Geometic vertex operators

Abstract

Vertex operators, being families of birational transformations of infinite-dimensional algebraic ``varieties'' M, act on appropriate line bundles on M. However, they act on (meromorphic) sections only as_partial operators_: they are defined on a subspace (in an appropriate_lattice of subspaces_ of Mer(M)), and send "smooth families" of vectors in such a subspace to smooth families. Axiomatizing this, we define_conformal fields_ as arbitrary families of partial operators in Mer(M) which satisfy both these properties. The ``variety'' M related to standard vertex operators is formed by rational functions of one variable z in Z=P^1, changing the variety Z one obtains different examples of M, and multidimensional analogues of vertex operators. One can cover Mer(M) by ``big smooth subsets''; these subsets are parameterized by appropriate projective bundles over Hilbert schemes of points on Z. We deduce conformal associativity relation for conformal fields, and conformal commutation relations for Laurent coefficients of commuting conformal fields from geometric properties of the Hilbert schemes. We start with providing examples of OPE in smooth families, i.e., smooth families a(s) and b(t) (of partial linear operators) such that a(s)b(t) has a pole when s=t. We also provide geometric description of boson-fermion correspondence, and relations of this correspondence to geometry of the set of meromorphic functions.