## Saman Habibi Esfahani : Non-linear Dirac operators and multi-valued harmonic forms

- Geometry and Topology ( 0 Views )This talk is based on joint work with Yang Li. I will discuss non-linear Dirac operators and related regularity questions, which arise in various problems in gauge theory, Floer theory, DT theory, and minimal submanifolds. These operators are used to define generalized Seiberg-Witten equations on 3- and 4-manifolds. Taubes proposed that counting harmonic spinors with respect to these operators on 3-manifolds could lead to new 3-manifold invariants, while Donaldson and Segal suggested counting spinors over special Lagrangians to define Calabi-Yau invariants. Similar counts appear in holomorphic Floer theory, where Doan and Rezchikov outlined a Fukaya 2-category for hyperkähler manifolds based on such counts. The central question in all of these proposals is whether the space of such harmonic spinors is compact. We address this question in certain cases, proving and disproving several conjectures in the field and, in particular, answering a question raised by Taubes in 1999. The key observation is that multivalued harmonic forms, in the sense of Almgren and De Lellis-Spadaro's Q-valued functions, play a crucial role in the problem.