pin group

**spin geometry**, **string geometry**, **fivebrane geometry** …

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

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.

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.

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

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

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 $v_1 \cdots v_{2k}$ 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

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

where we understand the standard quadratic form on $\mathbb{R}^n$ for either global sign

$\array{
\mathbb{R}^n &\overset{q_{\pm}}{\longrightarrow}& \mathbb{R}
\\
\vec x &\mapsto& \pm \underset{i}{\sum} (x^i)^2
}$

The corresponding two $Pin$-groups are denoted

$Pin_\pm(n)
\;\coloneqq\;
Pin\big( \mathbb{R}^n, q_\pm\big)$

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 |

A standard textbook account is

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

See also

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

The following article discusses which of the Pin groups are in fact compatible with general relativity

- Bas Janssens,
*The Pin Groups in General Relativity*(arXiv:1709.02742)

Last revised on March 16, 2019 at 08:53:01. See the history of this page for a list of all contributions to it.