The Algebra of Grand Unified Theories

The Algebra of Grand Unified Theories

John Huerta

Department of Mathematics UC Riverside

Oral Exam Presentation The Algebra of Grand Unified Theories Introduction

This talk is an introduction to the representation theory used in

I The Standard Model of Particle Physics (SM);

I Certain extensions of the SM, called Grand Unified Theories (GUTs). 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 is just one small piece. I Particles → basis vectors in a representation V of a Lie group 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.

The Algebra of Grand Unified Theories Introduction

There’s a loose correspondence between particle physics and representation theory: The Algebra of Grand Unified Theories Introduction

There’s a loose correspondence between particle physics and representation theory:

I Particles → basis vectors in a representation V of a Lie group 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. I The factor U(1) × SU(2) corresponds to the electroweak force. It represents a unification of electromagnetism and the weak force.

I Spontaneous symmetry breaking makes the electromagnetic and weak forces look different; at high energies, they’re the same.

I SU(3) corresponds to the strong force, which binds quarks together. No symmetry breaking here.

The Algebra of Grand Unified Theories The Standard Model The Group

The Standard Model group is

GSM = U(1) × SU(2) × SU(3) I Spontaneous symmetry breaking makes the electromagnetic and weak forces look different; at high energies, they’re the same.

I SU(3) corresponds to the strong force, which binds quarks together. No symmetry breaking here.

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 electroweak force. It represents a unification of electromagnetism and the weak force. I SU(3) corresponds to the strong force, which binds quarks together. No symmetry breaking here.

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 electroweak force. It represents a unification of electromagnetism and the weak force.

I Spontaneous symmetry breaking makes the electromagnetic and weak forces look different; at high energies, they’re the same. 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 electroweak force. It represents a unification of electromagnetism and the weak force.

I Spontaneous symmetry breaking makes the electromagnetic and weak forces look different; at high energies, they’re the same.

I SU(3) corresponds to the strong force, which binds quarks together. No symmetry breaking here. The Algebra of Grand Unified Theories The Standard Model The Particles

Standard Model Representation Name Symbol GSM-representation   νL 2 Left-handed leptons − C−1 ⊗ C ⊗ C eL

r g b ! uL, uL , uL 2 3 Left-handed quarks 1 ⊗ ⊗ r g b C 3 C C dL, dL , dL

Right-handed neutrino νR C0 ⊗ C ⊗ C − Right-handed electron eR C−2 ⊗ C ⊗ C

r g b 3 Right-handed up quarks u , u , u 4 ⊗ ⊗ R R R C 3 C C

r g b 3 Right-handed down quarks d , d , d 2 ⊗ ⊗ R R R C− 3 C C 1 I U is a U(1) irrep CY , where Y ∈ 3 Z. The underlying vector space is just C, and the action is given by

3Y α · z = α z, α ∈ U(1), z ∈ C

2 I V is an SU(2) irrep, either C or C . 3 I W is an SU(3) irrep, either C or C .

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 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 1 I U is a U(1) irrep CY , where Y ∈ 3 Z. The underlying vector space is just C, and the action is given by

3Y α · z = α z, α ∈ U(1), z ∈ C

2 I V is an SU(2) irrep, either C or C . 3 I W is an SU(3) irrep, either C or C . I The number Y in CY is called the hypercharge. 2 I C = hu, di; u and d are called isospin up and isospin down. 3 I C = hr, g, bi; r, g, and b are called red, green, and blue. For example: r 2 3 I uL = 1 ⊗ u ⊗ r ∈ C 1 ⊗ C ⊗ C , say “the red left-handed up 3 1 quark is the hypercharge 3 , isospin up, red particle.” − I eR = 1 ⊗ 1 ⊗ 1 ∈ C−2 ⊗ C ⊗ C, say “the right-handed electron is the hypercharge −2 isospin singlet which is colorless.”

The Algebra of Grand Unified Theories The Standard Model The Particles

Physicists use these irreps to classify the particles: For example: r 2 3 I uL = 1 ⊗ u ⊗ r ∈ C 1 ⊗ C ⊗ C , say “the red left-handed up 3 1 quark is the hypercharge 3 , isospin up, red particle.” − I eR = 1 ⊗ 1 ⊗ 1 ∈ C−2 ⊗ C ⊗ C, say “the right-handed electron is the hypercharge −2 isospin singlet which is colorless.”

