Conformal fields: a class of representations of Vect(N)

TL;DR

This paper introduces conformal fields as a new class of representations of the algebra of vector fields in N dimensions, linking local differential geometry with representation theory and constructing Fock modules.

Contribution

It defines conformal fields as a novel class of modules for Vect(N), with finite-dimensional restrictions to sl(N+1), and constructs associated Fock modules.

Findings

Conformal fields form a new class of Vect(N) modules.

Restrictions of conformal fields to sl(N+1) are finite-dimensional.

Infinities occur unless bosonic and fermionic degrees of freedom match.

Abstract

$Vect(N)$, the algebra of vector fields in $N$ dimensions, is studied. Some aspects of local differential geometry are formulated as $Vect(N)$ representation theory. There is a new class of modules, {\it conformal fields}, whose restrictions to the subalgebra $sl(N+1) \subset Vect(N)$ 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.