Download - Pati Salam Model
![Page 1: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/1.jpg)
The Algebra of Grand Unified Theories
The Algebra of Grand Unified Theories
John Huerta
Department of MathematicsUC Riverside
Oral Exam Presentation
![Page 2: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/2.jpg)
The Algebra of Grand Unified Theories
Introduction
This talk is an introduction to the representation theory used inI The Standard Model of Particle Physics (SM);I Certain extensions of the SM, called Grand Unified
Theories (GUTs).
![Page 3: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/3.jpg)
The Algebra of Grand Unified Theories
Introduction
There’s a lot I won’t talk about:I quantum field theory;I spontaneous symmetry breaking;I any sort of dynamics.
This stuff is essential to particle physics. What I discuss here isjust one small piece.
![Page 4: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/4.jpg)
The Algebra of Grand Unified Theories
Introduction
There’s a loose correspondence between particle physics andrepresentation theory:
I Particles → basis vectors in a representation V of a Liegroup G.
I Classification of particles → decomposition into irreps.I Unification → G ↪→ H; particles are “unified” into fewer
irreps.I Grand Unification → as above, but H is simple.I The Standard Model → a particular representation VSM of
a particular Lie group GSM.
![Page 5: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/5.jpg)
The Algebra of Grand Unified Theories
Introduction
There’s a loose correspondence between particle physics andrepresentation theory:
I Particles → basis vectors in a representation V of a Liegroup G.
I Classification of particles → decomposition into irreps.I Unification → G ↪→ H; particles are “unified” into fewer
irreps.I Grand Unification → as above, but H is simple.I The Standard Model → a particular representation VSM of
a particular Lie group GSM.
![Page 6: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/6.jpg)
The Algebra of Grand Unified Theories
The Standard Model
The Group
The Standard Model group is
GSM = U(1)× SU(2)× SU(3)
I The factor U(1)× SU(2) corresponds to the electroweakforce. It represents a unification of electromagnetism andthe weak force.
I Spontaneous symmetry breaking makes theelectromagnetic and weak forces look different; at highenergies, they’re the same.
I SU(3) corresponds to the strong force, which binds quarkstogether. No symmetry breaking here.
![Page 7: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/7.jpg)
The Algebra of Grand Unified Theories
The Standard Model
The Group
The Standard Model group is
GSM = U(1)× SU(2)× SU(3)
I The factor U(1)× SU(2) corresponds to the electroweakforce. It represents a unification of electromagnetism andthe weak force.
I Spontaneous symmetry breaking makes theelectromagnetic and weak forces look different; at highenergies, they’re the same.
I SU(3) corresponds to the strong force, which binds quarkstogether. No symmetry breaking here.
![Page 8: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/8.jpg)
The Algebra of Grand Unified Theories
The Standard Model
The Group
The Standard Model group is
GSM = U(1)× SU(2)× SU(3)
I The factor U(1)× SU(2) corresponds to the electroweakforce. It represents a unification of electromagnetism andthe weak force.
I Spontaneous symmetry breaking makes theelectromagnetic and weak forces look different; at highenergies, they’re the same.
I SU(3) corresponds to the strong force, which binds quarkstogether. No symmetry breaking here.
![Page 9: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/9.jpg)
The Algebra of Grand Unified Theories
The Standard Model
The Group
The Standard Model group is
GSM = U(1)× SU(2)× SU(3)
I The factor U(1)× SU(2) corresponds to the electroweakforce. It represents a unification of electromagnetism andthe weak force.
I Spontaneous symmetry breaking makes theelectromagnetic and weak forces look different; at highenergies, they’re the same.
I SU(3) corresponds to the strong force, which binds quarkstogether. No symmetry breaking here.
![Page 10: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/10.jpg)
The Algebra of Grand Unified Theories
The Standard Model
The Particles
Standard Model RepresentationName Symbol GSM-representation
Left-handed leptons(
νLe−L
)C−1⊗C2⊗C
Left-handed quarks
(ur
L, ugL , ub
L
d rL, dg
L , dbL
)C 1
3⊗C2⊗C3
Right-handed neutrino νR C0 ⊗C ⊗C
Right-handed electron e−R C−2⊗C ⊗C
Right-handed up quarks urR , ug
R , ubR C 4
3⊗C ⊗C3
Right-handed down quarks d rR , dg
R , dbR C− 2
3⊗C ⊗C3
![Page 11: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/11.jpg)
The Algebra of Grand Unified Theories
The Standard Model
The Particles
Here, we’ve written a bunch of GSM = U(1)× SU(2)× SU(3)irreps as U ⊗ V ⊗W , where
I U is a U(1) irrep CY , where Y ∈ 13Z. The underlying vector
space is just C, and the action is given by
α · z = α3Y z, α ∈ U(1), z ∈ C
I V is an SU(2) irrep, either C or C2.I W is an SU(3) irrep, either C or C3.
