Noncommutative Geometry models for Particle Physics and Cosmology, Lecture II Matilde Marcolli Villa de Leyva school, July 2011 Matilde Marcolli NCG models for particles and cosmology, II This lecture based on Ali Chamseddine, Alain Connes, Matilde Marcolli, Gravity and the Standard Model with neutrino mixing, ATMP 11 (2007) 991{1090, arXiv:hep-th/0610241 Symmetries and NCG Symmetries of gravity coupled to matter: G = U(1) × SU(2) × SU(3) G = Map(M; G) o Diff(M) Is it G = Diff(X )? Not for a manifold, yes for an NC space 1 Example: A = C (M; Mn(C)) G = PSU(n) 1 ! Inn(A) ! Aut(A) ! Out(A) ! 1 1 ! Map(M; G) !G! Diff(M) ! 1: Matilde Marcolli NCG models for particles and cosmology, II Symmetries viewpoint: can think of X = M × F noncommutative with G = Diff(X ) pure gravity symmetries for X combining gravity and gauge symmetries together (no a priori distinction between \base" and “fiber" directions) Want same with action functional for pure gravity on NC space X = M × F giving gravity coupled to matter on M Matilde Marcolli NCG models for particles and cosmology, II Product geometry M × F Two spectral triples (Ai ; Hi ; Di ; γi ; Ji ) of KO-dim 4 and 6: A = A1 ⊗ A2 H = H1 ⊗ H2 D = D1 ⊗ 1 + γ1 ⊗ D2 γ = γ1 ⊗ γ2 J = J1 ⊗ J2 Case of 4-dimensional spin manifold M and finite NC geometry F : 1 1 A = C (M) ⊗ AF = C (M; AF ) 2 2 H = L (M; S) ⊗ HF = L (M; S ⊗ HF ) D = =@M ⊗ 1 + γ5 ⊗ DF DF chosen in the moduli space described last time Matilde Marcolli NCG models for particles and cosmology, II Dimension of NC spaces: different notions of dimension for a spectral triple (A; H; D) Metric dimension: growth of eigenvalues of Dirac operator KO-dimension (mod 8): sign commutation relations of J, γ, D Dimension spectrum: poles of zeta functions −s ζa;D (s) = Tr(ajDj ) For manifolds first two agree and third contains usual dim; for NC spaces not same: DimSp ⊂ C can have non-integer and non-real points, KO not always metric dim mod 8, see F case X = M × F metrically four dim 4 = 4 + 0; KO-dim is 10 = 4 + 6 (equal 2 mod 8); DimSp k 2 Z≥0 with k ≤ 4 Matilde Marcolli NCG models for particles and cosmology, II Variant: almost commutative geometries 1 2 (C (M; E); L (M; E ⊗ S); DE ) M smooth manifold, E algebra bundle: fiber Ex finite dimensional algebra AF C 1(M; E) smooth sections of a algebra bundle E E S Dirac operator DE = c ◦ (r ⊗ 1 + 1 ⊗ r ) with spin connection rS and hermitian connection on bundle Compatible grading and real structure An equivalent intrinsic (abstract) characterization in: Branimir Ca´ci´c,´ A reconstruction theorem for almost-commutative spectral triples, arXiv:1101.5908 Here on assume for simplicity product M × F Matilde Marcolli NCG models for particles and cosmology, II Inner fluctuations and gauge fields Setup: Right A-module structure on H ξ b = b0 ξ; ξ 2 H ; b 2 A Unitary group, adjoint representation: ξ 2 H ! Ad(u) ξ = u ξ u∗ ξ 2 H Inner fluctuations: 0 −1 D ! DA = D + A + " JAJ with A = A∗ self-adjoint operator of the form X A = aj [D; bj ] ; aj ; bj 2 A Note: not an equivalence relation (finite geometry, can fluctuate D to zero) but like \self Morita equivalences" Matilde Marcolli NCG models for particles and cosmology, II Properties of