Gauge Theory and Differential Topology of 4-Manifolds

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Fathali, Hadi | 2024

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 57815 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Bahraini, Alireza
Abstract:
After discovery of exotic structures by John Milnor, mathematicains understood that the classification problem of differential structures up to diffeomorphims is fundamentaly different from classification up to homeomorphism and new invarients are required to study differential topology. One the most important breakthroughs in this area was Donaldson’s idea which obtains a differential topological invariant for 4-manifolds with gauge theoritic ideas known as Yang-Mills (which are non-Abelian generalizations of Maxwell’s theory). A decade after Donaldson’s revolutionary idea Edward Witten and Natan Seiberg presented a new gauge theory known as Sieberg-Witten theory. Later mathematicians discovered that many of Donaldson’s results can be achevied with this theory which has less technical complexities. In this thesis we begin with study of modular space of solutions to Seiberg-Witten equations and will deduce properties of base manifold by studying this space and introduce Seiberg-Witten invariants of 4-manifolds. Then will compute this invariant for special classes of 4-manifolds known as Kahler and symplictic manifolds
Keywords:
Seiberg-Witten Equations ; Kahler Manifold ; Dirac Operator ; Symplectic Manifold ; Solutions Modular Space ; Spin Structure ; Self-Dual Forms