nLab Spin Chern-Simons theory

Idea

What is called Spin Chern-Simons theory is a (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 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.

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.

Related concepts

• super-modular tensor category

• analytically continued Chern-Simons theory

• holomorphic Chern-Simons theory

• eta invariant

The following table lists classes of examples of square roots of line bundles

line bundle	square root	choice corresponds to
canonical bundle	Theta characteristic	over Riemann surface and Hermitian manifold (e.g.Kähler manifold): spin structure
density bundle	half-density bundle
canonical bundle of Lagrangian submanifold	metalinear structure	metaplectic correction
determinant line bundle	Pfaffian line bundle
quadratic secondary intersection pairing	partition function of self-dual higher gauge theory	integral Wu structure

References

The idea of spin Chern-Simons theory originates with

• 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, euclid:cmp/1104180750, pdf]

Indication that the effective abelian Chern-Simons theory describing 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) in: Nonabelions in the fractional quantum Hall effect, Nucl. Phys. B 360 (1991) 362 [doi:10.1016/0550-3213(91)90407-O, pdf]

• Zhenghan Wang on joint work with N. Read: Spin MTC and Fermionic QH States, talk at Workshop on Topological Phases in Condensed Matter 2008, University of Illinois (2008) [pdf, pdf]

Further discussion:

For general (compact) gauge Lie groups:

• 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, doi:10.1006/aima.2000.1935]

• Jerome Jenquin, Classical Chern-Simons on manifolds with spin structure [arXiv:0504524]

• Jerome Jenquin, Spin Chern-Simons and Spin TQFTs [arXiv:0605239]

Specifically for 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]

• Dmitriy Belov, Gregory Moore, Holographic Action for the Self-Dual Field [arXiv:hep-th/0605038]

  (in view ofself-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 [doi:10.1016/j.nuclphysb.2020.115272, arXiv:2005.03203]