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$\ell$-dimensional euclidean space $E$ corresponds to an identification of the Clifford algebra of $E$ with the canonical anticommutation relation algebra for $\ell$ ( 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

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 $\delta$ which allows to decompose the exterior space-time derivative $d$ as a $BRS$ 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

Strong Connections on Quantum Principal Bundles

A gauge invariant notion of a strong connection is presented and characterized. It is then used to justify the way in which a global curvature form is defined. Strong connections are interpreted as those that are induced from the base space of a quantum bundle. Examples of both strong and non-strong connections are provided. In particular, such connections are constructed on a quantum deformation of the fibration $S^2 -> RP^2$. A certain class of strong $U_q(2)$-connections on a trivial quantum principal bundle is shown to be equivalent to the class of connections on a free module that are compatible with the q-dependent hermitian metric. A particular form of the Yang-Mills action on a trivial U\sb q(2)-bundle is investigated. It is proved to coincide with the Yang-Mills action constructed by A.Connes and M.Rieffel. Furthermore, it is shown that the moduli space of critical points of this action functional is independent of q.Comment: AMS-LaTeX, 40 pages, major revision including examples of connections over a quantum real projective spac

Noncommutative Geometry of Finite Groups

A finite set can be supplied with a group structure which can then be used to select (classes of) differential calculi on it via the notions of left-, right- and bicovariance. A corresponding framework has been developed by Woronowicz, more generally for Hopf algebras including quantum groups. A differential calculus is regarded as the most basic structure needed for the introduction of further geometric notions like linear connections and, moreover, for the formulation of field theories and dynamics on finite sets. Associated with each bicovariant first order differential calculus on a finite group is a braid operator which plays an important role for the construction of distinguished geometric structures. For a covariant calculus, there are notions of invariance for linear connections and tensors. All these concepts are explored for finite groups and illustrated with examples. Some results are formulated more generally for arbitrary associative (Hopf) algebras. In particular, the problem of extension of a connection on a bimodule (over an associative algebra) to tensor products is investigated, leading to the class of `extensible connections'. It is shown that invariance properties of an extensible connection on a bimodule over a Hopf algebra are carried over to the extension. Furthermore, an invariance property of a connection is also shared by a `dual connection' which exists on the dual bimodule (as defined in this work).Comment: 34 pages, Late

BRS Symmetry in Connes' Non-commutative Geometry

We extend the BRS and anti-BRS symmetry to the two point space of Connes' non-commutative model building scheme. The constraint relations are derived and the quantum Lagrangian constructed. We find that the quantum Lagrangian can be written as a functional of the curvature for symmetric gauges with the BRS, anti-BRS auxiliary field finding a geometrical interepretation as the extension of the Higgs scalar.Comment: 28 pages, To appear in the Journal of Physics

Algebraic structure of gravity in Ashtekar variables

The BRST transformations for gravity in Ashtekar variables are obtained by using the Maurer-Cartan horizontality conditions. The BRST cohomology in Ashtekar variables is calculated with the help of an operator $\delta$ introduced by S.P. Sorella, which allows to decompose the exterior derivative as a BRST commutator. This BRST cohomology leads to the differential invariants for four-dimensional manifolds.Comment: 19 pages, report REF. TUW 94-1

Cosmological model with Born-Infeld type scalar field

The non-abelian generalization of the Born-Infeld non-linear lagrangian is extended to the non-commutative geometry of matrices on a manifold. In this case not only the usual SU(n) gauge fields appear, but also a natural generalization of the multiplet of scalar Higgs fields, with the double-well potential as a first approximation. The matrix realization of non-commutative geometry provides a natural framework in which the notion of a determinant can be easily generalized and used as the lowest-order term in a gravitational lagrangian of a new kind. As a result, we obtain a Born-Infeld-like lagrangian as a root of sufficiently high order of a combination of metric, gauge potentials and the scalar field interactions. We then analyze the behavior of cosmological models based on this lagrangian. It leads to primordial inflation with varying speed, with possibility of early deceleration ruled by the relative strength of the Higgs field.Comment: 27 page

Nonassociative geometry in quasi-Hopf representation categories II: connections and curvature

We continue our systematic development of noncommutative and nonassociative differential geometry internal to the representation category of a quasitriangular quasi-Hopf algebra. We describe derivations, differential operators, differential calculi and connections using universal categorical constructions to capture algebraic properties such as Leibniz rules. Our main result is the construction of morphisms which provide prescriptions for lifting connections to tensor products and to internal homomorphisms. We describe the curvatures of connections within our formalism, and also the formulation of Einstein-Cartan geometry as a putative framework for a nonassociative theory of gravity

Orientational instabilities in nematics with weak anchoring under combined action of steady flow and external fields

We study the homogeneous and the spatially periodic instabilities in a nematic liquid crystal layer subjected to steady plane {\em Couette} or {\em Poiseuille} flow. The initial director orientation is perpendicular to the flow plane. Weak anchoring at the confining plates and the influence of the external {\em electric} and/or {\em magnetic} field are taken into account. Approximate expressions for the critical shear rate are presented and compared with semi-analytical solutions in case of Couette flow and numerical solutions of the full set of nematodynamic equations for Poiseuille flow. In particular the dependence of the type of instability and the threshold on the azimuthal and the polar anchoring strength and external fields is analysed.Comment: 12 pages, 6 figure