Quantum field theories on algebraic curves. I. Additive bosons

Using Serre's adelic interpretation of cohomology, we develop a ‘differential and integral calculus’ on an algebraic curve $X$ over an algebraically closed field $k$ of constants of characteristic zero, define algebraic analogues of additive multi-valued functions on $X$ and prove the corresponding generalized residue theorem. Using the representation theory of the global Heisenberg algebra and lattice Lie algebra, 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.