2022/08/30 by A. Zuevsky, Zuevsky, A.
Mathematics · #Analytic and geometric function theory #FOS: Mathematics #Functional Analysis (math.FA) #Geometry and complex manifolds #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2208.14483
openalex publication_date 2022/08/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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 κ Riemann surface, we construct a κ-parameter family of extended coboundary operators \widetildeδnm(ρ1,…,ρκ) acting on the double complex of predetermined meromorphic functions on the configuration space Fn\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 κ 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 \widetildeδn+1m-2κ-1∘\widetildeδnm=0 and of the convergence of the defining power series in the sewing parameters ρp 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.