ar X iv : h ep - t h / 92 07 03 0 v 1 9 J ul 1 99 2 Vect ( N ) conformal fields and their exterior derivatives

Conformal fields are a recently discovered class of representations of the algebra of vector fields in N dimensions. Invariant first-order differential operators (exterior derivatives) for conformal fields are constructed. 1 1. Introduction In order to describe physical quantities, a coordinate system has to be introduced, but physics itself should not depend on how this is done. Therefore any sensible object must transform as a representation of the group of diffeomorphisms (coordinate transformations) in the relevant space, and conversely the representation theory of the diffeomorphism group amounts to a classification of inequivalent meaningful objects. A sensible starting point is the Lie algebra of the diffeomorphism group; this is the algebra of vector fields. Moreover, we must first deal with the local properties of this algebra. Since any manifold is locally diffeomorphic to R N , the only important parameter is the dimensionality N , and it makes sense to talk about the algebra of vector fields in N dimensions, V ect(N). This algebra has recently attracted some interest by physicists 1−6 ; references some earlier mathematical literature can be found in Ref. 1. The one-dimensional case is special, because it admits a central extension (Virasoro algebra), which is a cornerstone of modern theoretical physics 7. From the arguments above, it is clear that N-dimensional local differential geometry may be considered as the representation theory of V ect(N), and it is in fact straightforward to describe e.g. tensor fields and exterior derivatives in such terms. In a recent paper 1 , we discovered a new class of V ect(N) representations which seem more natural than tensor fields, namely conformal fields. A tensor field can be considered as a scalar field decorated with indices from the rigid gl(N) subalgebra. Similarly, a conformal field has indices from the " conformal " subalgebra sl(N + 1), which is obtained from gl(N) by adding translations and " conformal " transformations. Because gl(N) ⊂ sl(N + 1), a conformal field transforms nicely under a larger finite-dimensional subalgebra than a tensor field does. This paper is organized as follows. In section 2 we recollect some relevant facts about tensor fields and exterior derivatives, formulated in a fashion which emphasizes the representation theory aspects. Section 3 contains the definition of conformal fields, in a slightly more streamlined notation than in Ref. 1. It also contains some minor new results. In section 4 we construct first-order differential operators that are …