Superstring vacuum amplitude on the torus

Question

My question is how to obtain the superstring (Type II A and B) vacuum amplitudes on a torus. They are given in Polchinski's String Theory Vol. 2 equation (10.7.9):

$Z_\psi^{\pm}=\frac{1}{2}[Z^0_0(\tau)^4-Z^0_1(\tau)^4-Z^1_0(\tau)^4\mp Z^1_1(\tau)^4]$

I understand how each individual $Z^\alpha_\beta$ is obtained but do not understand how they are put together to get $Z_\psi^{\pm}$.

Answer 1

Each of the four terms corresponds to a choice of boundary conditions for the fermions on the torus i.e. to a choice of spin structure on the torus. To put together these terms is required by the GSO projection, which is necessary to obtain a tachyon free spectrum and spacetime supersymmetry (and so consistency because of the presence of a gravitino in the spectrum), or equivalently to satisfy the modular invariance property of the partition function. Modular transformations exchange the various fermionic boundary conditions and so all of them have to be combined to obtain a modular invariant quantity.