I'm looking to better understand Chern-Simons theory on a torus. We are given the action
$$
S(\phi) = \int_E (\partial \phi)(\overline\partial \phi) + \frac{\lambda}{6}(\partial \phi)^3
$$
which yields (apparently) the propagator
$$
P(z) = \frac{1}{4\pi}\wp(z;\tau) - \frac{1}{12}E_2(\tau)
$$
(at least for $z \neq 0$)

What I would like to understand is the meaning of the terms in the action, as well as how the propagator is derived.

Classically, the action
$$
\int_E (\partial \phi)(\overline\partial\phi)
$$
yields the equation of motion
$$
\partial\overline\partial\phi = 0
$$
or that $\phi(z, \overline z)$ is of the form $\phi_1(z) + \phi_2(\overline z)$. This then seems suggestive that the interaction term $(\partial \phi)^3$ tells us that we are only interested in *holomorphic* functions on $E$, which of course are going to be given by some combination of $\wp(z;\tau)$ and its derivatives. This seems reasonable, but other than an intuitive feel, I don't see why this is actually the case.

My questions are:

- Is this a reasonable interpretation of the terms in the action?
- Since my reasoning was based on classical ideas, how does this carry over to the quantum setting?
- Given the above, how does one use this to derive that the propagator is as given above and not, say,
$$
P(z) = \wp'(z)\big(\wp(z) + E_4(\tau)\big)
$$
or something else similar?

I should probably specify that I am a mathematician, not a physicist.

This post imported from StackExchange Physics at 2014-05-04 11:23 (UCT), posted by SE-user Simon Rose