Mirror Manifolds: A Brief Review and Progress Report

TL;DR

This paper reviews the discovery and development of mirror symmetry in string theory, highlighting its conceptual implications, computational aspects, and recent advances in proof techniques.

Contribution

It provides a comprehensive overview of mirror symmetry's discovery, consequences, and introduces new proof methods extending its domain.

Findings

Mirror symmetry links classical and quantum theories.

Recent work affirms duality and extends mirror symmetry construction.

Conceptual and computational implications of mirror symmetry are significant.

Abstract

We first give a complete, albeit brief, review of the discovery of mirror symmetry in $N=2$ string/conformal field theory. In particular, we describe the naturality arguments which led to the initial mirror symmetry conjectures and the subsequent work which established the existence of mirror symmetry through direct construction. We then review a number of striking consequences of mirror symmetry -- both conceptual and calculational. Finally, we describe recent work which introduces a variant on our original proof of the existence of mirror symmetry. This work affirms classical--quantum symmetry duality as well as extends the domain of our initial mirror symmetry construction.