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The goal of this paper is to study the representation theory of a classical infinite-dimensional Lie algebra - the Lie algebra of vector fields on an N-dimensional torus for N > 1. The case N=1 gives a famous Virasoro algebra (or its centerless version - the Witt algebra). The algebra of vector fields has an important class of tensor modules parametrized by finite-dimensional modules of gl(N). Tensor modules can be used in turn to construct bounded irreducible modules for the vector fields on N+1-dimensional torus, which are the central objects of our study. We solve two problems regarding these bounded modules: we construct their free field realizations and determine their characters. To solve these problems we analyze the structure of the irreducible modules for the semidirect product of vector fields with the quotient of 1-forms by the differentials of functions. These modules remain irreducible when restricted to the subalgebra of vector fields, unless they belongs to the chiral de Rham complex, introduced by Malikov-Schechtman-Vaintrob.

Representations of the Lie algebra of vector fields on a torus and the chiral de Rham complex