+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Cohomology +--{: .hide} [[!include cohomology - contents]] =-- #### Quantum field theory +--{: .hide} [[!include functorial quantum field theory - contents]] =-- #### $\infty$-Chern-Weil theory +--{: .hide} [[!include infinity-Chern-Weil theory - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea What is called _Spin Chern-Simons theory_ is a ([[prequantum field theory|pre-]])[[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 [[AdS3-CFT2 and CS-WZW correspondence|holographically dual]] theory in 2d be the correct [[self-dual higher gauge theory|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]]. ## Properties ### Relation to framed Chern-Simons theory In general, [[Chern-Simons theory]] requires [[3-manifolds]] equipped with [[2-framings]]. But the combination of a 2-framing with a [[spin structure]] is essentially an actual [[framing]], cf. [Sawin 2002 p 2](#Sawin02). ## Related concepts * [[super-modular tensor category]] * [[analytically continued Chern-Simons theory]] * [[holomorphic Chern-Simons theory]] * [[eta invariant]] [[!include square roots of line bundles - table]] ## References The idea of spin Chern-Simons theory originates with * {#DijkgraafWitten90} [[Robbert Dijkgraaf]], [[Edward Witten]]: *Topological Spin Theories*, section 5 in: *Topological Gauge Theories and Group Cohomology*, Commun. Math. Phys. **129** (1990) 393 [[doi:10.1007/BF02096988](https://doi.org/10.1007/BF02096988), [euclid:cmp/1104180750](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-129/issue-2/Topological-gauge-theories-and-group-cohomology/cmp/1104180750.full), [pdf](/nlab/files/DW.pdf)] Indication that the [[effective field theory|effective]] [[abelian Chern-Simons theory]] [describing](abelian+Chern-Simons+theory#ReferencesAbCSForFQHE) the [[fractional quantum Hall effect]] has to be understood, in general, as a spin Chern-Simons theory: * [[Gregory Moore]], [[Nicholas Read]], p. 381 ([20 of 35](https://www.physics.rutgers.edu/~gmoore/MooreReadNonabelions.pdf#page=20)) in: *Nonabelions in the fractional quantum Hall effect*, Nucl. Phys. B **360** (1991) 362 [<a href="https://doi.org/10.1016/0550-3213(91)90407-O">doi:10.1016/0550-3213(91)90407-O</a>, [pdf](http://www.physics.rutgers.edu/~gmoore/MooreReadNonabelions.pdf)] * [[Zhenghan Wang]] on joint work with [[Nicholas Read|N. Read]]: *Spin MTC and Fermionic QH States*, talk at *[Workshop on Topological Phases in Condensed Matter 2008](https://research.physics.illinois.edu/ICMT/Workshops/Topological2008/)*, University of Illinois (2008) [[pdf](https://research.physics.illinois.edu/ICMT/Workshops/Topological2008/Talks/wang.pdf), [[Wang-SpinMTCandFermQH.pdf:file]]] Further discussion: For general ([[compact Lie group|compact]]) [[gauge groups|gauge]] [[Lie groups]]: * {#Sawin02} [[Stephen F. Sawin]]: _Invariants of Spin Three-Manifolds From Chern-Simons Theory and Finite-Dimensional Hopf Algebras_, Advances in Mathematics **165** 1 (2002) 35-70 [[arXiv:math/9910106](http://arxiv.org/abs/math/9910106), [doi:10.1006/aima.2000.1935](https://doi.org/10.1006/aima.2000.1935)] * [[Jerome Jenquin]], _Classical Chern-Simons on manifolds with spin structure_ [[arXiv:0504524](http://arxiv.org/abs/math/0504524)] * [[Jerome Jenquin]], _Spin Chern-Simons and Spin TQFTs_ [[arXiv:0605239](http://arxiv.org/abs/math/0605239)] Specifically for [[abelian group|abelian]] [[gauge groups]] $\mathrm{U}(1)^n$ ([[abelian Chern-Simons theory]]): * Dmitriy Belov, [[Gregory Moore]]: _Classification of abelian spin Chern-Simons theories_ [[arXiv:hep-th/0505235](http://arxiv.org/abs/hep-th/0505235)] * {#BelovMoore06} Dmitriy Belov, [[Gregory Moore]], _Holographic Action for the Self-Dual Field_ [[arXiv:hep-th/0605038](http://arxiv.org/abs/hep-th/0605038)] > (in view of[[self-dual higher gauge theory]]) * Takuya Okuda, Koichi Saito, Shuichi Yokoyama: *$U(1)$ spin Chern-Simons theory and Arf invariants in two dimensions*, Nuclear Physics B **962** (2021) 115272 \[<a href="https://doi.org/10.1016/j.nuclphysb.2020.115272">doi:10.1016/j.nuclphysb.2020.115272</a>, [arXiv:2005.03203](https://arxiv.org/abs/2005.03203)\] [[!redirects spin Chern-Simons theory]]