nLab
SU(2)-structure

Contents

Context

Riemannian geometry

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

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)SU(2)-structure (e.g. CMD 14, Fino 18b)

Properties

As exceptional geometry

Spin(8)-subgroups and reductions to exceptional geometry

reductionfrom spin groupto maximal subgroup
Spin(7)-structureSpin(8)Spin(7)
G2-structureSpin(7)G2
CY3-structureSpin(6)SU(3)
SU(2)-structureSpin(5)SU(2)
generalized reductionfrom Narain groupto direct product group
generalized Spin(7)-structureSpin(8,8)Spin(8,8)Spin(7)×Spin(7)Spin(7) \times Spin(7)
generalized G2-structureSpin(7,7)Spin(7,7)G 2×G 2G_2 \times G_2
generalized CY3Spin(6,6)Spin(6,6)SU(3)×SU(3)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)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)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)SU(2)-structures (arXiv:1410.0942)

Last revised on March 31, 2019 at 05:52:09. See the history of this page for a list of all contributions to it.