Cosimplicial meromorphic functions cohomology on complex manifolds

Developing ideas of \cite{Fei}, we introduce canonical cosimplicial cohomology of meromorphic functions for infinite-dimensional Lie algebra formal series with prescribed analytic behavior on domains of a complex manifold $M$. Graded differential cohomology of a sheaf of Lie algebras $\mathcal G$ via the cosimplicial cohomology of $\mathcal G$-formal series for any covering by Stein spaces on $M$ is computed. A relation between cosimplicial cohomology (on a special set of open domains of $M$) of formal series of an infinite-dimensional Lie algebra $\mathcal G$ and singular cohomology of auxiliary manifold associated to a $\mathcal G$-module is found. Finally, multiple applications in conformal field theory, deformation theory, and in the theory of foliations are proposed.