nLab spin group

Idea

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

For special cases in low dimensions see at: Spin(2), Spin(3), Spin(4), Spin(5), Spin(6), Spin(7), Spin(8)

Definition

Definition

A quadratic vector space $(V, \langle -,-\rangle)$ is a vector space $V$ over finite dimension over a field $k$ of characteristic 0, and equipped with a symmetric bilinear form $\langle -,-\rangle \colon V \otimes V \to k$.

Conventions as in (Varadarajan 04, section 5.3).

We write $q\colon v \mapsto \langle v ,v \rangle$ for the corresponding quadratic form.

Definition

The Clifford algebra $CL(V,q)$ of a quadratic vector space, def. , is the associative algebra over $k$ which is the quotient

$$Cl(V,q) \coloneqq T(V)/I(V,q)$$

of the tensor algebra of $V$ by the ideal generated by the elements $v \otimes v - q(v)$.

Since the tensor algebra $T(V)$ is naturally $\mathbb{Z}$-graded, the Clifford algebra $Cl(V,q)$ is naturally $\mathbb{Z}/2\mathbb{Z}$-graded.

Let $(\mathbb{R}^n, q = {\vert -\vert})$ be the $n$-dimensional Cartesian space with its canonical scalar product. Write $Cl^\mathbb{C}(\mathbb{R}^n)$ for the complexification of its Clifford algebra.

Proposition

There exists a unique complex representation

$$Cl^{\mathbb{C}}(\mathbb{R}^n) \longrightarrow End(\Delta_n)$$

of the algebra $Cl^\mathbb{C}(\mathbb{R}^n)$ of smallest dimension

$$dim_{\mathbb{C}}(\Delta_n) = 2^{[n/2]} \,.$$

Definition

The Pin group $Pin(V;q)$ of a quadratic vector space, def. , is the subgroup of the group of units in the Clifford algebra $Cl(V,q)$

$$Pin(V,q) \hookrightarrow GL_1(Cl(V,q))$$

on those elements which are multiples $v_1 \cdots v_{n}$ of elements $v_i \in V$ with $q(v_i) = 1$.

The Spin group $Spin(V,q)$ is the further subgroup of $Pin(V;q)$ on those elements which are even number multiples $v_1 \cdots v_{2k}$ of elements $v_i \in V$ with $q(v_i) = 1$.

Specifically, “the” Spin group is

$$Spin(n) \coloneqq Spin(\mathbb{R}^n) \,.$$

A spin representation is a linear representation of the spin group, def. .

Properties

General

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

$$\mathbb{Z}_2 \to Spin \to SO \,.$$

Relation to Whitehead tower of orthogonal group

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

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

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

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

By co-killing these groups step by step one gets

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

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

	$n$	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Whitehead tower of orthogonal group		orientation	spin group		string group				fivebrane group				ninebrane group
higher versions		special orthogonal group	spin group		string 2-group				fivebrane 6-group				ninebrane 10-group
homotopy groups of stable orthogonal group	$\pi_n(O)$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	0	$\mathbb{Z}$	0	0	0	$\mathbb{Z}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	0	$\mathbb{Z}$	0	0	0	$\mathbb{Z}$	$\mathbb{Z}_2$
stable homotopy groups of spheres	$\pi_n(\mathbb{S})$	$\mathbb{Z}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	$\mathbb{Z}_{24}$	0	0	$\mathbb{Z}_2$	$\mathbb{Z}_{240}$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2$	$\mathbb{Z}_6$	$\mathbb{Z}_{504}$	0	$\mathbb{Z}_3$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2$	$\mathbb{Z}_{480} \oplus \mathbb{Z}_2$	$\mathbb{Z}_2 \oplus \mathbb{Z}_2$
image of J-homomorphism	$im(\pi_n(J))$	0	$\mathbb{Z}_2$	0	$\mathbb{Z}_{24}$	0	0	0	$\mathbb{Z}_{240}$	$\mathbb{Z}_2$	$\mathbb{Z}_2$	0	$\mathbb{Z}_{504}$	0	0	0	$\mathbb{Z}_{480}$	$\mathbb{Z}_2$

Exceptional isomorphisms

In low dimensions the spin groups happens to be isomorphic to various other classical Lie groups. One speaks of exceptional isomorphisms or sporadic isomorphisms.

