nLab Floer homology

Idea

Floer homology is a common name for several similar frameworks of infinite-dimensional analogues of Morse homology, related to certain Fredholm operators which arise as elliptic operators on certain manifolds. Some of these are related to symplectic topology/geometry or contact geometry and another group is related to aspects of the geometry of 3-manifolds. One of these uses the Chern-Simons theory action functional as a Morse function on the space of connections on $G$-principal bundles over the given 3-manifold.

The most important flavours are the Floer-Oh homology for action functionals which is important in the study of symplectic manifolds, its version for Lagrangean intersection theory and the instanton Floer homology studied by Andreas Floer and important in the study of 3-manifolds and the mathematics of gauge theories.

Floer homology for the action functional is a more systematic approach to the phenomena which were discovered several years before Floer in a historical article of Mikhail Gromov on J-holomorphic curves.

Examples and Variants

Instanton Floer homology

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 (vanishing curvature), whereas the flow lines of $S_{CS}$ correspond to the Yang-Mills instantons on $\Sigma \times [0,1]$.

For more see

• instanton Floer homology.

References

The original articles are

• Andreas Floer,

  • The unregularized gradient flow of the symplectic action , Comm. Pure Appl. Math. 41 (1988), 775–813.

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

  • Morse theory for Lagrangian intersections- , J. Differential Geom. 28 (1988), no. 3, 513–547.

  • Cuplength estimates on Lagrangian intersections , Comm. Pure Appl. Math. 42 (1989), no. 4, 335–356.

  • Symplectic fixed points and holomorphic spheres , Comm. Math. Phys. 120, no. 4 (1989), 575–611.

  • Witten’s complex and infinite dimensional Morse Theory , J. Diff. Geom. 30 (1989), p. 202–221.

Surveys are in

• Yong-Geun Oh, Symplectic topology and Floer homology, pdf

• Darko Milinković, Floer homology in classical mechanics and quantum field theory, ps

• Simon Donaldson, Floer homology groups in Yang-Mills theory Cambridge University Press (2002) (pdf)

• Qingtao Chen, Introduction to Floer Homology and its relation with TQFT (2005) (pdf)

On Holomorphic Floer Theory:

• Maxim Kontsevich, Yan Soibelman. Holomorphic Floer theory I: exponential integrals in finite and infinite dimensions. (2024) (arXiv:2402.07343)

See also

• wikipedia