nLab SU(2)-structure

Idea

reduction of a structure group to SU(2), of particular interest on 5-manifolds

In dimension 5 Sasaki-Einstein structure is a special case of $SU(2)$-structure (e.g. CMD 14, Fino 18b)

Properties

As exceptional geometry

Spin(8)-subgroups and reductions to exceptional geometry

reduction	from spin group	to maximal subgroup
Spin(7)-structure	Spin(8)	Spin(7)
G₂-structure	Spin(7)	G₂
CY3-structure	Spin(6)	SU(3)
SU(2)-structure	Spin(5)	SU(2)
generalized reduction	from Narain group	to direct product group
generalized Spin(7)-structure	$Spin(8,8)$	$Spin(7) \times Spin(7)$
generalized G₂-structure	$Spin(7,7)$	$G_2 \times G_2$
generalized CY3	$Spin(6,6)$	$SU(3) \times SU(3)$

see also: coset space structure on n-spheres

References

• Diego Conti, Simon Salamon, Generalized Killing spinors in dimension 5, Trans. Amer. Math. Soc. 359 (2007), no. 11, 5319-5343 (arXiv:math/0508375)

• Luis C. de Andrés, Marisa Fernández, Anna Fino, Luis Ugarte, Contact 5-manifolds with SU(2)-structure (arXiv:0706.0386)

• Daniël Prins, On flux vacua, SU(n)-structures and generalised complex geometry (arXiv:1602.05415)

• Anna Fino, Contact manifolds and $SU(2)$-structure, talk at Holonomy Groups and Applications in String Theory, Hamburg July 2008 (pdf)

• Anna Fino, Hypo contact and Sasakian structures on Lie groups, talk at Workshop on CR and Sasakian Geometry, Luxembourg– 24 - 26 March 2008 (pdf)

• Lucio Bedullia, Luigi Vezzoni, Torsion of $SU(2)$-structures and Ricci curvature in dimension 5, Differential Geometry and its Applications Volume 27, Issue 1, February 2009, Pages 85-99 (doi:10.1016/j.difgeo.2008.06.008)

• Beniamino Cappelletti-Montano, Giulia Dileo, Nearly Sasakian geometry and $SU(2)$-structures (arXiv:1410.0942)