Spin(6) (changes) in nLab
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Context
Group Theory
- group, ∞-group
- group object, group object in an (∞,1)-category
- abelian group, spectrum
- super abelian group
- group action, ∞-action
- representation, ∞-representation
- progroup
- homogeneous space
Classical groups
Finite groups
Group schemes
Topological groups
Lie groups
Super-Lie groups
Higher groups
Cohomology and Extensions
Related concepts
Spin geometry
spin geometry, string geometry, fivebrane geometry …
Ingredients
Spin geometry
rotation groups in low dimensions:
see also
String geometry
Fivebrane geometry
Ninebrane geometry
Contents
Idea
The spin group in dimension 6.
Properties
Exceptional isomorphism
Proposition
There is an exceptional isomorphism
\begin{tikzpicture}
\node (Spin6) at (0,1.4) {Spin(6)\mathrm{Spin}(6)}; \node (SU4) at (3.4,1.4) {SU(4)\mathrm{SU}(4)};
\node at (1.7,1.4) {≃\simeq};
\node (center) at (0,0) {}; \node (topright) at (30:1) {}; \node (left) at (180-30:1) {}; \node (botright) at (0,-1) {};
\drawfill=black circle (.1); \drawfill=black circle (.1); \drawdraw=lightgray, fill=lightgray circle (.1); \drawfill=black circle (.1);
\draw (center) to (topright); \drawlightgray to (botright); \draw (center) to (left);
\begin{scope}[shift={(3.4,0)}] \node (center) at (0,0) {}; \node (left) at (-1,0) {}; \node (right) at (+1,0) {};
\drawfill=black circle (.1); \drawfill=black circle (.1); \drawfill=black circle (.1);
\draw (center) to (left); \draw (center) to (right); \end{scope} \end{tikzpicture}
between \end{tikzpicture}Spin(6) and SU(4), reflecting, under the classification of simple Lie groups, the coincidence of Dynkin diagram of “D3” with A3.
between Spin(6) and SU(4), reflecting, under the classification of simple Lie groups, the coincidence of the Dynkin diagrams “D3” and A3.
(e.g. Figueroa-O’Farrill 10, Lemma 8.1)
One way to see the isomorphism Spin(6)≅SU(4)\mathrm{Spin}(6) \cong \mathrm{SU}(4) is as follows. Let VV be a 4-dimensional complex vector space with an inner product and a compatible complex volume form, meaning an element of the exterior product Λ 4V\Lambda^4 V whose norm is 1 in the norm coming from the inner product on VV. The inner product defines a conjugate-linear isomorphism V≅V *V \cong V^\ast (with the complex dual vector space) that together with the complex volume form can be used to define a conjugate-linear Hodge star operator on Λ 2V\Lambda^2 V. This Hodge star operator squares to the identity, and its +1+1 and −1-1 eigenspaces, say Λ ± 2V\Lambda_{\pm}^2 V, each become 6-dimensional real inner product spaces in a natural way. Thus, the group SU(V)\mathrm{SU}(V), consisting of all complex-linear transformations of VV that preserve the inner product and complex volume form, acts as linear transformations of Λ + 2V\Lambda_+^2 V that preserve the inner product, giving a homomorphism ρ:SU(V)→O(Λ + 2V)\rho: \mathrm{SU}(V) \to \mathrm{O}(\Lambda_+^2 V). Since SU(V)\mathrm{SU}(V) is connected we in fact have ρ:SU(V)→SO(Λ + 2V)\rho: \mathrm{SU}(V) \to \mathrm{SO}(\Lambda_+^2 V).
Specializing to the case V=ℂ 4V = \mathbb{C}^4 we get a Lie group homomorphism ρ:SU(4)→SO(6)\rho: \mathrm{SU}(4) \to \mathrm{SO}(6). Since dρd\rho is nonzero and SU(4)\mathrm{SU}(4) is simple, dρd\rho must be injective. Since
dim(SU(4))=15=dim(SO(6)), \mathrm{dim}(\mathrm{SU}(4)) = 15 = \mathrm{dim}(\mathrm{SO}(6)),
dρd\rho must also be surjective. Since SO(6)\mathrm{SO}(6) is connected and dρd\rho is a bijection, ρ\rho must be a covering map. Since ρ(±1)=1\rho(\pm 1) = 1, ρ\rho exhibits SU(4)\mathrm{SU}(4) as a connected cover of SO(6)\mathrm{SO}(6) that is at least a double cover. But the universal cover of SO(6)\mathrm{SO}(6), namely Spin(6)\mathrm{Spin}(6), is only a double cover. Thus SU(4)\mathrm{SU}(4) is a double cover of SO(6)\mathrm{SO}(6), and SU(4)≅Spin(6)\mathrm{SU}(4) \cong \mathrm{Spin}(6).
Coset spaces
coset space-structures on n-spheres:
| standard: | |
|---|---|
| S n−1≃ diffSO(n)/SO(n−1)S^{n-1} \simeq_{diff} SO(n)/SO(n-1) | this Prop. |
| S 2n−1≃ diffSU(n)/SU(n−1)S^{2n-1} \simeq_{diff} SU(n)/SU(n-1) | this Prop. |
| S 4n−1≃ diffSp(n)/Sp(n−1)S^{4n-1} \simeq_{diff} Sp(n)/Sp(n-1) | this Prop. |
| exceptional: | |
| S 7≃ diffSpin(7)/G 2S^7 \simeq_{diff} Spin(7)/G_2 | Spin(7)/G₂ is the 7-sphere |
| S 7≃ diffSpin(6)/SU(3)S^7 \simeq_{diff} Spin(6)/SU(3) | since Spin(6) ≃\simeq SU(4) |
| S 7≃ diffSpin(5)/SU(2)S^7 \simeq_{diff} Spin(5)/SU(2) | since Sp(2) is Spin(5) and Sp(1) is SU(2), see Spin(5)/SU(2) is the 7-sphere |
| S 6≃ diffG 2/SU(3)S^6 \simeq_{diff} G_2/SU(3) | G₂/SU(3) is the 6-sphere |
| S 15≃ diffSpin(9)/Spin(7)S^15 \simeq_{diff} Spin(9)/Spin(7) | Spin(9)/Spin(7) is the 15-sphere |
see also Spin(8)-subgroups and reductions
homotopy fibers of homotopy pullbacks of classifying spaces:
\begin{imagefromfile} “file_name”: “ExceptionalSpheres.jpg”, “width”: 730 \end{imagefromfile}
(from FSS 19, 3.4)
GG-Structure and 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(8,8) | Spin(7)×Spin(7)Spin(7) \times Spin(7) |
| generalized G₂-structure | Spin(7,7)Spin(7,7) | G 2×G 2G_2 \times G_2 |
| generalized CY3 | Spin(6,6)Spin(6,6) | SU(3)×SU(3)SU(3) \times SU(3) |
see also: coset space structure on n-spheres
\linebreak
rotation groups in low dimensions:
see also
\linebreak
References
- José Figueroa-O'Farrill, PG course on Spin Geometry lecture 8: Parallel and Killing spinors, 2010 (pdf)
Last revised on August 17, 2021 at 06:19:55. See the history of this page for a list of all contributions to it.