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

In the following $Sp(n)$ denotes the quaternionic unitary group in quaternionic dimension $n$.

We have

• in Euclidean signature

  • $Spin(1) \simeq O(1)$

  • Spin(2)$\simeq U(1) \simeq SO(2) \simeq S^1$ (SO(2), the circle group, see also at real Hopf fibration)

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

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

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

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

    this is given by identifying $\mathbb{R}^4 \simeq \mathbb{H}$ with the quaternions and $SU(2) \simeq S^3$ with the group of unit quternions. Then left and right quaternion multiplication gives a homomorphism

    $$SU(2) \times SU(2) \longrightarrow SO(4)$$
    $$(g,h) \mapsto ( x \mapsto \; g^{-1} x h )$$

    which is a double cover and hence exhibits the isomorphism.

    In particular therefore the inclusion $Spin(3) \hookrightarrow Spin(4)$ corresponds to the diagonal $S^3 \hookrightarrow S^3 \times S^3$.

    At the level of Lie algebras $\mathfrak{so}(4) \simeq \wedge^2 \mathbb{R}^4$ and the $\pm 1$-eigenspaces of the Hodge star operator $\star \colon \Wedge^2 \mathbb{R}^4 \to \mathbb{R}^4$ gives the direct sum decomposition $\mathfrak{so}(4) \simeq \mathfrak{su}(2) \oplus \mathfrak{su}(2) \simeq \mathfrak{so}(3) \oplus \mathfrak{so}(3)$

  • Spin(5)$\simeq Sp(2)$ (an indirect consequence of triality, see e.g. Čadek-Vanžura 97)

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

• in Lorentzian signature

  • $Spin(1,1) \simeq GL(1,\mathbb{R})$

  • $Spin(2,1) \simeq SL(2, \mathbb{R})$ – 2d special linear group of real numbers

  • $Spin(3,1) \simeq SL(2,\mathbb{C})$ – 2d special linear group of complex numbers

  • $Spin(4,1) \simeq Sp(1,1)$

  • $Spin(5,1) \simeq SL(2,\mathbb{H})$ – 2d special linear group of quaternions

  • $Spin(9,1) \simeq_{in\;some\;sense} SL(2, \mathbb{O})$ – 2d special linear group of octonions (see SL(2,O) for more on this would-be isomorphism)

• in anti de Sitter signature

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

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

  • $Spin(4,2) \simeq SU(2,2)$

• in mixed signature

  • $Spin(3,3) \simeq SL(4,\mathbb{R})$ (Garrett 13 (2.12))

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

exceptional spinors and real normed division algebras

Lorentzian spacetime dimension	$\phantom{AA}$spin group	normed division algebra	$\,\,$ brane scan entry
$3 = 2+1$	$Spin(2,1) \simeq SL(2,\mathbb{R})$	$\phantom{A}$ $\mathbb{R}$ the real numbers	super 1-brane in 3d
$4 = 3+1$	$Spin(3,1) \simeq SL(2, \mathbb{C})$	$\phantom{A}$ $\mathbb{C}$ the complex numbers	super 2-brane in 4d
$6 = 5+1$	$Spin(5,1) \simeq$ SL(2,H)	$\phantom{A}$ $\mathbb{H}$ the quaternions	little string
$10 = 9+1$	Spin(9,1) ${\simeq}$ “SL(2,O)”	$\phantom{A}$ $\mathbb{O}$ the octonions	heterotic/type II string

Examples

rotation groups in low dimensions:

