Talk 10: Spin(11,3), particles and octonions (Kirill Krasnov)

Workshop: Octonions and the Standard Model (Perimeter Institute, 2021)

Speaker: Kirill Krasnov

Title: Spin(11,3), particles and octonions

Abstract: 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.

Workshop overview: Over the years, various researchers have suggested connections between the octonions and the standard model of particle physics. The past few years, in particular, were marked by an upsurge of activity on this subject, stimulated by the recent observation that the standard model gauge group and fermion representation can be elegantly characterized in terms of the octonions. This workshop -- the first ever on this topic -- was intended to bring this new community together in an attempt to better understand these ideas, establish a common language, and stimulate further progress. (Organized by Latham Boyle and Kirill Krasnov.)

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