Higher spin geometry
Higher Lie theory
Formal Lie groupoids
The spin group is the universal covering space of the special orthogonal group . By the usual arguments it inherits a group structure for which the operations are smooth and so is a Lie group like .
Conventions as in (Varadarajan 04, section 5.3).
We write for the corresponding quadratic form.
Since the tensor algebra is naturally -graded, the Clifford algebra is naturally -graded.
Let be the -dimensional Cartesian space with its canonical scalar product?. Write for the complexification of its Clifford algebra.
There exists a unique complex representation
of the algebra of smallest dimension
The Spin group of a quadratic vector space, def. 1, is the subgroup of the group of units in the Clifford algebra
on those elements which are even number multiples of elements with .
Specifically, “the” Spin group is
A spin representation is a linear representation of the spin group, def. 3.
By definition the spin group sits in a short exact sequence of groups
Relation to Whitehead tower of orthogonal group
The spin group is one element in the Whitehead tower of , which starts out like
The homotopy groups of are for and for sufficiently large
By co-killing these groups step by step one gets
Via the J-homomorphism this is related to the stable homotopy groups of spheres:
|Whitehead tower of orthogonal group||orientation||spin||string||fivebrane||ninebrane|
|homotopy groups of stable orthogonal group||0||0||0||0||0||0||0||0|
|stable homotopy groups of spheres||0||0||0|
|image of J-homomorphism||0||0||0||0||0||0||0||0||0|
In low dimensions the spin group happens to be isomorphic (“sporadic isomorphisms”) to various other classical group (among them the general linear group for the real numbers , the complex numbers and the quaternions , the orthogonal group , the unitary group and the symplectic group ).
See for instance (Garrett 13). See also division algebra and supersymmetry.
Beyond these dimensions there are still some interesting identifications, but the situation becomes much more involved.
|Lorentzian spacetime dimension||spin group||normed division algebra||brane scan entry|
| the real numbers|
| the complex numbers|
| the quaternions||little string|
| the octonions||heterotic/type II string|
See spin geometry
The name arises due to the requirement that the structure group of the tangent bundle of spacetime lifts to 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
fivebrane group string group spin group special orthogonal group orthogonal group.
Another extension of is the spin^c group.
|group||symbol||universal cover||symbol||higher cover||symbol|
|orthogonal group||Pin group||Tring group|
|special orthogonal group||Spin group||String group|
|anti de Sitter group|
|Poincaré group||Poincaré spin group|
|super Poincaré group|
A standard textbook reference is
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 Bundles and Symplectic Cobordism, Publ. RIMS, Kyoto Univ. 28 (1992), 39-55 (pdf)