group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
What is called Spin Chern-Simons theory is a prequantum field theory/quantum field theory like Chern-Simons field theory but defined on/restricted to 3-manifolds equipped with spin structure and making use of that structure to divide the action functional (in the exponent) by 2.
(Beware that there is also ordinary $G$-Chern-Simons theory for gauge group $G = Spin(n)$ a spin group, which in traditional parlance one might also pronounce as “Spin Chern-Simons theory”, but which is different, in general, from Spin Chern-Simons theory in the sense discussed here.)
The division by 2 makes the holographically dual theory in 2d be the correct self-dual theory. The generalization of the Spin structure to higher dimensional Chern-Simons theory is that of integral Wu structure. In the next relevant case of 7d Chern-Simons theory this is related to the flux quantizaton condition on the supergravity C-field wholse holographically related self-dual higher gauge field is the 2-form-field in the 6d (2,0)-superconformal QFT on the M5-brane.
Notice that if a 3-manifold admits a spin structure then it also admits a framing. (…)
The following table lists classes of examples of square roots of line bundles
For general (compact) Lie groups as gauge groups spin Chern-Simons theory is discussed in
Jerome Jenquin, Classical Chern-Simons on manifolds with spin structure (arXiv:0504524)
Jerome Jenquin, Spin Chern-Simons and Spin TQFTs (arXiv:0605239)
For abelian gauge groups Spin Chern-Simons theory is discussed in
and with emphasis on the holographically dual self-dual higher gauge theory in
This is based on work by Witten and Hopkins-Singer, see the references at self-dual higher gauge theory.
Last revised on October 12, 2014 at 15:39:01. See the history of this page for a list of all contributions to it.