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SUMMARY:Finite quantum geometry\, octonions and the theory of fundamental
particles.
DTSTART;VALUE=DATE-TIME:20210208T170000Z
DTEND;VALUE=DATE-TIME:20210208T180000Z
DTSTAMP;VALUE=DATE-TIME:20210418T120726Z
UID:indico-contribution-5-55@events.perimeterinstitute.ca
DESCRIPTION:Speakers: Michel Dubois-Violette (CNRS\, Universite Paris-Sacl
ay)\nWe will describe an approach to the theory of fundamental particlesba
sed on finite-dimensional quantum algebras of observables. We will explain
why the unimodularity of the color group suggests an interpretation of th
e quarklepton symmetry which involves the octonions and leads to the quant
um spaces underlying the Jordan algebras of octonionic hermitian 2 × 2 an
d 3 × 3 matrices as internal geometry for fundamental particles. In the c
ourse of this talk\, we will remind shortly why the finite-dimensional alg
ebras of observables are the finite-dimensional euclidean Jordan agebras a
nd we will describe their classifications. We will also explain our differ
ential calculus on Jordan algebras and the theory of connections on Jordan
modules. It is pointed out that the above theory of connections implies p
otentially a lot of scalar particles.\n\nhttps://events.perimeterinstitute
.ca/event/5/contributions/55/
LOCATION:
URL:https://events.perimeterinstitute.ca/event/5/contributions/55/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Gravity as the square of gauge theory
DTSTART;VALUE=DATE-TIME:20210301T170000Z
DTEND;VALUE=DATE-TIME:20210301T180000Z
DTSTAMP;VALUE=DATE-TIME:20210418T120726Z
UID:indico-contribution-5-58@events.perimeterinstitute.ca
DESCRIPTION:Speakers: Leron Borsten (Imperial College London)\nCan gravity
\, in certain regards\, be the `product' of two gauge theories\, such as t
hose appearing in the Standard Model? I will begin by reviewing the Bern
—Carrasco—Johansson colour—kinematics duality conjecture\, which imp
lies that one can write the scattering amplitudes of Einstein-Hilbert grav
ity (coupled to a Kalb-Ramond 2-form and dilaton scalar) as the double cop
y of Yang—Mills amplitudes. Although the colour—kinematics duality\, a
nd 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. \n\nI will then describe how one can take
this `gravity = gauge x gauge' amplitude paradigm `off-shell’ as a prod
uct of spacetime fields: the Yang-Mills BRST-Lagrangian itself double copi
es into perturbatively quantised Einstein-Hilbert gravity coupled to a Kal
b-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 homot
opy algebras underpinning this result and the inclusion of supersymmetry\,
which reveals fascinating octonionic structures (some very well-known\, o
thers completely new) that will be the subject of Mia Hughes's talk in the
following week.\n\nhttps://events.perimeterinstitute.ca/event/5/contribut
ions/58/
LOCATION:
URL:https://events.perimeterinstitute.ca/event/5/contributions/58/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Supersymmetry and RCHO revisited
DTSTART;VALUE=DATE-TIME:20210215T170000Z
DTEND;VALUE=DATE-TIME:20210215T180000Z
DTSTAMP;VALUE=DATE-TIME:20210418T120726Z
UID:indico-contribution-5-56@events.perimeterinstitute.ca
DESCRIPTION:Speakers: Paul Townsend (University of Cambridge)\n"Various li
nks 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 resul
t 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 constr
aint on a K-valued worldline superfield of N=1\,2\,4\,8 worldline supersym
metry. This provide an explicit connection of octonions to the free-field
D=10 super-Maxwell theory."\n\nhttps://events.perimeterinstitute.ca/event/
5/contributions/56/
LOCATION:
URL:https://events.perimeterinstitute.ca/event/5/contributions/56/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Spin (8\,9\,10)\, Octonions and the Standard Model
DTSTART;VALUE=DATE-TIME:20210222T170000Z
DTEND;VALUE=DATE-TIME:20210222T180000Z
DTSTAMP;VALUE=DATE-TIME:20210418T120726Z
UID:indico-contribution-5-57@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/
END:VEVENT
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SUMMARY:A Magic Pyramid of Supergravity Theories from Yang-Mills Squared
DTSTART;VALUE=DATE-TIME:20210308T170000Z
DTEND;VALUE=DATE-TIME:20210308T180000Z
DTSTAMP;VALUE=DATE-TIME:20210418T120726Z
UID:indico-contribution-5-59@events.perimeterinstitute.ca
DESCRIPTION:Speakers: Mia Hughes (Imperial College London)\n"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. Dimension
ally reducing these initial theories into dimensions 3\, 4\, 5\, 6\, 7\, 8
