Hi,

I have a question regarding computation of central charges from SW differential. Let's say that the spectral curve has the following form:

$\lambda^2(z,u)=f(z,u) dz^2$

In which $f(z,u)$ is a polynomial of order for example three. The periods (which define the central charge of the theory) is given by the following integrals:

$a(u)\equiv\oint_{\gamma_1} dz\, \sqrt{f(z,u)} \qquad;\qquad a_D(u)\equiv\oint_{\gamma_2} dz\, \sqrt{f(z,u)}$

In which $\gamma_1$ and $\gamma_2$ are two cycles of torus. These cycles are defined by the branch cuts between branch points of the $\sqrt{f(z,u)}$. Here we have three branch points (and one at infinity) so we can consider three different combination of the (*finite*) cuts. If I denote the roots of $f(z,u)$ by $z_i(u)$, then the choice of the cuts are as follows:

- A cut from $z_1(u)$ and $z_2(u)$ and a cut from $z_3(u)$ to infinity;

- A cut from $z_2(u)$ and $z_3(u)$ and a cut from $z_1(u)$ to infinity;

- A cut from $z_1(u)$ and $z_3(u)$ and a cut from $z_2(u)$ to infinity;

My question is that *which one of these choices corresponds to computation of $a(u)$ and which one corresponds to computation of $a_D(u)$ and why?* In the original SW paper, it has been argued that the special choices were made to reproduce the results that are obtained previously in the paper, but what would be the choices if we want to compute central charges for a theory which may or may not have a Lagrangian description?