Operator formalism for b-c systems with $\lambda=1$ on general algebraic curves

TL;DR

This paper develops an operator formalism for b-c systems with conformal weight 1 on algebraic curves, enabling explicit calculations in string theory and quantum field theory on Riemann surfaces.

Contribution

It introduces a novel operator formalism for b-c systems on algebraic curves, facilitating explicit amplitude computations and vacuum state construction.

Findings

Explicit construction of vacuum states and operators.

Rigorous computation of amplitudes using normal ordering.

Application to string theory and field quantization on Riemann surfaces.

Abstract

In this letter we develope an operator formalism for the $b-c$ systems with conformal weight $\lambda=1$ defined on a general closed and orientable Riemann surface. The advantage of our approach is that the Riemann surface is represented as an affine algebraic curve. In this way it is possible to perform explicit calculations in string theory at any perturbative order. Besides the obvious applications in string theories and conformal field theories, (the $b-c$ systems at $\lambda=1$ are intimately related to the free scalar field theory), the operator formalism presented here sheds some light also on the quantization of field theories on Riemann surfaces. In fact, we are able to construct explicitly the vacuum state of the $b-c$ systems and to define creation and annihilation operators. All the amplitudes are rigorously computed using simple normal ordering prescriptions as in the flat case.