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.