Green's function on torus

Question

I have a question about the Green's function $G(z,w)$ on torus which takes the form (for example the first equation in the paper https://annals.math.princeton.edu/wp-content/uploads/annals-v172-n2-p03-p.pdf) \begin{equation} -\nabla^2_z G(z,w)=\delta_w(z)-1/|T| \end{equation} where the first term on the RHS is a delta function, and the second term on the RHS $-1/|T|$ is a constant term. This Green's function may have some relevance in CFT on a torus. My question is why this constant term come into play? What is the meaning of the constant term?

Answer 1

When both "space" ($x$) and "time" ($y$) directions are periodic, the Laplacian on torus with coodinate $z=x+iy$ has a normalized zero mode $$ \varphi_0(z) = \frac 1 {\sqrt{{\rm Im}(\tau)}} $$ (Here $\tau$ is the modular parameter defining the torus.)

As $$ -\nabla^2 \varphi_0=0. $$ the zero mode means that the Laplace operator is not 1-1 and so prevents the Laplacian with periodic boundary conditions from having an inverse. There is therefore no actual Green function. Instead we must therefore resort to a modified Green function. We can make use of a theta function with characteristics defined by $$ \theta\left[ \matrix{a\cr b}\right] (z|\tau)= \sum_{m=-\infty}^\infty \exp\{i\pi \tau(m+a)^2 +2\pi i (m+a)(z+b)\}, \quad {\rm Im}(\tau)>0\quad a,b \in {\mathbb R}. $$

Observe that $$ F(x,y) \equiv e^{-\pi y^2/ {\rm Im}(\tau)} \theta\left[ \matrix{\textstyle{\frac 12}\cr \textstyle{\frac 12}}\right] (z|\tau) $$ obeys $$ F(x+1,y) =-F(x,y), \quad F(x+ {\rm Re}(\tau), y+{\rm Im}(\tau)) =({\rm phase}) F(x,y) $$ so $$ G_0(x,y)) = -\frac{1}{2\pi} \ln |F| = -\frac{1}{2\pi} \ln\left| \theta\left[ \matrix{\textstyle{\frac 12}\cr \textstyle{\frac 12}}\right] (z|\tau)\right| + \frac 12 y^2 / {\rm Im}(\tau)\\ = -\frac 1{2\pi} \ln |E(z)| + \frac 12 y^2 / {\rm Im}(\tau)+const. $$ is both periodic on the torus and obeys $$ -\nabla^2 G_0(x,y) = \delta^2(x,y) - 1/{\rm Im}(\tau)\\ = \sum_n \varphi_n(z) \varphi^*_n(0)- \varphi_0(z)\varphi_0(0) $$ Here the sum is over all $n$ including $n=0$. The $\varphi_n(z)$, $n>0$ are the eigenfuctions of the Laplacian with non-zero eigenvalues.

The modified Green function can be used to solve $$ -\nabla^2 \phi= f(x,y) $$ as $$ \phi(x,y) = \int_{\rm torus} G_0(x-x', y-y'))f(x',y') dx' dy'=0. $$ provided that $f$ is perpendicular to the the zero mode, i.e. $$ \int_{\rm torus} f(x,y) dx dy=0. $$

This is consistent with the Fredholm alternative for linear operators.

Answer 2

The constant term is needed because on a compact manifold with periodic boundary conditions there is a zero mode in the spectrum of the laplacian. This is easier to see on the circle, where $\frac{d^2}{dt^2}f = \lambda f$ has the periodic constant solution $f(t) =c$. This makes the operator not-inveritble.

One should instead work in the space orthogonal to the zero modes. Let's look at the spectral decomposition of the Green function: $$G(t, t') = \sum_\lambda u_\lambda^*(t) u_\lambda(t') \times \lambda^{-1}, $$ where $u_\lambda(t) $are the eigenfunctions of the Operator. Note that $$\nabla^2 G = \sum_\lambda u_\lambda^*(t) u_\lambda(t') = \delta(t - t') $$ by completeness.

Now you see in the expression for the Green function why $\lambda = 0$ would be problematic. We remove this from the sum to work on the subspace orthogonal to this function, getting $$G'(t, t') = \sum_{\lambda \neq 0} u_\lambda^*(t) u_\lambda(t') \times \lambda^{-1}$$ Following which we get a modified Green equation $$\nabla^2 G = \sum_{\lambda \neq 0} u_\lambda^*(t) u_\lambda(t') = \delta(t - t') - u_0^*(t)u(t').$$ This last term corresponds to the constant piece you're subtracting in the case of periodic boundary conditions, because we saw that the zero eigenfunctions is constant. The factor of T is simply from some normalisation condition.

The other way of seeing it is to think in terms of electrostatics. On a compact manifold the periodicity is inconsistent with the Green function that represents the response to a point charge placed at some point: $$\int_{M} \delta(t, t') = \int_{M} \nabla^{2} G = \int_{\partial M}\nabla G \cdot dn = 0$$ since the Green function will be periodic. Yet the left hand side does not integrate to 0. To fix this we must add a constant negative "background charge" which modifies the LHS to $\int_{M}\delta(t, t') - |c|^{2}$ with $c$ chosen so that the integral vanishes. In other words this background charge is needed to ensure compatibility with the periodic boundary conditions.