Dynkin label	sp. orth. group	spin group	pin group	semi-spin group
	SO(2)	Spin(2)	Pin(2)
B1	SO(3)	Spin(3)	Pin(3)
D2	SO(4)	Spin(4)	Pin(4)
B2	SO(5)	Spin(5)	Pin(5)
D3	SO(6)	Spin(6)
B3	SO(7)	Spin(7)
D4	SO(8)	Spin(8)		SO(8)
B4	SO(9)	Spin(9)
D5	SO(10)	Spin(10)
B5	SO(11)	Spin(11)
D6	SO(12)	Spin(12)
	$\vdots$	$\vdots$
D8	SO(16)	Spin(16)		SemiSpin(16)
	$\vdots$	$\vdots$
D16	SO(32)	Spin(32)		SemiSpin(32)

see also

• Spin(5).Spin(3)

• finite rotation groups

• ADE-classification

• Spin(11,3)

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 $Spin(n)$ so as to ‘define particles with spin’… (Someone more awake and focused please put this into proper words!)

See spin structure.

Related concepts

• spinor, spin representation, spinor bundle

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.

• spin^c group, spin^h group

• metaplectic group

• semi-spin group

group	symbol	universal cover	symbol	higher cover	symbol
orthogonal group	$\mathrm{O}(n)$	Pin group	$Pin(n)$	Tring group	$Tring(n)$
special orthogonal group	$SO(n)$	Spin group	$Spin(n)$	String group	$String(n)$
Lorentz group	$\mathrm{O}(n,1)$	$\,$	$Spin(n,1)$	$\,$	$\,$
anti de Sitter group	$\mathrm{O}(n,2)$	$\,$	$Spin(n,2)$	$\,$	$\,$
conformal group	$\mathrm{O}(n+1,t+1)$	$\,$
Narain group	$O(n,n)$
Poincaré group	$ISO(n,1)$	Poincaré spin group	$\widehat {ISO}(n,1)$	$\,$	$\,$
super Poincaré group	$sISO(n,1)$	$\,$	$\,$	$\,$	$\,$
superconformal group

References

Textbook accounts:

• H. Blaine Lawson, Marie-Louise Michelsohn, chapter I, section 2 of Spin geometry, Princeton University Press (1989)

• Eckhard Meinrenken: Clifford algebras and Lie groups, Ergebn. Mathem. & Grenzgeb., Springer (2013) [doi:10.1007/978-3-642-36216-3]

• Howard Georgi, §21 & 22 in: Lie Algebras In Particle Physics, Westview Press (1999), CRC Press (2019) [doi:10.1201/9780429499210]

  (with an eye towards application to spinors in (the standard model of) particle physics)

See also

• Veeravalli Varadarajan, section 7 of Supersymmetry for mathematicians: An introduction, Courant lecture notes in mathematics, American Mathematical Society, Providence, R.I (2004)

• wikipedia Spin group

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, pdf ]

• Vassily Gorbounov, Nigel Ray, Orientations of $Spin$ Bundles and Symplectic Cobordism, Publ. RIMS, Kyoto Univ. 28 (1992), 39-55 (pdf, doi: 10.2977/prims/1195168855)

The exceptional isomorphism Spin(5) $\simeq$ Sp(2) is discussed via triality in

• Martin Čadek, Jiří Vanžura, On $Sp(2)$ and $Sp(2) \cdot Sp(1)$-structures in 8-dimensional vector bundles, Publicacions Matemàtiques Vol. 41, No. 2 (1997), pp. 383-401 (jstor:43737249)

Discussion of the cohomology of the classifying space $B Spin$ includes

• E. Thomas, On the cohomology groups of the classifying space for the stable spinor groups, Bol. Soc. Mat. Mexicana (2) 7 (1962) 57-69.

• Harsh Pittie, The integral homology and cohomology rings of SO(n) and Spin(n), Journal of Pure and Applied Algebra Volume 73, Issue 2, 19 August 1991, Pages 105–153 (doi:10.1016/0022-4049(91)90108-E)