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.
","abstract_html":"<p>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. </p>","abstract_has_math":false,"creators":["Nguyen, Minh Lam"],"institution":null,"degree_name":"Doctor of Philosophy in Mathematics (PhD)","degree_level":"Dissertation","degree_discipline":null,"degree_department":null,"school":null,"contributors":["Ryan, John","Harrington, Phillip S."],"advisors":["Van Horn-Morris, Jeremy"],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-05-01T07:00:00Z","date_published":"2022-05-01T07:00:00Z","updated_at":"2026-07-24T01:00:19Z","subjects":["11/8th conjecture","Gauge theory","geometric analysis","low dimensional topology","mathematical physics","Rarita-Schwinger operator","Seiberg-Witten theory","Elementary Particles and Fields and String Theory","Geometry and Topology","Partial Differential Equations"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarworks.uark.edu/etd/4429","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Ryan, John","Harrington, Phillip S."]},{"key":"dc:contributor.advisor","label":"Advisor","values":["Van Horn-Morris, Jeremy"]},{"key":"dc:creator","label":"Author","values":["Nguyen, Minh Lam"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2022"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2024-02-06T08:00:00Z"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy in Mathematics (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["11/8th conjecture","Gauge theory","geometric analysis","low dimensional topology","mathematical physics","Rarita-Schwinger operator","Seiberg-Witten theory","Elementary Particles and Fields and String Theory","Geometry and Topology","Partial Differential Equations"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarworks.uark.edu/etd/4429"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["Finite Dimensional Approximation and Pin(2)-equivariant Property for Rarita-Schwinger-Seiberg-Witten Equations"]}]}],"canonical_facts":{"dc:contributor":["Ryan, John","Harrington, Phillip S."],"dc:contributor.advisor":["Van Horn-Morris, Jeremy"],"dc:creator":["Nguyen, Minh Lam"],"dc:date":["2022"],"dc:date.available":["2024-02-06T08:00:00Z"],"dc:description.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.
"],"dc:identifier":["https://scholarworks.uark.edu/etd/4429"],"dc:subject":["11/8th conjecture","Gauge theory","geometric analysis","low dimensional topology","mathematical physics","Rarita-Schwinger operator","Seiberg-Witten theory","Elementary Particles and Fields and String Theory","Geometry and Topology","Partial Differential Equations"],"dc:title":["Finite Dimensional Approximation and Pin(2)-equivariant Property for Rarita-Schwinger-Seiberg-Witten Equations"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy in Mathematics (PhD)"]},"updated_at":"2026-07-24T01:00:19Z"}