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Spin Chern-Simons theory

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Quantum field theory

\infty-Chern-Weil theory

Contents

Idea

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 GG-Chern-Simons theory for gauge group G=Spin(n)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

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

line bundlesquare rootchoice corresponds to
canonical bundleTheta characteristicover Riemann surface and Hermitian manifold (e.g.Kähler manifold): spin structure
density bundlehalf-density bundle
canonical bundle of Lagrangian submanifoldmetalinear structuremetaplectic correction
determinant line bundlePfaffian line bundle
quadratic secondary intersection pairingpartition function of self-dual higher gauge theoryintegral Wu structure

References

For general (compact) Lie groups as gauge groups spin Chern-Simons theory is discussed in

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.