Factorization of formal exponentials and uniformization

Abstract

Let g be a Lie algebra over a field of characteristic zero equipped with a vector space decomposition g=g-⊕g+, and let s and t be commuting formal variables commuting with g. We prove that the map C: sg-[[s,t]]×tg+[[s,t]]→sg-[[s,t]]⊕tg+[[s,t]] defined by the Campbell-Baker-Hausdorff formula and given by esg-etg+=eC(sg-,tg+) for g±∈g±[[s,t]] is a bijection, as is well known when g is finite-dimensional over R or C, by geometry. It follows that there exist unique Ψ±∈g±[[s,t]] such that etg+esg-=esΨ-etΨ+ (also well known in the finite-dimensional geometric setting). We apply this to a Lie algebra g consisting of certain formal infinite series with coefficients in a Z-graded Lie algebra p, for instance, an affine Lie algebra, the Virasoro algebra, or a Grassmann envelope of the N=1 Neveu-Schwarz superalgebra. For p the Virasoro algebra, the result was first proved by Huang as a step in the construction of a geometric formulation of the notion of vertex operator algebra, and for p a Grassmann envelope of the Neveu-Schwarz superalgebra, it was first proved by Barron as a corresponding step in the construction of a supergeometric formulation of the notion of vertex operator superalgebra. In the special case of the Virasoro (resp., N=1 Neveu-Schwarz) algebra with zero central charge the result gives the precise expansion of the uniformizing function for a sphere (resp., supersphere) with tubes resulting from the sewing of two spheres (resp., superspheres) with tubes in two-dimensional genus-zero holomorphic conformal (resp., N=1 superconformal) field theory. The general result places such uniformization problems into a broad formal algebraic context. © 2000 Academic Press.