Application of abelian holonomy formalism to the elementary theory of numbers

TL;DR

This paper explores the connection between abelian holonomy operators in conformal field theory and number theory, particularly linking zero-mode holonomy to the Riemann zeta function and prime number scattering.

Contribution

It introduces a novel application of abelian holonomy formalism to elementary number theory, relating zero-mode holonomy operators to the Riemann zeta function and prime scattering amplitudes.

Findings

Zero-mode holonomy operators can be expressed using Riemann's zeta function.

A generalized linking number concept is developed via vacuum expectation values.

The framework offers a physical interpretation of the Riemann hypothesis.

Abstract

We consider an abelian holonomy operator in two-dimensional conformal field theory with zero-mode contributions. The analysis is made possible by use of a geometric-quantization scheme for abelian Chern-Simons theory on $S^1 \times S^1 \times {\bf R}$. We find that a purely zero-mode part of the holonomy operator can be expressed in terms of Riemann's zeta function. We also show that a generalization of linking numbers can be obtained in terms of the vacuum expectation values of the zero-mode holonomy operators. Inspired by mathematical analogies between linking numbers and Legendre symbols, we then apply these results to a space of ${\bf F}_p = {\bf Z}/ p {\bf Z}$ where $p$ is an odd prime number. This enables us to calculate "scattering amplitudes" of identical odd primes in the holonomy formalism. In this framework, the Riemann hypothesis can be interpreted by means of a physically obvious fact, i.e., there is no notion of "scattering" for a single-particle system. Abelian gauge theories described by the zero-mode holonomy operators will be useful for studies on quantum aspects of topology and number theory.