spin geometry

string geometry

# Contents

## Idea

The simply connected cover of the orthogonal group. Its restriction along the inclusion of the special orthogonal group is the Spin group. Hence the $Pin$-group is “like the Spin group, but including reflections”.

## 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

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