Conformal Modules

In this paper we study a class of modules over infinite-dimensional Lie (super)algebras, which we call conformal modules. In particular we classify and construct explicitly all irreducible conformal modules over the Virasoro and the N=1 Neveu-Schwarz algebras, and over the current algebras. 0. Introduction Conformal module is a basic tool for the construction of free field realization of infinitedimensional Lie (super)algebras in conformal field theory. This is one of the reasons to classify and construct such modules. In the present paper we solve this problem under the irreducibility assumption for the Virasoro and the Neveu-Schwarz algebras, for the current algebras and their semidirect sums. Since complete reducibility does not hold for conformal modules, one has to discuss the extension problem. This problem is solved in [1]. The basic idea of our approach is to use three (more or less) equivalent languages. The first is the language of local formal distributions, the second is the language of modules over conformal algebras, and the third is the language of conformal modules over the annihilation subalgebras. The problem is solved using the third language by means of the crucial Lemma 3.1. Note that conformal modules over Lie algebras of Cartan type were studied in [6], where, in particular, a proof of Corollary 3.3 is contained. † partially supported by NSC grant 86-2115-M-006-012 of the ROC †† partially supported by NSF grant DMS-9622870