ar X iv : h ep - t h / 92 07 02 9 v 1 9 J ul 1 99 2 Conformal fields : a class of representations of Vect ( N )

V ect(N), the algebra of vector fields in N dimensions, is studied. Some aspects of local differential geometry are formulated as V ect(N) representation theory. There is a new class of modules, conformal fields, whose restrictions to the subalgebra sl(N + 1) ⊂ V ect(N) are finite-dimensional sl(N + 1) representations. In this regard they are simpler than tensor fields. Fock modules are also constructed. Infinities, which are unremovable even by normal ordering, arise unless bosonic and fermionic degrees of freedom match. 1 1. Introduction It seems to be a reasonable assumption that the physical content of a theory is independent of the choice of coordinate system. This leads by definition to the conclusion that any physical quantity must be an intrinsic object in the sense of differential geometry, i.e. that it must transform as a representation of the group Dif f (M) of all coordinate transformations, or diffeomorphisms, on the base manifold M. Something which is not a representation of Dif f (M) is clearly an artefact of the choice of coordinate system and thus unsuitable for physics. Diffeomorphisms on arbitrary manifolds have been studied by mathematicians from various points of view for a long time 1−6 , and they have recently attracted some interest by physicists as natural generalizations of the one-dimensional case 7−12. Nevertheless, the representation theory of Dif f (M) is not very well understood when dim M > 1. In particular, no irreducible or lowest-weight representations are known to us 13. In order to study the diffeomorphism group it is reasonable to start with its Lie algebra of vector fields on M. Moreover, we only consider vector fields locally; we expect problems of homological nature to appear in a global approach, but an understanding of the local properties is certainly a prerequisite for global analysis. Such a program has of course been carried out in great detail on the circle, where it leads to the Virasoro algebra, i.e. the universal central extension of V ect(1) 13−15. The same algebra is also at the core of conformal field theory which is important to the theory of critical phenomena in two dimensions 16. In section 2 we define V ect(N), the algebra of vector fields in N-dimensional space. The algebra is given in a plane wave basis; this is always possible to do locally. We also show that it does not admit any central extension except when …