spin geometry

string geometry

∞-Lie theory

# Contents

## 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 $SO(n)$.

## 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. 1, 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. 1, 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 $v_1 \cdots v_{2k}$ of elements $v_i \in V$ with $q(V) = 1$.

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 $q(V) = 1$.

Specifically, “the” Spin group is

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

A spin representation is a linear representation of the spin group, def. 3.

## 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$012345678910111213141516
Whitehead tower of orthogonal grouporientationspinstringfivebraneninebrane
homotopy groups of stable orthogonal group$\pi_n(O)$$\mathbb{Z}_2$$\mathbb{Z}_2$0$\mathbb{Z}$000$\mathbb{Z}$$\mathbb{Z}_2$$\mathbb{Z}_2$0$\mathbb{Z}$000$\mathbb{Z}$$\mathbb{Z}_2$
stable homotopy groups of spheres$\pi_n(\mathbb{S})$$\mathbb{Z}$$\mathbb{Z}_2$$\mathbb{Z}_2$$\mathbb{Z}_{24}$00$\mathbb{Z}_2$$\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$$\mathbb{Z}_2 \oplus \mathbb{Z}_2$
image of J-homomorphism$im(\pi_n(J))$0$\mathbb{Z}_2$0$\mathbb{Z}_{24}$000$\mathbb{Z}_{240}$$\mathbb{Z}_2$$\mathbb{Z}_2$0$\mathbb{Z}_{504}$000$\mathbb{Z}_{480}$$\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 $GL(p,V)$ for $V$ the real numbers $\mathbb{R}$, the complex numbers $\mathbb{C}$ and the quaternions $\mathbb{Q}$, the orthogonal group $O(p,q)$, the unitary group $U(p,q)$ and the symplectic group $Sp(p,q)$).

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

We have

• in Riemannian signature

• $Spin(1) \simeq O(1)$

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

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. Gorbunov-Ray 92)

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

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

• $Spin(5) \simeq Sp(2)$

• $Spin(6) \simeq SU(4)$ (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

• in anti de Sitter signature

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

• $Spin(3,2) \simeq Sp(4,\mathbb{R})$

• $Spin(4,2) \simeq SU(2,2)$

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+1$$Spin(2,1) \simeq SL(2,\mathbb{R})$$\mathbb{R}$ the real numbers
$4 = 3+1$$Spin(3,1) \simeq SL(2, \mathbb{C})$$\mathbb{C}$ the complex numbers
$6 = 5+1$$Spin(5,1) \simeq SL(2, \mathbb{H})$$\mathbb{H}$ the quaternionslittle string
$10 = 9+1$$Spin(9,1) \underset{some\,sense}{\simeq} SL(2,\mathbb{O})$$\mathbb{O}$ the octonionsheterotic/type II string

## Applications

### 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.

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 $SO$ is the spin^c group.

groupsymboluniversal coversymbolhigher coversymbol
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

A standard textbook reference is

• Vassily Gorbunov, Nigel Ray, Orientations of $Spin$ Bundles and Symplectic Cobordism, Publ. RIMS, Kyoto Univ. 28 (1992), 39-55 (pdf)
Discussion of the cohomology of the classifying space $B Spin$ includes