Canonical torsor bundle of prescribed rational functions on complex curves

Prescribed rational functions constitute a subset of rational functions satisfying certain symmetry and analyticity conditions. We define and construct explicitly prescribed rational functions-valued bundle $\mathcal{W}_M$ over a smooth complex curve $M$. An intrinsic coordinate-independent formulation for such bundle is is given. The construction presented in this paper is useful for studies of the canonical cosimplicial cohomology of infinite-dimensional Lie algebras on smooth manifolds, as well as for purposed of conformal field theory, deformation theory, and the theory of foliations

Bigraded differential algebra for vertex algebra complexes

For the bicomplex structure of grading-restricted vertex algebra cohomology defined in [6], we show that the orthogonality and double grading conditions applied endow it with the structure of a bigraded differential algebra with respect to a natural multiplication. The generators and commutation relations of the bigraded differential algebra form a continual Lie algebra $\mathcal G(V)$ with the root space provided by a grading-restricted vertex algebra $V$. We prove that the differential algebra generates non-vanishing cohomological invariants associated to a vertex algebra $V$. Finaly, we provide examples associated to various choices of the vertex algebra bicomplex subspaces.Comment: arXiv admin note: substantial text overlap with arXiv:2012.07343, arXiv:2012.05904; text overlap with arXiv:1006.2516 by other author

Multiple products of meromorphic functions

Let $\mathfrak g$ be an infinite-dimensional Lie algebra, and $G$ be the algebraic completion of a $\mathfrak g$-module. Using the geometric model of Schottky uniformization of Riemann sphere to obtain a higher genus Riemann surface, we construct a family of parametric extensions of coboundary operators for the double complexes of meromorphic functions depending on elements of $G$, and possessing prescribed analytic properties