spin geometry, string geometry, fivebrane geometry …
rotation groups in low dimensions:
see also
Classical groups
Finite groups
Group schemes
Topological groups
Lie groups
Super-Lie groups
Higher groups
Cohomology and Extensions
Related concepts
∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
-Lie groupoids
-Lie groups
-Lie algebroids
-Lie algebras
What are called half-spin groups or semi-spin groups (McInnes 99a, McInnes 99b) are quotient groups of spin groups by a non-standard Z/2-subgroup:
Generally, every spin group is, essentially by definition, a -group extension of the corresponding special orthogonal group, so that the quotient group by the resulting canonical subgroup inclusion recovers SO(n)
But in the special case that the dimension is a positive multiple of 4 distinct from 8 (i.e. ), there is another -conjugacy class of subgroups , which is distinct from the canonical , and hence yields a quotient group
which is distinct from (i.e. not isomorphic to) SO(n).
This is called the semi-spin group or half-spin group in that dimension.
The semi-spin group in dimension 4 is just the direct product group of SU(2) with SO(3):
While also for Spin(8) it is the case that the center contains two copies of Z/2, , in this case the existence of triality automorphisms actually makes these two copies behave identically, so that here the would-be semi-spin groups happens to coincide with SO(8) after all:
(e.g. McInnes 99a, p. 9)
The subgroup of the exceptional Lie group E8 which corresponds to the Lie algebra-inclusion is the semi-spin group SemiSpin(16):
On the other hand, the special orthogonal group is not a subgroup of (e.g. McInnes 99a, p. 11).
In heterotic string theory with gauge group the direct product group it is typically this subgroup which is considered (but typically denoted , see also Distler-Sharpe 10, Sec. 1).
In heterotic string theory precisely two (isomorphism classes of) gauge groups are consistent (give quantum anomaly cancellation): one is the direct product group of the exceptional Lie group E8 with itself, the other is in fact the semi-spin group SemiSpin(32) (see McInnes 99a, p. 5).
Beware that the string theory literature often writes this as , which is at best ambiguous and misleading, or even as , which is wrong. Of course this follows the general tradition in the physics literature to write identifications of Lie groups that are really only identifications of their Lie algebras, see also “SO(10)-GUT theory”.
rotation groups in low dimensions:
see also
Brett McInnes, The Semispin Groups in String Theory, J. Math. Phys. 40 (1999) 4699-4712 [arXiv:hep-th/9906059, doi:10.1063/1.532999]
Brett McInnes, Gauge Spinors and String Duality, Nucl. Phys. B577:439-460, 2000 (arXiv:hep-th/9910100)
Mboyo Esole, Monica Jinwoo Kang, Flopping and Slicing: and -models (arXiv:1802.04802)
Last revised on October 29, 2023 at 16:26:26. See the history of this page for a list of all contributions to it.