Arithmetic of Calabi-Yau Varieties and Rational Conformal Field Theory

TL;DR

This paper explores a novel arithmetic algebraic geometric approach to connect Calabi-Yau manifold geometry with conformal field theory, focusing on number fields derived from fusion rules and zeta functions.

Contribution

It introduces a framework linking Calabi-Yau geometry and conformal field theory through algebraic number fields from cohomological L-functions and fusion rules.

Findings

Number fields from fusion rules can be derived from zeta functions.

Quantum dimensions are characterized by algebraic number fields.

A natural number theoretic characterization of quantum dimensions is proposed.

Abstract

It is proposed that certain techniques from arithmetic algebraic geometry provide a framework which is useful to formulate a direct and intrinsic link between the geometry of Calabi-Yau manifolds and the underlying conformal field theory. Specifically it is pointed out how the algebraic number field determined by the fusion rules of the conformal field theory can be derived from the number theoretic structure of the cohomological Hasse-Weil L-function determined by Artin's congruent zeta function of the algebraic variety. In this context a natural number theoretic characterization arises for the quantum dimensions in this geometrically determined algebraic number field.