# [Physics] Green’s function on torus

conformal-field-theorydifferential-geometrydirac-delta-distributionsgreens-functionsstring-theory

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)
$$-\nabla^2_z G(z,w)=\delta_w(z)-1/|T|$$
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?

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. $$\int_{\rm torus} f(x,y) dx dy=0.$$