Floer Homology in Gauge Theory

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 pi: E → B induces dpi: TE → TB, where TeE locally splits into Tpi(e)B ⊕ T1G by local triviality. Thus, ker dpi is T1G 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∗gθ = (adg) −1 ◦ θ(1.1) θ(v) = vg, v ∈ ker dpi.(1.2) Here the right action Rg acts by g on the right and vg refers to the identification of v with an element vg of g as mentioned before. In particular, this means that if θx denote the Maurer-Cartan form on Ex = G and ix: Ex ↪ → E, then i∗x(θ) = θx. Also recall the structure equation (1.3) dθx