Factorization of Formal Exponentials and Uniformization

Let be a Lie algebra over a field of characteristic zero equipped with a vector space decomposition = − + , and let s and t be commuting formal variables commuting with . We prove that the map C: s − [[s, t]] × t + [[s, t]] → s − [[s, t]] t + [[s, t]] defined by the Campbell–Baker–Hausdorff formula and given by esg − etg + = eC(sg − , tg + ) for g ± ± [[s, t]] is a bijection, as is well known when is finite-dimensional over or , by geometry. It follows that there exist unique Ψ ± ± [[s, t]] such that etg + esg − = esΨ − etΨ + (also well known in the finite-dimensional geometric setting). We apply this to a Lie algebra consisting of certain formal infinite series with coefficients in a -graded Lie algebra , for instance, an affine Lie algebra, the Virasoro algebra, or a Grassmann envelope of the N = 1 Neveu–Schwarz superalgebra. For 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 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.