Pure spinors, intrinsic torsion and curvature in odd dimensions

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14310%2F17%3A00094690" target="_blank" >RIV/00216224:14310/17:00094690 - isvavai.cz</a>

Výsledek na webu

<a href="http://dx.doi.org/10.1016/j.difgeo.2017.02.008" target="_blank" >http://dx.doi.org/10.1016/j.difgeo.2017.02.008</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.difgeo.2017.02.008" target="_blank" >10.1016/j.difgeo.2017.02.008</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Pure spinors, intrinsic torsion and curvature in odd dimensions

Popis výsledku v původním jazyce

We study the geometric properties of a $(2m + 1)$-dimensional complex manifold $M$ admitting a holomorphic reduction of the frame bundle to the structure group $P subset Spin(2m + 1, C)$, the stabiliser of the line spanned by a pure spinor at a point. Geometrically, $M$ is endowed with a holomorphic metric $g$, a holomorphic volume form, a spin structure compatible with $g$, and a holomorphic pure spinor field $xi$ up to scale. The defining property of $xi$ is that it determines an almost null structure, i.e. an $m$-plane distribution $N_xi$ along which $g$ is totally degenerate. We develop a spinor calculus, by means of which we encode the geometric properties of $N_xi$ and of its rank-$(m + 1)$ orthogonal complement $N_xi^perp$ corresponding to the algebraic properties of the intrinsic torsion of the $P$-structure. This is the failure of the Levi-Civita connection $nabla$ of $g$ to be compatible with the $P$ -structure. In a similar way, we examine the algebraic properties of the curvature of $nabla$. Applications to spinorial differential equations are given. Notably, we relate the integrability properties of $N_xi$ and $N_xi^perp$ to the existence of solutions of odd- dimensional versions of the zero-rest-mass field equation. We give necessary and sufficient conditions for the almost null structure associated to a pure conformal Killing spinor to be integrable. Finally, we conjecture a Goldberg–Sachs-type theorem on the existence of a certain class of almost null structures when $(M, g)$ has prescribed curvature. We discuss applications of this work to the study of real pseudo-Riemannian manifolds.

Název v anglickém jazyce

Pure spinors, intrinsic torsion and curvature in odd dimensions

Popis výsledku anglicky

We study the geometric properties of a $(2m + 1)$-dimensional complex manifold $M$ admitting a holomorphic reduction of the frame bundle to the structure group $P subset Spin(2m + 1, C)$, the stabiliser of the line spanned by a pure spinor at a point. Geometrically, $M$ is endowed with a holomorphic metric $g$, a holomorphic volume form, a spin structure compatible with $g$, and a holomorphic pure spinor field $xi$ up to scale. The defining property of $xi$ is that it determines an almost null structure, i.e. an $m$-plane distribution $N_xi$ along which $g$ is totally degenerate. We develop a spinor calculus, by means of which we encode the geometric properties of $N_xi$ and of its rank-$(m + 1)$ orthogonal complement $N_xi^perp$ corresponding to the algebraic properties of the intrinsic torsion of the $P$-structure. This is the failure of the Levi-Civita connection $nabla$ of $g$ to be compatible with the $P$ -structure. In a similar way, we examine the algebraic properties of the curvature of $nabla$. Applications to spinorial differential equations are given. Notably, we relate the integrability properties of $N_xi$ and $N_xi^perp$ to the existence of solutions of odd- dimensional versions of the zero-rest-mass field equation. We give necessary and sufficient conditions for the almost null structure associated to a pure conformal Killing spinor to be integrable. Finally, we conjecture a Goldberg–Sachs-type theorem on the existence of a certain class of almost null structures when $(M, g)$ has prescribed curvature. We discuss applications of this work to the study of real pseudo-Riemannian manifolds.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GP14-27885P" target="_blank" >GP14-27885P: Skoro izotropní struktury v pseudo-riemannovské geometrii</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2017

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Differential Geometry and its Applications

ISSN

0926-2245

e-ISSN

—

Svazek periodika

51

Číslo periodika v rámci svazku

April

Stát vydavatele periodika

NL - Nizozemsko

Počet stran výsledku

36

Strana od-do

117-152

Kód UT WoS článku

000399856700011

EID výsledku v databázi Scopus

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