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Spin(6)

Contents

Context

Group Theory

Spin geometry

Contents

Idea

The spin group in dimension 6.

Properties

Exceptional isomorphism

Proposition

There is an exceptional isomorphism

between Spin(6) and SU(4), reflecting, under the classification of simple Lie groups, the coincidence of Dynkin diagram of “D3” with 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 VV *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= 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 n1 diffSO(n)/SO(n1)S^{n-1} \simeq_{diff} SO(n)/SO(n-1)this Prop.
S 2n1 diffSU(n)/SU(n1)S^{2n-1} \simeq_{diff} SU(n)/SU(n-1)this Prop.
S 4n1 diffSp(n)/Sp(n1)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_2Spin(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:

(from FSS 19, 3.4)

GG-Structure and 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


rotation groups in low dimensions:

Dynkin labelsp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
B1SO(3)Spin(3)Pin(3)
D2SO(4)Spin(4)Pin(4)
B2SO(5)Spin(5)Pin(5)
D3SO(6)Spin(6)
B3SO(7)Spin(7)
D4SO(8)Spin(8)SO(8)
B4SO(9)Spin(9)
D5SO(10)Spin(10)
B5SO(11)Spin(11)
D6SO(12)Spin(12)
\vdots\vdots
D8SO(16)Spin(16)SemiSpin(16)
\vdots\vdots
D16SO(32)Spin(32)SemiSpin(32)

see also


References

Last revised on January 16, 2021 at 01:50:32. See the history of this page for a list of all contributions to it.