Spectral networks in Donaldson-Thomas theory

• Applicant: Dr. Johannes Horn
• Subject Area: Mathematics
• Term: since 2026
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 577902339

The goal of this proposal is to develop a full-fledged theory of spectral networks and abelianisation as envisioned by Gaiotto, Moore and Neitzke. In particular, treating spectral networks that accumulate on the surface and higher rank spectral networks. In a first work package, we aim to prove the abelianisation of local systems on a closed Riemann surface as an analogue to the spectral correspondence for Higgs bundles. This has several far-reaching applications. On one hand, abelianisation is a missing link to a conjectural solution of the Riemann-Hilbert problem in Bridgeland's program of geometrisation of Donaldson-Thomas invariants. On the other hand, it enables applications to cluster structures. In a second work package, we aim to provide a mathematical framework for higher rank spectral networks associated to points in the SL(n,C)-Hitchin base. As a main result, we aim to generalize the Kontsevich-Soibelman wall-crossing formula to higher rank BPS states. This is an important step towards an interpretation of the SL(n,C)-Hitchin base as the 'physicist slice' in some generalised moduli space of stability conditions.

• DFG Programme: Research Grants