This paper is devoted to the study of supergeometry of complex projective superspaces Pn|m. First, we provide formulas for the cohomology of invertible sheaves of the form OPn|m(ℓ), that are pullbacks of ordinary invertible sheaves on the reduced variety Pn. Next, by studying the even Picard group Pic0(Pn|m), classifying invertible sheaves of rank 1|0, we show that the sheaves OPn|m(ℓ) are not the only invertible sheaves on Pn|m, but there are also new genuinely supersymmetric invertible sheaves that are unipotent elements in the even Picard group. We study the Π-Picard group PicΠ(Pn|m), classifying Π-invertible sheaves of rank 1|1, proving that there are also non-split Π-invertible sheaves on supercurves P1|m. Further, we investigate infinitesimal automorphisms and first order deformations of Pn|m, by studying the cohomology of the tangent sheaf using a supersymmetric generalisation of the Euler exact sequence. A special attention is paid to the meaningful case of supercurves P1|mand of Calabi–Yau's Pn|n+1. Last, with an eye to applications to physics, we show in full detail how to endow P1|2with the structure of N=2 super Riemann surface and we obtain its SUSY-preserving infinitesimal automorphisms from first principles, that prove to be the Lie superalgebra osp(2|2). A particular effort has been devoted to keep the exposition as concrete and explicit as possible.

Projective superspaces in practice

Cacciatori, Sergio Luigi
;
NOJA, SIMONE
2018-01-01

Abstract

This paper is devoted to the study of supergeometry of complex projective superspaces Pn|m. First, we provide formulas for the cohomology of invertible sheaves of the form OPn|m(ℓ), that are pullbacks of ordinary invertible sheaves on the reduced variety Pn. Next, by studying the even Picard group Pic0(Pn|m), classifying invertible sheaves of rank 1|0, we show that the sheaves OPn|m(ℓ) are not the only invertible sheaves on Pn|m, but there are also new genuinely supersymmetric invertible sheaves that are unipotent elements in the even Picard group. We study the Π-Picard group PicΠ(Pn|m), classifying Π-invertible sheaves of rank 1|1, proving that there are also non-split Π-invertible sheaves on supercurves P1|m. Further, we investigate infinitesimal automorphisms and first order deformations of Pn|m, by studying the cohomology of the tangent sheaf using a supersymmetric generalisation of the Euler exact sequence. A special attention is paid to the meaningful case of supercurves P1|mand of Calabi–Yau's Pn|n+1. Last, with an eye to applications to physics, we show in full detail how to endow P1|2with the structure of N=2 super Riemann surface and we obtain its SUSY-preserving infinitesimal automorphisms from first principles, that prove to be the Lie superalgebra osp(2|2). A particular effort has been devoted to keep the exposition as concrete and explicit as possible.
2018
http://www.elsevier.com/locate/geomphys
Calabi–Yau supermanifolds; N=2 super Riemann surfaces; Supergeometry; Π-projective geometry; Mathematics - Algebraic Geometry; Mathematics - Algebraic Geometry; High Energy Physics - Theory; Mathematical Physics; Mathematics - Mathematical Physics; Mathematical Physics; Physics and Astronomy (all); Geometry and Topology
Cacciatori, Sergio Luigi; Noja, Simone
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/2075570
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