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Noncommutative Geometry models for ParticlePhysics and Cosmology, Lecture I

Matilde Marcolli

Villa de Leyva school, July 2011

Matilde Marcolli NCG models for particles and cosmology, I

Plan of lectures

1 Noncommutative geometry approach to elementary particlephysics; Noncommutative Riemannian geometry; finitenoncommutative geometries; moduli spaces; the finitegeometry of the Standard Model

2 The product geometry; the spectral action functional and itsasymptotic expansion; bosons and fermions; the StandardModel Lagrangian; renormalization group flow, geometricconstraints and low energy limits

3 Parameters: relations at unification and running; running ofthe gravitational terms; the RGE flow with right handedneutrinos; cosmological timeline and the inflation epoch;effective gravitational and cosmological constants and modelsof inflation

4 The spectral action and the problem of cosmic topology;cosmic topology and the CMB; slow-roll inflation; Poissonsummation formula and the nonperturbative spectral action;spherical and flat space forms


Geometrization of physics

Kaluza-Klein theory: electromagnetism described by circlebundle over spacetime manifold, connection = EM potential;

Yang–Mills gauge theories: bundle geometry over spacetime,connections = gauge potentials, sections = fermions;

String theory: 6 extra dimensions (Calabi-Yau) overspacetime, strings vibrations = types of particles

NCG models: extra dimensions are NC spaces, pure gravity onproduct space becomes gravity + matter on spacetime


Why a geometrization of physics?The standard model of elementary particles


Standard model Lagrangian: can compute from simpler data?

LSM = −12∂νg

aµ∂νg

aµ − gsf

abc∂µgaνg

bµg

cν − 1

4g2sf

abcfadegbµgcνg

dµg

eν − ∂νW

+µ ∂νW

−µ −

M2W+µ W−

µ − 12∂νZ

0µ∂νZ

0µ − 1

2c2wM2Z0

µZ0µ − 1

2∂µAν∂µAν − igcw(∂νZ

0µ(W

+µ W−

ν −W+

ν W−µ )− Z0

ν (W+µ ∂νW

−µ −W−

µ ∂νW+µ ) + Z0

µ(W+ν ∂νW

−µ −W−

ν ∂νW+µ ))−

igsw(∂νAµ(W+µ W−

ν −W+ν W−

µ )−Aν(W+µ ∂νW

−µ −W−

µ ∂νW+µ ) + Aµ(W

+ν ∂νW

−µ −

W−ν ∂νW

+µ ))− 1

2g2W+

µ W−µ W+

ν W−ν + 1

2g2W+

µ W−ν W+

µ W−ν + g2c2w(Z

0µW

+µ Z0

νW−ν −

Z0µZ

0µW

+ν W−

ν ) + g2s2w(AµW+µ AνW

−ν − AµAµW

+ν W−

ν ) + g2swcw(AµZ0ν (W

+µ W−

ν −W+

ν W−µ )− 2AµZ

0µW

+ν W−

ν )− 12∂µH∂µH − 2M2αhH

2 − ∂µφ+∂µφ

− − 12∂µφ

0∂µφ0 −

βh

(2M2

g2+ 2M

gH + 1

2(H2 + φ0φ0 + 2φ+φ−)

)+ 2M4

g2αh −

gαhM (H3 +Hφ0φ0 + 2Hφ+φ−)−18g2αh (H

4 + (φ0)4 + 4(φ+φ−)2 + 4(φ0)2φ+φ− + 4H2φ+φ− + 2(φ0)2H2)−gMW+

µ W−µ H − 1

2gMc2wZ0

µZ0µH −

12ig

(W+

µ (φ0∂µφ− − φ−∂µφ

0)−W−µ (φ0∂µφ

+ − φ+∂µφ0))+

12g(W+

µ (H∂µφ− − φ−∂µH) +W−

µ (H∂µφ+ − φ+∂µH)

