Pure spinors, intrinsic torsion and curvature in odd dimensions

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.

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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  -

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
Original language	English
Type of outcome	Article in a journal
Field of Study	10101 Pure mathematics
Country of publisher	Netherlands
Confidentiality degree	is not subject to a state or trade secret
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
Tags	NZ, rivok
Tags	International impact, Reviewed
Changed by	Changed by: Ing. Nicole Zrilić, učo 240776. Changed: 11/4/2018 09:36.

Abstract

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
GP14-27885P, research and development project	Investor: Czech Science Foundation

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Pure spinors, intrinsic torsion and curvature in odd dimensions

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