Is gravitational Chern-Simons action "topological" or not?

Question

Here are the 2+1D gravitational Chern-Simons action of the connection $\Gamma$ or spin-connection:

$$ S=\int\Gamma\wedge\mathrm{d}\Gamma + \frac{2}{3}\Gamma\wedge\Gamma\wedge\Gamma \tag{a} $$

$$ S=\int\omega\wedge\mathrm{d}\omega + \frac{2}{3}\omega\wedge\omega\wedge\omega \tag{b} $$

A usual Chern-Simons theory of 1-form gauge field is said to be topological, since $S=\int A\wedge\mathrm{d}A + \frac{2}{3}A\wedge A \wedge A$ does not depend on the spacetime metric.

(1) Are (a) and (b) topological or not?

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

(3) Do we have topological gravitational Chern-Simons theory then? Then, what do questions (1) and (2) mean in this context of being topological?

Answer 1

I think the statement in Witten's paper, "Quantum Field Theory and the Jones Polynomial" saying the term is topological in the sense it is indeed independent of metric. However, he also mentioned, in order to make sense of this integration, i.e. make it to be a number, you need to choose a trivialization of tangent bundle, i.e. choose a framing. The tricky thing is the action is not invariant under twisting of framing. In this sense, the gravitational Chern-Simons is in fact a topological invariants of 3-manifold with chosen framing.

Well, if you want a truly topological invariants, you can choose a proper coefficient to make it independent of framing. E.g., in the paper you mentioned, if you choose $c=24$, the partition function will be independent of framing.

Answer 2

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 121 (3) (1989) 351–399.

Answer 3

Whilst the question is not a resource request, I would recommend Edward Witten's paper on the topic published in 1988, titled, 2+1 Dimensional Gravity as a Soluble System. In the paper, Witten shows:

• $2+1$ dimensional gravity with or without $\Lambda$ is soluble classically and at the quantum level
• $2+1$ dimensional gravity is related to a Yang-Mills theory with only a Chern-Simons term
• At the quantum level, such a theory has a vanishing beta function

Witten also discusses other routes, such as the relation to representations of the Virasoro algebra which is related to conformal field theory and string theory. Finally, to answer your question directly, if we interpret the fields as gauge fields, yes, the action is a topological invariant, at least classically.

Answer 4

The gravitational Chern-Simons action is topological, yes. The gauge connection encodes the field of gravity and since it is being integrated over, the result does not depend on a metric. (In the expressions you write maybe the vielbein contribution is missing? Or maybe you mean to have absorbed it in the notation.) Notice that it's just the usual Chern-Simons term which may be written down for many gauge groups, here specialized to the the Poincaré group or an AdS groups.

What one needs to know to understand what's going on here is this:

1. The Einstein-Hilbert action functional always has a first-order formulation in terms of vielbeing and spin connections, which are nothing but the componentes of a 1-form with values in the Poincaré Lie algebra. More precisely, the field of gravity may always be written as a Cartan connection for the inclusion of the Lorentz group into the Poincaré group.

2. Now when one writes down this first-order version of the Einstein Hilbert action in 3-dimensions then a little miracle happens: it turs out to be equal to the Chern-Simons action functional with that gauge group. See at Chern-Simons gravity.