Noncommutative Geometry Models for Particle Physics and Cosmology, Lecture I

Noncommutative Geometry models for Particle Physics 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 particle physics; Noncommutative Riemannian geometry; finite noncommutative geometries; moduli spaces; the finite geometry of the Standard Model 2 The product geometry; the spectral action functional and its asymptotic expansion; bosons and fermions; the Standard Model Lagrangian; renormalization group flow, geometric constraints and low energy limits 3 Parameters: relations at unification and running; running of the gravitational terms; the RGE flow with right handed neutrinos; cosmological timeline and the inflation epoch; effective gravitational and cosmological constants and models of inflation 4 The spectral action and the problem of cosmic topology; cosmic topology and the CMB; slow-roll inflation; Poisson summation 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 circle bundle 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) over spacetime, strings vibrations = types of particles NCG models: extra dimensions are NC spaces, pure gravity on product 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?

1 a a abc a b c 1 2 abc ade b c d e + SM = 2 ∂ν gµ∂ν gµ gsf ∂µgν gµgν 4 gs f f gµgνgµgν ∂ν Wµ ∂νWµ− L 2 + − 1 0 − 0 1 2 0 −0 1 − 0 + − M W W − ∂ν Z ∂νZ 2 M Z Z ∂µAν∂µAν igcw(∂νZ (W W − µ µ 2 µ µ 2cw µ µ 2 µ µ ν + − 0 + − +− 0 + − + − Wν Wµ−) Zν (Wµ ∂ν Wµ− Wµ−∂ν Wµ )+ Zµ(Wν ∂νWµ− Wν−∂ν Wµ )) −+ + − + +− + − igsw(∂νAµ(Wµ Wν− Wν Wµ−) Aν (Wµ ∂ν Wµ− Wµ−∂ν Wµ )+ Aµ(Wν ∂ν Wµ− + 1 2 −+ + − 1 2 + − + 2 2 0 + 0 − Wν−∂ν Wµ )) 2 g Wµ Wµ−Wν Wν− + 2 g Wµ Wν−Wµ Wν− + g cw(ZµWµ Zν Wν− 0 0 + − 2 2 + + 2 0 + − ZµZµWν Wν−)+ g sw(AµWµ AνWν− AµAµWν Wν−)+ g swcw(AµZν (Wµ Wν− + 0 + 1 − 2 2 + 1 0 0− W W −) 2A Z W W −) ∂ H∂ H 2M α H ∂ φ ∂ φ− ∂ φ ∂ φ ν µ − µ µ ν ν − 2 µ µ − h − µ µ − 2 µ µ − 2M 2 2M 1 2 0 0 + 2M 4 β + H + (H + φ φ +2φ φ−) + α h g2 g 2 g2 h − 3 0 0 + gαhM (H + Hφ φ +2Hφ φ−) 1 2 4 0 4 + 2 0 2 + 2−+ 0 2 2 8 g αh (H +(φ ) + 4(φ φ−) + 4(φ ) φ φ− +4H φ φ− + 2(φ ) H ) + 1 M 0 0 − gMW W −H g 2 Z Z H µ µ 2 cw µ µ 1 + 0 0 − 0 +− + 0 2 ig Wµ (φ ∂µφ− φ−∂µφ ) Wµ−(φ ∂µφ φ ∂µφ ) + 1 + − + − + 1− 1 0 0 0 g W (H∂ φ− φ−∂ H)+ W −(H∂ φ φ ∂ H) + g (Z (H∂ φ φ ∂ H)+ 2 µ µ µ µ µ µ 2 cw µ µ µ − 2 − − 1 0 0 + + sw 0 + + + M ( Z ∂µφ +W ∂µφ− +W −∂µφ ) ig MZ (W φ− W −φ )+igswMAµ(W φ− cw µ µ µ cw µ µ µ µ 2 − − − + 1 2cw 0 + + + + W −φ ) ig − Z (φ ∂ φ− φ−∂ φ )+ igs A (φ ∂ φ− φ−∂ φ ) µ 2cw µ µ µ w µ µ µ 1 2 + −2 0 2 + − 1 2 1 0 0 2 0 2 − 2 2 +− g W W − (H +(φ ) +2φ φ−) g 2 Z Z (H +(φ ) + 2(2s 1) φ φ−) 4 µ µ 8 cw µ µ w 2 −2 − − 1 2 sw 0 0 + + 1 2 sw 0 + + 1 2 0 + g Z φ (W φ− + W −φ ) ig Z H(W φ− W −φ )+ g s A φ (W φ− + 2 cw µ µ µ 2 cw µ µ µ 2 w µ µ + 1 2 − + + −2 sw 2 0 + W −φ )+ ig s A H(W φ− W −φ ) g (2c 1)Z A φ φ− µ 2 w µ µ µ cw w µ µ 2 2 + 1 a σ µ σ a− λ − λ λ λ− λ λ −λ g s A A φ φ− + ig λ (¯q γ q )g e¯ (γ∂ + m )e ν¯ (γ∂ + m )ν u¯ (γ∂ + w µ µ 2 s ij i j µ − e − ν − j mλ)uλ d¯λ(γ∂ + mλ)dλ + igs A (¯eλγµeλ)+ 2 (¯uλγµuλ) 1 (d¯λγµdλ) + u j − j d j w µ − 3 j j − 3 j j ig Z0 (¯νλγµ(1 + γ5)νλ)+(¯eλγµ(4s2 1 γ5)eλ)+(d¯λγµ( 4 s2 1 γ5)dλ)+ 4cw µ{ w− − j 3 w − − j (¯uλγµ(1 8 s2 + γ5)uλ) + ig W + (¯νλγµ(1 + γ5)U lep eκ)+(¯uλγµ(1 + γ5)C dκ) + j − 3 w j } 2√2 µ λκ j λκ j ig κ lep µ 5 λ κ µ 5 λ W (¯e U † γ (1 + γ )ν )+(d¯ C† γ (1 + γ )u ) + 2√2 µ− κλ j κλ j ig φ+ mκ(¯νλU lep (1 γ5)eκ)+ mλ(¯νλU lep (1 + γ5)eκ + 2M√2 − e λκ − ν λκ ig g mλ φ mλ(¯eλU lep† (1 + γ5)νκ) mκ(¯eλU lep† (1 γ5)νκ ν H(¯νλνλ) 2M√2 − e λκ ν λκ 2 M λ λ − λ − − − g me H(¯eλeλ)+ ig mν φ0(¯νλγ5νλ) ig me φ0(¯eλγ5eλ) 1 ν¯ M R (1 γ )ˆν 2 M 2 M − 2 M − 4 λ λκ − 5 κ − 1 ν¯ M R (1 γ )ˆν + ig φ+ mκ(¯uλC (1 γ5)dκ)+ mλ(¯uλC (1 + γ5)dκ + 4 λ λκ 5 κ 2M√2 d j λκ j u j λκ j − − − λ ig λ ¯λ 5 κ κ ¯λ 5 κ g mu λ λ φ− m (d C† (1 + γ )u ) m (d C† (1 γ )u H(¯u u ) 2M√2 d j λκ j − u j λκ − j − 2 M j j − mλ λ mλ g d ¯λ λ ig mu 0 λ 5 λ ig d 0 ¯λ 5 λ ¯a 2 a abc ¯a b c 2 M H(dj dj )+ 2 M φ (¯uj γ uj ) 2 M φ (dj γ dj )+ G ∂ G + gsf ∂µG G gµ + − 2 ¯ + 2 2 + ¯ 2 2 ¯ 0 2 M 0 ¯ 2 + ¯ 0 X (∂ M )X + X−(∂ M )X−+X (∂ 2 )X + Y ∂ Y + igcwWµ (∂µX X− − − − cw − ¯ + 0 + ¯ ¯ + ¯ 0 ∂µX X )+igswWµ (∂µYX− ∂µX Y )+ igcwWµ−(∂µX−X ¯ 0 + ¯ − ¯ + 0 ¯ + + − ∂µX X )+igswWµ−(∂µX−Y ∂µYX )+ igcwZµ(∂µX X − + + − ∂µX¯ −X−)+igswAµ(∂µX¯ X − 2 ¯ 1 ¯ + + ¯ 1 ¯ 0 0 1 2cw ¯ + 0 + ¯ 0 ∂µX−X−) gM X X H + X−X−H + 2 X X H + − igM X X φ X−X φ− + − 2 cw 2cw − 1 0 + 0 + 0 + 0 + igMX¯ X−φ X¯ X φ− +igMs X¯X−φ X¯ X φ− + 2cw w 1− + + 0 0 − igM X¯ X φ X¯ −X−φ . 2 − 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) should follow 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 conceptual structure, links particle physics to cosmology; the bad: worse chances 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: 1 S (g ) = R √g d4x EH µν 16πG ZM Gravity minimally coupled to matter: S = SEH + SSM with SSM = particle physics Standard Model Lagrangian ⇒ Right handed neutrinos with Majorana masses µν Modiﬁed gravity model f (R, R , Cλµνκ) with conformal gravity: Weyl curvature tensor 1 C = R (g R g R g R + g R ) λµνκ λµνκ − 2 λν µκ − λκ µν − µν λκ µκ λν 1 + Rα(g g g g ) 6 α λν µκ − λκ µν Non-minimal coupling of Higgs to gravity

