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

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular 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}{}& \esh &\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 09:52:09. See the history of this page for a list of all contributions to it.