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OCTONIONS AND THE STANDARD MODEL
from
Monday, February 8, 2021 (12:00 AM)
to
Monday, May 17, 2021 (1:00 PM)
Monday, February 8, 2021
12:00 PM
Finite quantum geometry, octonions and the theory of fundamental particles.

Michel DuboisViolette
(
CNRS, Universite ParisSaclay
)
Finite quantum geometry, octonions and the theory of fundamental particles.
Michel DuboisViolette
(
CNRS, Universite ParisSaclay
)
12:00 PM  1:00 PM
We will describe an approach to the theory of fundamental particlesbased on finitedimensional quantum algebras of observables. We will explain why the unimodularity of the color group suggests an interpretation of the quarklepton symmetry which involves the octonions and leads to the quantum spaces underlying the Jordan algebras of octonionic hermitian 2 × 2 and 3 × 3 matrices as internal geometry for fundamental particles. In the course of this talk, we will remind shortly why the finitedimensional algebras of observables are the finitedimensional euclidean Jordan agebras and we will describe their classifications. We will also explain our differential calculus on Jordan algebras and the theory of connections on Jordan modules. It is pointed out that the above theory of connections implies potentially a lot of scalar particles.
Tuesday, February 9, 2021
Wednesday, February 10, 2021
Thursday, February 11, 2021
Friday, February 12, 2021
Saturday, February 13, 2021
Sunday, February 14, 2021
Monday, February 15, 2021
12:00 PM
Supersymmetry and RCHO revisited

Paul Townsend
(
University of Cambridge
)
Supersymmetry and RCHO revisited
Paul Townsend
(
University of Cambridge
)
12:00 PM  1:00 PM
"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 superYangMills theories. The second half will review a result from 1993 that connects, via a twistortype transform, the superfield equations of superMaxwell theory in D=3,4,6,10 to a Kchirality constraint on a Kvalued worldline superfield of N=1,2,4,8 worldline supersymmetry. This provide an explicit connection of octonions to the freefield D=10 superMaxwell theory."
Tuesday, February 16, 2021
Wednesday, February 17, 2021
Thursday, February 18, 2021
Friday, February 19, 2021
Saturday, February 20, 2021
Sunday, February 21, 2021
Monday, February 22, 2021
12:00 PM
Spin (8,9,10), Octonions and the Standard Model

Kirill Krasnov
(
University of Nottingham
)
Spin (8,9,10), Octonions and the Standard Model
Kirill Krasnov
(
University of Nottingham
)
12:00 PM  1:00 PM
"I will start by explaining how the (Weyl) spinor representations of the pseudoorthogonal group Spin(2r+s,s) are the spaces of even and odd polyforms on Cr x Rs. Then, the triality identifies the MajoranaWeyl 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. "
Tuesday, February 23, 2021
Wednesday, February 24, 2021
Thursday, February 25, 2021
Friday, February 26, 2021
Saturday, February 27, 2021
Sunday, February 28, 2021
Monday, March 1, 2021
12:00 PM
Gravity as the square of gauge theory

Leron Borsten
(
Imperial College London
)
Gravity as the square of gauge theory
Leron Borsten
(
Imperial College London
)
12:00 PM  1:00 PM
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 EinsteinHilbert gravity (coupled to a KalbRamond 2form 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 nontrivial explicit examples. I will then describe how one can take this `gravity = gauge x gauge' amplitude paradigm `offshell’ as a product of spacetime fields: the YangMills BRSTLagrangian itself double copies into perturbatively quantised EinsteinHilbert gravity coupled to a KalbRamond 2form 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 wellknown, others completely new) that will be the subject of Mia Hughes's talk in the following week.
Tuesday, March 2, 2021
Wednesday, March 3, 2021
Thursday, March 4, 2021
Friday, March 5, 2021
Saturday, March 6, 2021
Sunday, March 7, 2021
Monday, March 8, 2021
12:00 PM
A Magic Pyramid of Supergravity Theories from YangMills Squared

Mia Hughes
(
Imperial College London
)
A Magic Pyramid of Supergravity Theories from YangMills Squared
Mia Hughes
(
Imperial College London
)
12:00 PM  1:00 PM
"I will begin by reviewing the unified description of pure Super YangMills (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 YangMills". The idea in its most basic form is that a symmetric tensor (graviton) can be built from the symmetric tensor product of two vectors (YangMills fields), an idea which can be extended to obtain entire supergravity multiplets from tensor products of SYM multiplets. Having established a divisionalgebraic description of Super YangMills 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 YangMills". The global symmetries of these 16 divisionalgebraic SYMsquared supergravity theories are precisely those belonging to the 4 x 4 FreudenthalRosenfeldTits "magic square" of Lie algebras! Furthermore, the scalar fields in these supergravity theories describe nonlinear 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/Mtheory and supergravity."
Tuesday, March 9, 2021
Wednesday, March 10, 2021
Thursday, March 11, 2021
Friday, March 12, 2021
Saturday, March 13, 2021
Sunday, March 14, 2021
Monday, March 15, 2021
12:00 PM
Division algebraic symmetry breaking

