Differential Calculus and Discrete Structures.pdfVIP

a r X i v : h e p - t h / 9 4 0 1 1 5 0 v 1 2 8 J a n 1 9 9 4 GOET-TP 98/93 December 1993 Differential Calculus and Discrete Structures Aristophanes Dimakis Department of Mathematics, University of Crete, GR-71409 Iraklion and Folkert Mu?ller-Hoissen Institut fu?r Theoretische Physik, D-37073 Go?ttingen, Germany Abstract There is a deformation of the ordinary differential calculus which leads from the continuum to a lattice (and induces a corresponding deformation of physical theories). We recall some of its features and relate it to a general framework of differential calculus on discrete sets. This framework generalizes the usual (lattice) discretization. To appear in the proceedings of the International Sym- posium on “Generalized Symmetries in Physics”, ASI Clausthal, July 1993. 1 1 Introduction In the context of ‘noncommutative geometry’ [1, 2] the following structure – which gener- alizes the notion of differential forms (on a manifold) – plays a crucial role. A differen- tial calculus for an associative algebra A (over IR or C) is a ZZ-graded associative algebra Λ(A) = ⊕ ∞ r=0 Λ r(A) (where Λr(A) are A-bimodules and Λ0(A) = A) together with a linear operator d : Λr(A) → Λr+1(A) satisfying d2 = 0 and d(ωω′) = (dω)ω′ + (?1)rω dω′ where ω ∈ Λr(A). We will assume that Λ(A) has a unit 1I such that d1I = 0. By now there is a vast literature dealing with differential calculi on various types of mostly non-commutative algebras, in particular quantum groups. But even commutative algebras exhibited in this context rather unexpected features. In physics, models of elementary particle physics were built with a space-time of the form M × ZZ2 where M is a four-dimensional differentiable manifold [3]. Using differential calculus on (the algebra of functions on) the two-point space ZZ2, it was possible to extend the Yang-Mills action to M × ZZ2. Its ZZ2-part turned out to be the usual Higgs potential. Later, it was demonstrated that a certain deformation of the ordinary