Quantum field theories on algebraic curves. I. Additive bosons
Abstract
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 multivalued 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 kvector space of rational functions on X to the vector space of additive multivalued functions uniquely determines these quantum theories of additive and charged bosons.
 Publication:

Izvestiya: Mathematics
 Pub Date:
 April 2013
 DOI:
 10.1070/IM2013v077n02ABEH002640
 arXiv:
 arXiv:0812.0169
 Bibcode:
 2013IzMat..77..378T
 Keywords:

 Mathematics  Algebraic Geometry;
 High Energy Physics  Theory;
 Mathematics  Quantum Algebra;
 Mathematics  Representation Theory;
 81R10;
 14H81
 EPrint:
 31 pages, published version. Invariant formulation added, multiplicative section removed