In 1988 Witten used current algebras to develop the fundaments of quantum field theory for free fermions living on an algebraic curve. The global symmetries are then given by the rational maps of the curve \(X\) to a finite-dimensional semi-simple Lie algebra over an algebraically closed field \(k\). In a previous paper, titled “Quantum field theories on an algebraic curve” [Lett. Math. Phys. 52, No. 1, 79–91 (2000; Zbl 1024.81016)], the author provided a solution to the problem of determining the expectation value functional for scalar fields with an abelian Lie algebra. Although abelian integrals have been studied before, so far the integral calculus on curves has not been fully developed.
In the present paper Takhtajan tries to fill this gap when the field \(k\) has characteristic zero. He gives an explicit construction of quantum field theories of (additive) bosons on an algebraic curve. The paper is organized as follows. Section 1 provides a long survey of the history of the subject. In Section 2 the author recalls the necessary basic facts from the theory of algebraic curves. This material is standard. In Section 3 he recalls the details of the differential calculus on an algebraic curve and develops a corresponding integral calculus. It is now assumed that the field \(k\) has characteristic 0 and the algebraic curve has genus \(g\geq 1\). In Section 4 local quantum field theories of additive charged bosons are formulated. Finally, in Section 5 global versions of the local QFTs – as studied in Section 4 – are constructed. The paper provides a stimulating account of the mathematical principles and techniques using algebraic curves and some concepts from quantum field theory though the aim is certainly not to suggest possible applications in particle physics.