1 2 2 2 2 4 4 D H µ H ξ0 R H + λ0 H √g d x 2| µ | − 0| | − | | | | ZM

Matilde Marcolli NCG models for particles and cosmology, I What is Noncommutative Geometry? X compact Hausdorﬀ topological space C(X ) abelian ⇔ C ∗-algebra of continuous functions (Gelfand–Naimark)

Noncommutative C ∗-algebra : think of as “continuous functions on an NC space” A Describe geometry in terms of algebra C(X ) and dense subalgebras like C ∞(X ) if X manifold Diﬀerential forms, bundles, connections, cohomology: all continue to make sense without commutativity (NC geometry) Can describe “bad quotients” as good spaces: functions on the 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( , , D) A H involutive algebra A representation π : ( ) A → L H self adjoint operator D on , dense domain H compact resolvent (1 + D2) 1/2 − ∈ K [a, D] bounded a ∀ ∈ A even if Z/2- grading γ on H [γ, a] = 0, a , Dγ = γD ∀ ∈ A − 2 Main example( C ∞(M), L (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 M compact and Euclidean signature (Euclidean gravity)

Matilde Marcolli NCG models for particles and cosmology, I Real structure KO-dimension n Z/8Z antilinear isometry J : ∈ H → H 2 J = ε, JD = ε0DJ, and Jγ = ε00γJ

n 0 1 2 3 4 5 6 7 ε 1 1 -1 -1 -1 -1 1 1 ε0 1 -1 1 1 1 -1 1 1 ε00 1 -1 1 -1

Commutation:[a, b0] = 0 a, b ∀ ∈ A where b0 = Jb∗J−1 b ∀ ∈ A Order 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-dimensional spacetime manifold M by a “ﬁnite× NC space” F (extra dimensions) Alain Connes, Gravity coupled with matter and the foundation of non-commutative geometry, Comm. Math. Phys. 182 (1996), no. 1, 155–176.

Almost commutative geometries: more general form, fibration (not product) over manifold, fiber finite NC space Branimir Caćić,´ A reconstruction theorem for almost-commutative spectral triples, arXiv:1101.5908 Jord Boeijink, Walter D. van Suijlekom, The noncommutative geometry of Yang-Mills fields, arXiv:1008.5101.