![Page 12: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/12.jpg)
The Algebra of Grand Unified Theories
The Standard Model
The Particles
Here, we’ve written a bunch of GSM = U(1)× SU(2)× SU(3)irreps as U ⊗ V ⊗W , where
I U is a U(1) irrep CY , where Y ∈ 13Z. The underlying vector
space is just C, and the action is given by
α · z = α3Y z, α ∈ U(1), z ∈ C
I V is an SU(2) irrep, either C or C2.I W is an SU(3) irrep, either C or C3.
![Page 13: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/13.jpg)
The Algebra of Grand Unified Theories
The Standard Model
The Particles
Physicists use these irreps to classify the particles:
I The number Y in CY is called the hypercharge.I C2 = 〈u, d〉; u and d are called isospin up and isospin
down.I C3 = 〈r , g, b〉; r , g, and b are called red, green, and blue.
For example:I ur
L = 1⊗ u ⊗ r ∈ C 13⊗C2 ⊗C3, say “the red left-handed up
quark is the hypercharge 13 , isospin up, red particle.”
I e−R = 1⊗ 1⊗ 1 ∈ C−2 ⊗ C⊗ C, say “the right-handedelectron is the hypercharge −2 isospin singlet which iscolorless.”
![Page 14: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/14.jpg)
The Algebra of Grand Unified Theories
The Standard Model
The Particles
Physicists use these irreps to classify the particles:I The number Y in CY is called the hypercharge.I C2 = 〈u, d〉; u and d are called isospin up and isospin
down.I C3 = 〈r , g, b〉; r , g, and b are called red, green, and blue.
For example:I ur
L = 1⊗ u ⊗ r ∈ C 13⊗C2 ⊗C3, say “the red left-handed up
quark is the hypercharge 13 , isospin up, red particle.”
I e−R = 1⊗ 1⊗ 1 ∈ C−2 ⊗ C⊗ C, say “the right-handedelectron is the hypercharge −2 isospin singlet which iscolorless.”
![Page 15: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/15.jpg)
The Algebra of Grand Unified Theories
The Standard Model
The Particles
Physicists use these irreps to classify the particles:I The number Y in CY is called the hypercharge.I C2 = 〈u, d〉; u and d are called isospin up and isospin
down.I C3 = 〈r , g, b〉; r , g, and b are called red, green, and blue.
For example:I ur
L = 1⊗ u ⊗ r ∈ C 13⊗C2 ⊗C3, say “the red left-handed up
quark is the hypercharge 13 , isospin up, red particle.”
I e−R = 1⊗ 1⊗ 1 ∈ C−2 ⊗ C⊗ C, say “the right-handedelectron is the hypercharge −2 isospin singlet which iscolorless.”
![Page 16: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/16.jpg)
The Algebra of Grand Unified Theories
The Standard Model
The Representation
I We take the direct sum of all these irreps, defining thereducible representation,
F = C−1 ⊗ C2 ⊗ C ⊕ · · · ⊕ C− 23⊗ C⊗ C3
which we’ll call the fermions.
I We also have the antifermions, F ∗, which is just the dual ofF .
I Direct summing these, we get the Standard Modelrepresentation
VSM = F ⊕ F ∗
![Page 17: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/17.jpg)
The Algebra of Grand Unified Theories
The Standard Model
The Representation
I We take the direct sum of all these irreps, defining thereducible representation,
F = C−1 ⊗ C2 ⊗ C ⊕ · · · ⊕ C− 23⊗ C⊗ C3
which we’ll call the fermions.I We also have the antifermions, F ∗, which is just the dual of
F .
I Direct summing these, we get the Standard Modelrepresentation
VSM = F ⊕ F ∗
![Page 18: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/18.jpg)
The Algebra of Grand Unified Theories
The Standard Model
The Representation
I We take the direct sum of all these irreps, defining thereducible representation,
F = C−1 ⊗ C2 ⊗ C ⊕ · · · ⊕ C− 23⊗ C⊗ C3
which we’ll call the fermions.I We also have the antifermions, F ∗, which is just the dual of
F .I Direct summing these, we get the Standard Model
representationVSM = F ⊕ F ∗
![Page 19: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/19.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The GUTs Goal:I GSM = U(1)× SU(2)× SU(3) is a mess!
I VSM = C−1 ⊗ C2 ⊗ C ⊕ · · · ⊕ C 23⊗ C⊗ C3∗ is a
mess!I Explain the hypercharges!I Explain other patterns:
I dim VSM = 32 = 25;I symmetry between quarks and leptons;I asymmetry between left and right.
![Page 20: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/20.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The GUTs Goal:I GSM = U(1)× SU(2)× SU(3) is a mess!I VSM = C−1 ⊗ C2 ⊗ C ⊕ · · · ⊕ C 2
3⊗ C⊗ C3∗ is a
mess!
I Explain the hypercharges!I Explain other patterns:
I dim VSM = 32 = 25;I symmetry between quarks and leptons;I asymmetry between left and right.
