Semi-infinite Induction and Wakimoto Modules

The purpose of this paper is to suggest the construction and study properties of semi-infinite induction, which relates to semi-infinite cohomology the same way induction relates to homology and coinduction to cohomology. We prove a version of the Shapiro Lemma, showing that the semi-infinite cohomology of a module is isomorphic to that of the semi-infinitely induced module. A practical outcome of our construction is a simple construction of the Wakimoto modules, highest-weight modules used in double-sided BGG resolutions of irreducible modules. Semi-infinite cohomology of Lie algebras, introduced as the appropriate mathematical setting for BRST theory by B. L. Feigin [6] (see also [10, 14] regarding basic facts on semi-infinite cohomology), is a cohomology theory that has properties in common with both cohomology and homology. Semi-infinite cohomology has become an important tool in representation theory of Lie algebras and quantum groups and string theory, see, for instance, [2, 4, 3, 5, 7, 8, 12, 13]. The purpose of this paper is to suggest the construction and study properties of semi-infinite induction, which relates to semi-infinite cohomology the same way induction relates to homology and coinduction to cohomology. The proof of our main theorem (Theorem 1.4, the semi-infinite Shapiro Lemma) is based on the independence of the choice of a resolution, which follows from the machinery of semi-infinite homological algebra developed in [14]. A practical outcome of our construction is a simple construction of Wakimoto modules, which were constructed by Feigin and E. Frenkel [7] in rather roundabout terms: using bosonization and also asH ∞/2+0 U (X,Lλ), a hypothetical semi-infinite cohomology, with support on the big Schubert cell, of an invertible sheaf over a semi-infinite flag manifold. The idea that Wakimoto modules might be obtained by some kind of semi-infinite induction goes back to the original paper of Feigin and Frenkel [7]. This idea was implemented by S. M. Arkhipov [2], who suggested an indirect construction of Wakimoto modules, which, in fact, may be considered as representing a different approach to semiinfinite induction. Wakimoto modules play an intermediate role between Verma and contragredient Verma modules: they all have a very similar behavior with respect to semi-infinite cohomology, usual homology, and cohomology, respectively. The three types of modules have the same character (see Proposition 2.2) but a different layout of irreducible pieces. This work was motivated in part by a construction of N. Berkovits and C. Vafa [5] of N = 1 and N = 2 string theories out of a given N = 0 (bosonic) string theory. In that construction, the bosonic string arose as a particular class of vacua for the N = 1 string and the N = 1 as a particular class of vacua for the N = 2 string. Berkovits and Vafa also suggested that there must be a universal string theory comprising all possible string theories, including, for instance, WN strings, Research supported in part by an AMS Centennial Fellowship and NSF grant DMS-9402076.