K3 en route From Geometry to Conformal Field Theory

TL;DR

This paper explores the connection between Calabi-Yau geometry and conformal field theory (CFT), focusing on K3 surfaces, toroidal theories, and the elliptic genus to illustrate the deep links between geometry and CFT.

Contribution

It provides a detailed background on Calabi-Yau geometry and discusses CFT constructions based on these geometries, highlighting K3 surfaces and elliptic genus as key examples.

Findings

Calabi-Yau geometry underpins certain CFT constructions.

K3 surfaces are central to understanding non-linear sigma models.

Elliptic genus links geometry with conformal field theory.

Abstract

To pave the way for the journey from geometry to conformal field theory (CFT), these notes present the background for some basic CFT constructions from Calabi-Yau geometry. Topics include the complex and Kaehler geometry of Calabi-Yau manifolds and their classification in low dimensions. I furthermore discuss CFT constructions for the simplest known examples that are based in Calabi-Yau geometry, namely for the toroidal superconformal field theories and their Z2-orbifolds. En route from geometry to CFT, I offer a discussion of K3 surfaces as the simplest class of Calabi-Yau manifolds where non-linear sigma model constructions bear mysteries to the very day. The elliptic genus in CFT and in geometry is recalled as an instructional piece of evidence in favor of a deep connection between geometry and conformal field theory.