plane, and can also be expressed in terms of spinor coordinates providing a mapping to the spherical geometry [@haldane_sphere]. However, the Laughlin state can also be extended to the torus geometry [@laughlin_torus], where continuous rotation symmetry is broken down and survives at most in form of a discrete subgroup. In this case, $\Psi_L^{\nu=1/m}$ is defined by its short-distance correlations which assume the form of the (odd) Jacobi $\vartheta_1$ theta function of $z_i - z_j$ [@laughlin_torus].

As a trial wavefunction, $\Psi_L^{\nu=1/3}$ provides an excellent description of the physical system at $\nu=1/3$ in the limit of strong cyclotron energy with respect to the Coulomb repulsion in the 2DES plane, $\hbar \omega_c \gg e^2/\epsilon \ell_B$, when excitations to higher LLs are prohibited. In this limit, the operators $a,\tilde{a}$ act trivially and the only dynamical degrees of freedom are the (non-commuting) guiding centers. Up to the normalization, we can then view the wavefunction (\[laughlinwf\]) as follows $$\label{laughlinwfg}
\Psi_L^{\nu=1/m} (g) \propto \prod_{i<j} \left( b_i^\dagger (g) - b_j^\dagger (g) \right)^m |0\rangle,$$ where $b_i^\dagger (g)$ explicitly depend on the metric $g$ [@duncan]. For general $g$, $b_i^\dagger (g)$ is obtained by a Bogoliubov transformation from the $b_i, b_i^\dagger$ in the rotationally invariant case. Equivalently, the wavefunction can be 