Mia Hughes
(
Imperial College London
)
Nicohl Furey
(
Humboldt University of Berlin
)
Division algebraic symmetry breaking
Mia Hughes
(
Imperial College London
)
Nicohl Furey
(
Humboldt University of Berlin
)
12:00 PM  1:00 PM
Can the 32Cdimensional 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) > PatiSalam > LeftRight symmetric > Standard model + BL (both pre and postHiggsmechanism). These complex structures derive from the octonions, then from the quaternions, then from the complex numbers. Finally, we describe a leftright symmetric Higgs system which exhibits, we believe for the first time, an explicit demonstration of quaternionic triality.
Tuesday, March 16, 2021
Wednesday, March 17, 2021
Thursday, March 18, 2021
Friday, March 19, 2021
Saturday, March 20, 2021
Sunday, March 21, 2021
Monday, March 22, 2021
12:00 PM
Clifford algebra of the Standard Model

Ivan Todorov
(
Bulgarian Academy of Sciences, Institute for Nuclear Research
)
Clifford algebra of the Standard Model
Ivan Todorov
(
Bulgarian Academy of Sciences, Institute for Nuclear Research
)
12:00 PM  1:00 PM
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 quarklepton splitting) is the PatiSalam 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 righthanded (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 spacetime) 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 Wboson masses in terms of the cosine of the theoretical Weinberg angle. The talk is based on the paper arXiv:2010.15621v3
Tuesday, March 23, 2021
Wednesday, March 24, 2021
Thursday, March 25, 2021
Friday, March 26, 2021
Saturday, March 27, 2021
Sunday, March 28, 2021
Monday, March 29, 2021
12:00 PM
The Higher Algebra of Supersymmetry

John Huerta
(
University of Lisbon
)
The Higher Algebra of Supersymmetry
John Huerta
(
University of Lisbon
)
12:00 PM  1:00 PM
We have already met the octonionic Fierz identity satisfied by spinors in 10dimensional spacetime. This identity makes superYangMills "super" and allows the GreenSchwarz string to be kappa symmetric. But it is also the defining equation of a "higher" algebraic structure: an Linfinity algebra extending the supersymmetry algebra. We introduce this Linfinity 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 FiorenzaSatiSchreiber.
Tuesday, March 30, 2021
Wednesday, March 31, 2021
Thursday, April 1, 2021
Friday, April 2, 2021
Saturday, April 3, 2021
Sunday, April 4, 2021
Monday, April 5, 2021
12:00 PM
Can We Understand the Standard Model?

John Baez
(
University of California, Riverside
)
Can We Understand the Standard Model?
John Baez
(
University of California, Riverside
)
12:00 PM  1:00 PM
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.
Tuesday, April 6, 2021
Wednesday, April 7, 2021
Thursday, April 8, 2021
Friday, April 9, 2021
Saturday, April 10, 2021
Sunday, April 11, 2021
Monday, April 12, 2021
12:00 PM
Can We Understand the Standard Model Using Octonions?

John Baez
(
University of California, Riverside
)
Can We Understand the Standard Model Using Octonions?
John Baez
(
University of California, Riverside
)
12:00 PM  1:00 PM
DuboisViolette and Todorov have shown that the Standard Model gauge group can be constructed using the exceptional Jordan algebra, consisting of 3×3 selfadjoint 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.
Tuesday, April 13, 2021
Wednesday, April 14, 2021
Thursday, April 15, 2021
Friday, April 16, 2021
Saturday, April 17, 2021
Sunday, April 18, 2021
Monday, April 19, 2021
12:00 PM
Spin(11,3), particles and octonions

Kirill Krasnov
(
University of Nottingham
)
Spin(11,3), particles and octonions
Kirill Krasnov
(
University of Nottingham
)
12:00 PM  1:00 PM
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 64dimensional semispinor representation S of the group Spin(11,3). I will describe an octonionic model for Spin(11,3) in which the semispinor representation gets identified with S=OxO', where O,O' are the usual and split octonions respectively. It is then wellknown 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 antiparticles. The splitting of S into eigenspaces of J' corresponds to splitting of Lorentz Dirac spinors into two different chiralities.
Tuesday, April 20, 2021
Wednesday, April 21, 2021
Thursday, April 22, 2021
Friday, April 23, 2021
Saturday, April 24, 2021
Sunday, April 25, 2021
Monday, April 26, 2021
12:00 PM
Potentially realistic SO(10) GUTs and their phenomenology

