spin geometry, string geometry, fivebrane geometry …
rotation groups in low dimensions:
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Lie groups
Super-Lie groups
Higher groups
Cohomology and Extensions
Related concepts
∞-Lie theory (higher geometry)
Background
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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 .
For special cases in low dimensions see at: Spin(2), Spin(3), Spin(4), Spin(5), Spin(6), Spin(7), Spin(8)
A quadratic vector space is a vector space over finite dimension over a field of characteristic 0, and equipped with a symmetric bilinear form .
Conventions as in (Varadarajan 04, section 5.3).
We write for the corresponding quadratic form.
The Clifford algebra of a quadratic vector space, def. , is the associative algebra over which is the quotient
of the tensor algebra of by the ideal generated by the elements .
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.
The Pin group of a quadratic vector space, def. , is the subgroup of the group of units in the Clifford algebra
on those elements which are multiples of elements with .
The Spin group is the further subgroup of 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. .
By definition the spin group sits in a short exact sequence of groups
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:
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 | 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 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 denotes the quaternionic unitary group in quaternionic dimension .
We have
in Euclidean signature
Spin(2) (SO(2), the circle group, see also at real Hopf fibration)
the projection corresponds to , see also at Theta characteristic
Spin(3) (the special unitary group SU(2)
the inclusion corresponds to the canonical (see e.g. Gorbounov-Ray 92)
this is given by identifying with the quaternions and with the group of unit quternions. Then left and right quaternion multiplication gives a homomorphism
which is a double cover and hence exhibits the isomorphism.
In particular therefore the inclusion corresponds to the diagonal .
At the level of Lie algebras and the -eigenspaces of the Hodge star operator gives the direct sum decomposition
Spin(5) (an indirect consequence of triality, see e.g. Čadek-Vanžura 97)
in Lorentzian signature
– 2d special linear group of real numbers
– 2d special linear group of complex numbers
– 2d special linear group of quaternions
– 2d special linear group of octonions (see SL(2,O) for more on this would-be isomorphism)
in anti de Sitter signature
in mixed signature
Beyond these dimensions there are still some interesting identifications, but the situation becomes much more involved.
exceptional spinors and real normed division algebras
rotation groups in low dimensions:
see also
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.
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 Bundles and Symplectic Cobordism, Publ. RIMS, Kyoto Univ. 28 (1992), 39-55 (pdf, doi: 10.2977/prims/1195168855)
The exceptional isomorphism Spin(5) Sp(2) is discussed via triality in
Discussion of the cohomology of the classifying space 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)
Last revised on September 11, 2024 at 09:45:16. See the history of this page for a list of all contributions to it.