Quantum field theories on algebraic curves. I.~Additive bosons

L. A. Takhtajan

Izvestiya. Mathematics , Tome 77 (2013) no. 2, pp. 378-406

Résumé

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.

Détail

Keywords: algebraic curves and algebraic functions, additive multi-valued functions, additive Ward identities, Heisenberg algebra, current algebra on an algebraic curve, generalized residue theorem, Fock spaces, quantum theories of free bosons on an algebraic curve, expectation value functional.
Mots-clés : adèles