Spectral Networks and Stability Conditions for Fukaya Categories with Coefficients.

Area of Science:

• Algebraic Geometry
• Mathematical Physics
• Category Theory

Background:

• Bridgeland stability conditions provide a framework for studying moduli spaces of objects in triangulated categories.
• Fukaya categories, derived from symplectic geometry, encode topological and geometric information of manifolds.
• Spectral networks, introduced by Gaiotto-Moore-Neitzke, relate to QFT and geometric structures.

Purpose of the Study:

• To define and study a novel notion of spectral networks within Fukaya categories with DG-category coefficients.
• To establish uniqueness results for these spectral network representatives.
• To explore the connection between spectral networks and semistable objects in derived categories.

Main Methods:

• Construction of spectral networks as objects in the Fukaya category, incorporating graph structures and algebraic data.
• Development of a general framework for proving uniqueness of spectral network representatives.
• Application of the theory to a specific case: a disk with six marked points and a quiver representation category.

Main Results:

• A new definition of spectral networks is introduced, generalizing existing concepts.
• A general uniqueness result for spectral network representatives is established.
• The conjecture is verified for a specific example involving a disk, quiver representations, and homological mirror symmetry.

Conclusions:

• The introduced spectral networks offer a powerful tool for understanding stability conditions and semistable objects in Fukaya categories.
• The established uniqueness provides a robust foundation for further theoretical development.
• The verified example highlights deep connections between different areas of mathematics, including symplectic geometry, representation theory, and algebraic geometry.