# nLab SU(2)-structure

Contents

### Context

#### Riemannian geometry

Riemannian geometry

## Applications

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

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

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 $SU(2)$-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)-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)$

## 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)

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