inner fluctuations( A; H; D; J) 1 ∗ Gauge potential A 2 ΩD , A = A Unitary u 2 A, then Ad(u)(D + A + "0 JAJ−1)Ad(u∗) = 0 −1 D + γu(A) + " J γu(A) J ∗ ∗ where γu(A) = u [D; u ] + u A u 0 1 ∗ D = D + A (with A 2 ΩD , A = A ) then 0 0 0 1 D + B = D + A ; A = A + B 2 ΩD 1 ∗ 8B 2 ΩD0 B = B D0 = D + A + "0 JAJ−1 then 0 0 −1 0 0 0 −1 0 1 D + B + " JBJ = D + A + " JA J A = A + B 2 ΩD 1 ∗ 8B 2 ΩD0 B = B Matilde Marcolli NCG models for particles and cosmology, II Gauge bosons and Higgs boson Unitary U(A) = fu 2 A j uu∗ = u∗u = 1g Special unitary SU(AF ) = fu 2 U(AF ) j det(u) = 1g det of action of u on HF Up to a finite abelian group SU(AF ) ∼ U(1) × SU(2) × SU(3) Unimod subgr of U(A) adjoint rep Ad(u) on H is gauge group of SM Unimodular inner fluctuations (in M directions) ) gauge bosons of SM: U(1), SU(2) and SU(3) gauge bosons Inner fluctuations in F direction ) Higgs field Matilde Marcolli NCG models for particles and cosmology, II More on Gauge bosons (1;0) P 0 Inner fluctuations A = i ai [=@M ⊗ 1; ai ] with with 0 0 0 0 1 ai = (λi ; qi ; mi ), ai = (λi ; qi ; mi ) in A = C (M; AF ) P 0 P 0 U(1) gauge field Λ = i λi dλi = i λi [=@M ⊗ 1; λi ] P 0 P α SU(2) gauge field Q = i qi dqi , with q = f0 + α ifασ and P 0 α Q = α fα[=@M ⊗ 1; ifασ ] 0 P 0 P 0 U(3) gauge field V = i mi dmi = i mi [=@M ⊗ 1; mi ] reduce the gauge field V 0 to SU(3) passing to unimodular subgroup SU(AF ) and unimodular gauge potential Tr(A) = 0 0 Λ 0 0 1 1 1 V 0 = −V − 0 Λ 0 = −V − Λ1 3 @ A 3 3 0 0 Λ Matilde Marcolli NCG models for particles and cosmology, II Gauge bosons and hypercharges The (1; 0) part of A + JAJ−1 acts on quarks and leptons by 0 4 1 3 Λ + V 0 0 0 2 B 0 − 3 Λ + V 0 0 C B 1 C @ 0 0 Q11 + 3 Λ + VQ12 A 1 0 0 Q21 Q22 + 3 Λ + V 0 0 0 0 0 1 B 0 −2Λ 0 0 C B C @ 0 0 Q11 − Λ Q12 A 0 0 Q21 Q22 − Λ ) correct hypercharges! Matilde Marcolli NCG models for particles and cosmology, II More on Higgs boson Inner fluctuations A(0;1) in the F -space direction X 0 (0;1) (0;1) ai [γ5 ⊗ DF ; ai ](x)jHf = γ5 ⊗ (Aq + A` ) i 0 X 0 Y A(0;1) = ⊗ 1 A(0;1) = q X 0 0 3 1 Y 0 0 ∗ ∗ 0 0 Υu'1 Υu'2 0 Υu'1 Υd '2 X = ∗ ∗ and X = 0 0 −Υd '¯2 Υd '¯1 −Υu'¯2 Υd '¯1 ∗ ∗ 0 0 Υν'1 Υν'2 0 Υν'1 Υe '2 Y = ∗ ∗ and Y = 0 0 −Υe '¯2 Υe '¯1 −Υν'¯2 Υe '¯1 P 0 0 P 0 0 P 0 0 ¯0 '1 = λi (αi − λi ), '2 = λi βi '1 = αi (λi − αi ) + βi βi and 0 P 0 ¯0 0 '2 = (−αi βi + βi (λi − α¯i )), for ai (x) = (λi ; qi ; mi ) and α β a0(x) = (λ0; q0; m0) and q = i i i i −β¯ α¯ Matilde Marcolli NCG models for particles and cosmology, II More on Higgs boson Discrete part of inner fluctuations: quaternion valued function H = '1 + '2j or ' = ('1;'2) 2 1;0 2 0;1 2 1;0 0;1 DA = (D ) + 14 ⊗ (D ) − γ5 [D ; 14 ⊗ D ] p 1;0 0;1 µ s 0;1 [D ; 14 ⊗ D ] = −1 γ [(rµ + Aµ); 14 ⊗ D ] 2 ∗ ∗ s This gives DA = r r − E where r r Laplacian of r = r + A 1 X −E = s ⊗ id + γµγν ⊗ − i γ γµ ⊗ (D ') + 1 ⊗ (D0;1)2 4 Fµν 5 M µ 4 µ<ν with s = −R scalar curvature and Fµν curvature of A i i D ' = @ ' + g W α' σα − g B ' µ µ 2 2 µ 2 1 µ SU(2) and U(1) gauge potentials Matilde Marcolli NCG models for particles and cosmology, II The spectral action functional Ali Chamseddine, Alain Connes, The spectral action principle, Comm. Math. Phys. 186 (1997), no. 3, 731{750. A good action functional for noncommutative geometries Tr(f (D=Λ)) D Dirac, Λ mass scale, f > 0 even smooth function (cutoff approx) Simple dimension spectrum ) expansion for Λ ! 