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. 2 I C = hu, di; u and d are called isospin up and isospin down. 3 I C = hr, g, bi; r, g, and b are called red, green, and blue. 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. 2 I C = hu, di; u and d are called isospin up and isospin down. 3 I C = hr, g, bi; r, g, and b are called red, green, and blue. For example: r 2 3 I uL = 1 ⊗ u ⊗ r ∈ C 1 ⊗ C ⊗ C , say “the red left-handed up 3 1 quark is the hypercharge 3 , isospin up, red particle.” − I eR = 1 ⊗ 1 ⊗ 1 ∈ C−2 ⊗ C ⊗ C, say “the right-handed electron is the hypercharge −2 isospin singlet which is colorless.” ∗ I We also have the antifermions, F , which is just the dual of F.

I Direct summing these, we get the Standard Model representation ∗ VSM = F ⊕ F

The Algebra of Grand Unified Theories The Standard Model The Representation

I We take the direct sum of all these irreps, defining the reducible representation,

2 3 F = C−1 ⊗ C ⊗ C ⊕ · · · ⊕ C 2 ⊗ C ⊗ C − 3 which we’ll call the fermions. I Direct summing these, we get the Standard Model representation ∗ VSM = F ⊕ F

The Algebra of Grand Unified Theories The Standard Model The Representation

I We take the direct sum of all these irreps, defining the reducible representation,

2 3 F = C−1 ⊗ C ⊗ C ⊕ · · · ⊕ C 2 ⊗ C ⊗ C − 3 which we’ll call the fermions. ∗ I We also have the antifermions, F , which is just the dual of F. The Algebra of Grand Unified Theories The Standard Model The Representation

I We take the direct sum of all these irreps, defining the reducible representation,

2 3 F = C−1 ⊗ C ⊗ C ⊕ · · · ⊕ C 2 ⊗ C ⊗ C − 3 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 representation ∗ VSM = F ⊕ F 2 3∗ I V = C−1 ⊗ C ⊗ C ⊕ · · · ⊕ C 2 ⊗ C ⊗ C is a SM 3 mess!

I Explain the hypercharges! I Explain other patterns: 5 I dim VSM = 32 = 2 ; I symmetry between quarks and leptons; I asymmetry between left and right.

The Algebra of Grand Unified Theories Grand Unified Theories

The GUTs Goal: I GSM = U(1) × SU(2) × SU(3) is a mess! I Explain the hypercharges! I Explain other patterns: 5 I dim VSM = 32 = 2 ; I symmetry between quarks and leptons; I asymmetry between left and right.

The Algebra of Grand Unified Theories Grand Unified Theories

The GUTs Goal: I GSM = U(1) × SU(2) × SU(3) is a mess! 2 3∗ I V = C−1 ⊗ C ⊗ C ⊕ · · · ⊕ C 2 ⊗ C ⊗ C is a SM 3 mess! I Explain other patterns: 5 I dim VSM = 32 = 2 ; I symmetry between quarks and leptons; I asymmetry between left and right.

The Algebra of Grand Unified Theories Grand Unified Theories

The GUTs Goal: I GSM = U(1) × SU(2) × SU(3) is a mess! 2 3∗ I V = C−1 ⊗ C ⊗ C ⊕ · · · ⊕ C 2 ⊗ C ⊗ C is a SM 3 mess!

I Explain the hypercharges! The Algebra of Grand Unified Theories Grand Unified Theories

The GUTs Goal: I GSM = U(1) × SU(2) × SU(3) is a mess! 2 3∗ I V = C−1 ⊗ C ⊗ C ⊕ · · · ⊕ C 2 ⊗ C ⊗ C is a SM 3 mess!

I Explain the hypercharges! I Explain other patterns: 5 I dim VSM = 32 = 2 ; I symmetry between quarks and leptons; I asymmetry between left and right. I V is also representation of GSM; I V may break apart into more GSM-irreps than G-irreps.

The Algebra of Grand Unified Theories Grand Unified Theories

The GUTs trick: if V is a representation of G and GSM ⊆ G, then 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. In short: V becomes isomorphic to VSM when we restrict from G to GSM.