Michal Malinsky
(
Charles University in Prague
)
Potentially realistic SO(10) GUTs and their phenomenology
Michal Malinsky
(
Charles University in Prague
)
12:00 PM  1:00 PM
I will take a look at the SO(10) grand unified theories from the perspective of the typical phenomenology constraints imposed on their structure. The current status of the minimal potentially realistic models will be briefly commented upon.
Tuesday, April 27, 2021
Wednesday, April 28, 2021
Thursday, April 29, 2021
Friday, April 30, 2021
Saturday, May 1, 2021
Sunday, May 2, 2021
Monday, May 3, 2021
12:00 PM
Jordan algebras: from QM to 5D supergravity to … Standard Model?

Paul Townsend
(
University of Cambridge
)
Jordan algebras: from QM to 5D supergravity to … Standard Model?
Paul Townsend
(
University of Cambridge
)
12:00 PM  1:00 PM
This talk will be about two applications of Jordan algebras. The first, to quantum mechanics, follows on from the talk of John Baez. I will explain how time dependence makes use of the associator, and how this is related to the commutator in the standard density matrix formulation. The associator of a Jordan algebra also determines the curvature of a Riemannian metric on its positive cone, invariant under the symmetry group of the norm (mentioned in the talk of John Baez); the cone is foliated by hypersurfaces of constant norm. This geometry is relevant to a class of N=2 5D supergravity theories (from the early 1980s) which arise (in some cases, at least) from CalabiYau compactification of 11D supergravity. The 5D interactions are determined by the structure constants of a euclidean Jordan algebra with cubic norm. The exceptional JA of 3x3 octonionic matrices yields an ``exceptional’’ 5D supergravity which yields, on reduction to 4D, an ``exceptional’’ N=2 supergravity with many similarities to N=8 supergravity, such as a noncompact global E7 symmetry. However, it has a compact `composite’ E6 gauge invariance (in contrast to the SU(8) of N=8 supergravity). An old speculation is that nonperturbative effects break the N=2 supersymmetry and cause the E6 gauge potentials to become the dynamical fields of an E6 GUT. Potentially (albeit improbably) this provides a connection between Mtheory, the exceptional Jordan algebra, and the Standard Model.
Tuesday, May 4, 2021
Wednesday, May 5, 2021
Thursday, May 6, 2021
Friday, May 7, 2021
Saturday, May 8, 2021
Sunday, May 9, 2021
Monday, May 10, 2021
12:00 PM
Discussion Session

David Jackson
(
Independent
)
Shane Farnsworth
(
Albert Einstein Institute
)
Tejinder Singh
(
Tata Institute for Fundamental Research
)
Jochen Szangolies
Nicohl Furey
(
Humboldt University of Berlin
)
Discussion Session
David Jackson
(
Independent
)
Shane Farnsworth
(
Albert Einstein Institute
)
Tejinder Singh
(
Tata Institute for Fundamental Research
)
Jochen Szangolies
Nicohl Furey
(
Humboldt University of Berlin
)
12:00 PM  1:00 PM
Tuesday, May 11, 2021
Wednesday, May 12, 2021
Thursday, May 13, 2021
Friday, May 14, 2021
Saturday, May 15, 2021
Sunday, May 16, 2021
Monday, May 17, 2021
12:00 PM
The standard model, left/right symmetry, and the "magic square"

Latham Boyle
(
Perimeter Institute
)
The standard model, left/right symmetry, and the "magic square"
Latham Boyle
(
Perimeter Institute
)
12:00 PM  1:00 PM
Recently, an intriguing connection between the exceptional Jordan algebra h_3(O) and the standard model of particle physics was noticed by DuboisViolette and Todorov (with further interpretation by Baez). How do the standard model fermions fit into this story? I will explain how they may be neatly incorporated by complexifying h_3(O) or, relatedly, by passing from RxO to CxO in the socalled "magic square" of normed division algebras. This, in turn, suggests that the standard model, with gauge group SU(3)xSU(2)xU(1), is embedded in a left/rightsymmetric theory, with gauge group SU(3)xSU(2)xSU(2)xU(1). This theory is not only experimentally viable, but offers some explanatory advantages over the standard model (including an elegant solution to the standard model's "strong CP problem"). Ramond's formulation of the magic square, based on triality, provides further insights, and possible hints about where to go next.