\, 9\, 10 gives a plethora of SYM theories written over the division algeb
ras\, 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\, respe
ctively. In each spacetime dimension\, maximally supersymmetric theories a
re written over the octonions.\n\nIn apparently completely different devel
opments\, 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 fr
om the symmetric tensor product of two vectors (Yang-Mills fields)\, an id
ea which can be extended to obtain entire supergravity multiplets from ten
sor products of SYM multiplets. Having established a division-algebraic de
scription of Super Yang-Mills theories\, I will then demonstrate how tenso
ring these multiplets together results in supergravity theories valued ove
r tensor products of division algebras.\n\nIn D = 3\, there are 4 SYM theo
ries (N = 1\, 2\, 4\, 8 over R\, C\, H\, O) and so there are 4 x 4 = 16 po
ssible supergravity theories to obtain by "squaring Yang-Mills". The globa
l symmetries of these 16 division-algebraic SYM-squared supergravity theor
ies are precisely those belonging to the 4 x 4 Freudenthal-Rosenfeld-Tits
"magic square" of Lie algebras! Furthermore\, the scalar fields in these s
upergravity theories describe non-linear sigma models\, whose target space
manifolds are division algebraic projective planes! Performing the same t
ensoring 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 giv
es an explicit octonionic explanation of many of the mysterious appearance
s of exceptional groups within string/M-theory and supergravity."\n\nhttps
://events.perimeterinstitute.ca/event/5/contributions/59/
LOCATION:
URL:https://events.perimeterinstitute.ca/event/5/contributions/59/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Division algebraic symmetry breaking
DTSTART;VALUE=DATE-TIME:20210315T160000Z
DTEND;VALUE=DATE-TIME:20210315T170000Z
DTSTAMP;VALUE=DATE-TIME:20210418T120726Z
UID:indico-contribution-5-60@events.perimeterinstitute.ca
DESCRIPTION:Speakers: Cohl Furey (Humboldt University of Berlin)\, Mia Hug
hes (Imperial College London)\nCan the 32C-dimensional algebra R(x)C(x)H(x
)O offer anything new for particle physics? Indeed it can. Here we identif
y a sequence of complex structures within R(x)C(x)H(x)O which sets in moti
on a cascade of breaking symmetries: Spin(10) -> Pati-Salam -> Left-Right
symmetric -> Standard model + B-L (both pre- and post-Higgs-mechanism). Th
ese complex structures derive from the octonions\, then from the quaternio
ns\, then from the complex numbers. Finally\, we describe a left-right sy
mmetric Higgs system which exhibits\, we believe for the first time\, an e
xplicit demonstration of quaternionic triality.\n\nhttps://events.perimete
rinstitute.ca/event/5/contributions/60/
LOCATION:
URL:https://events.perimeterinstitute.ca/event/5/contributions/60/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Clifford algebra of the Standard Model
DTSTART;VALUE=DATE-TIME:20210322T160000Z
DTEND;VALUE=DATE-TIME:20210322T170000Z
DTSTAMP;VALUE=DATE-TIME:20210418T120726Z
UID:indico-contribution-5-62@events.perimeterinstitute.ca
DESCRIPTION:Speakers: Ivan Todorov (Bulgarian Academy of Sciences\, Instit
ute for Nuclear Research)\nWe explore the Z2 graded product C`10 = C`4⊗
ˆC`6 (introduced by Furey) as a finite quantum algebra of the Standard Mo
del of particle physics. The gamma matrices generating C`10 are expressed
in terms of left multiplication by the imaginary octonion units and the Pa
uli matrices. The subgroup of Spin(10) that fixes an imaginary unit (and t
hus 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 iden
tify the preserved imaginary unit with the C`6 pseudoscalar ω6 = γ1...γ
6\, 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 sub
space including the right-handed (sterile) neutrino. We express the genera
tors 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\, r
espectively. The internal space observable algebra (an analog of the algeb
ra of real functions on space-time) is then defined as the Jordan subalgeb
ra of hermitian elements of the complexified Clifford algebra C ⊗ C`10 t
hat 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 anti
neutrino (with Y = 0) are allowed to mix into Majorana neutrinos. Restrict
ing 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 m
H mW of the Higgs and the W-boson masses in terms of the cosine of the the
oretical Weinberg angle.\nThe talk is based on the paper arXiv:2010.15621v
3\n\nhttps://events.perimeterinstitute.ca/event/5/contributions/62/
LOCATION:
URL:https://events.perimeterinstitute.ca/event/5/contributions/62/
