Spectral networks and stability conditions for Fukaya categories with coefficients

TL;DR

This paper introduces spectral networks in Fukaya categories with coefficients, relating them to stability conditions and special Lagrangians, and verifies their conjectured correspondence in a specific geometric case.

Contribution

It defines spectral networks with coefficients in a Fukaya category, proves uniqueness results, and verifies the conjecture in a specific example involving an elliptic curve quotient.

Findings

Spectral networks are analogs of special Lagrangian submanifolds with algebraic data.

Uniqueness of spectral network representatives is established.

The conjecture is verified for a disk with six marked points and an $A_2$ quiver category.

Abstract

Given a holomorphic family of Bridgeland stability conditions over a surface, we define a notion of spectral network which is an object in a Fukaya category of the surface with coefficients in a triangulated DG-category. These spectral networks are analogs of special Lagrangian submanifolds, combining a graph with additional algebraic data, and conjecturally correspond to semistable objects of a suitable stability condition on the Fukaya category with coefficients. They are closely related to the spectral networks of Gaiotto--Moore--Neitzke. One novelty of our approach is that we establish a general uniqueness results for spectral network representatives. We also verify the conjecture in the case when the surface is disk with six marked points on the boundary and the coefficients category is the derived category of representations of an $A_2$ quiver. This example is related, via homological mirror symmetry, to the stacky quotient of an elliptic curve by the cyclic group of order six.