Line fields on punctured surfaces and twisted derived categories

TL;DR

This paper extends the formal construction of the Fukaya category for punctured surfaces to include line fields, leading to a new class of twisted derived categories and establishing their invariance under decorated graph transformations via mirror symmetry.

Contribution

It introduces a novel extension of the Fukaya category incorporating line fields and demonstrates the invariance of the resulting twisted derived categories through B-model mirror symmetry methods.

Findings

The extended category depends on a decorated graph with a line field.

The twisted derived categories are invariant under graph decorations.

Different mirror constructions are proven equivalent using B-model techniques.

Abstract

The Fukaya category of a punctured surface can be reconstructed from a pair-of-pants decomposition using a formal construction that attaches a category to a trivalent graph. We extend this formal construction to include a choice of line field on the surface, this requires a certain decoration on the graph. On the mirror side we show that this leads to a kind of twisted derived category which has not been widely studied. Mirror symmetry predicts that our category should be an invariant of decorated graphs and we prove that this is indeed the case, using only B-model methods. We also give B-model proofs that a few different mirror constructions are equivalent.