)+ 1

2g 1cw(Z0

µ(H∂µφ0 − φ0∂µH) +

M ( 1cwZ0

µ∂µφ0+W+

µ ∂µφ−+W−

µ ∂µφ+)−ig s2w

cwMZ0

µ(W+µ φ−−W−

µ φ+)+igswMAµ(W+µ φ−−

W−µ φ+)− ig 1−2c2w

2cwZ0

µ(φ+∂µφ

− − φ−∂µφ+) + igswAµ(φ+∂µφ

− − φ−∂µφ+)−14g2W+

µ W−µ (H2 + (φ0)2 + 2φ+φ−)− 1

8g2 1

c2wZ0

µZ0µ (H

2 + (φ0)2 + 2(2s2w − 1)2φ+φ−)−12g2 s

2w

cwZ0

µφ0(W+

µ φ− +W−µ φ+)− 1

2ig2 s

2w

cwZ0

µH(W+µ φ− −W−

µ φ+) + 12g2swAµφ

0(W+µ φ− +

W−µ φ+) + 1

2ig2swAµH(W+

µ φ− −W−µ φ+)− g2 sw

cw(2c2w − 1)Z0

µAµφ+φ− −

g2s2wAµAµφ+φ− + 1

2igs λ

aij(q

σi γ

µqσj )gaµ − eλ(γ∂ +mλ

e )eλ − νλ(γ∂ +mλ

ν)νλ − uλ

j (γ∂ +

mλu)u

λj − dλj (γ∂ +mλ

d)dλj + igswAµ

(−(eλγµeλ) + 2

3(uλ

j γµuλ

j )− 13(dλj γ

µdλj ))+

ig4cw

Z0µ{(νλγµ(1 + γ5)νλ) + (eλγµ(4s2w − 1− γ5)eλ) + (dλj γ

µ(43s2w − 1− γ5)dλj ) +

(uλj γ

µ(1− 83s2w + γ5)uλ

j )}+ ig

2√2W+

µ

((νλγµ(1 + γ5)U lep

λκeκ) + (uλ

j γµ(1 + γ5)Cλκd

κj ))+

ig

2√2W−

µ

((eκU lep†

κλγµ(1 + γ5)νλ) + (dκjC

†κλγ

µ(1 + γ5)uλj ))+

ig

2M√2φ+

(−mκ

e (νλU lep

λκ(1− γ5)eκ) +mλν(ν

λU lepλκ(1 + γ5)eκ

)+

ig

2M√2φ−

(mλ

e (eλU lep†

λκ(1 + γ5)νκ)−mκν(e

λU lep†λκ(1− γ5)νκ

)− g

2mλ

ν

MH(νλνλ)−

g2mλ

e

MH(eλeλ) + ig

2mλ

ν

Mφ0(νλγ5νλ)− ig

2mλ

e

Mφ0(eλγ5eλ)− 1

4νλ M

Rλκ (1− γ5)νκ −

14νλ M

Rλκ (1− γ5)νκ +

ig

2M√2φ+

(−mκ

d(uλjCλκ(1− γ5)dκj ) +mλ

u(uλjCλκ(1 + γ5)dκj

)+

ig

2M√2φ−

(mλ

d(dλjC

†λκ(1 + γ5)uκ

j )−mκu(d

λjC

†λκ(1− γ5)uκ

j

)− g

2mλ

u

MH(uλ

juλj )−

g2

mλd

MH(dλj d

λj ) +

ig2

mλu

Mφ0(uλ

j γ5uλ

j )− ig2

mλd

Mφ0(dλj γ

5dλj ) + Ga∂2Ga + gsfabc∂µG

aGbgcµ +

X+(∂2 −M2)X+ + X−(∂2 −M2)X−+X0(∂2 − M2

c2w)X0 + Y ∂2Y + igcwW

+µ (∂µX

0X− −∂µX

+X0)+igswW+µ (∂µY X− − ∂µX

+Y ) + igcwW−µ (∂µX

−X0 −∂µX

0X+)+igswW−µ (∂µX

−Y − ∂µY X+) + igcwZ0µ(∂µX

+X+ −∂µX

−X−)+igswAµ(∂µX+X+ −

∂µX−X−)−1

2gM

(X+X+H + X−X−H + 1

c2wX0X0H

)+1−2c2w

2cwigM

(X+X0φ+ − X−X0φ−)+

12cw

igM(X0X−φ+ − X0X+φ−)+igMsw

(X0X−φ+ − X0X+φ−)+

12igM

(X+X+φ0 − X−X−φ0

).

1


Standard model parameters: are there relations? why these values?


Building mathematical models: essential requirements

Conceptualize: complicated things (SM Lagrangian) shouldfollow from simple things (geometry)

Enrich: extend existing models with new features

Predict: Higgs mass? new particles? new parameter relations?new phenomena?

The elephant in the room: Gravity!

