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Geometric Analysis Seminar

Variations of the Z2-Dirac operator

Gregory Jacob Parker (MIT)

Location:  Zoom
Date & time: Tuesday, 26 April 2022 at 2:50PM - 3:50PM

Abstract: It is a classical result that on a compact 3-manifold \(Y\), the set of metrics for which there exists a harmonic spinor of the spin Dirac operator is a codimension 1 subset in the space of all metrics. In this talk, I will discuss an extension of this result to the case of the \(mathbb Z_2\)-Dirac operator, which is defined as the Dirac operator on the complement of a codimension 2 submanifold \(mathcal Zsubseteq Y\) twisted by a flat connection on \(Y-mathcal Z\) whose holonomy lies in \(mathbb Z_2\). The deformation problem for solutions of this operator carries an infinite-dimensional obstruction for a fixed \(mathcal Z\). Coupling the operator to the geometry of \(mathcal Z\)  by considering the infinite-dimensional family of Dirac operators parameterized by embedded submanifolds gives rise to a Fredholm problem up to a loss of regularity phenomenon. The proof of the result then requires the use of the Nash-Moser Implicit Function Theorem or related techniques.

Zoom link:
Meeting ID: 979 5349 0430
Password: 515087

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