![Page 21: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/21.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The GUTs Goal:I GSM = U(1)× SU(2)× SU(3) is a mess!I VSM = C−1 ⊗ C2 ⊗ C ⊕ · · · ⊕ C 2
3⊗ C⊗ C3∗ is a
mess!I Explain the hypercharges!
I Explain other patterns:I dim VSM = 32 = 25;I symmetry between quarks and leptons;I asymmetry between left and right.
![Page 22: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/22.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The GUTs Goal:I GSM = U(1)× SU(2)× SU(3) is a mess!I VSM = C−1 ⊗ C2 ⊗ C ⊕ · · · ⊕ C 2
3⊗ C⊗ C3∗ is a
mess!I Explain the hypercharges!I Explain other patterns:
I dim VSM = 32 = 25;I symmetry between quarks and leptons;I asymmetry between left and right.
![Page 23: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/23.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The GUTs trick: if V is a representation of G and GSM ⊆ G,then
I V is also representation of GSM;I V may break apart into more GSM-irreps than G-irreps.
![Page 24: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/24.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The GUTs trick: if V is a representation of G and GSM ⊆ G,then
I V is also representation of GSM;I V may break apart into more GSM-irreps than G-irreps.
![Page 25: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/25.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
More precisely, we want:I A homomorphism φ : GSM → G.I A unitary representation ρ : G → U(V ).I An isomorphism of vector spaces f : VSM → V .I Such that
GSMφ //
��
G
ρ
��U(VSM)
U(f ) // U(V )
commutes.
In short: V becomes isomorphic to VSM when we restrict fromG to GSM.
![Page 26: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/26.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
More precisely, we want:I A homomorphism φ : GSM → G.I A unitary representation ρ : G → U(V ).I An isomorphism of vector spaces f : VSM → V .I Such that
GSMφ //
��
G
ρ
��U(VSM)
U(f ) // U(V )
commutes.In short: V becomes isomorphic to VSM when we restrict fromG to GSM.
![Page 27: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/27.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The SU(5) Theory
The SU(5) GUT, due to Georgi and Glashow, is all about “2isospins + 3 colors = 5 things”:
I Take C5 = 〈u, d , r , g, b〉.I C5 is a representation of SU(5), as is the 32-dimensional
exterior algebra:
ΛC5 ∼= Λ0C5 ⊕ Λ1C5 ⊕ Λ2C5 ⊕ Λ3C5 ⊕ Λ4C5 ⊕ Λ5C5
I Theorem There’s a homomorphism φ : GSM → SU(5) anda linear isomorphism h : VSM → ΛC5 making
GSMφ //
��
SU(5)
��U(VSM)
U(h) // U(ΛC5)
commute.
![Page 28: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/28.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The SU(5) Theory
The SU(5) GUT, due to Georgi and Glashow, is all about “2isospins + 3 colors = 5 things”:
I Take C5 = 〈u, d , r , g, b〉.I C5 is a representation of SU(5), as is the 32-dimensional
exterior algebra:
ΛC5 ∼= Λ0C5 ⊕ Λ1C5 ⊕ Λ2C5 ⊕ Λ3C5 ⊕ Λ4C5 ⊕ Λ5C5
I Theorem There’s a homomorphism φ : GSM → SU(5) anda linear isomorphism h : VSM → ΛC5 making
GSMφ //
��
SU(5)
��U(VSM)
U(h) // U(ΛC5)
commute.
![Page 29: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/29.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The SU(5) Theory
The SU(5) GUT, due to Georgi and Glashow, is all about “2isospins + 3 colors = 5 things”:
I Take C5 = 〈u, d , r , g, b〉.I C5 is a representation of SU(5), as is the 32-dimensional
exterior algebra:
ΛC5 ∼= Λ0C5 ⊕ Λ1C5 ⊕ Λ2C5 ⊕ Λ3C5 ⊕ Λ4C5 ⊕ Λ5C5
I Theorem There’s a homomorphism φ : GSM → SU(5) anda linear isomorphism h : VSM → ΛC5 making
GSMφ //
��
SU(5)
��U(VSM)
U(h) // U(ΛC5)
commute.
![Page 30: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/30.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The SU(5) Theory
ProofI Let S(U(2)× U(3)) ⊆ SU(5) be the subgroup preserving
the 2 + 3 splitting C2 ⊕ C3 ∼= C5.
I Can find φ : GSM → S(U(2)× U(3)) ⊆ SU(5).I The representation ΛC5 of SU(5) is isomorphic to VSM
when pulled back to GSM.
We define φ by
φ : (α, g, h) ∈ U(1)×SU(2)×SU(3) 7−→(
α3g 00 α−2h
)∈ SU(5)
![Page 31: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/31.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The SU(5) Theory
ProofI Let S(U(2)× U(3)) ⊆ SU(5) be the subgroup preserving
the 2 + 3 splitting C2 ⊕ C3 ∼= C5.I Can find φ : GSM → S(U(2)× U(3)) ⊆ SU(5).
