nLab SO(10)

Idea

$SO(10)$ is the special orthogonal group in dimension 10. $Spin(10)$ is the spin group in dimension 10: this is the universal cover of $SO(10)$, which is its only connected double cover. In the classification of simple Lie groups, either of these groups falls under the entry D5.

$Spin(10)$ the gauge group of an important grand unified theory: there is an inclusion $SU(5) \to Spin(10)$ and a 2-1 map $Spin(4) \times Spin(6) \to Spin(10)$, and the pullback of these is the Standard Model gauge group $S(U(2) \times U(3))$ (BaezHuerta10).

In this grand unified theory, all the right-handed fermions and antifermions in one generation form of the Standard Model including a right-handed neutrino, lie in a single representation of $Spin(10)$, called the right-handed Weyl spinor representation. This is sometimes denoted by $\mathbf{16}$ because it is 16-dimensional. Similarly, the left-handed fermions and antifermions lie in the left-handed Weyl spinor representation, which is the dual $\mathbf{16}\ast$.

These representations can be constructed explicitly in a number of ways. For example, they are commonly described as the even and odd subspaces of $\Lambda \mathbb{C}^5$. A less commonly used octonionic description is below.

Octonionic construction of representation

The group $Spin(10)$ has two 16-dimensional faithful irreducible complex representations called its left-handed and right-handed Weyl spinor representation. (Bryant 20) constructs one of these using octonions as follows.

For $(r,x) \in \mathbb{R} \oplus \mathbb{O}$ define a linear map $m_{r,x} \colon \mathbb{C} \otimes \mathbb{O}^2 \to \mathbb{C} \otimes \mathbb{O}^2$ as follows:

$$m_{r,x} = i \left( \begin{array}{cc} r & C R_x \\ -C L_x & -r \end{array} \right)$$

where $r$ here denotes multiplication by the real number $r$ and $L_x, R_x, C \colon \mathbb{O} \to \mathbb{O}$ are left multiplication by the octonion $x$, right multiplication by $x$ and octonionic conjugation, respectively. Since $m_{r,x}^2 = - r^2 -{|x|}^2$ this map induces an action of the Clifford algebra $Cliff(\mathbb{R} \oplus \mathbb{O})$ on $\mathbb{C} \otimes \mathbb{O}^2$. (To be precise, this is Clifford algebra generated by 9 anticommuting square roots of minus 1). One can show that this action gives an isomorphism of complex associative algebras

$$Cliff(\mathbb{R} \oplus \mathbb{O}) \cong End(\mathbb{C} \otimes \mathbb{O}^2)$$

where $End(\mathbb{C} \otimes \mathbb{O}^2)$, the algebra of all complex-linear transformations of $\mathbb{C} \otimes \mathbb{O}^2$, is isomorphic to the algebra of $16 \times 16$ complex matrices.

The group $Spin(n+1)$ is contained in the even part of the Clifford algebra on $n+1$ anticommuting square roots of $-1$, which is isomorphic to the Clifford algebra on $n$ anticommuting square roots of $-1$. Thus, $Spin(10)$ can be seen as a subgroup of $End(\mathbb{C} \otimes \mathbb{O}^2)$. Bryant shows that in fact $Spin(10)$ is isomorphic to the group of linear transformations generated by elements

$$\left( \begin{array}{cc} r & C R_x \\ -C L_x & r \end{array} \right)$$

where $r \in \mathbb{R}$ and $x \in \mathbb{O}$ have $r^2 + {|x|}^2 = 1$, together with elements

$$\left( \begin{array}{cc} \alpha & 0 \\ 0 & \alpha^{-1} \end{array} \right)$$

where $\alpha \in \mathbb{C}$ has $|\alpha| = 1$.

This description of $Spin(10)$ makes manifest its 16-dimensional right-handed Weyl spinor representation, which is a faithful complex representation. The left-handed Weyl spinor representation is the dual of this, which is not isomorphic.

Related concepts

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

General

• Robert Bryant, Notes on spinors in low dimension. (arXiv:2011.05568)

As a grand unified gauge group

Discussion as a gauge group in grand unified theory (see there):

• Stefano Bertolini, Luca Di Luzio, Michal Malinský, Intermediate mass scales in the non-supersymmetric SO(10) grand unification: a reappraisal, Phys. Rev. D80:015013, 2009 (arXiv:0903.4049)

review:

• Howard Georgi, §24 in: Lie Algebras In Particle Physics, Westview Press (1999), CRC Press (2019) [doi:10.1201/9780429499210]

• Michal Malinský, 35 years of GUTs - where do we stand?, 2009 (pdf)

• John Baez and John Huerta, The algebra of grand unified theories, Bull. Amer. Math. Soc. 47 (2010), 483-552. (arxiv:0904.1556)

for non-superymmetric models:

• L. Lavoura and Lincoln Wolfenstein, Resuscitation of minimal $SO(10)$ grand unification, Phys. Rev. D 48, 264 (doi:10.1103/PhysRevD.48.264)

• Guido Altarelli, Davide Meloni, A non Supersymmetric SO(10) Grand Unified Model for All the Physics below $M_{GUT}$ (arXiv:1305.1001)

• Alexander Dueck, Werner Rodejohann, Fits to $SO(10)$ Grand Unified Models (arXiv:1306.4468)

• Chee Sheng Fong, Davide Meloni, Aurora Meroni, Enrico Nardi, Leptogenesis in $SO(10)$ (arXiv:1412.4776)

  (in view of leptogenesis)

• Tommy Ohlsson, Marcus Pernow, Fits to Non-Supersymmetric SO(10) Models with Type I and II Seesaw Mechanisms Using Renormalization Group Evolution (arXiv:1903.08241)

• Mainak Chakraborty, M.K. Parida, Biswonath Sahoo, Triplet Leptogenesis, Type-II Seesaw Dominance, Intrinsic Dark Matter, Vacuum Stability and Proton Decay in Minimal SO(10) Breakings (arXiv:1906.05601)

  Results indicating non-SUSY $SO(10)$ as self sufficient theory for neutrino masses, baryon asymmetry, dark matter, vacuum stability of SM scalar potential, origin of three gauge forces, and observed proton stability.

• Nobuchika Okada, Digesh Raut, Qaisar Shafi, Inflation, Proton Decay, and Higgs-Portal Dark Matter in $SO(10) \times U(1)_\psi$ (arXiv:1906.06869)

for supersymmetric models:

• Archana Anandakrishnan, B. Charles Bryant, Stuart Raby, LHC Phenomenology of $SO(10)$ Models with Yukawa Unification II (arXiv:1404.5628)

• Ila Garg, New minimal supersymmetric $SO(10)$ GUT phenomenology and its cosmological implications (arXiv:1506.05204)

On symmetry breaking in Spin(10)-grand unified theory to the exact standard model gauge group, via real normed division algebra (octonions, quaternions):

• Kirill Krasnov, Geometry of $Spin(10)$ Symmetry Breaking [arXiv:2209.05088]

• Nichol Furey, M. J. Hughes, Division algebraic symmetry breaking [arXiv:2210.10126]