Matilde Marcolli NCG models for particles and cosmology, I Finite real spectral triples: F = ( F , F , DF ) A H ﬁnite dimensional (real) C -algebra A ∗ N = i=1Mni (Ki ) A ⊕ Ki = R or C or H quaternions (Wedderburn) Representation on ﬁnite dimensional Hilbert space , with bimodule structure given by J (condition [a, b0] = 0)H

D∗ = D with order one condition

[[D, a], b0] = 0

Moduli spaces (under unitary equivalence) ⇒ Ali Chamseddine, Alain Connes, Matilde Marcolli, Gravity and the standard model with neutrino mixing, arXiv:hep-th/0610241 Branimir Caćić,´ Moduli spaces of Dirac operators for finite spectral 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 and the standard model with neutrino mixing, arXiv:hep-th/0610241 Minimal input ansatz: left-right symmetric algebra

LR = C HL HR M3(C) A ⊕ ⊕ ⊕

involution (λ, qL, qR , m) (λ,¯ q¯L, q¯R , m ) 7→ ∗ 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 have very few representations (very constrained choice) Get gauge groups later from inner automorphisms

Matilde Marcolli NCG models for particles and cosmology, I Adjoint action: bimodule over , u ( ) unitary M A ∈ U A

Ad(u)ξ = uξu∗ ξ ∀ ∈ M

Odd bimodule: bimodule for LR odd iﬀ s = (1, 1, 1, 1)M acts by Ad(s)A = 1 − − − op Rep of = ( LR R LR )p as C-algebra p⇔= 1 (1 Bs s0A), with⊗ A0 = op 2 − ⊗ A A 4 times = − M2(C) M6(C) B ⊕ ⊕ Contragredient bimodule of M 0 = ξ¯ ; ξ , a ξ¯b = b ξ a M { ∈ M} ∗ ∗

Matilde Marcolli NCG models for particles and cosmology, I The bimodule F M F = sum of all inequivalent irreducible odd LR -bimodules M A dimC F = 32 M 0 F = M E ⊕ E 0 0 0 0 = 2L 1 2R 1 2L 3 2R 3 E ⊗ ⊕ ⊗ ⊕ ⊗ ⊕ ⊗ 0 F = by antilinear JF (ξ, η¯) = (η, ξ¯) for ξ , η M ∼ MF ∈ E 2 J = 1 , ξ b = JF b∗JF ξ ξ F , b LR F ∈ M ∈ A Sum irreducible representations of B 0 0 0 0 2L 1 2R 1 2L 3 2R 3 ⊗ ⊕ ⊗ ⊕ ⊗ ⊕ ⊗ 1 20 1 20 3 20 3 20 ⊕ ⊗ L ⊕ ⊗ R ⊕ ⊗ L ⊕ ⊗ R Grading: γF = c JF c JF with c = (0, 1, 1, 0) LR − − ∈ A 2 J = 1 , JF γF = γF JF F − Grading and KO-dimension: commutations KO-dim 6 mod 8 ⇒ Matilde Marcolli NCG models for particles and cosmology, I Interpretation as particles (Fermions)

N = 3 generations (input): F = F F F H M ⊕ M ⊕ M Left action of sum of representations π π with LR f 0 f¯ A 0 0 0 |H ⊕ |H f = and f¯ = and with no equivalent HsubrepresentationsE ⊕ E ⊕ E (disjoint)H E ⊕ E ⊕ E

If D mixes f and ¯ no order one condition for LR H Hf ⇒ A Problem for coupled pair: LR and D with oﬀ diagonal terms A ⊂ A maximal where order one condition holds A

F = (λ, qL, λ, m) λ C , qL H , m M3(C) A { | ∈ ∈ ∈ } C H M3(C). ∼ ⊕ ⊕ unique up to Aut( LR ) spontaneous breakingA of LR symmetry ⇒

Matilde Marcolli NCG models for particles and cosmology, I Subalgebras with oﬀ diagonal Dirac and order one condition Operator T : f ¯ H → Hf

(T ) = b LR π0(b)T = T π(b) , A { ∈ A |

π0(b∗)T = T π(b∗) } involutive unital subalgebra of LR A LR involutive unital subalgebra of LR A ⊂ A A restriction of π and π to disjoint no oﬀ diag D for 0 A ⇒ A oﬀ diag D for pair e, e min proj in commutants of ∃ A ⇒ 0 π( LR ) and π ( LR ) and operator T A 0 A

e0Te = T = 0 and (T ) 6 A ⊂ A Then case by case analysis to identify max dimensional

