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Title Seiberg-Witten theory for symplectic manifolds
Publication Date
Degree Level masters
University/Publisher Universiteit Utrecht
Abstract In this thesis we give an introduction to Seiberg-Witten gauge theory used to study compact oriented four-dimensional manifolds X. Seiberg-Witten theory uses a Spin c structure to create two vector bundles over X called the spinor bundle and determinant line bundle. One then considers the set of solutions to the Seiberg-Witten equations, which are expressed in terms of a section of the spinor bundle and a Dirac operator formed out of a connection on the determinant line bundle. After taking the quotient by an action of a U(1)-gauge group, one constructs an invariant by integrating cohomology classes over the resulting moduli space. In this thesis we show these Seiberg-Witten invariants can be used to find obstructions to the existence of a symplectic structure on X.
Subjects/Keywords Seiberg-Witten theory, four-manifolds, symplectic manifolds, Spin c structures, Dirac operators
Contributors Cavalcanti, G.R.
Language en
Rights info:eu-repo/semantics/OpenAccess
Country of Publication nl
Format text/plain
Record ID oai:dspace.library.uu.nl:1874/282753
Repository utrecht
Date Indexed 2016-09-27

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…Appendix C]. 2.3 Almost complex and symplectic manifolds In this section we recall some of the theory on almost complex and symplectic manifolds. In particular we recall the notion of compatibility between these structures giving almost KĻ ahler…

…need not be unique. Many structures on the tangent bundle of a manifold can be expressed in terms of reductions of the structure group. Recall that the tangent bundle is a priori a GL(n)-bundle. â€Ē The reduction to O(n) < GL(n…

…reduction is not always possible. Furthermore, note that orientable manifolds admit exactly two orientations, or two different reductions of the structure group; â€Ē The reduction GL(n, C) < GL(2n, R) is the same as choosing an almost…

…bundles of rank k and n − k. We will see two other important examples of reduction of the structure group in the next section, when we discuss the Spin(n) and Spinc (n) groups. Let B be a manifold and consider again the set PG (B…

…U(1). This immediately implies the following result. Lemma 2.1.21. For G = U(1), the gauge group of any principal U(1)-bundle over X can be identified with the space C ∞ (X; U(1)) of smooth maps from…

…all f ∈ C ∞ (X) and s ∈ â„Ķ0 (X; E) we have ∇A (f s) = df ⊗ s + f ∇A (s). (2.2.6) Remark 2.2.6. The second relation is also referred to as the Leibniz identity. Out of a connection ∇A and a vector field…

…X ∈ X (X) we can form its covariant derivative ∇A (X) : Γ(E) → Γ(E). It is C ∞ (X)-linear with respect to X and linear with respect to the sections it acts on. If we denote ∇A (X) by ∇X , it…

…furthermore satisfies ∇X (f s) = f ∇X s + X(f )s for f ∈ C ∞ (X) and s ∈ Γ(E). Connections on vector bundles always exist through a partition of unity argument. We denote the space of all connections on a given vector…