  For an irreducible affine variety $X$ over an algebraically closed field of
characteristic zero we define two new classes of modules over the Lie algebra
of vector fields on $X$ - gauge modules and Rudakov modules, which admit a
compatible action of the algebra of functions. Gauge modules are
generalizations of modules of tensor densities whose construction was inspired
by non-abelian gauge theory. Rudakov modules are generalizations of a family of
induced modules over the Lie algebra of derivations of a polynomial ring
studied by Rudakov. We prove general simplicity theorems for these two types of
modules and establish a pairing between them.
