Octonions and the Standard Model

Various links between supersymmetry and the normed division algebras R,C,H,O were found in the 1980s. This talk will focus on the link between K=R,C,H,0 and supersymmetric field theories in a Minkowski spacetime of dimension D=3,4,6,10. The first half will survey the history starting with a 1944/5 paper of Dirac and heading towards the links found in 1986/7 between R,C,H,O and super-Yang-Mills theories. The second half will review a result from 1993 that connects, via a twistor-type transform, the superfield equations of super-Maxwell theory in D=3,4,6,10 to a K-chirality constraint on a K-valued worldline superfield of N=1,2,4,8 worldline supersymmetry. This provide an explicit connection of octonions to the free-field D=10 super-Maxwell theory.

"I will start by explaining how the (Weyl) spinor representations of the pseudo-orthogonal group Spin(2r+s,s) are the spaces of even and odd polyforms on Cr x Rs. Then, the triality identifies the Majorana-Weyl spinors of Spin(8) with octonions. Combining the two constructions one finds that the groups Spin(8+s,s) all have an octonionic description, with Weyl spinors of this group being a copy of O^(2^s). This also gives an octonionic description of the groups that can be embedded into Spin(8+s,s). Applying this construction to Spin(10,2) gives an octonionic description of Spin(10). The latter arises as the subgroup of Spin(10,2) that commutes with a certain complex structure on the space of its Weyl spinors O4. This gives a description of Weyl spinors of Spin(10) as O2_C, and an explicit description of the Lie algebra of Spin(10) as that of 2x2 matrices (of a special type) with complex (and octonionic) entries. It is well known from the SO(10) GUT that fermions of one generation of the SM can be described as components of a single Weyl spinor of Spin(10). Combining this with the previous construction one gets an explanation of why it is natural to identify elementary particles with components of two copies of complexified octonions. I explicitly describe the dictionary that provides this identification. I also describe how a choice of a unit imaginary octonion induces some natural complex structures on the space of Spin(10) Weyl spinors. For one of these complex structures, its commutant in Spin(10) is SU(2)_L x SU(2)_R x SU(3) x U(1). One thus gets a surprisingly large number of structures seen in the SM from very little input - a choice of a unit imaginary octonion. "

Can gravity, in certain regards, be the `product' of two gauge theories, such as those appearing in the Standard Model? I will begin by reviewing the Bern—Carrasco—Johansson colour—kinematics duality conjecture, which implies that one can write the scattering amplitudes of Einstein-Hilbert gravity (coupled to a Kalb-Ramond 2-form and dilaton scalar) as the double copy of Yang—Mills amplitudes. Although the colour—kinematics duality, and therefore the double copy, was quickly established at the tree level, it remains a longstanding open problem at the loop level, despite highly non-trivial explicit examples. I will then describe how one can take this `gravity = gauge x gauge' amplitude paradigm `off-shell’ as a product of spacetime fields: the Yang-Mills BRST-Lagrangian itself double copies into perturbatively quantised Einstein-Hilbert gravity coupled to a Kalb-Ramond 2-form and dilaton, establishing the validity of the double copy to all orders, tree and loop. I will end by briefly discussing the homotopy algebras underpinning this result and the inclusion of supersymmetry, which reveals fascinating octonionic structures (some very well-known, others completely new) that will be the subject of Mia Hughes's talk in the following week.

"I will begin by reviewing the unified description of pure Super Yang-Mills (SYM) Theory (consisting of just a gauge field and gaugino) in dimensions 3, 4, 6, and 10 over the four normed division algebras R, C, H, and O. Dimensionally reducing these initial theories into dimensions 3, 4, 5, 6, 7, 8, 9, 10 gives a plethora of SYM theories written over the division algebras, with a single master Lagrangian to rule them all. In particular, in D = 3 spacetime dimensions, the SYM theories with N = 1, 2, 4, and 8 supersymmetries enjoy a unified description over R, C, H, and O, respectively. In each spacetime dimension, maximally supersymmetric theories are written over the octonions. In apparently completely different developments, a popular thread in attempts to understand the quantum theory of gravity is the idea of "gravity as the square of Yang-Mills". The idea in its most basic form is that a symmetric tensor (graviton) can be built from the symmetric tensor product of two vectors (Yang-Mills fields), an idea which can be extended to obtain entire supergravity multiplets from tensor products of SYM multiplets. Having established a division-algebraic description of Super Yang-Mills theories, I will then demonstrate how tensoring these multiplets together results in supergravity theories valued over tensor products of division algebras. In D = 3, there are 4 SYM theories (N = 1, 2, 4, 8 over R, C, H, O) and so there are 4 x 4 = 16 possible supergravity theories to obtain by "squaring Yang-Mills". The global symmetries of these 16 division-algebraic SYM-squared supergravity theories are precisely those belonging to the 4 x 4 Freudenthal-Rosenfeld-Tits "magic square" of Lie algebras! Furthermore, the scalar fields in these supergravity theories describe non-linear sigma models, whose target space manifolds are division algebraic projective planes! Performing the same tensoring of SYM theories in spacetime dimensions D > 3 results in a whole "magic pyramid" of supergravities, with the magic square at the base in D = 3 and Type II supergravity at the apex in D = 10. This construction gives an explicit octonionic explanation of many of the mysterious appearances of exceptional groups within string/M-theory and supergravity."

