Arithmetic Spacetime Geometry from String Theory

TL;DR

This paper introduces an arithmetic framework for string compactification, linking conformal field theory characters to spacetime geometry and revealing connections to moonshine phenomena involving Mathieu and Conway groups.

Contribution

It formulates a novel approach to derive spacetime geometry from exactly solvable string theories and connects modular forms to arithmetic moonshine.

Findings

Conformal characters are derived from spacetime geometry.

Geometry is uniquely determined by world sheet field theory.

Modular forms exhibit complex multiplication and relate to moonshine groups.

Abstract

An arithmetic framework to string compactification is described. The approach is exemplified by formulating a strategy that allows to construct geometric compactifications from exactly solvable theories at $c=3$. It is shown that the conformal field theoretic characters can be derived from the geometry of spacetime, and that the geometry is uniquely determined by the two-dimensional field theory on the world sheet. The modular forms that appear in these constructions admit complex multiplication, and allow an interpretation as generalized McKay-Thompson series associated to the Mathieu and Conway groups. This leads to a string motivated notion of arithmetic moonshine.