Mirror Symmetry in Physics: The Basics

TL;DR

This paper introduces mathematicians to the physical origins of mirror symmetry, explaining its basis through conformal field theory and illustrating the connection with Calabi-Yau manifolds, using the torus as an example.

Contribution

It develops foundational concepts linking mirror symmetry, conformal field theory, and Calabi-Yau manifolds, making the physical origins accessible to mathematicians.

Findings

Mirror symmetry explained via conformal field theory.

Deep connection between (2,2) supersymmetric theories and Calabi-Yau manifolds.

Illustrative example using the torus to demonstrate duality.

Abstract

These notes are aimed at mathematicians working on topics related to mirror symmetry, but are unfamiliar with the physical origins of this subject. We explain the physical concepts that enable this surprising duality to exist, using the torus as an illustrative example. Then, we develop the basic foundations of conformal field theory so that we can explain how mirror symmetry was first discovered in that context. Along the way we will uncover a deep connection between conformal field theories with (2,2) supersymmetry and Calabi-Yau manifolds. (Based on lectures given during the "Thematic Program on Calabi-Yau Varieties: Arithmetic, Geometry and Physics" at the Fields Institute in Toronto, October 10-11, 2013.)