Quantum Field Theories on Algebraic Curves. I. Additive bosons

TL;DR

This paper develops a differential calculus on algebraic curves over algebraically closed fields, introduces algebraic analogs of multi-valued functions, and formulates quantum field theories of additive and charged bosons using Lie algebra representations.

Contribution

It introduces a novel algebraic differential calculus on curves and constructs quantum field theories of bosons via Lie algebra representations, connecting to the algebraic de Rham theorem.

Findings

Established a generalized residue theorem for algebraic curves.

Formulated quantum field theories of additive and charged bosons algebraically.

Proved the uniqueness of these theories via extended global symmetries.

Abstract

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.