Super Riemann surface and RNS puncture data

Let $S$ be a complex supermanifold of dimension $1\mathrm{\mid }1$; its reduced space $C=S_{\mathrm{r}\mathrm{e}\mathrm{d}}$ is a compact connected Riemann surface equipped with finitely many ordered distinct marks $p_1,\dots ,p_N$, where $N\ge 0$. For every mark $p_i$, specify an open neighborhood $U_i\subset C$, a radius $r_i>0$, and a biholomorphism $z_i:U_i\to \{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }<r_i\}$ with $z_i(p_i)=0$; require the $U_i$ to be pairwise disjoint. These are the pointed compact analytic worldsheet data. A (non-Ramond) super Riemann surface is such an $S$ together with an odd locally free rank-$0\mathrm{\mid }1$ direct summand $\mathcal{D}\subset T_S$ for which the bracket map ${\mathcal{D}}^{\otimes 2}\to T_S\mathrm{/}\mathcal{D}$, $Q\otimes Q\mapsto [Q,Q]\mathrm{/}2<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>\mathcal{D}$, is an isomorphism. In a local superconformal coordinate $z\mathrm{\mid }\theta$, $\mathcal{D}$ is generated by $Q={\mathrm{\partial }}_{\theta }+\theta {\mathrm{\partial }}_z$; with left odd derivatives, $Q^2={\mathrm{\partial }}_z$ and $[Q,Q]=2{\mathrm{\partial }}_z$. A superconformal coordinate change is a parity-preserving holomorphic change of coordinates whose push-forward preserves the line $\mathcal{D}$; in local coordinates this is $Q\tilde{z}-\tilde{\theta }Q\tilde{\theta }=0$.

An NS (Neveu–Schwarz) puncture is a marked superpoint with local coordinates $(z,\theta )=(0,0)$ and the nondegenerate local generator $Q$. A Ramond puncture is recorded as a Ramond divisor $F=\{z=0\}$ in a local superdomain: the distribution is generated by $Q_R={\mathrm{\partial }}_{\theta }+z\theta {\mathrm{\partial }}_z$, so $Q_R^2=z{\mathrm{\partial }}_z$ and its bracket has a simple zero exactly on $F$. Thus the bracket is an isomorphism off $F$, and the Ramond extension satisfies ${\mathcal{D}}^{\otimes 2}\cong (T_S\mathrm{/}\mathcal{D})\otimes \mathcal{O}(-F)$, where $\mathcal{O}(-F)$ is the ideal line locally generated by the even coordinate $z$ vanishing on $F$. On a simply connected sector of the punctured disk, $\psi =\sqrt{z}\theta$ gives $Q_R=\sqrt{z}({\mathrm{\partial }}_{\psi }+\psi {\mathrm{\partial }}_z)$; continuation once around $F$ changes $\psi$ to $-\psi$. This sign is the local Ramond spin monodromy. For a closed surface without Ramond divisors, a spin structure is a holomorphic line $S$ with $S^{\otimes 2}\cong K_C$, and the reduced odd line of $\mathcal{D}$ is $S^{\ast }$.

Complex supermanifold and Berezinian conventions

Pointed analytic worldsheet and family

Witten, Notes on Super Riemann Surfaces and Their Moduli, §§2.1–2.2, 4.1.3

Wang–Yin, On the Equivalence between SRS and PCO Formulations, §2.1

Here “SRS with Ramond data” means the nondegenerate distribution on the punctured complement together with the specified simple-degeneracy extension along each Ramond divisor. It does not assert that the bracket is invertible at a Ramond divisor. A supermoduli object parameterizing these data is supergeometric and is not identified with the ordinary moduli space of reduced Riemann surfaces; no global compactification is asserted.