# Spin(11,3), particles and octonions

### APA

Krasnov, K. (2021). Spin(11,3), particles and octonions. Perimeter Institute. https://pirsa.org/21040006

### MLA

Krasnov, Kirill. Spin(11,3), particles and octonions. Perimeter Institute, Apr. 19, 2021, https://pirsa.org/21040006

### BibTex

@misc{ pirsa_PIRSA:21040006, doi = {10.48660/21040006}, url = {https://pirsa.org/21040006}, author = {Krasnov, Kirill}, keywords = {Mathematical physics, Particle Physics, Quantum Fields and Strings}, language = {en}, title = {Spin(11,3), particles and octonions}, publisher = {Perimeter Institute}, year = {2021}, month = {apr}, note = {PIRSA:21040006 see, \url{https://pirsa.org}} }

University of Nottingham

**Collection**

Talk Type

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.