I The representation ΛC5 of SU(5) is isomorphic to VSMwhen pulled back to GSM.
We define φ by
φ : (α, g, h) ∈ U(1)×SU(2)×SU(3) 7−→(
α3g 00 α−2h
)∈ SU(5)
![Page 32: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/32.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The SU(5) Theory
ProofI Let S(U(2)× U(3)) ⊆ SU(5) be the subgroup preserving
the 2 + 3 splitting C2 ⊕ C3 ∼= C5.I Can find φ : GSM → S(U(2)× U(3)) ⊆ SU(5).I The representation ΛC5 of SU(5) is isomorphic to VSM
when pulled back to GSM.
We define φ by
φ : (α, g, h) ∈ U(1)×SU(2)×SU(3) 7−→(
α3g 00 α−2h
)∈ SU(5)
![Page 33: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/33.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The SU(5) Theory
ProofI Let S(U(2)× U(3)) ⊆ SU(5) be the subgroup preserving
the 2 + 3 splitting C2 ⊕ C3 ∼= C5.I Can find φ : GSM → S(U(2)× U(3)) ⊆ SU(5).I The representation ΛC5 of SU(5) is isomorphic to VSM
when pulled back to GSM.We define φ by
φ : (α, g, h) ∈ U(1)×SU(2)×SU(3) 7−→(
α3g 00 α−2h
)∈ SU(5)
![Page 34: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/34.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The SU(5) Theory
φ maps GSM onto S(U(2)× U(3)), but it has a kernel:
ker φ = {(α, α−3, α2)|α6 = 1} ∼= Z6
ThusGSM/Z6
∼= S(U(2)× U(3))
The subgroup Z6 ⊆ GSM acts trivially on VSM.
![Page 35: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/35.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The SU(5) Theory
φ maps GSM onto S(U(2)× U(3)), but it has a kernel:
ker φ = {(α, α−3, α2)|α6 = 1} ∼= Z6
ThusGSM/Z6
∼= S(U(2)× U(3))
The subgroup Z6 ⊆ GSM acts trivially on VSM.
![Page 36: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/36.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The SU(5) Theory
Because GSM respects the 2 + 3 splitting
ΛC5 ∼= Λ(C2 ⊕ C3) ∼= ΛC2 ⊗ ΛC3
As a GSM-representation,
ΛC2 ∼= C0 ⊗ Λ0C2 ⊕ C1 ⊗ Λ1C2 ⊕ C2 ⊗ Λ2C2
As a GSM-representation,
ΛC3 ∼= C0⊗Λ0C3 ⊕ C− 23⊗Λ1C3 ⊕ C− 4
3⊗Λ2C3 ⊕ C−2⊗Λ3C3
![Page 37: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/37.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The SU(5) Theory
Because GSM respects the 2 + 3 splitting
ΛC5 ∼= Λ(C2 ⊕ C3) ∼= ΛC2 ⊗ ΛC3
As a GSM-representation,
ΛC2 ∼= C0 ⊗ Λ0C2 ⊕ C1 ⊗ Λ1C2 ⊕ C2 ⊗ Λ2C2
As a GSM-representation,
ΛC3 ∼= C0⊗Λ0C3 ⊕ C− 23⊗Λ1C3 ⊕ C− 4
3⊗Λ2C3 ⊕ C−2⊗Λ3C3
![Page 38: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/38.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The SU(5) Theory
Because GSM respects the 2 + 3 splitting
ΛC5 ∼= Λ(C2 ⊕ C3) ∼= ΛC2 ⊗ ΛC3
As a GSM-representation,
ΛC2 ∼= C0 ⊗ Λ0C2 ⊕ C1 ⊗ Λ1C2 ⊕ C2 ⊗ Λ2C2
As a GSM-representation,
ΛC3 ∼= C0⊗Λ0C3 ⊕ C− 23⊗Λ1C3 ⊕ C− 4
3⊗Λ2C3 ⊕ C−2⊗Λ3C3
![Page 39: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/39.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The SU(5) Theory
Then tensor them together, use C2 ∼= C2∗ andCY1 ⊗ CY2
∼= CY1+Y2 to see how
VSM∼= ΛC5
as GSM-representations.
Thus there’s a linear isomorphism h : VSM → ΛC5 making
GSMφ //
��
SU(5)
��U(VSM)
U(h) // U(ΛC5)
commute.
![Page 40: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/40.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The SU(5) Theory
Then tensor them together, use C2 ∼= C2∗ andCY1 ⊗ CY2
∼= CY1+Y2 to see how
VSM∼= ΛC5
as GSM-representations.Thus there’s a linear isomorphism h : VSM → ΛC5 making
GSMφ //
��
SU(5)
��U(VSM)
U(h) // U(ΛC5)
commute.