1 Z X k −k Tr(f (D=Λ)) ∼ fk Λ − jDj + f (0) ζD (0) + o(1); k R 1 k−1 with fk = 0 f (v) v dv momenta of f −s where DimSp(A; H; D) = poles of ζb;D (s) = Tr(bjDj ) Matilde Marcolli NCG models for particles and cosmology, II Asymptotic expansion of the spectral action −t∆ X α Tr(e ) ∼ aα t (t ! 0) and the ζ function −s=2 ζD (s) = Tr(∆ ) Non-zero term aα with α < 0 ) pole of ζD at −2α with 2 a Res ζ (s) = α s=−2α D Γ(−α) No log t terms ) regularity at 0 for ζD with ζD (0) = a0 Matilde Marcolli NCG models for particles and cosmology, II Get first statement from Z 1 −s −s=2 1 −t∆ s=2−1 jDj = ∆ = s e t dt Γ 2 0 R 1 α+s=2−1 −1 with 0 t dt = (α + s=2) . Second statement from 1 s s ∼ as s ! 0 Γ 2 2 contrib to ζD (0) from pole part at s = 0 of Z 1 Tr(e−t∆) ts=2−1 dt 0 R 1 s=2−1 2 given by a0 0 t dt = a0 s Matilde Marcolli NCG models for particles and cosmology, II Spectral action with fermionic terms 1 S = Tr(f (D =Λ)) + h J ξ;~ D ξ~i ; ξ~ 2 H+; A 2 A cl DA = Dirac with unimodular inner fluctuations, J = real structure, + Hcl = classical spinors, Grassmann variables Fermionic terms 1 hJξ;~ D ξ~i 2 A antisymmetric bilinear form A(ξ~) on + Hcl = fξ 2 Hcl j γξ = ξg ) nonzero on Grassmann variables Euclidean functional integral ) Pfaffian Z − 1 A(ξ~) Pf (A) = e 2 D[ξ~] avoids Fermion doubling problem of previous models based on symmetric hξ; DAξi for NC space with KO-dim=0 Matilde Marcolli NCG models for particles and cosmology, II Grassmann variables Anticommuting variables with basic integration rule Z ξ dξ = 1 An antisymmetric bilinear form A(ξ1; ξ2): if ordinary commuting variables A(ξ; ξ) = 0 but not on Grassmann variables 0 0 0 Example: 2-dim case A(ξ ; ξ) = a(ξ1ξ2 − ξ2ξ1), if ξ1 and ξ2 anticommute, with integration rule as above Z 1 Z − A(ξ,ξ) −aξ1ξ2 e 2 D[ξ] = e dξ1dξ2 = a Pfaffian as functional integral: antisymmetric quadratic form Z − 1 A(ξ,ξ) Pf (A) = e 2 D[ξ] Method to treat Majorana fermions in the Euclidean setting Matilde Marcolli NCG models for particles and cosmology, II Fermionic part of SM Lagrangian Explicit computation of 1 hJξ;~ D ξ~i 2 A gives part of SM Larangian with •LHf = coupling of Higgs to fermions •Lgf = coupling of gauge bosons to fermions •Lf = fermion terms Matilde Marcolli NCG models for particles and cosmology, II Bosonic part of the spectral action 1 f Z p S = (48 f Λ4 − f Λ2 c + 0 d) g d 4x π2 4 2 4 96 f Λ2 − f c Z p + 2 0 R g d 4x 24π2 f Z 11 p + 0 ( R∗R∗ − 3 C C µνρσ) g d 4x 10 π2 6 µνρσ (−2 a f Λ2 + e f ) Z p + 2 0 j'j2 g d 4x π2 f a Z p + 0 jD 'j2 g d 4x 2 π2 µ f a Z p − 0 R j'j2 g d 4x 12 π2 f b Z p + 0 j'j4 g d 4x 2 π2 f Z 5 p + 0 (g 2 G i G µνi + g 2 F α F µνα + g 2 B Bµν ) g d 4x; 2 π2 3 µν 2 µν 3 1 µν Matilde Marcolli NCG models for particles and cosmology, II Parameters: f0, f2, f4 free parameters, f0 = f (0) and, for k > 0, Z 1 k−1 fk = f (v)v dv: 0 a; b; c; d; e functions of Yukawa parameters of νMSM y y y y a = Tr(Yν Yν + Ye Ye + 3(Yu Yu + Yd Yd )) y 2 y 2 y 2 y 2 b = Tr((Yν Yν) + (Ye Ye ) + 3(Yu Yu) + 3(Yd Yd ) ) c = Tr(MMy) d = Tr((MMy)2) y y