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 φ GSM / G ρ  U(f )  U(VSM) / U(V ) commutes. 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 φ GSM / G ρ  U(f )  U(VSM) / U(V ) commutes. In short: V becomes isomorphic to VSM when we restrict from G to GSM. 5 I Take C = hu, d, r, g, bi. 5 I C is a representation of SU(5), as is the 32-dimensional exterior algebra: 5 ∼ 0 5 1 5 2 5 3 5 4 5 5 5 ΛC = Λ C ⊕ Λ C ⊕ Λ C ⊕ Λ C ⊕ Λ C ⊕ Λ C

I Theorem There’s a homomorphism φ: GSM → SU(5) and 5 a linear isomorphism h: VSM → ΛC making φ GSM / SU(5)

 U(h)  5 U(VSM) / U(ΛC ) commute.

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 “2 isospins + 3 colors = 5 things”: I Theorem There’s a homomorphism φ: GSM → SU(5) and 5 a linear isomorphism h: VSM → ΛC making φ GSM / SU(5)

 U(h)  5 U(VSM) / U(ΛC ) commute.

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 “2 isospins + 3 colors = 5 things”: 5 I Take C = hu, d, r, g, bi. 5 I C is a representation of SU(5), as is the 32-dimensional exterior algebra: 5 ∼ 0 5 1 5 2 5 3 5 4 5 5 5 ΛC = Λ C ⊕ Λ C ⊕ Λ C ⊕ Λ C ⊕ Λ C ⊕ Λ C 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 “2 isospins + 3 colors = 5 things”: 5 I Take C = hu, d, r, g, bi. 5 I C is a representation of SU(5), as is the 32-dimensional exterior algebra: 5 ∼ 0 5 1 5 2 5 3 5 4 5 5 5 ΛC = Λ C ⊕ Λ C ⊕ Λ C ⊕ Λ C ⊕ Λ C ⊕ Λ C

I Theorem There’s a homomorphism φ: GSM → SU(5) and 5 a linear isomorphism h: VSM → ΛC making φ GSM / SU(5)

 U(h)  5 U(VSM) / U(ΛC ) commute. We define φ by

 α3g 0  φ:(α, g, h) ∈ U(1)×SU(2)×SU(3) 7−→ ∈ SU(5) 0 α−2h

I Can find φ: GSM → S(U(2) × U(3)) ⊆ SU(5). 5 I The representation ΛC of SU(5) is isomorphic to VSM when pulled back to GSM.

The Algebra of Grand Unified Theories Grand Unified Theories The SU(5) Theory

Proof

I Let S(U(2) × U(3)) ⊆ SU(5) be the subgroup preserving 2 3 ∼ 5 the 2 + 3 splitting C ⊕ C = C . We define φ by

 α3g 0  φ:(α, g, h) ∈ U(1)×SU(2)×SU(3) 7−→ ∈ SU(5) 0 α−2h

5 I The representation ΛC of SU(5) is isomorphic to VSM when pulled back to GSM.

The Algebra of Grand Unified Theories Grand Unified Theories The SU(5) Theory

Proof

I Let S(U(2) × U(3)) ⊆ SU(5) be the subgroup preserving 2 3 ∼ 5 the 2 + 3 splitting C ⊕ C = C . I Can find φ: GSM → S(U(2) × U(3)) ⊆ SU(5). We define φ by

 α3g 0  φ:(α, g, h) ∈ U(1)×SU(2)×SU(3) 7−→ ∈ SU(5) 0 α−2h

The Algebra of Grand Unified Theories Grand Unified Theories The SU(5) Theory

Proof

I Let S(U(2) × U(3)) ⊆ SU(5) be the subgroup preserving 2 3 ∼ 5 the 2 + 3 splitting C ⊕ C = C . I Can find φ: GSM → S(U(2) × U(3)) ⊆ SU(5). 5 I The representation ΛC of SU(5) is isomorphic to VSM when pulled back to GSM. The Algebra of Grand Unified Theories Grand Unified Theories The SU(5) Theory

Proof

I Let S(U(2) × U(3)) ⊆ SU(5) be the subgroup preserving 2 3 ∼ 5 the 2 + 3 splitting C ⊕ C = C . I Can find φ: GSM → S(U(2) × U(3)) ⊆ SU(5). 5 I The representation ΛC of SU(5) is isomorphic to VSM when pulled back to GSM. We define φ by

 α3g 0  φ:(α, g, h) ∈ U(1)×SU(2)×SU(3) 7−→ ∈ SU(5) 0 α−2h Thus ∼ GSM/Z6 = S(U(2) × U(3)) The subgroup Z6 ⊆ GSM acts trivially on VSM.