![Page 41: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/41.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The Pati–Salam Model
The idea of the Pati–Salam model, due to Pati and Salam:
I Unify the C3 ⊕ C representation of SU(3) into the irrep C4
of SU(4).I This creates explicit symmetry between quarks and
leptons.I Unify the C2 ⊕ C⊕ C representations of SU(2) into the
representation C2 ⊗ C ⊕ C⊗ C2 of SU(2)× SU(2).I This treats left and right more symmetrically.
![Page 42: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/42.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The Pati–Salam Model
The idea of the Pati–Salam model, due to Pati and Salam:I Unify the C3 ⊕ C representation of SU(3) into the irrep C4
of SU(4).I This creates explicit symmetry between quarks and
leptons.
I Unify the C2 ⊕ C⊕ C representations of SU(2) into therepresentation C2 ⊗ C ⊕ C⊗ C2 of SU(2)× SU(2).
I This treats left and right more symmetrically.
![Page 43: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/43.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The Pati–Salam Model
The idea of the Pati–Salam model, due to Pati and Salam:I Unify the C3 ⊕ C representation of SU(3) into the irrep C4
of SU(4).I This creates explicit symmetry between quarks and
leptons.I Unify the C2 ⊕ C⊕ C representations of SU(2) into the
representation C2 ⊗ C ⊕ C⊗ C2 of SU(2)× SU(2).I This treats left and right more symmetrically.
![Page 44: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/44.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The Pati–Salam Model
Standard Model RepresentationName Symbol GSM-representation
Left-handed leptons(
νLe−L
)C−1⊗C2⊗C
Left-handed quarks
(ur
L, ugL , ub
L
d rL, dg
L , dbL
)C 1
3⊗C2⊗C3
Right-handed neutrino νR C0 ⊗C ⊗C
Right-handed electron e−R C−2⊗C ⊗C
Right-handed up quarks urR , ug
R , ubR C 4
3⊗C ⊗C3
Right-handed down quarks d rR , dg
R , dbR C− 2
3⊗C ⊗C3
![Page 45: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/45.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The Pati–Salam Model
The Pati–Salam representationName Symbol SU(2)× SU(2)× SU(4)-
representation
Left-handed fermions
(νL, ur
L, ugL , ub
L
e−L , d rL, dg
L , dbL
)C2⊗C ⊗C4
Right-handed fermions
(νR , ur
R , ugR , ub
R
e−R , d rR , dg
R , dbR
)C ⊗C2⊗C4
I Write GPS = SU(2)× SU(2)× SU(4).I Write VPS = C2 ⊗ C⊗ C4 ⊕ C⊗ C2 ⊗ C4 ⊕ dual.
![Page 46: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/46.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The Pati–Salam Model
The Pati–Salam representationName Symbol SU(2)× SU(2)× SU(4)-
representation
Left-handed fermions
(νL, ur
L, ugL , ub
L
e−L , d rL, dg
L , dbL
)C2⊗C ⊗C4
Right-handed fermions
(νR , ur
R , ugR , ub
R
e−R , d rR , dg
R , dbR
)C ⊗C2⊗C4
I Write GPS = SU(2)× SU(2)× SU(4).
I Write VPS = C2 ⊗ C⊗ C4 ⊕ C⊗ C2 ⊗ C4 ⊕ dual.
![Page 47: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/47.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The Pati–Salam Model
The Pati–Salam representationName Symbol SU(2)× SU(2)× SU(4)-
representation
Left-handed fermions
(νL, ur
L, ugL , ub
L
e−L , d rL, dg
L , dbL
)C2⊗C ⊗C4
Right-handed fermions
(νR , ur
R , ugR , ub
R
e−R , d rR , dg
R , dbR
)C ⊗C2⊗C4
I Write GPS = SU(2)× SU(2)× SU(4).I Write VPS = C2 ⊗ C⊗ C4 ⊕ C⊗ C2 ⊗ C4 ⊕ dual.
![Page 48: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/48.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The Pati–Salam Model
To make the Pati–Salam model work, we need to proveTheorem There exists maps θ : GSM → GPS andf : VSM → VPS which make the diagram
GSMθ //
��
GPS
��U(VSM)
U(f ) // U(VPS)
commute.
![Page 49: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/49.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The Pati–Salam Model
Proof
I Want θ : GSM → SU(2)× SU(2)× SU(4):
I Pick θ so GSM maps to a subgroup ofSU(2)× SU(2)× SU(4) that preserves the 3 + 1 splitting
C4 ∼= C3 ⊕ C
and the 1 + 1 splitting
C⊗ C2 ∼= C⊕ C
![Page 50: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/50.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The Pati–Salam Model
Proof
I Want θ : GSM → SU(2)× SU(2)× SU(4):I Pick θ so GSM maps to a subgroup of
SU(2)× SU(2)× SU(4) that preserves the 3 + 1 splitting
C4 ∼= C3 ⊕ C
and the 1 + 1 splitting
C⊗ C2 ∼= C⊕ C
![Page 51: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/51.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The Pati–Salam Model
We need some facts:
I Spin(2n) has a representation ΛCn, called the Diracspinors.
