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

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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?

This post imported from StackExchange Physics at 2014-12-17 16:12 (UTC), posted by SE-user mysteriousness
asked Dec 17, 2014 in Theoretical Physics by mysteriousness (145 points) [ no revision ]
Closely related: physics.stackexchange.com/q/56211, physics.stackexchange.com/q/28888

This post imported from StackExchange Physics at 2014-12-17 16:12 (UTC), posted by SE-user joshphysics

1 Answer

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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.

This post imported from StackExchange Physics at 2014-12-17 16:12 (UTC), posted by SE-user Urs Schreiber
answered Dec 17, 2014 by Urs Schreiber (5,765 points) [ no revision ]
Maybe to highlight further: gravity is always "topological" in that the background structure that it is defined on is only that of a smooth manifold, not including a background metric. That's how topological field theories were originally introduced: as theories sharing this property with gravity but being possibly simpler and hence easier to analyse.

One should really be speaking of gravity as a "topological field theory", too. At least classically (pre-quantumly). What happens after quantization is famously open. Maybe after quantization it's not a field theory anymore, but a string theory.

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