Yang-Mills instanton Floer homology
Special and general types
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 a 3-dimensional compact smooth manifold and a simply connected compact Lie group let be the space of -connections on , which is equivalently the groupoid of Lie algebra valued forms on in this case.
The instanton Floer homology groups of are something like the “mid-dimensional” singular homology groups of the configuration space .
More precisely, there is canonically the Chern-Simons action functional
on this space of connections, and one can form the corresponding Morse homology.
The critical locus of is the space of flat -connections (those with vanishing curvature), whereas the flow lines of correspond to the Yang-Mills instantons on .
The original reference is
- Andreas Floer, An instanton-invariant for 3-manifolds , Comm. Math. Phys. 118 (1988), no. 2, 215–240, euclid
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)
Revised on August 30, 2011 19:45:04
by Zoran Škoda