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: −3 2 6 ∼ ker φ = {(α, α , α )|α = 1} = Z6 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: −3 2 6 ∼ ker φ = {(α, α , α )|α = 1} = Z6

Thus ∼ GSM/Z6 = S(U(2) × U(3)) The subgroup Z6 ⊆ GSM acts trivially on VSM. As a GSM-representation, 2 ∼ 0 2 1 2 2 2 ΛC = C0 ⊗ Λ C ⊕ C1 ⊗ Λ C ⊕ C2 ⊗ Λ C

As a GSM-representation, 3 ∼ 0 3 1 3 2 3 3 3 ΛC = C0⊗Λ C ⊕ C 2 ⊗Λ C ⊕ C 4 ⊗Λ C ⊕ C−2⊗Λ C − 3 − 3

The Algebra of Grand Unified Theories Grand Unified Theories The SU(5) Theory

Because GSM respects the 2 + 3 splitting 5 ∼ 2 3 ∼ 2 3 ΛC = Λ(C ⊕ C ) = ΛC ⊗ ΛC As a GSM-representation, 3 ∼ 0 3 1 3 2 3 3 3 ΛC = C0⊗Λ C ⊕ C 2 ⊗Λ C ⊕ C 4 ⊗Λ C ⊕ C−2⊗Λ C − 3 − 3

The Algebra of Grand Unified Theories Grand Unified Theories The SU(5) Theory

Because GSM respects the 2 + 3 splitting 5 ∼ 2 3 ∼ 2 3 ΛC = Λ(C ⊕ C ) = ΛC ⊗ ΛC

As a GSM-representation, 2 ∼ 0 2 1 2 2 2 ΛC = C0 ⊗ Λ C ⊕ C1 ⊗ Λ C ⊕ C2 ⊗ Λ C The Algebra of Grand Unified Theories Grand Unified Theories The SU(5) Theory

Because GSM respects the 2 + 3 splitting 5 ∼ 2 3 ∼ 2 3 ΛC = Λ(C ⊕ C ) = ΛC ⊗ ΛC

As a GSM-representation, 2 ∼ 0 2 1 2 2 2 ΛC = C0 ⊗ Λ C ⊕ C1 ⊗ Λ C ⊕ C2 ⊗ Λ C

As a GSM-representation, 3 ∼ 0 3 1 3 2 3 3 3 ΛC = C0⊗Λ C ⊕ C 2 ⊗Λ C ⊕ C 4 ⊗Λ C ⊕ C−2⊗Λ C − 3 − 3 5 Thus there’s a linear isomorphism h: VSM → ΛC making

φ GSM / SU(5)

 U(h)  5 U(VSM) / U(ΛC )

commute.

The Algebra of Grand Unified Theories Grand Unified Theories The SU(5) Theory

2 ∼ 2∗ Then tensor them together, use C = C and ∼ CY1 ⊗ CY2 = CY1+Y2 to see how ∼ 5 VSM = ΛC

as GSM-representations. The Algebra of Grand Unified Theories Grand Unified Theories The SU(5) Theory

2 ∼ 2∗ Then tensor them together, use C = C and ∼ CY1 ⊗ CY2 = CY1+Y2 to see how ∼ 5 VSM = ΛC

as GSM-representations. 5 Thus there’s a linear isomorphism h: VSM → ΛC making

φ GSM / SU(5)

 U(h)  5 U(VSM) / U(ΛC )

commute. 3 4 I Unify the C ⊕ C representation of SU(3) into the irrep C of SU(4).

I This creates explicit symmetry between quarks and leptons. 2 I Unify the C ⊕ 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.

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: 2 I Unify the C ⊕ 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.

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: 3 4 I Unify the C ⊕ C representation of SU(3) into the irrep C of SU(4).

I This creates explicit symmetry between quarks and leptons. 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: 3 4 I Unify the C ⊕ C representation of SU(3) into the irrep C of SU(4).

I This creates explicit symmetry between quarks and leptons. 2 I Unify the C ⊕ 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. The Algebra of Grand Unified Theories Grand Unified Theories The Pati–Salam Model

Standard Model Representation Name Symbol GSM-representation   νL 2 Left-handed leptons − C−1 ⊗ C ⊗ C eL

r g b ! uL, uL , uL 2 3 Left-handed quarks 1 ⊗ ⊗ r g b C 3 C C dL, dL , dL

Right-handed neutrino νR C0 ⊗ C ⊗ C − Right-handed electron eR C−2 ⊗ C ⊗ C

r g b 3 Right-handed up quarks u , u , u 4 ⊗ ⊗ R R R C 3 C C

r g b 3 Right-handed down quarks d , d , d 2 ⊗ ⊗ R R R C− 3 C C I Write GPS = SU(2) × SU(2) × SU(4). 2 4 2 4 I Write VPS = C ⊗ C ⊗ C ⊕ C ⊗ C ⊗ C ⊕ dual.

