By Liviu I. Nicolaescu
During this quantity the writer offers, in nice aspect and with many examples, a easy selection of rules, innovations, and purposes had to behavior self sufficient examine in gauge concept and its use in geometry and topology. whole and self-contained computations of the Seiberg-Witten invariants of most easily attached algebraic surfaces utilizing merely Witten's factorization procedure are integrated. additionally given is a brand new method of slicing and pasting Seiberg-Witten invariants, that is illustrated by means of examples similar to the hooked up sum theorem, the blow-up formulation, and an evidence of a vanishing results of Fintushel and Stern. The e-book is an acceptable textbook for complicated graduate classes in differential geometry, algebraic topology, simple PDEs and sensible research
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Extra resources for Notes on Seiberg-Witten theory
Sample text
4 s E ock remainder of the The endomorphism R = 4 + c(F (∇ )) is the Weitzenb¨ generalized Laplacian D2M,E . At this point we want to discuss some features of the above formula when dim M is even. In this case SM is Z2 -graded − SM = S+ M ⊕ SM and in particular we obtain a splitting − SM ⊗ E = S+ M ⊗ E ⊕ SM ⊗ E. With respect to the above grading the operator DM,E has the block decomposition 0 D∗M,E DM,E = DM,E 0 ∞ − where DM,E : C ∞ (S+ M ⊗ E) → C (SM ⊗ E). Then D2M,E = 0 D∗M,E DM,E 0 DM,E D∗M,E .
12) D∗M,E DM,E = ∇∗ ∇ + + c F + (∇E ) , 4 s + c F − (∇E ) 4 where F ± (∇E ) denotes the self/antiself-dual part of the curvature of ∇E . 3. Clifford algebras and Dirac operators 49 Assume now that (M, g) is an oriented, n-dimensional Riemannian manifold equipped with a spinc structure σ ∈ Spinc (M ). Denote by (gαβ ) a collection of gluing data defining the SO structure PSO(M ) on M with respect to some good open cover (Uα ). Moreover, we assume σ is defined by the data hαβ : Uαβ → Spinc (n). Denote by ρc the fundamental complex spinorial representation ρc : Spinc (n) → Aut (Sn ).
BL (u, v)dv(g0 ) + ∂M M Prove the following. e. L is a bundle morphism) then BL = 0. (b) If L1 ∈ PDO (F, G) and L2 ∈ PDO (E, F ) then BL1 L2 (u, v) = BL1 (L2 u, v) + BL2 (u, L∗1 v). 26 1. 1. Riemannian manifold with boundary (c) BL∗ (v, u) = −BL (u, v). (d) Suppose ∇ is a Hermitian connection on E and X ∈ Vect (M ). Set L = ∇X : C ∞ (E) → C ∞ (E). Then BL (u, v) = u, v g(X, n). (e) Let L = ∇ : C ∞ (E) → C ∞ (T ∗ M ⊗ E). Then BL (u, v) = u, in v E where in denotes the contraction by n. (f) Denote by ν the section of T ∗ M |∂M g-dual to n.