On Curvature in Noncommutative Geometry

A general definition of a bimodule connection in noncommutative geometry has been recently proposed. For a given algebra this definition is compared with the ordinary definition of a connection on a left module over the associated enveloping algebra. The corresponding curvatures are also compared.Comment: 16 pages, PlainTe

Linear Connections on Fuzzy Manifolds

Linear connections are introduced on a series of noncommutative geometries which have commutative limits. Quasicommutative corrections are calculated.Comment: 10 pages PlainTex; LPTHE Orsay 95/42; ESI Vienna 23

Linear connections on matrix geometries

A general definition of a linear connection in noncommutative geometry has been recently proposed. Two examples are given of linear connections in noncommutative geometries which are based on matrix algebras. They both possess a unique metric connection.Comment: 14p, LPTHE-ORSAY 94/9

Linear Connections in Non-Commutative Geometry

A construction is proposed for linear connections on non-commutative algebras. The construction relies on a generalisation of the Leibnitz rules of commutative geometry and uses the bimodule structure of $\Omega^1$. A special role is played by the extension to the framework of non-commutative geometry of the permutation of two copies of $\Omega^1$. The construction of the linear connection as well as the definition of torsion and curvature is first proposed in the setting of the derivations based differential calculus of Dubois- Violette and then a generalisation to the framework proposed by Connes as well as other non-commutative differential calculi is suggested. The covariant derivative obtained admits an extension to the tensor product of several copies of $\Omega^1$. These constructions are illustrated with the example of the algebra of $n \times n$ matrices.Comment: 15 pages, LMPM ../94 (uses phyzzx

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

Fuzzy Surfaces of Genus Zero

A fuzzy version of the ordinary round 2-sphere has been constructed with an invariant curvature. We here consider linear connections on arbitrary fuzzy surfaces of genus zero. We shall find as before that they are more or less rigidly dependent on the differential calculus used but that a large number of the latter can be constructed which are not covariant under the action of the rotation group. For technical reasons we have been forced to limit our considerations to fuzzy surfaces which are small perturbations of the fuzzy sphere.Comment: 11 pages, Late

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

Curvature and geometric modules of noncommutative spheres and tori

When considered as submanifolds of Euclidean space, the Riemannian geometry of the round sphere and the Clifford torus may be formulated in terms of Poisson algebraic expressions involving the embedding coordinates, and a central object is the projection operator, projecting tangent vectors in the ambient space onto the tangent space of the submanifold. In this note, we point out that there exist noncommutative analogues of these projection operators, which implies a very natural definition of noncommutative tangent spaces as particular projective modules. These modules carry an induced connection from Euclidean space, and we compute its scalar curvature

Properties of Phase transitions of a Higher Order

The following is a thermodynamic analysis of a III order (and some aspects of a IV order) phase transition. Such a transition can occur in a superconductor if the normal state is a diamagnet. The equation for a phase boundary in an H-T (H is the magnetic field, T, the temperature) plane is derived. by considering two possible forms of the gradient energy, it is possible to construct a field theory which describes a III or a IV order transition and permits a study of thermal fluctuations and inhomogeneous order parameters.Comment: 13 pages, revtex, no figure

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