Notes On Super Riemann Surfaces And Their Moduli

TL;DR

This work consolidates a holomorphic and a smooth perspective on super Riemann surfaces (SRs), building the SR moduli space 𝔐 and the supermoduli space 𝔐 ̄ via cohomology and deformation theory, and connects these geometric foundations to the perturbative framework of superstrings. It develops the Berezinian Ber(Σ) as the natural holomorphic volume form, identifies Ber(Σ) ≅ 𝒟^{−1}, and uses this to formulate Lagrangians, ghost systems, and current algebras in a coordinate-free manner, while also detailing the deformation theory via fields and embeddings. The text then extends these holomorphic insights to the smooth picture of worldsheet physics, integrating over carefully defined cycles Γ (and their compactifications) and addressing the intricate behavior at infinity, degenerations, and punctures (NS and Ramond) in both open/ unoriented and Type II settings. A major portion analyzes the modular geometry of SRs at low genus, the structure of punctures, and the canonical parameters governing degenerations, culminating in a comprehensive treatment of contour integrals and the super period matrix. Finally, it surveys N=2 SRs and dualities, tying these mathematical structures to broader string theory contexts and duality relations, with explicit implications for perturbative amplitudes and infrared behavior.

Abstract

These are notes on the theory of super Riemann surfaces and their moduli spaces, aiming to collect results that are useful for a better understanding of superstring perturbation theory in the RNS formalism.

Figures (17)

• Figure 1: The moduli space ${\mathcal{M}}_{0,4}$ that parametrizes a genus 0 Riemann surface $\Sigma_0$ with four punctures $z_1,\dots,z_4$ is compactified by adding "points at infinity" in which $\Sigma_0$ splits into a union of two components, each of which contains two of the punctures. There are three such points at infinity, depending on how $z_1,\dots,z_4$ are divided pairwise between the two components.
• Figure 2: (a) A Riemann surface $\Sigma_0$ with a long tube. (b) A conformally equivalent picture in which $\Sigma_0$ has a narrow neck. The example sketched here represents a separating degeneration. (c) A similar picture with a nonseparating degeneration, indicated by the arrow.
• Figure 3: (a) Along the divisor ${\fam\eusmfam D}_{\mathrm{nonsep}}\subset\widehat{{\mathcal{M}}}_{\text{ g},\text{ n}}$ that parametrizes nonseparating degenerations of a Riemann surface $\Sigma_0$, the genus of $\Sigma_0$ is reduced by 1, but the two points that are glued together at the node (marked by the arrow) count as additional punctures, one on each branch. So this divisor is a copy of $\widehat{{\mathcal{M}}}_{\text{ g}-1,\text{ n}+2}$. This is sketched here for $\text{ g}=\text{ n}=2$. (b) Divisors in $\widehat{{\mathcal{M}}}_{\text{ g},\text{ n}}$ that correspond to separating degenerations correspond to decompositions $\text{ g}=\text{ g}_1+\text{ g}_2$, $\text{ n}=\text{ n}_1+\text{ n}_2$. The divisor corresponding to such a decomposition is a copy of $\widehat{{\mathcal{M}}}_{\text{ g}_1,\text{ n}_1+1}\times \widehat{{\mathcal{M}}}_{\text{ g}_2,\text{ n}_2+1}$, where the points glued together count as one extra puncture on each side. This is sketched here for $\text{ g}_1=2$, $\text{ g}_2=1$, $\text{ n}_ 1=\text{ n}_ 2=2$.
• Figure 4: In the Deligne-Mumford compactification, a process in which two or more punctures on a Riemann surface $\Sigma_0$ approach each other as in (a), or equivalently are connected to the rest of the surface through a long tube, as in (b), leads to a separating divisor in which $\Sigma_0$ splits into two components, one of which has genus 0, as shown in (c). The number of punctures on the genus 0 component, counting the extra puncture at the node, is always at least 3. In fact, we started in (a) with at least 2 punctures coming together, and there is 1 more where the two components of $\Sigma_0$ are glued together. So the Deligne-Mumford compactification can be constructed while never allowing $\Sigma_0$ to have a genus 0 component with fewer than three punctures.
• Figure 5: If $z_1,\dots,z_s$ (with $s\geq 2$) is any number of punctures in $\Sigma_0$, there is a divisor at infinity in $\widehat{{\mathcal{M}}}_{\text{ g},\text{ n}}$ along which $z_1,\dots,z_s$ are contained in a separate genus 0 component of $\Sigma_0$. This locus is of complex codimension 1, and not of codimension $s-1$ as one might guess from the fact that naively it corresponds to the condition $z_1=\dots = z_s$.