Moduli spaces and D-brane categories of tori using SCFT

TL;DR

This paper explores the moduli spaces of superconformal field theories on tori, revealing enhanced structures, and connects algebraic D-brane descriptions with geometric categories and mirror symmetry conjectures.

Contribution

It provides a detailed analysis of N=2 moduli spaces, introduces algebraic D-brane descriptions, and links these to geometric categories and mirror symmetry.

Findings

Enhanced moduli space for N=2 theories with two complex structures

Algebraic description of D-branes using gluing matrices

Connection between D-brane categories and homological mirror symmetry

Abstract

We analyse the moduli spaces of superconformal field theories (SCFTs). For N=2 we find an enhanced moduli space which in geometrical terms corresponds to tori with two independent complex structures. To explain the precise relation with the moduli space of SCFTs on K3 surfaces as described by Aspinwall and Morrison, we discuss some subtleties with the precise interpretation of the N=2 and N=4 moduli spaces. We also explain why in some cases the SYZ-description of mirror symmetry as fibrewise T-duality seems to break down. Using gluing matrices we give an algebraic description of D-branes and construct the corresponding boundary states. We study how isomorphisms of the SCFTs act on D-branes. Finally we give a geometrical interpretation of our algebraic constructions and make contact with the geometrical D-brane categories and Kontsevich's homological mirror symmetry conjecture.