END:VEVENT
BEGIN:VEVENT
SUMMARY:The Higher Algebra of Supersymmetry
DTSTART;VALUE=DATE-TIME:20210329T160000Z
DTEND;VALUE=DATE-TIME:20210329T170000Z
DTSTAMP;VALUE=DATE-TIME:20210418T120726Z
UID:indico-contribution-5-79@events.perimeterinstitute.ca
DESCRIPTION:Speakers: John Huerta (University of Lisbon)\nWe have already
met the octonionic Fierz identity satisfied by spinors in 10-dimensional s
pacetime. This identity makes super-Yang-Mills "super" and allows the Gree
n-Schwarz string to be kappa symmetric. But it is also the defining equati
on 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 Fior
enza-Sati-Schreiber.\n\nhttps://events.perimeterinstitute.ca/event/5/contr
ibutions/79/
LOCATION:
URL:https://events.perimeterinstitute.ca/event/5/contributions/79/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Spin(11\,3)\, particles and octonions
DTSTART;VALUE=DATE-TIME:20210419T160000Z
DTEND;VALUE=DATE-TIME:20210419T170000Z
DTSTAMP;VALUE=DATE-TIME:20210418T120726Z
UID:indico-contribution-5-82@events.perimeterinstitute.ca
DESCRIPTION:Speakers: Kirill Krasnov (University of Nottingham)\nThe ferm
ionic fields of one generation of the Standard Model\, including the Loren
tz spinor degrees of freedom\, can be identified with components of a sing
le 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 us
ual 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 o
ther of a spacelike unit u'. In either case\, the identification S=OxO' im
plies 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 s
ubgroup 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 symmetri
c extension of the SM and Lorentz group. The splitting of S into eigenspac
es of J corresponds to splitting into particles and anti-particles. The sp
litting of S into eigenspaces of J' corresponds to splitting of Lorentz Di
rac spinors into two different chiralities.\n\nhttps://events.perimeterins
titute.ca/event/5/contributions/82/
LOCATION:
URL:https://events.perimeterinstitute.ca/event/5/contributions/82/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Can We Understand the Standard Model Using Octonions?
DTSTART;VALUE=DATE-TIME:20210412T160000Z
DTEND;VALUE=DATE-TIME:20210412T170000Z
DTSTAMP;VALUE=DATE-TIME:20210418T120726Z
UID:indico-contribution-5-81@events.perimeterinstitute.ca
DESCRIPTION:Speakers: John Baez (University of California\, Riverside)\nDu
bois-Violette and Todorov have shown that the Standard Model gauge group c
an 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 exa
mple\, it implies that the Standard Model gauge group consists of the symm
etries of an octonionic qutrit that restrict to symmetries of an octonioni
c qubit and preserve all the structure arising from a choice of unit imagi
nary octonion. It also sheds light on why the Standard Model gauge group a
cts on 10d Euclidean space\, or Minkowski spacetime\, while preserving a 4
+6 splitting.\n\nhttps://events.perimeterinstitute.ca/event/5/contribution
s/81/
LOCATION:
URL:https://events.perimeterinstitute.ca/event/5/contributions/81/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Can We Understand the Standard Model?
DTSTART;VALUE=DATE-TIME:20210405T160000Z
DTEND;VALUE=DATE-TIME:20210405T170000Z
DTSTAMP;VALUE=DATE-TIME:20210418T120726Z
UID:indico-contribution-5-80@events.perimeterinstitute.ca
DESCRIPTION:Speakers: John Baez (University of California\, Riverside)\n40
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 M
odel is the way it is. In this talk we review some lessons from grand unif
ied theories and also from recent work using the octonions. The gauge grou
p of the Standard Model and its representation on one generation of fermio
ns arises naturally from a process that involves splitting 10d Euclidean s
pace into 4+6 dimensions\, but also from a process that involves splitting
10d Minkowski spacetime into 4d Minkowski space and 6 spacelike dimension
s. We explain both these approaches\, and how to reconcile them.\n\nhttps:
//events.perimeterinstitute.ca/event/5/contributions/80/
LOCATION:
URL:https://events.perimeterinstitute.ca/event/5/contributions/80/
END:VEVENT
BEGIN:VEVENT
SUMMARY:TBA
DTSTART;VALUE=DATE-TIME:20210426T160000Z
DTEND;VALUE=DATE-TIME:20210426T170000Z
DTSTAMP;VALUE=DATE-TIME:20210418T120726Z
UID:indico-contribution-5-83@events.perimeterinstitute.ca
DESCRIPTION:Speakers: Michal Malinsky (Charles University in Prague)\nhttp
s://events.perimeterinstitute.ca/event/5/contributions/83/
LOCATION:
URL:https://events.perimeterinstitute.ca/event/5/contributions/83/
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