Factorization of formal exponentials and uniformization

Abstract

Let $\mathfrak{g}$ be a Lie algebra in characteristic zero equipped with a vector space decomposition $\mathfrak{g}={\mathfrak{g}}^{-}\oplus {\mathfrak{g}}^{+}$, and let $s$ and $t$ be commuting formal variables. We prove that the Campbell-Baker-Hausdorff map $C:s{\mathfrak{g}}^{-}[[s,t]]\times t{\mathfrak{g}}^{+}[[s,t]]\to s{\mathfrak{g}}^{-}[[s,t]]\oplus t{\mathfrak{g}}^{+}[[s,t]]$ given by $e^{s{g}^{-}}{e}^{t{g}^{+}}=e^{C(s{g}^{-},t{g}^{+})}$ for $g^{\pm }\in {\mathfrak{g}}^{\pm }[[s,t]]$ is a bijection, as is well known when $\mathfrak{g}$ is finite-dimensional over $\mathbb{R}$ or $\mathbb{C}$, by geometry. It follows that there exist unique ${\Psi }^{\pm }\in {\mathfrak{g}}^{\pm }[[s,t]]$ such that $e^{t{g}^{+}}{e}^{s{g}^{-}}=e^{s{\Psi }^{-}}{e}^{t{\Psi }^{+}}$ (also well known in the finite-dimensional geometric setting). We apply this to $\mathfrak{g}$ consisting of certain formal infinite series with coefficients in a Lie algebra $\mathfrak{p}$. For $\mathfrak{p}$ the Virasoro algebra (resp., a Grassmann envelope of the Neveu-Schwarz superalgebra), the result was first proved by Huang (resp., Barron) as a step in the construction of a (super)geometric formulation of the notion of vertex operator (super)algebra. For 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.

LaTex file, 31 pages

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