The Dirac cohomology from the Dirac operator in the equation of the same name has been rencently studied, but by mathematicians (in particular Vogan) as a tool to solve problems of representation theory of Lie groups. It is the analog of the De Rham cohomology that has an application for electromagnetism. I'm looking for an account that is accessible for a physicist having a good knowledge about differential geometry and topology, including homology and cohomology. I'm especially interested in the dual homology.
You can read the following references :
Voigt work and comments for introduction : http://www.math.columbia.edu/~woit/wordpress/?p=2189
and
http://www.ams.org/journals/jams/2002-15-01/S0894-0347-01-00383-6/S0894-0347-01-00383-6.pdf
http://www.ams.org/journals/ert/2006-10-12/S1088-4165-06-00267-6/S1088-4165-06-00267-6.pdf
You can read the following references :
Voigt work and comments for introduction : http://www.math.columbia.edu/~woit/wordpress/?p=2189
and
http://www.ams.org/journals/jams/2002-15-01/S0894-0347-01-00383-6/S0894-0347-01-00383-6.pdf
http://www.ams.org/journals/ert/2006-10-12/S1088-4165-06-00267-6/S1088-4165-06-00267-6.pdf
Interesting question. Dirac cohomology like BRST cohomology is in mathematical terms a Lie algebra cohomology theory. I think you are looking for Lie group cohomology which is in some sense dual to Lie algebra cohomology.
But if you really want to construct a homology theory then you have to formulate the theory as homotopy classes of the space into a spectrum (cohomology). This can be dually formulated via smash products to obtain the homology theory.
But what you wrote above you need the Lie group cohomology.
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