nLab semi-spin group

Idea

What are called half-spin groups or semi-spin groups (McInnes 99a, McInnes 99b) are quotient groups of spin groups $Spin(n)$ by a non-standard Z/2-subgroup:

Generally, every spin group $Spin(n)$ is, essentially by definition, a $\mathbb{Z}/2$-group extension of the corresponding special orthogonal group, so that the quotient group by the resulting canonical subgroup inclusion $\mathbb{Z}/2 \overset{\iota}{\hookrightarrow} Spin(n)$ recovers SO(n)

$$Spin(n)/_{\iota}(\mathbb{Z}/2) \simeq SO(n) \,.$$

But in the special case that the dimension $n = 4k$ is a positive multiple of 4 distinct from 8 (i.e. $k \in \mathbb{N}_{\gt 0}, k \neq 2$), there is another $\mathbb{Z}/2$-conjugacy class of subgroups $\mathbb{Z}/2 \overset{\iota_{s}}{\hookrightarrow} Spin(4k)$, which is distinct from the canonical $\iota$, and hence yields a quotient group

$$SemiSpin\big(4k \big) \;\coloneqq\; Spin\big(4k\big)/_{\iota_s} (\mathbb{Z}/2)$$

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.

Examples

SemiSpin(4)

The semi-spin group in dimension 4 is just the direct product group of SU(2) with SO(3):

$$SemiSpin(4) \;\simeq\; SU(2) \times SO(3)$$

SemiSpin(8)

While also for Spin(8) it is the case that the center contains two copies of Z/2, $Z\big( Spin(8)\big) \simeq \mathbb{Z}_2 \times \mathbb{Z}_2$, 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:

$$Spin(8)/_{\iota_s} \mathbb{Z}_2 \;\simeq\; SO(8) \;\simeq\; Spin(8)/_{\iota} \mathbb{Z}_2$$

(e.g. McInnes 99a, p. 9)

SemiSpin(16)

The subgroup of the exceptional Lie group E8 which corresponds to the Lie algebra-inclusion $\mathfrak{so}(16) \hookrightarrow \mathfrak{e}_8$ is the semi-spin group SemiSpin(16):

$$SemiSpin(16) \;\subset\; E_8$$

On the other hand, the special orthogonal group $SO(16)$ is not a subgroup of $E_8$ (e.g. McInnes 99a, p. 11).

In heterotic string theory with gauge group the direct product group $E_8 \times E_8$ it is typically this subgroup $Semispin(16) \times SemiSpin(16)$ which is considered (but typically denoted $Spin(16)/\mathbb{Z}_2$, see also Distler-Sharpe 10, Sec. 1).

SemiSpin(32)

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 (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: $SO(4)$ and $Spin(4)$-models (arXiv:1802.04802)