nLab Yang-Mills instanton Floer homology

Idea

Instanton Floer homology is a variant of Floer homology which applies to 3-dimensional manifolds. It is effectively the Morse homology of the Chern-Simons theory action functional.

For $\Sigma$ a 3-dimensional compact smooth manifold and $G$ a simply connected compact Lie group let $[\Sigma,\mathbf{B}G_{conn}]$ be the space of $G$-connections on $\Sigma$, which is equivalently the groupoid of Lie algebra valued forms on $\Sigma$ in this case.

The instanton Floer homology groups of $\Sigma$ are something like the “mid-dimensional” singular homology groups of the configuration space $[\Sigma,\mathbf{B}G_{conn}]$.

More precisely, there is canonically the Chern-Simons action functional

$$S_{CS} : [\Sigma,\mathbf{B}G_{conn}] \to U(1)$$

on this space of connections, and one can form the corresponding Morse homology.

The critical locus of $S_{CS}$ is the space of flat $G$-connections (those with vanishing curvature), whereas the flow lines of $S_{CS}$ correspond to the Yang-Mills instantons on $\Sigma \times [0,1]$.

References

The original reference is

• Andreas Floer, An instanton-invariant for 3-manifolds , Comm. Math. Phys. 118 (1988), no. 2, 215–240, euclid

Reviews:

• Simon Donaldson, Floer homology groups in Yang-Mills theory Cambridge Tracts in Mathematics 147 (2002), pdf

• Tomasz S. Mrowka, Introduction to Instanton Floer Homology at Introductory Workshop: Homology Theories of Knots and Links , MSRI (video)

Generalizations to 3-manifolds with boundary are discussed in

• Dietmar Salamon, Katrin Wehrheim, Instanton Floer homology with Lagrangian boundary conditions (arXiv:0607318)