semi-spin group in nLab

Contents

Contents

Idea

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

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

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

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

SemiSpin(4k)≔Spin(4k)/ ι s(ℤ/2) 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)≃SU(2)×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(Spin(8))≃ℤ 2×ℤ 2Z\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)/ ι sℤ 2≃SO(8)≃Spin(8)/ ιℤ 2 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 𝔰𝔬(16)↪𝔢 8\mathfrak{so}(16) \hookrightarrow \mathfrak{e}_8 is the semi-spin group SemiSpin(16):

SemiSpin(16)⊂E 8 SemiSpin(16) \;\subset\; E_8

On the other hand, the special orthogonal group SO(16)SO(16) is not a subgroup of E 8E_8 (e.g. McInnes 99a, p. 11).

In heterotic string theory with gauge group the direct product group E 8×E 8E_8 \times E_8 it is typically this subgroup Semispin(16)×SemiSpin(16)Semispin(16) \times SemiSpin(16) which is considered (but typically denoted Spin(16)/ℤ 2Spin(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×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) (see McInnes 99a, p. 5).

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

• spin group

• Spin(n1).Spin(n2)

• heterotic string theory, Green-Schwarz anomaly cancellation

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