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

@article{Takhtajan2008QuantumFT, title={Quantum field theories on algebraic curves. I. Additive bosons}, author={Leon A. Takhtajan}, journal={Izvestiya: Mathematics}, year={2008}, volume={77}, pages={378 - 406} }

Using Serre's adelic interpretation of cohomology, we develop a ‘differential and integral calculus’ on an algebraic curve over an algebraically closed field of constants of characteristic zero, define algebraic analogues of additive multi-valued functions on 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…

## 2 Citations

### On Integrable Field Theories as Dihedral Affine Gaudin Models

- MathematicsInternational Mathematics Research Notices
- 2018

We introduce the notion of a classical dihedral affine Gaudin model, associated with an untwisted affine Kac–Moody algebra $\widetilde{\mathfrak{g}}$ equipped with an action of the dihedral group…

### Reciprocity Laws on Algebraic Surfaces via Iterated Integrals

- Mathematics
- 2012

This paper presents a proof of reciprocity laws for the Parshin symbol and for two new local symbols, defined here, which we call 4-function local symbols. The reciprocity laws for the Parshin symbol…

## 27 References

### Quantum field theory, Grassmannians, and algebraic curves

- Mathematics
- 1988

This paper is devoted in part to clarifying some aspects of the relation between quantum field theory and infinite Grassmannians, and in part to pointing out the existence of a close analogy between…

### Introduction to the theory of algebraic functions of one variable

- Mathematics
- 1951

This classical book, written by a famous French mathematician in the early 1950s, presents an approach to algebraic geometry of curves treated as the theory of algebraic functions on the curve. Among…

### Vertex Algebras and Algebraic Curves

- Mathematics
- 2000

Introduction Definition of vertex algebras Vertex algebras associated to Lie algebras Associativity and operator product expansion Applications of the operator product expansion Modules over vertex…

### Poincaré biextension and idèles on an algebraic curve

- Mathematics
- 2006

The Weil pairing of two elements of the torsion of the Jacobian of an algebraic curve can be expressed in terms of the product of the local Hilbert symbols of two special ideles associated with the…

### The Mathematical Heritage of Hermann Weyl

- Mathematics
- 1988

On induced representations by R. Bott Differentiable structures on fractal-like sets, determined by intrinsic scaling functions on dual Cantor sets by D. Sullivan Representation theory and arithmetic…

### Projective structures on a Riemann surface, II

- Mathematics
- 1999

This is a continuation of an earlier work [BR] (referred to as Part I),where we studied some algebraic-geometric aspects of projective structures on a compact Riemann surface. For a compact Riemann…

### A holomorphic version of the Tate-Iwasawa method for unramified -functions. I

- Mathematics
- 2014

Using the Tate-Iwasawa method the problem of meromorphic continuation and of the existence of a functional equation can be solved for the zeta and -functions of one-dimensional arithmetical schemes.…

### Quantum Field Theories on an Algebraic Curve

- Physics, Mathematics
- 2000

We formulate quantum field theories on an algebraic curve and outline a 'paradigm' interpreting Ward identities as reciprocity laws.

### Geometric realization of conformal field theory on Riemann surfaces

- Mathematics
- 1988

Conformal field theory on a family of Riemann surfaces is formulated. We derive equations of motion of vacua which are parametrized by moduli of Riemann surfaces and show that these vacua are…

### Principles of Algebraic Geometry

- Mathematics
- 1978

A comprehensive, self-contained treatment presenting general results of the theory. Establishes a geometric intuition and a working facility with specific geometric practices. Emphasizes applications…