined functor on the stable category. For left truncated N-complexes, we show
that amplitude cohomology discriminates the isomorphism class up to a
projective functor summand. Moreover amplitude cohomology of positive
N-complexes is proved to be isomorphic to an Ext functor of an indecomposable
N-complex inside the abelian functor category. Finally we show that for the
monoidal structure of N-complexes a Clebsch-Gordan formula holds, in other
words the fusion rules for N-complexes can be determined.Comment: Final versio
Complex structures and the Elie Cartan approach to the theory of spinors
Each isometric complex structure on a 2-dimensional euclidean space
corresponds to an identification of the Clifford algebra of with the
canonical anticommutation relation algebra for ( fermionic) degrees of
freedom. The simple spinors in the terminology of E.~Cartan or the pure spinors
in the one of C. Chevalley are the associated vacua. The corresponding states
are the Fock states (i.e. pure free states), therefore, none of the above
terminologies is very good.Comment: 10
Examples of derivation-based differential calculi related to noncommutative gauge theories
Some derivation-based differential calculi which have been used to construct
models of noncommutative gauge theories are presented and commented. Some
comparisons between them are made.Comment: 22 pages, conference given at the "International Workshop in honour
of Michel Dubois-Violette, Differential Geometry, Noncommutative Geometry,
Homology and Fundamental Interactions". To appear in a special issue of
International Journal of Geometric Methods in Modern Physic
BRS Cohomology of the Supertranslations in D=4
Supersymmetry transformations are a kind of square root of spacetime
translations. The corresponding Lie superalgebra always contains the
supertranslation operator . We find that the
cohomology of this operator depends on a spin-orbit coupling in an SU(2) group
and has a quite complicated structure. This spin-orbit type coupling will turn
out to be basic in the cohomology of supersymmetric field theories in general.Comment: 14 pages, CTP-TAMU-13/9
From Koszul duality to Poincar\'e duality
We discuss the notion of Poincar\'e duality for graded algebras and its
connections with the Koszul duality for quadratic Koszul algebras. The
relevance of the Poincar\'e duality is pointed out for the existence of twisted
potentials associated to Koszul algebras as well as for the extraction of a
good generalization of Lie algebras among the quadratic-linear algebras.Comment: Dedicated to Raymond Stora. 27 page
The Origin of Chiral Anomaly and the Noncommutative Geometry
We describe the scalar and spinor fields on noncommutative sphere starting
from canonical realizations of the enveloping algebra . The gauge extension of a free spinor model, the Schwinger model on
a noncommutative sphere, is defined and the model is quantized. The
noncommutative version of the model contains only a finite number of dynamical
modes and is non-perturbatively UV-regular. An exact expresion for the chiral
anomaly is found. In the commutative limit the standard formula is recovered.Comment: 30 page
Abundance of local actions for the vacuum Einstein equations
We exhibit large classes of local actions for the vacuum Einstein equations.
In presence of fermions, or more generally of matter which couple to the
connection, these actions lead to inequivalent equations revealing an arbitrary
number of parameters. Even in the pure gravitational sector, any corresponding
quantum theory would depend on these parameters.Comment: 10 pages. Final version to appear in Letters in Mathematical Physic
Chirality and Dirac Operator on Noncommutative Sphere
We give a derivation of the Dirac operator on the noncommutative -sphere
within the framework of the bosonic fuzzy sphere and define Connes' triple. It
turns out that there are two different types of spectra of the Dirac operator
and correspondingly there are two classes of quantized algebras. As a result we
obtain a new restriction on the Planck constant in Berezin's quantization. The
map to the local frame in noncommutative geometry is also discussed.Comment: 24 pages, latex, no figure
Algebraic characterization of the Wess-Zumino consistency conditions in gauge theories
A new way of solving the descent equations corresponding to the Wess-Zumino
consistency conditions is presented. The method relies on the introduction of
an operator which allows to decompose the exterior space-time
derivative as a commutator. The case of the Yang-Mills theories is
treated in detail.Comment: 16 pages, UGVA-DPT 1992/08-781 to appear in Comm. Math. Phy
Lectures on graded differential algebras and noncommutative geometry
These notes contain a survey of some aspects of the theory of graded
differential algebras and of noncommutative differential calculi as well as of
some applications connected with physics. They also give a description of
several new developments.Comment: 71 pages; minor typo correction
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