Full Record

Author | Klaasse, R.L. |

Title | Seiberg-Witten theory for symplectic manifolds |

URL | http://dspace.library.uu.nl:8080/handle/1874/282753 |

Publication Date | 2013 |

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 |

Sample Search Hits | Sample Images

…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.
âĒ 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;
âĒ The reduction GL(n, *C*) < GL(2n, R) is the same as choosing an almost…

…bundles of rank k and n â 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* â (X; U(1)) of smooth maps from…

…all f â *C* â (X) and s â âĶ0 (X; E) we have
âA (f s) = df â s + f âA (s).
(2.2.6)
Remark 2.2.6. The second relation is also referred to as the Leibniz identity.
Out of a connection âA and a vector field…

…X â X (X) we can form its covariant derivative
âA (X) : Î(E) â Î(E). It is *C* â (X)-linear with respect to X and linear with respect
to the sections it acts on. If we denote âA (X) by âX , it…

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