nLab
Yang-Mills instanton Floer homology

Context

\infty-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

Ingredients

Definition

Examples

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

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

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

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

S CS:[Σ,BG conn]U(1) 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 CSS_{CS} is the space of flat GG-connections (those with vanishing curvature), whereas the flow lines of S CSS_{CS} correspond to the Yang-Mills instantons on Σ×[0,1]\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)

Revised on August 30, 2011 19:45:04 by Zoran Škoda (161.53.130.104)