Is gravitational Chern-Simons action “topological” or not?

[Ganesh Ujwal]

Here are the 2+1D gravitational Chern-Simons action of the connection [latex]\Gamma[/latex] or spin-connection:
[latex]S=\int\Gamma\wedge\mathrm{d}\Gamma + \frac{2}{3}\Gamma\wedge\Gamma\wedge\Gamma \tag{1}[/latex]

 

[latex]S=\int\omega\wedge\mathrm{d}\omega + \frac{2}{3}\omega\wedge\omega\wedge\omega \tag{2}[/latex]

 

Usual Chern-Simons theory is said to be topological, since [latex]S=\int A\wedge\mathrm{d}A + \frac{2}{3}A\wedge A \wedge A[/latex] does not depend on the spacetime metric.

(1) Are they topological or not?

(2) Do they depend on the spacetime metric (the action including the integrand)?

(3) Do we have topological gravitational Chern-Simons theory then? What do (1) and (2) mean in this context?

 

 

[ajb]

Classically they are clearly topological. The metric does not appear, and you don't need a metric for integration on manifolds to make sense. Now in dimension 3 you can cast the Einstein-Hilbert action into a Chern-Simons theory as you say. The connection takes it values in the Lie algebra of the Poincare group.

In higher dimensions you need to use higher invariant polynomials, remember you need the integration to make sense. In this way you can get higher dimensional theories and this includes the Einstein-Hilbert action, but also with higher curvature terms.

There is no experimental evidence for the inclusion of this higher curvature terms in gravity. They are however, interesting from a non-perturbative quantum gravity perspective using the renormalisation group flow and the asymptotic safety.


Now, in perturbation quantisation of the Chern-Simons theory you do need a metric to define the path integrals. Witten in 1989 did this [1]. You get a expressions that do depend on this choice of metric, but he then showed how to make this all metric independent by adding another term.

References

[1] Edward Witten, Quantum Field Theory and the Jones Polynomialm, Commun. Math. Phys. 121 (3) (1989) 351–399.

Edited December 17, 2014 by ajb