Coupling gravity to matter: the good: better conceptualstructure, links particle physics to cosmology; the bad: worsechances of passing from classical to quantum theory

In NCG models: “all forces become gravity” (on an NC space)


What the NCG model provides:

Einstein–Hilbert action:

SEH(gµν) =1

16πG

∫

MR√

g d4x

Gravity minimally coupled to matter: S = SEH + SSM withSSM = particle physics ⇒ Standard Model Lagrangian

Right handed neutrinos with Majorana masses

Modified gravity model f (R,Rµν ,Cλµνκ) with conformalgravity: Weyl curvature tensor

Cλµνκ = Rλµνκ −1

2(gλνRµκ − gλκRµν − gµνRλκ + gµκRλν)

+1

6Rαα (gλνgµκ − gλκgµν)

Non-minimal coupling of Higgs to gravity∫

M

(1

2|DµH|2 − µ2

0|H|2 − ξ0 R |H|2 + λ0|H|4)√

g d4x


What is Noncommutative Geometry?

X compact Hausdorff topological space ⇔ C (X ) abelianC ∗-algebra of continuous functions (Gelfand–Naimark)

Noncommutative C ∗-algebra A: think of as “continuousfunctions on an NC space”

Describe geometry in terms of algebra C (X ) and densesubalgebras like C∞(X ) if X manifold

Differential forms, bundles, connections, cohomology: allcontinue to make sense without commutativity (NC geometry)

Can describe “bad quotients” as good spaces: functions onthe graph of the equivalence relation with convolution product

To do physics: need the analog of Riemannian geometry


Spectral triples and NC Riemannian manifolds (A,H,D)

involutive algebra Arepresentation π : A → L(H)

self adjoint operator D on H, dense domain

compact resolvent (1 + D2)−1/2 ∈ K[a,D] bounded ∀a ∈ Aeven if Z/2- grading γ on H

[γ, a] = 0, ∀a ∈ A, Dγ = −γD

Main example (C∞(M), L2(M, S), /∂M) with chirality γ5 in 4-dim

• Alain Connes, Geometry from the spectral point of view, Lett.Math. Phys. 34 (1995), no. 3, 203–238.

Note: to apply spectral triples methods need to work with Mcompact and Euclidean signature (Euclidean gravity)


Real structure KO-dimension n ∈ Z/8Zantilinear isometry J : H → H

J2 = ε, JD = ε′DJ, and Jγ = ε′′γJ

n 0 1 2 3 4 5 6 7

ε 1 1 -1 -1 -1 -1 1 1ε′ 1 -1 1 1 1 -1 1 1ε′′ 1 -1 1 -1

Commutation: [a, b0] = 0 ∀ a, b ∈ Awhere b0 = Jb∗J−1 ∀b ∈ AOrder one condition:

[[D, a], b0] = 0 ∀ a, b ∈ A


Spectral triples in NCG need not be manifolds:

Quantum groups

Fractals

NC tori

For particle physics models M × F , product of a 4-dimensionalspacetime manifold M by a “finite NC space” F (extra dimensions)

Alain Connes, Gravity coupled with matter and the foundationof non-commutative geometry, Comm. Math. Phys. 182(1996), no. 1, 155–176.

Almost commutative geometries: more general form, fibration (notproduct) over manifold, fiber finite NC space

Branimir Cacic, A reconstruction theorem foralmost-commutative spectral triples, arXiv:1101.5908

Jord Boeijink, Walter D. van Suijlekom, The noncommutativegeometry of Yang-Mills fields, arXiv:1008.5101.


Finite real spectral triples: F = (AF ,HF ,DF )

A finite dimensional (real) C ∗-algebra

A = ⊕Ni=1Mni (Ki )

Ki = R or C or H quaternions (Wedderburn)

Representation on finite dimensional Hilbert space H, withbimodule structure given by J (condition [a, b0] = 0)

D∗ = D with order one condition

[[D, a], b0] = 0

⇒ Moduli spaces (under unitary equivalence)

Ali Chamseddine, Alain Connes, Matilde Marcolli, Gravity andthe standard model with neutrino mixing,arXiv:hep-th/0610241

Branimir Cacic, Moduli spaces of Dirac operators for finitespectral triples, arXiv:0902.2068


Building a particle physics model ...this lecture based on:

Ali Chamseddine, Alain Connes, Matilde Marcolli, Gravity andthe standard model with neutrino mixing,arXiv:hep-th/0610241