I SU(2)× SU(2) ∼= Spin(4), and C2 ⊗ C ⊕ C⊗ C2 ∼= ΛC2
I SU(4) ∼= Spin(6), and C4 ⊕ C4∗ ∼= ΛC3.I VPS
∼= ΛC2 ⊗ ΛC3 as a representation ofGPS
∼= Spin(4)× Spin(6).I C4 ∼= ΛoddC3 ∼= Λ1C3 ⊕ Λ3C3 has a 3 + 1 splitting — the
grading!I C⊗ C2 ∼= ΛevC2 ∼= Λ0C2 ⊕ Λ2C2 has a 1 + 1 splitting — the
grading!
![Page 52: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/52.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The Pati–Salam Model
We need some facts:I Spin(2n) has a representation ΛCn, called the Dirac
spinors.I SU(2)× SU(2) ∼= Spin(4), and C2 ⊗ C ⊕ C⊗ C2 ∼= ΛC2
I SU(4) ∼= Spin(6), and C4 ⊕ C4∗ ∼= ΛC3.
I VPS∼= ΛC2 ⊗ ΛC3 as a representation of
GPS∼= Spin(4)× Spin(6).
I C4 ∼= ΛoddC3 ∼= Λ1C3 ⊕ Λ3C3 has a 3 + 1 splitting — thegrading!
I C⊗ C2 ∼= ΛevC2 ∼= Λ0C2 ⊕ Λ2C2 has a 1 + 1 splitting — thegrading!
![Page 53: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/53.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The Pati–Salam Model
We need some facts:I Spin(2n) has a representation ΛCn, called the Dirac
spinors.I SU(2)× SU(2) ∼= Spin(4), and C2 ⊗ C ⊕ C⊗ C2 ∼= ΛC2
I SU(4) ∼= Spin(6), and C4 ⊕ C4∗ ∼= ΛC3.I VPS
∼= ΛC2 ⊗ ΛC3 as a representation ofGPS
∼= Spin(4)× Spin(6).
I C4 ∼= ΛoddC3 ∼= Λ1C3 ⊕ Λ3C3 has a 3 + 1 splitting — thegrading!
I C⊗ C2 ∼= ΛevC2 ∼= Λ0C2 ⊕ Λ2C2 has a 1 + 1 splitting — thegrading!
![Page 54: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/54.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The Pati–Salam Model
We need some facts:I Spin(2n) has a representation ΛCn, called the Dirac
spinors.I SU(2)× SU(2) ∼= Spin(4), and C2 ⊗ C ⊕ C⊗ C2 ∼= ΛC2
I SU(4) ∼= Spin(6), and C4 ⊕ C4∗ ∼= ΛC3.I VPS
∼= ΛC2 ⊗ ΛC3 as a representation ofGPS
∼= Spin(4)× Spin(6).I C4 ∼= ΛoddC3 ∼= Λ1C3 ⊕ Λ3C3 has a 3 + 1 splitting — the
grading!
I C⊗ C2 ∼= ΛevC2 ∼= Λ0C2 ⊕ Λ2C2 has a 1 + 1 splitting — thegrading!
![Page 55: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/55.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The Pati–Salam Model
We need some facts:I Spin(2n) has a representation ΛCn, called the Dirac
spinors.I SU(2)× SU(2) ∼= Spin(4), and C2 ⊗ C ⊕ C⊗ C2 ∼= ΛC2
I SU(4) ∼= Spin(6), and C4 ⊕ C4∗ ∼= ΛC3.I VPS
∼= ΛC2 ⊗ ΛC3 as a representation ofGPS
∼= Spin(4)× Spin(6).I C4 ∼= ΛoddC3 ∼= Λ1C3 ⊕ Λ3C3 has a 3 + 1 splitting — the
grading!I C⊗ C2 ∼= ΛevC2 ∼= Λ0C2 ⊕ Λ2C2 has a 1 + 1 splitting — the
grading!
