For semisimple Lie algebra targets
For discrete group targets
For discrete 2-group targets
For Lie 2-algebra targets
For targets extending the super Poincare Lie algebra
(such as the supergravity Lie 3-algebra, the supergravity Lie 6-algebra)
Chern-Simons-supergravity
for higher abelian targets
for symplectic Lie n-algebroid targets
for the -structure on the BRST complex of the closed string:
higher dimensional Chern-Simons theory
topological AdS7/CFT6-sector
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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
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
Last revised on August 30, 2011 at 19:45:04. See the history of this page for a list of all contributions to it.