- #1

binbagsss

- 1,259

- 11

##S_{CS}=\frac{k}{4\pi}\int d^3 x \epsilon^{\mu \nu \rho} tr(a_{\mu}\partial_{\nu}a_{\rho} -\frac{2i}{3}a_{\mu}a_{\nu}a_{\rho})##.

manifold ## \bf{R} \times \Sigma ## where ##\bf{R}## is time and ##\Sigma## is a spatial compact manifold.

##S_{CS}=\frac{k}{4\pi}\int dt \int\limits_{\sigma} d^2x tr(\epsilon^{ij} a_i \frac{\partial}{\partial t} a_j + a_0 f_{12)}##

I'm very stuck , I'm not sure where to begin. Any hint or explanation very much appreciated.

For example how have we gone from levi - civita tensor in '3-d to 2-d', how have we gone to only a time derivative in the first term- all spatial derivatives vanish? why would this be?

also very confused about the last term. I can't see no reason why spatial derivatives would vanish so I guess instead it's made use of compactness combined with the levi-civita symbol causing something to vanish? I guess such vanishing might also be the reason we able to write in terms of the '2-d' levi-civita symbol instead, but I'm pretty clueless.

Many thanks in advance.