  Using Serre's adelic interpretation of cohomology, we develop a `differential
and integral calculus' on an algebraic curve X over an algebraically closed
filed k of constants of characteristic zero, define algebraic analogs of
additive multi-valued functions on X and prove corresponding generalized
residue theorem. Using the representation theory of the global Heisenberg and
lattice Lie algebras, we formulate quantum field theories of additive and
charged bosons on an algebraic curve X. These theories are naturally connected
with the algebraic de Rham theorem. We prove that an extension of global
symmetries (Witten's additive Ward identities) from the k-vector space of
rational functions on X to the vector space of additive multi-valued functions
uniquely determines these quantum theories of additive and charged bosons.
