Quantum Field Theories on Algebraic Curves and A. Weil Reciprocity Law

Using Serre’s adelic interpretation of the cohomology, we develop “differential and integral calculus” on an algebraic curve X over an algebraically closed constant field k of characteristic zero, define an algebraic analogs of additive and multiplicative multi-valued functions on X, and prove corresponding generalized residue theorem and A. Weil reciprocity law. Using the representation theory of global Heisenberg and lattice Lie algebras and the Heisenberg system, we formulate quantum field theories of additive, charged, and multiplicative bosons on an algebraic curve X. We prove that extension of the respected global symmetries — Witten’s additive and multiplicative Ward identities — from the k-vector space of rational functions on X to the k-vector space of additive multi-valued functions, and from the multiplicative group of rational functions on X to the group of multiplicative multi-valued functions on X, defines these theories uniquely. The quantum field theory of additive bosons is naturally associated with the algebraic de Rham theorem and the generalized residue theorem, and the quantum field theory of multiplicative bosons — with the generalized A. Weil reciprocity law.