Multiple products of meromorphic functions

Abstract

Let $\mathfrak g$ be an infinite-dimensional Lie algebra and let $G$ be the algebraic completion of a graded $\mathfrak g$-module $W$. Using the Schottky uniformization of the Riemann sphere as a geometric model for a genus $\kappa$ Riemann surface, we construct a $\kappa$-parameter family of extended coboundary operators $\widetilde\delta^n_m(\rho_1,\ldots,\rho_\kappa)$ acting on the double complex of predetermined meromorphic functions on the configuration space $F_n\C$ with values determined by $G$. The extension is realized as a graded trace, defined coordinate-freely as the trace of a canonically associated finite-rank endomorphism of each homogeneous component $W_{(k)}$ of $W$, of the classical coboundary operator, the $\kappa$ sewing loci being held disjoint from the free marked points at which the classical differential acts. The sewing operator is exhibited as a chain map between explicitly defined complexes. We give a complete proof of the resulting chain property $\widetilde\delta^{n+1}_{m-2\kappa-1}\circ\widetilde\delta^n_m=0$ and of the convergence of the defining power series in the sewing parameters $\rho_p$ under an explicit growth hypothesis, and we determine precisely how the construction depends on the auxiliary choice of local coordinates and sewing annuli. Applications of the resulting cohomology theory - to the sheaf of conformal blocks on the Deligne-Mumford moduli space of stable curves, to secondary characteristic classes of holomorphic foliations, to graded trace functions arising in the description of topological phases of matter, and to integrable hierarchies of Toda type - are proposed and discussed as motivation, without claiming these correspondences as theorems of the present paper.