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Contents

Idea

The spin group Spin(n)Spin(n) is the universal covering space of the special orthogonal group SO(n)SO(n). By the usual arguments it inherits a group structure for which the operations are smooth and so is a Lie group like SO(n)SO(n).

Properties

General

By definition the spin group sits in a short exact sequence of groups

2SpinSO. \mathbb{Z}_2 \to Spin \to SO \,.

Relation to Whitehead tower of orthogonal group

The spin group is one element in the Whitehead tower of O(n)O(n), which starts out like

Fivebrane(n)String(n)Spin(n)SO(n)O(n). \cdots \to Fivebrane(n) \to String(n) \to Spin(n) \to SO(n) \to \mathrm{O}(n) \,.

The homotopy groups of O(n)O(n) are for kk \in \mathbb{N} and for sufficiently large nn

π 8k+0(O) = 2 π 8k+1(O) = 2 π 8k+2(O) =0 π 8k+3(O) = π 8k+4(O) =0 π 8k+5(O) =0 π 8k+6(O) =0 π 8k+7(O) =. \array{ \pi_{8k+0}(O) & = \mathbb{Z}_2 \\ \pi_{8k+1}(O) & = \mathbb{Z}_2 \\ \pi_{8k+2}(O) & = 0 \\ \pi_{8k+3}(O) & = \mathbb{Z} \\ \pi_{8k+4}(O) & = 0 \\ \pi_{8k+5}(O) & = 0 \\ \pi_{8k+6}(O) & = 0 \\ \pi_{8k+7}(O) & = \mathbb{Z} } \,.

By co-killing these groups step by step one gets

cokillthis toget π 0(O) = 2 SO π 1(O) = 2 Spin π 2(O) =0 π 3(O) = String π 4(O) =0 π 5(O) =0 π 6(O) =0 π 7(O) = Fivebrane. \array{ cokill\, this &&&& to\,get \\ \\ \pi_{0}(O) & = \mathbb{Z}_2 &&& SO \\ \pi_{1}(O) & = \mathbb{Z}_2 &&& Spin \\ \pi_{2}(O) & = 0 \\ \pi_{3}(O) & = \mathbb{Z} &&& String \\ \pi_{4}(O) & = 0 \\ \pi_{5}(O) & = 0 \\ \pi_{6}(O) & = 0 \\ \pi_{7}(O) & = \mathbb{Z} &&& Fivebrane } \,.

Via the J-homomorphism this is related to the stable homotopy groups of spheres:

nn012345678910111213141516
Whitehead tower of orthogonal grouporientationspinstringfivebraneninebrane
homotopy groups of stable orthogonal groupπ n(O)\pi_n(O) 2\mathbb{Z}_2 2\mathbb{Z}_20\mathbb{Z}000\mathbb{Z} 2\mathbb{Z}_2 2\mathbb{Z}_20\mathbb{Z}000\mathbb{Z} 2\mathbb{Z}_2
stable homotopy groups of spheresπ n(𝕊)\pi_n(\mathbb{S})\mathbb{Z} 2\mathbb{Z}_2 2\mathbb{Z}_2 24\mathbb{Z}_{24}00 2\mathbb{Z}_2 240\mathbb{Z}_{240} 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 2 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2 6\mathbb{Z}_6 504\mathbb{Z}_{504}0 3\mathbb{Z}_3 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2 480 2\mathbb{Z}_{480} \oplus \mathbb{Z}_2 2 2\mathbb{Z}_2 \oplus \mathbb{Z}_2
image of J-homomorphismim(π n(J))im(\pi_n(J))0 2\mathbb{Z}_20 24\mathbb{Z}_{24}000 240\mathbb{Z}_{240} 2\mathbb{Z}_2 2\mathbb{Z}_20 504\mathbb{Z}_{504}000 480\mathbb{Z}_{480} 2\mathbb{Z}_2

Exceptional isomorphisms

In low dimensions the spin group happens to be isomorphic (“sporadic isomorphisms”) to various other classical group (among them the general linear group GL(p,V)GL(p,V) for VV the real numbers \mathbb{R}, the complex numbers \mathbb{C} and the quaternions \mathbb{Q}, the orthogonal group O(p,q)O(p,q), the unitary group U(p,q)U(p,q) and the symplectic group Sp(p,q)Sp(p,q)).

See for instance (Garrett 13). See also division algebra and supersymmetry.

