nLab SemiSpin(32)

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 $E_8 \times E_8$ 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 Spin(32)$/\mathbb{Z}_2$, 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”.

Related entries

• spin group

• Spin(n1).Spin(n2)

• heterotic string theory, Green-Schwarz anomaly cancellation

rotation groups in low dimensions:

Dynkin label	sp. orth. group	spin group	pin group	semi-spin group
	SO(2)	Spin(2)	Pin(2)
B1	SO(3)	Spin(3)	Pin(3)
D2	SO(4)	Spin(4)	Pin(4)
B2	SO(5)	Spin(5)	Pin(5)
D3	SO(6)	Spin(6)
B3	SO(7)	Spin(7)
D4	SO(8)	Spin(8)		SO(8)
B4	SO(9)	Spin(9)
D5	SO(10)	Spin(10)
B5	SO(11)	Spin(11)
D6	SO(12)	Spin(12)
	$\vdots$	$\vdots$
D8	SO(16)	Spin(16)		SemiSpin(16)
	$\vdots$	$\vdots$
D16	SO(32)	Spin(32)		SemiSpin(32)

see also

• Spin(5).Spin(3)

• finite rotation groups

• ADE-classification

References

• Brett McInnes, The Semispin Groups in String Theory, J. Math. Phys. 40:4699-4712, 1999 (arXiv:hep-th/9906059)

• Brett McInnes, Gauge Spinors and String Duality, Nucl. Phys. B577:439-460, 2000 (arXiv:hep-th/9910100)