The Algebra of Grand Unified Theories Grand Unified Theories The Pati–Salam Model

The Pati–Salam representation Name Symbol SU(2) × SU(2) × SU(4)- representation ! ν , ur , ug, ub L L L L 2⊗ ⊗ 4 Left-handed fermions − r g b C C C eL , dL, dL , dL ! ν , ur , ug , ub R R R R ⊗ 2⊗ 4 Right-handed fermions − r g b C C C eR , dR, dR, dR 2 4 2 4 I Write VPS = C ⊗ C ⊗ C ⊕ C ⊗ C ⊗ C ⊕ dual.

The Algebra of Grand Unified Theories Grand Unified Theories The Pati–Salam Model

The Pati–Salam representation Name Symbol SU(2) × SU(2) × SU(4)- representation ! ν , ur , ug, ub L L L L 2⊗ ⊗ 4 Left-handed fermions − r g b C C C eL , dL, dL , dL ! ν , ur , ug , ub R R R R ⊗ 2⊗ 4 Right-handed fermions − r g b C C C eR , dR, dR, dR

I Write GPS = SU(2) × SU(2) × SU(4). The Algebra of Grand Unified Theories Grand Unified Theories The Pati–Salam Model

The Pati–Salam representation Name Symbol SU(2) × SU(2) × SU(4)- representation ! ν , ur , ug, ub L L L L 2⊗ ⊗ 4 Left-handed fermions − r g b C C C eL , dL, dL , dL ! ν , ur , ug , ub R R R R ⊗ 2⊗ 4 Right-handed fermions − r g b C C C eR , dR, dR, dR

I Write GPS = SU(2) × SU(2) × SU(4). 2 4 2 4 I Write VPS = C ⊗ C ⊗ C ⊕ C ⊗ C ⊗ C ⊕ dual. The Algebra of Grand Unified Theories Grand Unified Theories The Pati–Salam Model

To make the Pati–Salam model work, we need to prove Theorem There exists maps θ : GSM → GPS and f : VSM → VPS which make the diagram

θ GSM / GPS

 U(f )  U(VSM) / U(VPS)

commute. I Pick θ so GSM maps to a subgroup of SU(2) × SU(2) × SU(4) that preserves the 3 + 1 splitting

4 ∼ 3 C = C ⊕ C

and the 1 + 1 splitting

2 ∼ C ⊗ C = C ⊕ C

The Algebra of Grand Unified Theories Grand Unified Theories The Pati–Salam Model

Proof

I Want θ : GSM → SU(2) × SU(2) × SU(4): 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

4 ∼ 3 C = C ⊕ C

and the 1 + 1 splitting

2 ∼ C ⊗ C = C ⊕ C n I Spin(2n) has a representation ΛC , called the Dirac spinors. ∼ 2 2 ∼ 2 I SU(2) × SU(2) = Spin(4), and C ⊗ C ⊕ C ⊗ C = ΛC ∼ 4 4∗ ∼ 3 I SU(4) = Spin(6), and C ⊕ C = ΛC . ∼ 2 3 I VPS = ΛC ⊗ ΛC as a representation of ∼ GPS = Spin(4) × Spin(6). 4 ∼ odd 3 ∼ 1 3 3 3 I C = Λ C = Λ C ⊕ Λ C has a 3 + 1 splitting — the grading! 2 ∼ ev 2 ∼ 0 2 2 2 I C ⊗ C = Λ C = Λ C ⊕ Λ C has a 1 + 1 splitting — the grading!

The Algebra of Grand Unified Theories Grand Unified Theories The Pati–Salam Model

We need some facts: ∼ 2 3 I VPS = ΛC ⊗ ΛC as a representation of ∼ GPS = Spin(4) × Spin(6). 4 ∼ odd 3 ∼ 1 3 3 3 I C = Λ C = Λ C ⊕ Λ C has a 3 + 1 splitting — the grading! 2 ∼ ev 2 ∼ 0 2 2 2 I C ⊗ C = Λ C = Λ C ⊕ Λ C has a 1 + 1 splitting — the grading!

