Geometry & Topology Seminar: Floer theories from non-linear Dirac operators

Abstract: This talk is based on joint work with Yang Li. I will discuss certain non-linear Dirac operators and related questions in geometric analysis that arise in higher-dimensional gauge theory. Taubes proposed that counting harmonic spinors associated with these operators, called Fueter sections, on 3-manifolds could lead to new 3-manifold invariants. Furthermore, homological Floer-theoretic invariants have been proposed, defined using a count of Fueter sections. Similarly, Donaldson and Segal suggested counting Fueter sections over special Lagrangians to define Calabi-Yau 3-fold invariants. A key question in all these proposals is whether the space of Fueter sections is compact. In this talk, we address this question in specific cases, proving and disproving several conjectures in the field, and answer some related questions raised by Taubes and Donaldson-Segal.

Host: Ali Daemi