Can the 32C-dimensional algebra R(x)C(x)H(x)O offer anything new for particle physics? Indeed it can. Here we identify a sequence of complex structures within R(x)C(x)H(x)O which sets in motion a cascade of breaking symmetries: Spin(10) -> Pati-Salam -> Left-Right symmetric -> Standard model + B-L (both pre- and post-Higgs-mechanism). These complex structures derive from the octonions, then from the quaternions, then from the complex numbers. Finally, we describe a left-right symmetric Higgs system which exhibits, we believe for the first time, an explicit demonstration of quaternionic triality.

We explore the Z2 graded product C`10 = C`4⊗ˆC`6 (introduced by Furey) as a finite quantum algebra of the Standard Model of particle physics. The gamma matrices generating C`10 are expressed in terms of left multiplication by the imaginary octonion units and the Pauli matrices. The subgroup of Spin(10) that fixes an imaginary unit (and thus allows to write O = C⊗C 3 expressing the quark-lepton splitting) is the Pati-Salam group GP S = Spin(4) × Spin(6)/Z2 ⊂ Spin(10). If we identify the preserved imaginary unit with the C`6 pseudoscalar ω6 = γ1...γ6, ω2 6 = −1 (cf. the talk of Furey and Hughes), then Pex = 1 2 (1 − iω6) will play the role of the projector on the extended particle subspace including the right-handed (sterile) neutrino. We express the generators of C`4 and C`6 in terms of fermionic oscillators aα, a∗ α, α = 1, 2 and bj , b∗ j , j = 1, 2, 3 describing flavour and colour, respectively. The internal space observable algebra (an analog of the algebra of real functions on space-time) is then defined as the Jordan subalgebra of hermitian elements of the complexified Clifford algebra C ⊗ C`10 that commute with the weak hypercharge 1 2 Y = 1 3 P3 j=1 b ∗ j bj − 1 2 P2 α=1 a ∗ αaα. We only distinguish particles from antiparticles if they have different eigenvalues of Y . Thus the sterile neutrino and antineutrino (with Y = 0) are allowed to mix into Majorana neutrinos. Restricting C`10 to the particle subspace which consists of leptons with Y < 0 and quarks with Y > 0 allows a natural definition of the Higgs field Φ, the scalar of Quillen’s superconnection, as an element of C`1 4, the odd part of the first factor in C`10. As an application we express the ratio mH mW of the Higgs and the W-boson masses in terms of the cosine of the theoretical Weinberg angle. The talk is based on the paper arXiv:2010.15621v3 a copy of which including minor corrections is attached.

We have already met the octonionic Fierz identity satisfied by spinors in 10-dimensional spacetime. This identity makes super-Yang-Mills "super" and allows the Green-Schwarz string to be kappa symmetric. But it is also the defining equation of a "higher" algebraic structure: an L-infinity algebra extending the supersymmetry algebra. We introduce this L-infinity algebra in octonionic language, and describe its cousins in various dimensions. We then survey various consequences of its existence, such as the brane bouquet of Fiorenza-Sati-Schreiber.

40 years trying to go beyond the Standard Model hasn't yet led to any clear success. As an alternative, we could try to understand why the Standard Model is the way it is. In this talk we review some lessons from grand unified theories and also from recent work using the octonions. The gauge group of the Standard Model and its representation on one generation of fermions arises naturally from a process that involves splitting 10d Euclidean space into 4+6 dimensions, but also from a process that involves splitting 10d Minkowski spacetime into 4d Minkowski space and 6 spacelike dimensions. We explain both these approaches, and how to reconcile them.

Dubois-Violette and Todorov have shown that the Standard Model gauge group can be constructed using the exceptional Jordan algebra, consisting of 3×3 self-adjoint matrices of octonions. After an introduction to the physics of Jordan algebras, we ponder the meaning of their construction. For example, it implies that the Standard Model gauge group consists of the symmetries of an octonionic qutrit that restrict to symmetries of an octonionic qubit and preserve all the structure arising from a choice of unit imaginary octonion. It also sheds light on why the Standard Model gauge group acts on 10d Euclidean space, or Minkowski spacetime, while preserving a 4+6 splitting.

The fermionic fields of one generation of the Standard Model, including the Lorentz spinor degrees of freedom, can be identified with components of a single real 64-dimensional semi-spinor representation S of the group Spin(11,3). I will describe an octonionic model for Spin(11,3) in which the semi-spinor representation gets identified with S=OxO', where O,O' are the usual and split octonions respectively. It is then well-known that choosing a unit imaginary octonion u in Im(O) equips O with a complex structure J. Similarly, choosing a unit imaginary split octonion u' in Im(O') equips O' with a complex structure J', except that there are now two inequivalent complex structures, one parametrised by a choice of a timelike and the other of a spacelike unit u'. In either case, the identification S=OxO' implies that there are two natural commuting complex structures J, J' on S. Our main new observation is that there is a choice of J,J' so that the subgroup of Spin(11,3) that commutes with both is the direct product SU(3)xU(1)xSU(2)_LxSU(2)_R x Spin(1,3) of the group of the left/right symmetric extension of the SM and Lorentz group. The splitting of S into eigenspaces of J corresponds to splitting into particles and anti-particles. The splitting of S into eigenspaces of J' corresponds to splitting of Lorentz Dirac spinors into two different chiralities.