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 |

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…d2
(d dB s − i ∗ (ω ∧ Fb s)) .
b s) =
2
2
2 B
(2.3.33)
Looking at the definition of the Hodge star operator, we now see that ∗ω∧ = Λ on two-forms,
so that we are done.
2.4
Elliptic *operators*
In this section we…

…If this is true,
its index ind(T ) is defined by
ind(T ) = dim ker(T ) − dim coker(T ).
22
(2.4.2)
We now consider the analogous notion for *operators* between sections of vector bundles.
Definition…

…Hence any self-adjoint Fredholm operator has index 0.
An important fact of Fredholm *operators* is that their index is invariant under perturbations.
In other words, if one were to take a family of Fredholm *operators* parameterized by some
connected…

…topological space, all of their indices would agree. There is a broad theory on
determining Fredholmness of *operators* and calculating their index. A key concept is that of
ellipticity. Let π : T ∗ X → X be the natural projection. Let E and F be vector bundles…

…without going to a local description of D. One of the main reasons for introducing the notion of ellipticity is that most
differential *operators* encountered in geometry satisfy this condition, but moreover that we
have the following result.
Theorem 2.4.5…

…result which will be of importance to us, having to do with elliptic *operators*
between Sobolev spaces.
p
Theorem 2.5.10 (Elliptic regularity). Let D : Lp
k (E) → Lk (F ) be a first order elliptic
differential operator between…