![Page 56: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/56.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The Pati–Salam Model
Build θ so that
I θ maps GSM onto the subgroup S(U(3)× U(1)) ⊆ Spin(6)that preserves the 3 + 1 splitting:
(α, x , y) ∈ U(1)× SU(2)× SU(3) 7→(
αy 00 α−3
)
I θ maps GSM onto the subgroupSU(2)× S(U(1)× U(1)) ⊆ Spin(4) that preserves the1 + 1 splitting:
(α, x , y) ∈ U(1)× SU(2)× SU(3) 7→(
x ,
(α3 00 α−3
))
![Page 57: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/57.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The Pati–Salam Model
Build θ so that
I θ maps GSM onto the subgroup S(U(3)× U(1)) ⊆ Spin(6)that preserves the 3 + 1 splitting:
(α, x , y) ∈ U(1)× SU(2)× SU(3) 7→(
αy 00 α−3
)
I θ maps GSM onto the subgroupSU(2)× S(U(1)× U(1)) ⊆ Spin(4) that preserves the1 + 1 splitting:
(α, x , y) ∈ U(1)× SU(2)× SU(3) 7→(
x ,
(α3 00 α−3
))
![Page 58: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/58.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The Pati–Salam Model
The payoff:As a GSM-representation,
ΛC2 ∼= C−1 ⊗ Λ0C2 ⊕ C0 ⊗ Λ1C2 ⊕ C1 ⊗ Λ2C2
Earlier, we had
ΛC2 ∼= C0 ⊗ Λ0C2 ⊕ C1 ⊗ Λ1C2 ⊕ C2 ⊗ Λ2C2
As a GSM-representation,
ΛC3 ∼= C1⊗Λ0C3 ⊕ C 13⊗Λ1C3 ⊕ C− 1
3⊗Λ2C3 ⊕ C−1⊗Λ3C3
Earlier, we had
ΛC3 ∼= C0⊗Λ0C3 ⊕ C− 23⊗Λ1C3 ⊕ C− 4
3⊗Λ2C3 ⊕ C−2⊗Λ3C3
![Page 59: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/59.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The Pati–Salam Model
The payoff:As a GSM-representation,
ΛC2 ∼= C−1 ⊗ Λ0C2 ⊕ C0 ⊗ Λ1C2 ⊕ C1 ⊗ Λ2C2
Earlier, we had
ΛC2 ∼= C0 ⊗ Λ0C2 ⊕ C1 ⊗ Λ1C2 ⊕ C2 ⊗ Λ2C2
As a GSM-representation,
ΛC3 ∼= C1⊗Λ0C3 ⊕ C 13⊗Λ1C3 ⊕ C− 1
3⊗Λ2C3 ⊕ C−1⊗Λ3C3
Earlier, we had
ΛC3 ∼= C0⊗Λ0C3 ⊕ C− 23⊗Λ1C3 ⊕ C− 4
3⊗Λ2C3 ⊕ C−2⊗Λ3C3
![Page 60: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/60.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The Pati–Salam Model
The payoff:As a GSM-representation,
ΛC2 ∼= C−1 ⊗ Λ0C2 ⊕ C0 ⊗ Λ1C2 ⊕ C1 ⊗ Λ2C2
Earlier, we had
ΛC2 ∼= C0 ⊗ Λ0C2 ⊕ C1 ⊗ Λ1C2 ⊕ C2 ⊗ Λ2C2
As a GSM-representation,
ΛC3 ∼= C1⊗Λ0C3 ⊕ C 13⊗Λ1C3 ⊕ C− 1
3⊗Λ2C3 ⊕ C−1⊗Λ3C3
Earlier, we had
ΛC3 ∼= C0⊗Λ0C3 ⊕ C− 23⊗Λ1C3 ⊕ C− 4
3⊗Λ2C3 ⊕ C−2⊗Λ3C3
![Page 61: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/61.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The Pati–Salam Model
The payoff:As a GSM-representation,
ΛC2 ∼= C−1 ⊗ Λ0C2 ⊕ C0 ⊗ Λ1C2 ⊕ C1 ⊗ Λ2C2
Earlier, we had
ΛC2 ∼= C0 ⊗ Λ0C2 ⊕ C1 ⊗ Λ1C2 ⊕ C2 ⊗ Λ2C2
As a GSM-representation,
ΛC3 ∼= C1⊗Λ0C3 ⊕ C 13⊗Λ1C3 ⊕ C− 1
3⊗Λ2C3 ⊕ C−1⊗Λ3C3
Earlier, we had
ΛC3 ∼= C0⊗Λ0C3 ⊕ C− 23⊗Λ1C3 ⊕ C− 4
3⊗Λ2C3 ⊕ C−2⊗Λ3C3
![Page 62: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/62.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The Pati–Salam Model
We can recycle the fact that VSM∼= ΛC2 ⊗ ΛC3 from the SU(5)
theory.
Thus there’s an isomorphism of vector spacesf : VSM → ΛC2 ⊗ ΛC3 such that
GSMθ //
��
Spin(4)× Spin(6)
��U(VSM)
U(f ) // U(ΛC2 ⊗ ΛC3)
commutes.
![Page 63: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/63.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The Pati–Salam Model
We can recycle the fact that VSM∼= ΛC2 ⊗ ΛC3 from the SU(5)
theory.Thus there’s an isomorphism of vector spacesf : VSM → ΛC2 ⊗ ΛC3 such that
GSMθ //
��
Spin(4)× Spin(6)
��U(VSM)
U(f ) // U(ΛC2 ⊗ ΛC3)
commutes.