The Algebra of Grand Unified Theories Grand Unified Theories The Pati–Salam Model

We need some facts: n I Spin(2n) has a representation ΛC , called the Dirac spinors. ∼ 2 2 ∼ 2 I SU(2) × SU(2) = Spin(4), and C ⊗ C ⊕ C ⊗ C = ΛC ∼ 4 4∗ ∼ 3 I SU(4) = Spin(6), and C ⊕ C = ΛC . 4 ∼ odd 3 ∼ 1 3 3 3 I C = Λ C = Λ C ⊕ Λ C has a 3 + 1 splitting — the grading! 2 ∼ ev 2 ∼ 0 2 2 2 I C ⊗ C = Λ C = Λ C ⊕ Λ C has a 1 + 1 splitting — the grading!

The Algebra of Grand Unified Theories Grand Unified Theories The Pati–Salam Model

We need some facts: n I Spin(2n) has a representation ΛC , called the Dirac spinors. ∼ 2 2 ∼ 2 I SU(2) × SU(2) = Spin(4), and C ⊗ C ⊕ C ⊗ C = ΛC ∼ 4 4∗ ∼ 3 I SU(4) = Spin(6), and C ⊕ C = ΛC . ∼ 2 3 I VPS = ΛC ⊗ ΛC as a representation of ∼ GPS = Spin(4) × Spin(6). 2 ∼ ev 2 ∼ 0 2 2 2 I C ⊗ C = Λ C = Λ C ⊕ Λ C has a 1 + 1 splitting — the grading!

The Algebra of Grand Unified Theories Grand Unified Theories The Pati–Salam Model

We need some facts: n I Spin(2n) has a representation ΛC , called the Dirac spinors. ∼ 2 2 ∼ 2 I SU(2) × SU(2) = Spin(4), and C ⊗ C ⊕ C ⊗ C = ΛC ∼ 4 4∗ ∼ 3 I SU(4) = Spin(6), and C ⊕ C = ΛC . ∼ 2 3 I VPS = ΛC ⊗ ΛC as a representation of ∼ GPS = Spin(4) × Spin(6). 4 ∼ odd 3 ∼ 1 3 3 3 I C = Λ C = Λ C ⊕ Λ C has a 3 + 1 splitting — the grading! The Algebra of Grand Unified Theories Grand Unified Theories The Pati–Salam Model

We need some facts: n I Spin(2n) has a representation ΛC , called the Dirac spinors. ∼ 2 2 ∼ 2 I SU(2) × SU(2) = Spin(4), and C ⊗ C ⊕ C ⊗ C = ΛC ∼ 4 4∗ ∼ 3 I SU(4) = Spin(6), and C ⊕ C = ΛC . ∼ 2 3 I VPS = ΛC ⊗ ΛC as a representation of ∼ GPS = Spin(4) × Spin(6). 4 ∼ odd 3 ∼ 1 3 3 3 I C = Λ C = Λ C ⊕ Λ C has a 3 + 1 splitting — the grading! 2 ∼ ev 2 ∼ 0 2 2 2 I C ⊗ C = Λ C = Λ C ⊕ Λ C has a 1 + 1 splitting — the grading! I θ maps GSM onto the subgroup SU(2) × S(U(1) × U(1)) ⊆ Spin(4) that preserves the 1 + 1 splitting:

  α3 0  (α, x, y) ∈ U(1) × SU(2) × SU(3) 7→ x, 0 α−3

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:

 αy 0  (α, x, y) ∈ U(1) × SU(2) × SU(3) 7→ 0 α−3 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:

 αy 0  (α, x, y) ∈ U(1) × SU(2) × SU(3) 7→ 0 α−3

I θ maps GSM onto the subgroup SU(2) × S(U(1) × U(1)) ⊆ Spin(4) that preserves the 1 + 1 splitting:

  α3 0  (α, x, y) ∈ U(1) × SU(2) × SU(3) 7→ x, 0 α−3 Earlier, we had 2 ∼ 0 2 1 2 2 2 ΛC = C0 ⊗ Λ C ⊕ C1 ⊗ Λ C ⊕ C2 ⊗ Λ C

As a GSM-representation, 3 ∼ 0 3 1 3 2 3 3 3 ΛC = C1⊗Λ C ⊕ C 1 ⊗Λ C ⊕ C 1 ⊗Λ C ⊕ C−1⊗Λ C 3 − 3 Earlier, we had 3 ∼ 0 3 1 3 2 3 3 3 ΛC = C0⊗Λ C ⊕ C 2 ⊗Λ C ⊕ C 4 ⊗Λ C ⊕ C−2⊗Λ C − 3 − 3

