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
Matilde Marcolli NCG models for particles and cosmology, I
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
Matilde Marcolli NCG models for particles and cosmology, I
Why a geometrization of physics?The standard model of elementary particles
Matilde Marcolli NCG models for particles and cosmology, I
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
Matilde Marcolli NCG models for particles and cosmology, I
Standard model parameters: are there relations? why these values?
Matilde Marcolli NCG models for particles and cosmology, I
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)
Matilde Marcolli NCG models for particles and cosmology, I
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
Matilde Marcolli NCG models for particles and cosmology, I
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
Matilde Marcolli NCG models for particles and cosmology, I
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)
Matilde Marcolli NCG models for particles and cosmology, I
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
Matilde Marcolli NCG models for particles and cosmology, I
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.
Matilde Marcolli NCG models for particles and cosmology, I
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
Matilde Marcolli NCG models for particles and cosmology, I
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
Matilde Marcolli NCG models for particles and cosmology, I
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
Matilde Marcolli NCG models for particles and cosmology, I
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∗
Matilde Marcolli NCG models for particles and cosmology, I
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
Interpretation as particles (Fermions)
q(λ) =
(λ 00 λ
)q(λ)| ↑〉 = λ| ↑〉, q(λ)| ↓〉 = λ| ↓〉
2L ⊗ 10: neutrinos νL ∈ | ↑〉L ⊗ 10 and charged leptonseL ∈ | ↓〉L ⊗ 10
2R ⊗ 10: right-handed neutrinos νR ∈ | ↑〉R ⊗ 10 and chargedleptons eR ∈ | ↓〉R ⊗ 10
2L ⊗ 30 (color indices): u/c/t quarks uL ∈ | ↑〉L ⊗ 30 andd/s/b quarks dL ∈ | ↓〉L ⊗ 30
2R ⊗ 30 (color indices): u/c/t quarks uR ∈ | ↑〉R ⊗ 30 andd/s/b quarks dR ∈ | ↓〉R ⊗ 30
1⊗ 20L,R : antineutrinos νL,R ∈ 1⊗ ↑〉0L,R , and charged
antileptons eL,R ∈ 1⊗ | ↓〉0L,R3⊗ 20
L,R (color indices): antiquarks uL,R ∈ 3⊗ | ↑〉0L,R and
dL,R ∈ 3⊗ | ↓〉0L,RMatilde 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
Matilde Marcolli NCG models for particles and cosmology, I
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
Matilde Marcolli NCG models for particles and cosmology, I
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)
Matilde Marcolli NCG models for particles and cosmology, I
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
Matilde Marcolli NCG models for particles and cosmology, I
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)
Matilde Marcolli NCG models for particles and cosmology, I
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
Matilde Marcolli NCG models for particles and cosmology, I
CKM matrix very strict experimental constraints (SM compatible)
unitarity triangle (offdiag elements of V ∗V adding to 0)constraints also on neutrino mixing matrix
Matilde Marcolli NCG models for particles and cosmology, I
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”?
Matilde Marcolli NCG models for particles and cosmology, I
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...?)
Matilde Marcolli NCG models for particles and cosmology, I
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
Matilde Marcolli NCG models for particles and cosmology, I
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