![Page 64: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/64.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The Spin(10) Theory
Extend the SU(5) theory to get the Spin(10) theory, due toGeorgi:
I In general,
SU(n)ψ //
%%KKKKKKKKKSpin(2n)
��U(ΛCn)
I Set n = 5:
SU(5)ψ //
��
Spin(10)
��U(ΛC5)
1 // U(ΛC5)
![Page 65: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/65.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The Spin(10) Theory
Extend the SU(5) theory to get the Spin(10) theory, due toGeorgi:
I In general,
SU(n)ψ //
%%KKKKKKKKKSpin(2n)
��U(ΛCn)
I Set n = 5:
SU(5)ψ //
��
Spin(10)
��U(ΛC5)
1 // U(ΛC5)
![Page 66: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/66.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The Spin(10) Theory
Or extend the Pati–Salam model:I In general,
Spin(2n)× Spin(2m)η //
��
Spin(2n + 2m)
��U(ΛCn ⊗ ΛCm)
U(g) // U(ΛCn+m)
I Set n = 2 and m = 3:
Spin(4)× Spin(6)η //
��
Spin(10)
��U(ΛC2 ⊗ ΛC3)
U(g) // U(ΛC5)
![Page 67: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/67.jpg)
The Algebra of Grand Unified Theories
Grand Unified Theories
The Spin(10) Theory
Or extend the Pati–Salam model:I In general,
Spin(2n)× Spin(2m)η //
��
Spin(2n + 2m)
��U(ΛCn ⊗ ΛCm)
U(g) // U(ΛCn+m)
I Set n = 2 and m = 3:
Spin(4)× Spin(6)η //
��
Spin(10)
��U(ΛC2 ⊗ ΛC3)
U(g) // U(ΛC5)
![Page 68: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/68.jpg)
The Algebra of Grand Unified Theories
Conclusion
Theorem The cube of GUTs
GSMφ //
θ
vvnnnnnnnnnnnnn
��
SU(5)ψ
yyssssssssss
��
Spin(4)× Spin(6)η //
��
Spin(10)
��
U(VSM)U(h) //
U(f )
vvnnnnnnnnnnnnU(ΛC5)
1
yyssssssssss
U(ΛC2 ⊗ ΛC3)U(g) // U(ΛC5)
commutes.
![Page 69: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/69.jpg)
The Algebra of Grand Unified Theories
Conclusion
ProofI The vertical faces of the cube commute.
I The two maps from GSM to U(ΛC5) are equal:
GSMφ //
�
SU(5)
ψ
��Spin(4)× Spin(6)
η // Spin(10) // U(ΛC5)
I ΛC5 is a faithful representation of Spin(10) — the top facecommutes:
GSMφ //
�
SU(5)
ψ
��Spin(4)× Spin(6)
η // Spin(10)
![Page 70: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/70.jpg)
The Algebra of Grand Unified Theories
Conclusion
ProofI The vertical faces of the cube commute.I The two maps from GSM to U(ΛC5) are equal:
GSMφ //
�
SU(5)
ψ
��Spin(4)× Spin(6)
η // Spin(10) // U(ΛC5)
I ΛC5 is a faithful representation of Spin(10) — the top facecommutes:
GSMφ //
�
SU(5)
ψ
��Spin(4)× Spin(6)
η // Spin(10)
![Page 71: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/71.jpg)
The Algebra of Grand Unified Theories
Conclusion
ProofI The vertical faces of the cube commute.I The two maps from GSM to U(ΛC5) are equal:
GSMφ //
�
SU(5)
ψ
��Spin(4)× Spin(6)
η // Spin(10) // U(ΛC5)
I ΛC5 is a faithful representation of Spin(10) — the top facecommutes:
GSMφ //
�
SU(5)
ψ
��Spin(4)× Spin(6)
η // Spin(10)
![Page 72: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/72.jpg)
The Algebra of Grand Unified Theories
Conclusion
I The intertwiners commute:
VSMh //
f��
ΛC5
ΛC2 ⊗ ΛC3
g
99ssssssssss
I The bottom face commutes:
U(VSM)U(h) //
U(f )��
U(ΛC5)
1��
U(ΛC2 ⊗ ΛC3)U(g) // U(ΛC5)
![Page 73: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/73.jpg)
The Algebra of Grand Unified Theories
Conclusion
I The intertwiners commute:
VSMh //
f��
ΛC5
ΛC2 ⊗ ΛC3
g
99ssssssssss
I The bottom face commutes:
U(VSM)U(h) //
U(f )��
U(ΛC5)
1��
U(ΛC2 ⊗ ΛC3)U(g) // U(ΛC5)
![Page 74: Pati Salam Model](https://reader033.vdocument.in/reader033/viewer/2022061616/55cf9a92550346d033a269a8/html5/thumbnails/74.jpg)
The Algebra of Grand Unified Theories
Conclusion
Thus the cube
GSMφ //
θ
vvnnnnnnnnnnnnn
��
SU(5)ψ
yyssssssssss
��
Spin(4)× Spin(6)η //
��
Spin(10)
��
U(VSM)U(h) //
U(f )
vvnnnnnnnnnnnnU(ΛC5)
1
yyssssssssss
U(ΛC2 ⊗ ΛC3)U(g) // U(ΛC5)
commutes.