I'm looking to better understand Chern-Simons theory on a torus. We are given the action
S(ϕ)=∫E(∂ϕ)(¯∂ϕ)+λ6(∂ϕ)3
which yields (apparently) the propagator
P(z)=14π℘(z;τ)−112E2(τ)
(at least for
z≠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
∫E(∂ϕ)(¯∂ϕ)
yields the equation of motion
∂¯∂ϕ=0
or that
ϕ(z,¯z) is of the form
ϕ1(z)+ϕ2(¯z). This then seems suggestive that the interaction term
(∂ϕ)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
℘(z;τ) 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)=℘′(z)(℘(z)+E4(τ))
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