Minimal input ansatz:

left-right symmetric algebra

ALR = C⊕HL ⊕HR ⊕M3(C)

involution (λ, qL, qR ,m) 7→ (λ, qL, qR ,m∗)

subalgebra C⊕M3(C) integer spin C-alg

subalgebra HL ⊕HR half-integer spin R-alg

More general choices of initial ansatz:

A.Chamseddine, A.Connes, Why the Standard Model,J.Geom.Phys. 58 (2008) 38–47

Slogan: algebras better than Lie algebras, more constraints on reps


Comment: associative algebras versus Lie algebras

In geometry of gauge theories: bundle over spacetime,connections and sections, automorphisms gauge group:Lie group

Decomposing composite particles into elementary particles:Lie group representations (hadrons and quarks)

If want only elementary particles: associative algebras havevery few representations (very constrained choice)

Get gauge groups later from inner automorphisms


Adjoint action:M bimodule over A, u ∈ U(A) unitary

Ad(u)ξ = uξu∗ ∀ξ ∈M

Odd bimodule: M bimodule for ALR odd iffs = (1,−1,−1, 1) acts by Ad(s) = −1

⇔ Rep of B = (ALR ⊗R AopLR)p as C-algebra

p = 12 (1− s ⊗ s0), with A0 = Aop

B = ⊕4−timesM2(C)⊕M6(C)

Contragredient bimodule of M

M0 = {ξ ; ξ ∈M} , a ξ b = b∗ξ a∗


The bimodule MF

MF = sum of all inequivalent irreducible odd ALR -bimodules

dimCMF = 32

MF = E ⊕ E0

E = 2L ⊗ 10 ⊕ 2R ⊗ 10 ⊕ 2L ⊗ 30 ⊕ 2R ⊗ 30

MF∼=M0

F by antilinear JF (ξ, η) = (η, ξ) for ξ , η ∈ E

J2F = 1 , ξ b = JFb∗JF ξ ξ ∈MF , b ∈ ALR

Sum irreducible representations of B

2L ⊗ 10 ⊕ 2R ⊗ 10 ⊕ 2L ⊗ 30 ⊕ 2R ⊗ 30

⊕1⊗ 20L ⊕ 1⊗ 20

R ⊕ 3⊗ 20L ⊕ 3⊗ 20

R

Grading: γF = c − JF c JF with c = (0, 1,−1, 0) ∈ ALR

J2F = 1 , JF γF = − γF JF

Grading and KO-dimension: commutations ⇒ KO-dim 6 mod 8Matilde Marcolli NCG models for particles and cosmology, I

Subalgebra and order one condition:

N = 3 generations (input): HF =MF ⊕MF ⊕MF

Left action of ALR sum of representations π|Hf⊕ π′|Hf

withHf = E ⊕ E ⊕ E and Hf = E0 ⊕ E0 ⊕ E0 and with no equivalentsubrepresentations (disjoint)

If D mixes Hf and Hf ⇒ no order one condition for ALR

Problem for coupled pair: A ⊂ ALR and D with off diagonal terms

maximal A where order one condition holds

AF = {(λ, qL, λ,m) | λ ∈ C , qL ∈ H , m ∈ M3(C)}

∼ C⊕H⊕M3(C).

unique up to Aut(ALR)⇒ spontaneous breaking of LR symmetry


Subalgebras with off diagonal Dirac and order one conditionOperator T : Hf → Hf

A(T ) = {b ∈ ALR | π′(b)T = Tπ(b) ,

π′(b∗)T = Tπ(b∗)}involutive unital subalgebra of ALR

A ⊂ ALR involutive unital subalgebra of ALR

restriction of π and π′ to A disjoint ⇒ no off diag D for A∃ off diag D for A ⇒ pair e, e ′ min proj in commutants ofπ(ALR) and π′(ALR) and operator T

e ′Te = T 6= 0 and A ⊂ A(T )

Then case by case analysis to identify max dimensional


SymmetriesUp to a finite abelian group

SU(AF ) ∼ U(1)× SU(2)× SU(3)

Adjoint action of U(1) (in powers of λ ∈ U(1))

↑ ⊗10 ↓ ⊗10 ↑ ⊗30 ↓ ⊗30

2L −1 −1 13

13

2R 0 −2 43 −2

3

⇒ correct hypercharges of fermions (confirms identification of HF

basis with fermions)


