in Functional Analysis, Geometry, Topology

Instanton Floer homology

Floer homology is in essence the extension of the usual Morse homology of closed finite-dimensional manifolds to certain infinite-dimensional situations, where the naive Morse index of a critical point is not necessarily finite, and compactness becomes a subtle issue. In these notes we focus on the instanton Floer homology \mathbf{HF}^\bullet(N) of a homology three-sphere N, which is the Morse homology of the Chern-Simons functional \mathbf{CS}. Defined on the infinite dimensional space of gauge equivalence classes of \mathrm{SU}(2)-connections on N and being circle-valued, this functional brings in further features not present in classical Morse homology. For one, due to the presence of reducible connections, the gauge equivalence classes of connections only form a manifold away from a collection of singular points. Furthermore, since \mathbf{CS} is circle-valued it can occur that gradient flow lines form loops, making the trajectory spaces more subtle to describe.

In order to address the difficulties arising from the definition of the Morse index, we will introduce the spectral flow of a family of self-adjoint operators between separable Hilbert spaces, a tool common to all Floer-type theories. Using it, we will be able to define a relative index between two critical points of \mathbf{CS}. This index will only be \Z_8-valued however, due to looping gradient flow lines. To circumvent the issues arising from reducible connections, we will restrict ourselves to homology three-spheres, for which achieving transversality is easier than for other classes of three-manifolds. A feature of \mathbf{CS} which makes it particularly suited for Morse homology is that its critical points and gradient trajectories have geometrical interpretations. The critical points of \mathbf{CS} are flat connections on N, and the flow lines are instantons over the Riemannian tube \R \times N which join two connections. As we will see, instantons are a special case of Yang-Mills connections, which allows us to use Uhlenbeck’s compactness theorem to derive the compactness up to broken trajectories of the trajectory spaces. The invariant we will obtain in the end takes the form of a \Z_8-graded vector space over the field \Z_2. For the usual three-sphere S^3 and the PoincarĂ© homology sphere P they are given by:

    \begin{equation*} \boxed {\mathbf{HF}^\bullet(S^3) = 0, \quad \mathbf{HF}^\bullet(P) = (0, \Z_2, 0, 0, 0, \Z_2, 0, 0).}  \end{equation*}

Thus the Floer homology groups will be unrelated to the usual homology groups, and provide a stronger invariant. To conclude, we will investigate further properties of the Floer groups, such as their relation to the representation-theoretic Casson invariant, which appears as the Euler characteristic of the Floer groups, and (3+1)-dimensional topological quantum field theories.

We will move according to the following outline. In the first chapter, we introduce electromagnetism as an example of an abelian gauge theory, familiarise ourselves with the concepts we will later encounter in the non-abelian setting and provide a link between gauge theory and topology of three-manifolds via Hodge theory. In the second chapter, we will then recall principal bundles properly, fix the notation, and derive the relation between three- and four-dimensional \mathrm{SU}(2)-gauge theory. Next, we will introduce the Chern-Simons and Yang-Mills functionals \mathbf{CS} and \mathbf{YM} in the third chapter and analyse the relation between \mathbf{CS}-gradient flow lines and instantons, which are the local minimizers of \mathbf{YM}. We will also investigate the local picture around a critical point of these functionals. In the last chapter we will construct the trajectory spaces and the Floer homology groups from a chain complex generated by flat connections, emphasising the Fredholm analysis. We will show independence of auxiliary data and of the perturbation chosen, where it will be important to restrict to the case of homology three-spheres.

We assume that the reader is familiar with differential geometry (bundles and connections, Riemannian geometry, de Rham cohomology, Morse homology), algebraic topology (especially computational tools in (co)homology theories, Poincaré duality, intersection theory, some homotopy theory), as well as functional analysis (Sobolev spaces on open domains, differential operators on \R^n).

These notes do not contain original work, but merely adapt and combine the literature on the subject, while supplementing explanations where deemed useful. We mostly follow Donaldson’s monograph on the topic, and apply the treatment of Morgan’s lecture notes to give the analytical foundations for the Chern-Simons and Yang-Mills functionals. The first chapter on electromagnetism reshuffles the first part of the excellent lecture notes by Evans on the Yang-Mills equations, and the appendix bundles statements from multiple sources. For references see the PDF.

I would like to thank my supervisor Prof. Will Merry for guiding me through this endeavour with frequent and useful meetings. His (virtual) office door was always open to me, and I appreciated both his technical knowledge in geometry and analysis and his expertise in the wider world of Floer theories, which often provided helpful analogies an additional angles of attack during my study of the instanton flavoured theory.

The entire thesis can be found here.

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