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SemiSpin(32)

Contents

Context

Higher spin geometry

Group Theory

Higher Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

The semi-spin group in dimension 32.

Properties

In heterotic string theory

In heterotic string theory precisely two (isomorphism classes of) gauge groups are consistent (give quantum anomaly cancellation): one is the direct product group E 8×E 8E_8 \times E_8 of the exceptional Lie group E8 with itself, the other is in fact the semi-spin group SemiSpin(32)SemiSpin(32) (see McInnes 99a, p. 5).

Beware that the string theory literature often writes this as Spin(32)/ 2/\mathbb{Z}_2, which is at best ambiguous and misleading, or even as SO(32), 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:

sp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
SO(3)Spin(3)
SO(4)Spin(4)
SO(5)Spin(5)Pin(5)
Spin(6)
Spin(7)
SO(8)Spin(8)SO(8)
SO(9)Spin(9)
\vdots\vdots
SO(16)Spin(16)SemiSpin(16)
SO(32)Spin(32)SemiSpin(32)

see also

References

Last revised on May 15, 2019 at 11:41:38. See the history of this page for a list of all contributions to it.