Factorization of Formal Exponentials and Uniformization

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 e − e + = 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 e + e − = 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 Supported in part by an NSF Mathematical Sciences Postdoctoral Research Fellowship and by a University of California President’s Postdoctoral Fellowship Supported in part by NSF grant DMS-9622961 Supported in part by NSF grants DMS-9401851 and DMS-9701150