On the Nonprojectedness of Supermoduli with Neveu-Schwarz and Ramond Punctures

TL;DR

The paper proves that the supermoduli space $\mathfrak M_{g,n,2r}$ is not projected when $g\geq n+5r+3$, extending previous results to configurations with both NS and Ramond punctures. It builds a framework using branched covers from a genus-2 base, an immersion $\Psi$ of the branched-cover moduli into the target supermoduli, and a split bosonic normal bundle to transfer nonprojectedness from a base moduli space to the target. Central to the argument is obstruction theory, particularly the second obstruction class $\omega_2$, and a lifting/immersion strategy supported by corollaries on finite coverings. A Hurwitz-theoretic analysis yields the minimal genus and a constructive approach to realize relevant triples $(g,n,2r)$, which underpins the explicit bound $g\geq n+5r+3$ ensuring nonprojectedness.

Abstract

We study the supermoduli space $\mathfrak {M}_{g,n,2r}$ of Super Riemann Surfaces (SRS) of genus $g$, with $n$ Neveu-Schwarz punctures and $2r$ Ramond punctures. We improve the result of Donagi, Witten, and Ott by showing that the supermoduli space $\mathfrak M_{g,n,2r}$ is not projected if $g\geq n+5r+3$.

Figures (1)

• Figure 1: With particles replaced by strings, a 1-loop Feynman diagram (left) becomes a Super Riemann surface of genus 1 (right) in superstring perturbation theory.

Theorems & Definitions (27)

• Definition 2.1: split supermanifold
• Example 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4: tangent bundle
• Definition 2.5: split and projected supermanifold
• Lemma 2.1
• proof
• Theorem 2.1
• Theorem 2.2