Matilde Marcolli NCG models for particles and cosmology, I Symmetries Up to a ﬁnite abelian group

SU( F ) U(1) SU(2) SU(3) A ∼ × × Adjoint action of U(1) (in powers of λ U(1)) ∈ 10 10 30 30 ↑ ⊗ ↓ ⊗ ↑ ⊗ ↓ ⊗ 1 1 2L 1 1 − − 3 3 4 2 2R 0 2 − 3 − 3

correct hypercharges of fermions (conﬁrms identiﬁcation of F basis⇒ with fermions) H

Matilde Marcolli NCG models for particles and cosmology, I Classifying Dirac operators for ( F , F , γF , JF ) all possible DF A H self adjoint on F , commuting with JF , anticommuting with γF 0 H and [[D, a], b ] = 0, a, b F Input conditions (massless∀ ∈ photon): A commuting with subalgebra

CF F , CF = (λ, λ, 0) , λ C ⊂ A { ∈ }

ST ∗ then D(Y ) = with S = S1 (S3 13) T S¯ ⊕ ⊗

0 0 Y(∗ 1) 0 ↑ 0 0 0 Y(∗ 1) S1 =  ↓  Y( 1) 0 0 0  ↑   0 Y( 1) 0 0   ↓    same for S3, with Y( 1), Y( 1), Y( 3), Y( 3) GL3(C) and YR symmetric: ↓ ↑ ↓ ↑ ∈ 0 T : ER = R 1 JF ER ↑ ⊗ →

Matilde Marcolli NCG models for particles and cosmology, I Moduli space 3 1 3 = pairs (Y( 3), Y( 3)) modulo C × C C ↓ ↑

Y(0 3) = W1 Y( 3) W3∗, Y(0 3) = W2 Y( 3) W3∗ ↓ ↓ ↑ ↑

Wj unitary matrices

3 = (K K) (G G)/K C × \ ×

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

1 = triplets (Y( 1), Y( 1), YR ) with YR symmetric modulo C ↓ ↑ ¯ Y(0 1) = V1 Y( 1)V3∗, Y(0 1) = V2 Y( 1) V3∗, YR0 = V2 YR V2∗ ↓ ↓ ↑ ↑

π : 1 3 surjection forgets YR ﬁber symmetric matrices mod C →2 C YR λ YR dim ( 3 1) = 31 (dim ﬁber 12-1=11) 7→ R C × C

Matilde Marcolli NCG models for particles and cosmology, I Physical interpretation: Yukawa parameters and Majorana masses Representatives in 3 1: C × C

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

Y( 1) = UPMNS∗ δ( 1) UPMNS Y( 1) = δ( 1) ↑ ↑ ↓ ↓ δ , δ diagonal: Dirac masses ↑ ↓

c1 s1c3 s1s3 − − U = s1c2 c1c2c3 s2s3e c1c2s3 + s2c3e  − δ δ  s1s2 c1s2c3 + c2s3e c1s2s3 c2c3e δ − δ   angles and phase ci = cos θi , si = sin θi , eδ = exp(iδ) UCKM = Cabibbo–Kobayashi–Maskawa UPMNS = Pontecorvo–Maki–Nakagawa–Sakata neutrino mixing ⇒ YR = 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 space of Dirac operators Experimental constraints deﬁne subvarieties in the moduli space 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 model νMSM: Minimal Standard Model with additional right handed neutrinos with Majorana mass terms Free 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 complex phase 3 neutrino masses 11 Majorana mass matrix parameters 1 QCD vacuum angle Moduli space of Dirac operators on the ﬁnite NC space F : all masses, mixing angles, phases, Majorana mass terms Other 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 models Changing the ﬁnite geometry produces other particle models: Minimal Standard Model Supersymmetric QCD QED Respecively: Alain Connes, Gravity coupled with matter and the foundation of 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 Next episode: Product geometries Almost commutative geometries The spectral action Asymptotic expansion at large energies The Lagrangian Renormalization group equations and low energy limit

Matilde Marcolli NCG models for particles and cosmology, I