Local NS and Ramond superdisc models

For a local superfunction $f(z,\theta )=f_0(z)+\theta f_1(z)$, direct application gives $Q_{NS}f=f_1+\theta f_0^{\mathrm{\prime }}$ and $Q_{NS}^{2}f=f_0^{\mathrm{\prime }}+\theta f_1^{\mathrm{\prime }}={\mathrm{\partial }}_{z}f$. Thus $[Q_{NS},Q_{NS}]=2{\mathrm{\partial }}_z$ and the rank-$0\mathrm{\mid }1$ distribution is locally free with quotient tangent line generated by ${\mathrm{\partial }}_z<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>Q_{NS}$.

The same calculation for $Q_R$ gives $Q_R^{2}f=z{\mathrm{\partial }}_{z}f$ and $[Q_R,Q_R]=2z{\mathrm{\partial }}_z$. Its bracket map is an isomorphism on $U\setminus \{0\}$ and has a simple zero along $F=\{0\}$. Since $z{\mathrm{\partial }}_z$ generates the ideal line $T_S\mathrm{/}\mathcal{D}\otimes \mathcal{O}(-F)$, the Ramond divisor relation in [F2] holds locally.

The induced map ${\mathcal{D}}^{\otimes 2}\to T_S\mathrm{/}\mathcal{D}$ sends $Q_{NS}\otimes Q_{NS}$ to ${\mathrm{\partial }}_z<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>Q_{NS}$, a nowhere-zero frame, so it is an isomorphism on $U$. Marking $(z,\theta )=(0,0)$ therefore gives an NS puncture in the stated local model.

On any simply connected sector of $U\setminus \{0\}$ choose a branch $\sqrt{z}$ and set $\psi =\sqrt{z}\theta$. Holding $\theta$ fixed, ${\mathrm{\partial }}_z={\mathrm{\partial }}_z{\mathrm{\mid }}_{\psi }+\frac{\psi }{2z}{\mathrm{\partial }}_{\psi }$, while ${\mathrm{\partial }}_{\theta }=\sqrt{z}{\mathrm{\partial }}_{\psi }$. Hence $z\theta {\mathrm{\partial }}_z=\sqrt{z}\psi {\mathrm{\partial }}_z+\frac{{\psi }^2}{2\sqrt{z}}{\mathrm{\partial }}_{\psi }=\sqrt{z}\psi {\mathrm{\partial }}_z$, since ${\psi }^2=0$, and therefore $Q_R=\sqrt{z}({\mathrm{\partial }}_{\psi }+\psi {\mathrm{\partial }}_z)$. After one circuit around $0$, the branch changes sign and so does $\psi$; this is the Ramond monodromy.