We have

  • in Riemannian signature

    • Spin(1)O(1)Spin(1) \simeq O(1)

    • Spin(2)U(1)SO(2)S 1Spin(2) \simeq U(1) \simeq SO(2) \simeq S^1 (the circle group)

      the projection Spin(2)SO(2)Spin(2)\to SO(2) corresponds to S 12S 1S^1\stackrel{\cdot 2}{\longrightarrow} S^1, see also at Theta characteristic

    • Spin(3)Sp(1)SU(2)S 3Spin(3) \simeq Sp(1) \simeq SU(2) \simeq S^3 (the special unitary group SU(2))

      the inclusion Spin(2)Spin(3)Spin(2) \hookrightarrow Spin(3) corresponds to the canonical S 1S 3S^1 \hookrightarrow S^3 (see e.g. Gorbunov-Ray 92)

    • Spin(4)Sp(1)×Sp(1)S 3×S 3Spin(4) \simeq Sp(1)\times Sp(1) \simeq S^3 \times S^3

      the inclusion Spin(3)Spin(4)Spin(3) \hookrightarrow Spin(4) corresponds to the diagonal S 3S 3×S 3S^3 \hookrightarrow S^3 \times S^3

    • Spin(5)Sp(2)Spin(5) \simeq Sp(2)

    • Spin(6)SU(4)Spin(6) \simeq SU(4) (the special unitary group SU(4))

  • in Lorentzian signature

  • in anti de Sitter signature

    • Spin(2,2)SL(2,)×SL(2,)Spin(2,2) \simeq SL(2,\mathbb{R}) \times SL(2,\mathbb{R})

    • Spin(3,2)Sp(4,)Spin(3,2) \simeq Sp(4,\mathbb{R})

    • Spin(4,2)SU(2,2)Spin(4,2) \simeq SU(2,2)

Beyond these dimensions there are still some interesting identifications, but the situation becomes much more involved.

Lorentzian spacetime dimensionspin groupnormed division algebrabrane scan entry
3=2+13 = 2+1Spin(2,1)SL(2,)Spin(2,1) \simeq SL(2,\mathbb{R})\mathbb{R} the real numbers
4=3+14 = 3+1Spin(3,1)SL(2,)Spin(3,1) \simeq SL(2, \mathbb{C})\mathbb{C} the complex numbers
6=5+16 = 5+1Spin(5,1)SL(2,)Spin(5,1) \simeq SL(2, \mathbb{H})\mathbb{H} the quaternionslittle string
10=9+110 = 9+1Spin(9,1) somesenseSL(2,𝕆)Spin(9,1) \simeq_{some\,sense} SL(2,\mathbb{O})𝕆\mathbb{O} the octonionsheterotic/type II string

Applications

Spin geometry

See spin geometry

In physics

The name arises due to the requirement that the structure group of the tangent bundle of spacetime lifts to Spin(n)Spin(n) so as to ‘define particles with spin’… (Someone more awake and focused please put this into proper words!)

See spin structure.

The Whitehead tower of the orthogonal group looks like

\cdots \to fivebrane group \to string group \to spin group \to special orthogonal group \to orthogonal group.

Another extension of SOSO is the spin^c group.

groupsymboluniversal coversymbolhigher coversymbol
orthogonal groupO(n)\mathrm{O}(n)Pin groupPin(n)Pin(n)Tring groupTring(n)Tring(n)
special orthogonal groupSO(n)SO(n)Spin groupSpin(n)Spin(n)String groupString(n)String(n)
Lorentz groupO(n,1)\mathrm{O}(n,1)\,Spin(n,1)Spin(n,1)\,\,
anti de Sitter groupO(n,2)\mathrm{O}(n,2)\,Spin(n,2)Spin(n,2)\,\,
Narain groupO(n,n)O(n,n)
Poincaré groupISO(n,1)ISO(n,1)Poincaré spin groupISO^(n,1)\widehat {ISO}(n,1)\,\,
super Poincaré groupsISO(n,1)sISO(n,1)\,\,\,\,

References

A standard textbook reference is

See also

Examples of sporadic (exceptional) spin group isomorphisms incarnated as isogenies onto orthogonal groups are discussed in

  • Paul Garrett, Sporadic isogenies to orthogonal groups, July 2013 (pdf)

  • Vassily Gorbunov, Nigel Ray, Orientations of SpinSpin Bundles and Symplectic Cobordism, Publ. RIMS, Kyoto Univ. 28 (1992), 39-55 (pdf)

Revised on September 9, 2014 20:25:15 by Urs Schreiber (185.26.182.27)