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SUMMARY:Spin (8\,9\,10)\, Octonions and the Standard Model
DTSTART;VALUE=DATE-TIME:20210222T170000Z
DTEND;VALUE=DATE-TIME:20210222T180000Z
DTSTAMP;VALUE=DATE-TIME:20220518T225251Z
UID:indico-contribution-57@events.perimeterinstitute.ca
DESCRIPTION:Speakers: Kirill Krasnov (University of Nottingham)\n"I will s
tart by explaining how the (Weyl) spinor representations of the pseudo-ort
hogonal 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 th
is group being a copy of O^(2^s). This also gives an octonionic descriptio
n of the groups that can be embedded into Spin(8+s\,s).\n\nApplying this c
onstruction to Spin(10\,2) gives an octonionic description of Spin(10). Th
e latter arises as the subgroup of Spin(10\,2) that commutes with a certai
n complex structure on the space of its Weyl spinors O4. This gives a desc
ription 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.\n\nIt is well known from the SO(10
) GUT that fermions of one generation of the SM can be described as compon
ents 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 ele
mentary particles with components of two copies of complexified octonions.
I explicitly describe the dictionary that provides this identification.\n
\nI also describe how a choice of a unit imaginary octonion induces some n
atural complex structures on the space of Spin(10) Weyl spinors. For one o
f 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 octon
ion. "\n\nhttps://events.perimeterinstitute.ca/event/5/contributions/57/
LOCATION:
URL:https://events.perimeterinstitute.ca/event/5/contributions/57/
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