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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 categoryof the surface with coefficients in a triangulated DG-category. These spectralnetworks are analogs of special Lagrangian submanifolds, combining a graph withadditional algebraic data, and conjecturally correspond to semistable objectsof 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 resultsfor spectral network representatives. We also verify the conjecture in the casewhen the surface is disk with six marked points on the boundary and thecoefficients category is the derived category of representations of an $A_2$quiver. This example is related, via homological mirror symmetry, to the stackyquotient of an elliptic curve by the cyclic group of order six.

Spectral networks and stability conditions for Fukaya categories with coefficients