How is the propagator computed on an elliptic curve?

I've been struggling for a while now understanding why the propagator for the action $$S(\varphi) = \int_E \partial \varphi \bar\partial\varphi + \frac{\lambda}{6}(\partial\varphi)^3$$ on an elliptic curve is given by $$P(z) = \begin{cases}\frac{1}{4\pi^2}\wp(z) + \frac{1}{12}E_2^*(\tau,\bar\tau) & z \neq 0 \\ \frac{1}{12}E_2^*(\tau,\bar\tau) & z = 0\end{cases}$$ I get that this is computed by looking at the "inverse" of the operator $\partial\bar\partial$ on the space of functions on $E$---that is, we are trying to solve the equation $$\partial\bar\partial G(x-y) = \delta(x-y)$$ and that one should compute this by looking at the Fourier transform.

This sounds all well and good: We will get some expression for the transformed Green's function, and then since we live on a compact space, the original Green's function will be a sum (not integral) over allowable momenta, which should be something analogous to a sum over the lattice points that we see in $\wp(z)$. So far so good.

The problem comes when I try to actually compute it. For concreteness, let us consider the case of $E$ given by $\mathbb{C}/\mathbb{Z}[i]$. That is, we look at $$\hat{G}(n,m) = \iint_E e^{-2\pi i (nx+my)} \partial\bar\partial G(x,y)\, dx\,dy$$ where $\partial = \partial_x - i\partial_y$, etc., and so $\partial\bar\partial = \partial_x^2 + \partial_y^2$. If I understand correctly, we should have (by the usual details with Fourier transforms) that $$\hat{G}(n,m) = \frac{-1}{4\pi^2(n^2+m^2)}$$ and so the propagator would be the inverse Fourier transform of this, i.e. something (I think?) to the effect of $$G(x,y) = -\frac{1}{4\pi^2}\sum_{m,n}\frac{e^{2\pi i (nx+my)}}{n^2+m^2}$$ ...which looks nothing like it ought to. What am I doing wrong?

• For a meaningful answer I think you really need to give the relevant background from conformal field theory, along the lines of your previous posting --- physics.stackexchange.com/questions/110655/… --- For a similar calculation, see Eq. 3.7 of arxiv.org/abs/hep-th/9311130 May 13, 2014 at 22:14
• @CarloBeenakker - I chose not to reference that because it seems that this question doesn't really need the physical context. It seems (to me?) that it is a purely mathematical one, regarding a computation that I'm finding difficult. Perhaps I'm wrong though. May 14, 2014 at 2:55

You can find a fully worked-out derivation of $P(z)$ on an elliptic curve in Appendix A of Feynman Graph Integrals and Almost Modular Forms by S. Li (2011). Basically, the result you quote appears upon evaluation of $\sum_{n,m=-\infty}^{\infty}(z-m-n)^{-2}$. This gives the Weierstrass elliptic function $\wp$ and Eisenstein series $E_2$ in the second equation of your posting.

You ask "What am I doing wrong?"

There is some confusing terminology here (mentioned also in response to your earlier physics.stackexchange posting). The object $P(z)$ is called the "propagator" in the conformal field theory context, but it is actually the second derivative of your propagator. More specifically, the function $P(z)$ is not the inverse of the Laplacian $\Delta_z$, which is the function you are computing in your posting, but instead

$$P(z)=\int_0^\infty \frac{d^2}{dz^2}e^{-t\Delta_z}\,dt.$$

• Ah, I feel silly for not looking through that paper more. I'd had it on my desk for quite some time, and just not got around to looking through it... May 14, 2014 at 17:33