Date of Graduation

5-2022

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics (PhD)

Degree Level

Graduate

Department

Mathematical Sciences

Advisor/Mentor

Jeremy Van Horn-Morris

Committee Member

John Ryan

Second Committee Member

Phillip Harrington

Keywords

11/8th conjecture, Gauge theory, geometric analysis, low dimensional topology, mathematical physics, Rarita-Schwinger operator, Seiberg-Witten theory

Abstract

The Rarita-Schwinger operator Q was initially proposed in the 1941 paper by Rarita and Schwinger to study wave functions of particles of spin 3/2, and there is a vast amount of physics literature on its properties. Roughly speaking, 3/2−spinors are spinor-valued 1-forms that also happen to be in the kernel of the Clifford multiplication. Let X be a simply connected Riemannian spin 4−manifold. Associated to a fixed spin structure on X, we define a Seiberg-Witten-like system of non-linear PDEs using Q and the Hodge-Dirac operator d∗ + d+ after suitable gauge-fixing. The moduli space of solutions M contains (3/2-spinors, purely imaginary 1-forms). Unlike in the case of Seiberg-Witten equations, solutions are hard to find or construct. However, by adapting the finite dimensional technique of Furuta, we provide a topological condition of X to ensure that M is non-compact; and thus, M contains infinitely many elements.

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