How is the propagator computed on an elliptic curve?

Question

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?

Answer 1

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

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