Classifying Dirac operators for (AF ,HF , γF , JF ) all possible DF

self adjoint on HF , commuting with JF , anticommuting with γFand [[D, a], b0] = 0, ∀a, b ∈ AF

Input conditions (massless photon): commuting with subalgebra

CF ⊂ AF , CF = {(λ, λ, 0) , λ ∈ C}

then D(Y ) =

(S T ∗

T S

)with S = S1 ⊕ (S3 ⊗ 13)

S1 =

0 0 Y ∗(↑1) 0

0 0 0 Y ∗(↓1)

Y(↑1) 0 0 00 Y(↓1) 0 0

same for S3, with Y(↓1), Y(↑1), Y(↓3), Y(↑3) ∈ GL3(C) and YR

symmetric:T : ER =↑R ⊗10 → JF ER


Moduli space C3 × C1 C3 = pairs (Y(↓3),Y(↑3)) modulo

Y ′(↓3) = W1 Y(↓3) W ∗3 , Y ′(↑3) = W2 Y(↑3) W ∗

3

Wj unitary matrices

C3 = (K × K )\(G × G )/K

G = GL3(C) and K = U(3) dimR C3 = 10 = 3 + 3 + 4(3 + 3 eigenvalues, 3 angles, 1 phase)

C1 = triplets (Y(↓1),Y(↑1),YR) with YR symmetric modulo

Y ′(↓1) = V1 Y(↓1)V ∗3 , Y ′(↑1) = V2 Y(↑1) V ∗3 , Y ′R = V2 YR V ∗2

π : C1 → C3 surjection forgets YR fiber symmetric matrices modYR 7→ λ2YR dimR(C3 × C1) = 31 (dim fiber 12-1=11)


Physical interpretation: Yukawa parameters and Majorana massesRepresentatives in C3 × C1:

Y(↑3) = δ(↑3) Y(↓3) = UCKM δ(↓3) U∗CKM

Y(↑1) = U∗PMNS δ(↑1) UPMNS Y(↓1) = δ(↓1)

δ↑, δ↓ diagonal: Dirac masses

U =

c1 −s1c3 −s1s3

s1c2 c1c2c3 − s2s3eδ c1c2s3 + s2c3eδs1s2 c1s2c3 + c2s3eδ c1s2s3 − c2c3eδ

angles and phase ci = cos θi , si = sin θi , eδ = exp(iδ)UCKM = Cabibbo–Kobayashi–MaskawaUPMNS = Pontecorvo–Maki–Nakagawa–Sakata⇒ neutrino mixingYR = Majorana mass terms for right-handed neutrinos


CKM matrix very strict experimental constraints (SM compatible)

unitarity triangle (offdiag elements of V ∗V adding to 0)constraints also on neutrino mixing matrix


Geometric point of view:

CKM and PMNS matrices data: coordinates on moduli spaceof Dirac operators

Experimental constraints define subvarieties in the modulispace

Symmetric spaces (K × K )\(G × G )/K interesting geometry

Get parameter relations from “interesting subvarieties”?


Summary: matter content of the NCG modelνMSM: Minimal Standard Model with additional right handedneutrinos with Majorana mass termsFree parameters in the model:

3 coupling constants

6 quark masses, 3 mixing angles, 1 complex phase

3 charged lepton masses, 3 lepton mixing angles, 1 complexphase

3 neutrino masses

11 Majorana mass matrix parameters

1 QCD vacuum angle

Moduli space of Dirac operators on the finite NC space F : allmasses, mixing angles, phases, Majorana mass termsOther parameters:

coupling constants: product geometry and action functional

vacuum angle not there (but quantum corrections...?)


Other particle modelsChanging the finite geometry produces other particle models:

Minimal Standard Model

Supersymmetric QCD

QED

Respecively:

Alain Connes, Gravity coupled with matter and the foundationof non-commutative geometry, Comm. Math. Phys. 182(1996), no. 1, 155–176

Thijs van den Broek, Walter D. van Suijlekom,Supersymmetric QCD and noncommutative geometry,arXiv:1003.3788

Koen van den Dungen, Walter D. van Suijlekom,Electrodynamics from Noncommutative Geometry,arXiv:1103.2928


Next episode:

Product geometries

Almost commutative geometries

The spectral action

Asymptotic expansion at large energies

The Lagrangian

Renormalization group equations and low energy limit


noncommutative geometry models for particle …matilde/vdl1.pdfnoncommutative geometry models for...

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