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cs - Floer Homology in Gauge Theory Somnath Basu Abstract...

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Floer Homology in Gauge Theory Somnath Basu August 10, 2010 Abstract This is an expository account of the Chern-Simons functional, leading to various invariants on homology 3-spheres. It is based mainly on a review article by K. Fukaya [F], D. Freed’s [DF1] on classical Chern-Simons theory, A. Floer’s instanton invariant [AF] and K. Walker’s [W] book on Casson’s invariant. Certain parts are taken verbatim from these sources with most of the basic details filled in. This essay is also inspired by the seminal paper of S. S. Chern and J. Simons [CS] ??? . 1 The Chern-Simons Functional A principal G -bundle E over B is a fibration with fibre a Lie group G and there is a free right (resp. left) action on the total space by elements of G such that this is free and transitive on each fibre. The map π : E B induces : TE TB , where T e E locally splits into T π ( e ) B T 1 G by local triviality. Thus, ker is T 1 G which can be canonically identified with g . For such a bundle, a connection is a 1-form θ on E taking values in g and satisfying R * g θ = (ad g ) - 1 θ (1.1) θ ( v ) = v g , v ker dπ. (1.2) Here the right action R g acts by g on the right and v g refers to the identification of v with an element v g of g as mentioned before. In particular, this means that if θ x denote the Maurer-Cartan form on E x = G and i x : E x , E , then i * x ( θ ) = θ x . Also recall the structure equation (1.3) x + 1 2 [ θ x , θ x ] = 0 . Proposition 1.1. Let E B be as above such that G is connected and simply connected. Let B be a manifold of dimension at most 3 . Then the bundle is trivial. Proof It suffices to prove that a section s : B E exists; then φ : B × G E defined by φ ( b, g ) = s ( b ) g gives an isomorphism of bundles. The obstructions to finding a section s : B E are the cohomology classes θ i H i ( B, ^ π i - 1 ( G )). It is well known that π 2 ( G ) = 0. Therefore, the primary obstruction is at least of degree 4 and there is a section, whence E is trivial. In what follows, let M be a closed, connected and oriented 3-manifold and let E be a principal SU (2)-bundle on M and we have chosen a trivialization for E . Let Ω i ( M, su 2 ) = Γ( M, i T * M ( M × su 2 )) 1
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denote the space of forms on M with values in su 2 . There are two obvious ways to extend a Ω 1 ( M, su 2 ) to a 1-form on E = M × SU (2) : (1) using (1.1) and declaring it to be the Maurer-Cartan form on each fibre, (2) just extend to each copy of M × { g } by (1.1). Thus, the space of connections on any (principal) SU (2)-bundle over M (making use of (1) above) is just A ( M ) = Ω 1 ( M, su 2 ) - the space of su 2 -valued 1-forms on M . A ( M ) is an affine space but when thought of as Ω 1 ( M, su 2 ), it can be treated as a vector space. The curvature of a connection a ∈ A ( M ) is defined to be (1.4) F a = da + a a Ω 2 ( M, su 2 ) . We observe here that ( a a )( X, Y ) = 1 2 ( a ( X ) a ( Y ) - a ( Y ) a ( X ) ) = 1 2 [ a ( X ) , a ( Y )] and R * g F a = ad g - 1 F a (1.5) i * x F a = 0 . (1.6) The last equation follows from (1.3). The Bianchi identity follows by differentiating F a : (1.7) dF a + [ a, F a ] = 0 .
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