Application of abelian holonomy formalism to the elementary theory of numbers

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 $S^1\times S^1\times \mathbf{R}$. 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 ${\mathbf{F}}_p=\mathbf{Z}/p\mathbf{Z}$ 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.

50 pages; v2,3. minor corrections; v4. minor revisions, published version

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Cited by in corpus (3)

• Holonomies of gauge fields in twistor space 4: functional MHV rules and one-loop amplitudes
• Abelian Chern-Simons theory on the torus and physical views on the Hecke operators
• Some properties of zero-mode wave functions in abelian Chern-Simons theory on the torus