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M. Nguyen 12-2-2021

Title:  Finite dimensional approximation and Pin(2)-equivariant properties for the Rarita-Schwinger -Seiberg-Witten equations.
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 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 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, contains
infinitely many elements.

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