Supergeometry of $\Pi$-Projective Spaces
| Autor: | Simone Noja |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
High Energy Physics - Theory
Pure mathematics Complex projective space 010102 general mathematics General Physics and Astronomy 01 natural sciences Mathematics - Algebraic Geometry Projective line 0103 physical sciences Supermanifold Supergeometry Sheaf Projective space 010307 mathematical physics Geometry and Topology 0101 mathematics Connection (algebraic framework) Complex number Mathematics::Symplectic Geometry Mathematical Physics Mathematics |
| Popis: | In this paper we prove that $\Pi$-projective spaces $\mathbb{P}^n_\Pi$ arise naturally in supergeometry upon considering a non-projected thickening of $\mathbb{P}^n$ related to the cotangent sheaf $\Omega^1_{\mathbb{P}^n}$. In particular, we prove that for $n \geq 2$ the $\Pi$-projective space $\mathbb{P}^n_\Pi$ can be constructed as the non-projected supermanifold determined by three elements $(\mathbb{P}^n, \Omega^1_{\mathbb{P}^n}, \lambda)$, where $\mathbb{P}^n$ is the ordinary complex projective space, $\Omega^1_{\mathbb{P}^n}$ is its cotangent sheaf and $\lambda $ is a non-zero complex number, representative of the fundamental obstruction class $\omega \in H^1 (\mathcal{T}_{\mathbb{P}^n} \otimes \bigwedge^2 \Omega^1_{\mathbb{P}^n}) \cong \mathbb{C}.$ Likewise, in the case $n=1$ the $\Pi$-projective line $\mathbb{P}^1_\Pi$ is the split supermanifold determined by the pair $(\mathbb{P}^1, \Omega^1_{\mathbb{P}^1} \cong \mathcal{O}_{\mathbb{P}^1} (-2)).$ Moreover we show that in any dimension $\Pi$-projective spaces are Calabi-Yau supermanifolds. To conclude, we offer pieces of evidence that, more in general, also $\Pi$-Grassmannians can be constructed the same way using the cotangent sheaf of their underlying reduced Grassmannians, provided that also higher, possibly fermionic, obstruction classes are taken into account. This suggests that this unexpected connection with the cotangent sheaf is characteristic of $\Pi$-geometry. Comment: 15 pages. Misprints fixed and exposition improved. Some of the main propositions of section 4 got rewritten in a more precise form. Main results are unaffected |
| Databáze: | OpenAIRE |
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