Pure spinors, intrinsic torsion and curvature in odd dimensions

J 2017

Pure spinors, intrinsic torsion and curvature in odd dimensions

TAGHAVI-CHABERT, Arman

Basic information

Original name

Pure spinors, intrinsic torsion and curvature in odd dimensions

Authors

TAGHAVI-CHABERT, Arman (250 France, guarantor, belonging to the institution)

Edition

Differential Geometry and its Applications, Amsterdam, Elsevier Science, 2017, 0926-2245

Other information

Language

English

Type of outcome

Článek v odborném periodiku

Field of Study

10101 Pure mathematics

Country of publisher

Netherlands

Confidentiality degree

není předmětem státního či obchodního tajemství

Impact factor

Impact factor: 0.760

RIV identification code

RIV/00216224:14310/17:00094690

Organization unit

Faculty of Science

DOI

http://dx.doi.org/10.1016/j.difgeo.2017.02.008

UT WoS

000399856700011

Keywords in English

Complex Riemannian geometry; Pure spinors; Distributions; Intrinsic torsion; Curvature prescription; Spinorial equations

Abstract

V originále

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.

Links

GP14-27885P, research and development project

Name: Skoro izotropní struktury v pseudo-riemannovské geometrii

Investor: Czech Science Foundation

Citovat

TAGHAVI-CHABERT, Arman. Pure spinors, intrinsic torsion and curvature in odd dimensions. Differential Geometry and its Applications. Amsterdam: Elsevier Science, 2017, vol. 51, April, p. 117-152. ISSN 0926-2245. Available from: https://dx.doi.org/10.1016/j.difgeo.2017.02.008.

@article{1376701,
   author = {TaghaviandChabert, Arman},
   article_location = {Amsterdam},
   article_number = {April},
   doi = {http://dx.doi.org/10.1016/j.difgeo.2017.02.008},
   keywords = {Complex Riemannian geometry; Pure spinors; Distributions; Intrinsic torsion; Curvature prescription; Spinorial equations},
   language = {eng},
   issn = {0926-2245},
   journal = {Differential Geometry and its Applications},
   title = {Pure spinors, intrinsic torsion and curvature in odd dimensions},
   volume = {51},
   year = {2017}
}

TY  - JOUR
ID  - 1376701
AU  - Taghavi-Chabert, Arman
PY  - 2017
TI  - Pure spinors, intrinsic torsion and curvature in odd dimensions
JF  - Differential Geometry and its Applications
VL  - 51
IS  - April
SP  - 117-152
EP  - 117-152
PB  - Elsevier Science
SN  - 09262245
KW  - Complex Riemannian geometry
KW  - Pure spinors
KW  - Distributions
KW  - Intrinsic torsion
KW  - Curvature prescription
KW  - Spinorial equations
N2  - 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.
ER  -

TAGHAVI-CHABERT, Arman. Pure spinors, intrinsic torsion and curvature in odd dimensions. \textit{Differential Geometry and its Applications}. Amsterdam: Elsevier Science, 2017, vol.~51, April, p.~117-152. ISSN~0926-2245. Available from: https://dx.doi.org/10.1016/j.difgeo.2017.02.008.

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