How to compute the propogator for Chern-Simons on a torus? 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.
 A: Comments to the question (v1): 


*

*The theory that Ref. 1-3 are studying is not Chern-Simons theory, but an interacting scalar boson $\phi: S^1\times S^1\to \mathbb{R}$, living on the torus $S^1\times S^1$, i.e. a Riemann surface of genus one.

*The propagator $\langle \phi(z)\phi(w)\rangle $ is here meant to be the free propagator. Thus we are considering a free scalar boson $\phi: S^1\times S^1\to \mathbb{R}$. This is a standard exercise, which can be found in many string theory textbooks, see e.g. Ref. 4-5.

*In mathematical terms, the propagator $G(z,w)=\langle \phi(z)\phi(w)\rangle$ is just the Greens function for the Laplacian $\partial \bar{\partial}$ on the torus. In other words, we need to solve a double-periodic Dirichlet problem.

*The double-periodic Greens function $G(z,w)$ can in principle be uniquely determined from knowledge of the pole structure. A perhaps more constructive type of derivation would involve the formula 
$$G(z,w)~=~\lim_{\varepsilon\searrow 0^+} \sum_n{}^{\prime} \frac{\psi_n(z)\psi^{\ast}_n(w)}{\lambda_n}e^{-\lambda_n\varepsilon },$$
where $\psi_n(z)$ denotes the eigenfunction for the eigenvalue $\lambda_n\geq 0$ of the Laplace operator $-\partial \bar{\partial}$, cf. e.g. Ref. 6. Here $\varepsilon>0$ is a regularization parameter. (Alternatively, one may use other regularization schemes.) The prime in the sum indicates that zeromodes should be excluded.

*It turns out that the second derivative of the propagator $\partial_z\partial_w G(z,w)$ is a meromorphic function, i.e. it is holomorphic away from its poles. [Perhaps confusingly, the second derivative of the propagator  $\partial_z\partial_w G(z,w)$ is referred to as 'the propagator' in Ref. 1 eq. (4.44).] 
References:


*

*R. Dijkgraaf, Chiral Deformations of Conformal Field Theories, Nucl. Phys. B493 (1997) 588, arXiv:hep-th/9609022.

*R. Dijkgraaf, Mirror Symmetry and Elliptic Curves.

*M.R. Douglas, Conformal Field Theory Techniques in Large N Yang-Mills Theory, arXiv:hep-th/9311130.

*J. Polchinski, String Theory Vol. 1, 1998; Section 7.2.

*E. Kiritsis, String Theory in a Nutshell, 2007; Section 4.18.3.

*E. Cohen, H. Kluberg-Stern, H. Navelet, and R. Peschanski, Regulated propagator on the flat torus, CERN preprint.
