Projective Superspaces in Practice

Abstract

We study the supergeometry of complex projective superspaces ${\mathbb{P}}^{n|m}$. First, we provide formulas for the cohomology of invertible sheaves of the form ${\mathcal{O}}_{{\mathbb{P}}^{n|m}}(\ell )$, that are pull-back of ordinary invertible sheaves on the reduced variety ${\mathbb{P}}^n$. Next, by studying the even Picard group $\mbox{Pic}_0 (\mathbb{P}^{n|m})$, classifying invertible sheaves of rank $1|0$, we show that the sheaves ${\mathcal{O}}_{{\mathbb{P}}^{n|m}}(\ell )$ are not the only invertible sheaves on ${\mathbb{P}}^{n|m}$, but there are also new genuinely supersymmetric invertible sheaves that are unipotent elements in the even Picard group. We study the $\Pi$-Picard group $\mbox{Pic}_Π(\mathbb{P}^{n|m})$, classifying $\Pi$-invertible sheaves of rank $1|1$, proving that there are also non-split $\Pi$-invertible sheaves on supercurves ${\mathbb{P}}^{1|m}$. Further, we investigate infinitesimal automorphisms and first order deformations of ${\mathbb{P}}^{n|m}$, by studying the cohomology of the tangent sheaf using a supersymmetric generalisation of the Euler exact sequence. A special special attention is paid to the meaningful case of supercurves ${\mathbb{P}}^{1|m}$ and of Calabi-Yau's ${\mathbb{P}}^{n|n+1}$. Last, with an eye to applications to physics, we show in full detail how to endow ${\mathbb{P}}^{1|2}$ with the structure of $\mathcal{N}=2$ super Riemann surface and we obtain its SUSY-preserving infinitesimal automorphisms from first principles, that prove to be the Lie superalgebra $\mathfrak{osp}(2|2)$. A particular effort has been devoted to keep the exposition as concrete and explicit as possible.

24 pages

References in corpus (5)

• Mirror Symmetry and Supermanifolds
• Supergeometry of $\Pi$-Projective Spaces
• One-Dimensional Super Calabi-Yau Manifolds and their Mirrors
• Topological Sigma Models On Supermanifolds
• Non Projected Calabi-Yau Supermanifolds over ${\mathbb{P}}^2$

Cited by in corpus (8)

• Pictures from Super Chern-Simons Theory
• $A_{\infty }$-Algebra from Supermanifolds
• Super Chern-Simons Theory: BV-formalism and $A_{\infty }$-algebras
• On Forms, Cohomology, and BV Laplacians in Odd Symplectic Geometry
• Superstring Field Theory, Superforms and Supergeometry
• On BV Supermanifolds and the Super Atiyah Class
• Non-Projected Supermanifolds and Embeddings in Super Grassmannians
• Navigating Collinear Superspace