The Algebra of Grand Unified Theories Grand Unified Theories The Pati–Salam Model The payoff: As a GSM-representation, 2 ∼ 0 2 1 2 2 2 ΛC = C−1 ⊗ Λ C ⊕ C0 ⊗ Λ C ⊕ C1 ⊗ Λ C As a GSM-representation, 3 ∼ 0 3 1 3 2 3 3 3 ΛC = C1⊗Λ C ⊕ C 1 ⊗Λ C ⊕ C 1 ⊗Λ C ⊕ C−1⊗Λ C 3 − 3 Earlier, we had 3 ∼ 0 3 1 3 2 3 3 3 ΛC = C0⊗Λ C ⊕ C 2 ⊗Λ C ⊕ C 4 ⊗Λ C ⊕ C−2⊗Λ C − 3 − 3

The Algebra of Grand Unified Theories Grand Unified Theories The Pati–Salam Model The payoff: As a GSM-representation, 2 ∼ 0 2 1 2 2 2 ΛC = C−1 ⊗ Λ C ⊕ C0 ⊗ Λ C ⊕ C1 ⊗ Λ C Earlier, we had 2 ∼ 0 2 1 2 2 2 ΛC = C0 ⊗ Λ C ⊕ C1 ⊗ Λ C ⊕ C2 ⊗ Λ C Earlier, we had 3 ∼ 0 3 1 3 2 3 3 3 ΛC = C0⊗Λ C ⊕ C 2 ⊗Λ C ⊕ C 4 ⊗Λ C ⊕ C−2⊗Λ C − 3 − 3

The Algebra of Grand Unified Theories Grand Unified Theories The Pati–Salam Model The payoff: As a GSM-representation, 2 ∼ 0 2 1 2 2 2 ΛC = C−1 ⊗ Λ C ⊕ C0 ⊗ Λ C ⊕ C1 ⊗ Λ C Earlier, we had 2 ∼ 0 2 1 2 2 2 ΛC = C0 ⊗ Λ C ⊕ C1 ⊗ Λ C ⊕ C2 ⊗ Λ C

As a GSM-representation, 3 ∼ 0 3 1 3 2 3 3 3 ΛC = C1⊗Λ C ⊕ C 1 ⊗Λ C ⊕ C 1 ⊗Λ C ⊕ C−1⊗Λ C 3 − 3 The Algebra of Grand Unified Theories Grand Unified Theories The Pati–Salam Model The payoff: As a GSM-representation, 2 ∼ 0 2 1 2 2 2 ΛC = C−1 ⊗ Λ C ⊕ C0 ⊗ Λ C ⊕ C1 ⊗ Λ C Earlier, we had 2 ∼ 0 2 1 2 2 2 ΛC = C0 ⊗ Λ C ⊕ C1 ⊗ Λ C ⊕ C2 ⊗ Λ C

As a GSM-representation, 3 ∼ 0 3 1 3 2 3 3 3 ΛC = C1⊗Λ C ⊕ C 1 ⊗Λ C ⊕ C 1 ⊗Λ C ⊕ C−1⊗Λ C 3 − 3 Earlier, we had 3 ∼ 0 3 1 3 2 3 3 3 ΛC = C0⊗Λ C ⊕ C 2 ⊗Λ C ⊕ C 4 ⊗Λ C ⊕ C−2⊗Λ C − 3 − 3 Thus there’s an isomorphism of vector spaces 2 3 f : VSM → ΛC ⊗ ΛC such that

θ GSM / Spin(4) × Spin(6)

 U(f )  2 3 U(VSM) / U(ΛC ⊗ ΛC )

commutes.

The Algebra of Grand Unified Theories Grand Unified Theories The Pati–Salam Model

∼ 2 3 We can recycle the fact that VSM = ΛC ⊗ ΛC from the SU(5) theory. The Algebra of Grand Unified Theories Grand Unified Theories The Pati–Salam Model

∼ 2 3 We can recycle the fact that VSM = ΛC ⊗ ΛC from the SU(5) theory. Thus there’s an isomorphism of vector spaces 2 3 f : VSM → ΛC ⊗ ΛC such that

θ GSM / Spin(4) × Spin(6)

 U(f )  2 3 U(VSM) / U(ΛC ⊗ ΛC )

commutes. I Set n = 5: ψ SU(5) / Spin(10)

