By Liviu I. Nicolaescu
During this quantity the writer provides, in nice element and with many examples, a easy choice of rules, concepts, and purposes had to behavior self sustaining study in gauge thought and its use in geometry and topology. entire and self-contained computations of the Seiberg-Witten invariants of most easily hooked up algebraic surfaces utilizing in simple terms Witten's factorization process are incorporated. additionally given is a brand new method of slicing and pasting Seiberg-Witten invariants, that's illustrated through examples equivalent to the hooked up sum theorem, the blow-up formulation, and an evidence of a vanishing results of Fintushel and Stern. The booklet is an appropriate textbook for complex graduate classes in differential geometry, algebraic topology, uncomplicated PDEs and useful research
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Extra info for Notes on Seiberg-Witten theory
Example text
E. 4) M E ∇E X ( c(α)u) = c(∇X α)u + c(α)∇X u, ∀u ∈ C ∞ (E), α ∈ Ω1 (M ), X ∈ Vect (M ). 28 1. Preliminaries When ∇M is the Levi-Civita connection we say that (E, c, ∇E ) is a geometric Dirac structure on M . The Dirac operator associated to a geometric Dirac structure will be called a geometric Dirac operator. 2. (Weitzenb¨ ock formula for geometric Dirac operators) If D is a geometric Dirac operator associated to the geometric Dirac structure (E, c, ∇E ) then D = D∗ and D2 = (∇E )∗ ∇E + c(F (∇E )).
The The curvature of the connection ∇ Riemannian curvature R of M (or equivalently, the curvature of the LeviCivita connection on T M ) is given by the collection of so(n)-valued 2-forms 1 Rα = dωα + ωα ∧ ωα = 2 ek ∧ e Rk k< where Rk : Uα → so(n) is given by i ei = Rijk ei . 7) ˜ = ρ−1 R ∗ (R) = ek ∧ e ρ−1 ∗ (Rk ) = − k< ˜ M is then The curvature of ∇ ˜ M) = −1 F (∇ 4 ek ∧ e k< 1 4 ek ∧ e k< i Rjk ei · ej . i D) The operator (d + d∗ )2 = dd∗ + d∗ d : Ω∗ (M ) → Ω∗ (M ) is a generalized Laplacian called the Hodge Laplacian. The covariant Laplacian in the above example is in some sense the basic example of generalized Laplacian. More precisely, we have the following result. We refer to [12] for a different proof. 2. 8. Suppose L : C ∞ (E) → C ∞ (E) is a formally selfadjoint generalized Laplacian. Then there exists a Hermitian connection on E and a symmetric endomorphism R : E → E such that L = ∇∗ ∇ + R.