Towards Homological Mirror Symmetry

TL;DR

This paper provides an expository overview of the A-side of Kontsevich's Homological Mirror Symmetry conjecture, focusing on $A_ olinebreak_ olinebreak ext{infty}$-categories, Floer theory, and their role in mirror symmetry.

Contribution

It offers a comprehensive, self-contained introduction to $A_ olinebreak_ olinebreak ext{infty}$-categories and Fukaya categories, emphasizing their geometric and homological aspects within mirror symmetry.

Findings

Clarifies the structure of Fukaya categories and Floer theory

Connects $A_ olinebreak_ olinebreak ext{infty}$-categories to mirror symmetry predictions

Provides foundational understanding for further research

Abstract

This is an expository article on the A-side of Kontsevich's Homological Mirror Symmetry conjecture. We give first a self-contained study of $A_\infty$-categories and their homological algebra, and later restrict to Fukaya categories, with particular emphasis on the basics of the underlying Floer theory, and the geometric features therein. Finally, we place the theory in the context of mirror symmetry, building towards its main predictions.