 1  U(ΛC5) / U(ΛC5)

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 to Georgi:

I In general, ψ SU(n) / Spin(2n) KK KK KK KK K%  U(ΛCn) 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 to Georgi:

I In general, ψ SU(n) / Spin(2n) KK KK KK KK K%  U(ΛCn)

I Set n = 5: ψ SU(5) / Spin(10)

 1  U(ΛC5) / U(ΛC5) I Set n = 2 and m = 3:

η Spin(4) × Spin(6) / Spin(10)

 U(g)  5 U(ΛC2 ⊗ ΛC3) / U(ΛC )

The Algebra of Grand Unified Theories Grand Unified Theories The Spin(10) Theory

Or extend the Pati–Salam model:

I In general,

η Spin(2n) × Spin(2m) / Spin(2n + 2m)

( ) n  m U g  + U(ΛC ⊗ ΛC ) / U(ΛCn m) The Algebra of Grand Unified Theories Grand Unified Theories The Spin(10) Theory

Or extend the Pati–Salam model:

I In general,

η Spin(2n) × Spin(2m) / Spin(2n + 2m)

( ) n  m U g  + U(ΛC ⊗ ΛC ) / U(ΛCn m)

I Set n = 2 and m = 3:

η Spin(4) × Spin(6) / Spin(10)

 U(g)  5 U(ΛC2 ⊗ ΛC3) / U(ΛC ) The Algebra of Grand Unified Theories Conclusion

Theorem The cube of GUTs

φ GSM / SU(5) nn s θ nnn ψ ss nnn sss nnn ss vnn η yss Spin(4) × Spin(6) / Spin(10)

 U(h)  5 U(VSM) / U(ΛC ) n s U(f ) nnn 1 ss nnn sss nnn ss  vnn U(g)  yss 5 U(ΛC2 ⊗ ΛC3) / U(ΛC )

commutes. 5 I The two maps from GSM to U(ΛC ) are equal: φ GSM / SU(5)

θ ψ  η  Spin(4) × Spin(6) / Spin(10) / U(ΛC5)

5 I ΛC is a faithful representation of Spin(10) — the top face commutes: φ GSM / SU(5)

θ ψ  η  Spin(4) × Spin(6) / Spin(10)

The Algebra of Grand Unified Theories Conclusion

Proof I The vertical faces of the cube commute. 5 I ΛC is a faithful representation of Spin(10) — the top face commutes: φ GSM / SU(5)

θ ψ  η  Spin(4) × Spin(6) / Spin(10)

The Algebra of Grand Unified Theories Conclusion

Proof I The vertical faces of the cube commute. 5 I The two maps from GSM to U(ΛC ) are equal: φ GSM / SU(5)

θ ψ  η  Spin(4) × Spin(6) / Spin(10) / U(ΛC5) The Algebra of Grand Unified Theories Conclusion

Proof I The vertical faces of the cube commute. 5 I The two maps from GSM to U(ΛC ) are equal: φ GSM / SU(5)

θ ψ  η  Spin(4) × Spin(6) / Spin(10) / U(ΛC5)

5 I ΛC is a faithful representation of Spin(10) — the top face commutes: φ GSM / SU(5)

θ ψ  η  Spin(4) × Spin(6) / Spin(10) I The bottom face commutes:

U(h) 5 U(VSM) / U(ΛC )

U(f ) 1  U(g)  5 U(ΛC2 ⊗ ΛC3) / U(ΛC )

The Algebra of Grand Unified Theories Conclusion

I The intertwiners commute:

h VSM / ΛC5 ss9 ss f ss ss g  ss ΛC2 ⊗ ΛC3 The Algebra of Grand Unified Theories Conclusion

I The intertwiners commute:

h VSM / ΛC5 ss9 ss f ss ss g  ss ΛC2 ⊗ ΛC3

I The bottom face commutes:

U(h) 5 U(VSM) / U(ΛC )

U(f ) 1  U(g)  5 U(ΛC2 ⊗ ΛC3) / U(ΛC ) The Algebra of Grand Unified Theories Conclusion

Thus the cube

φ GSM / SU(5) nn s θ nnn ψ ss nnn sss nnn ss vnn η yss Spin(4) × Spin(6) / Spin(10)

 U(h)  5 U(VSM) / U(ΛC ) n s U(f ) nnn 1 ss nnn sss nnn ss  vnn U(g)  yss 5 U(ΛC2 ⊗ ΛC3) / U(ΛC )

commutes.