The Spectral Geometry of the Standard Model

Transcription

1 Ma148b Spring 2016 Topics in Mathematical Physics

2 References A.H. Chamseddine, A. Connes, M. Marcolli, Gravity and the Standard Model with Neutrino Mixing, Adv. Theor. Math. Phys., Vol.11 (2007)

3 Building a particle physics model Minimal input ansatz: left-right symmetric algebra A LR = C H L H R M 3 (C) involution (λ, q L, q R, m) ( λ, q L, q R, m ) subalgebra C M 3 (C) integer spin C-alg subalgebra H L H R half-integer spin R-alg More general choices of initial ansatz: A.Chamseddine, A.Connes, Why the Standard Model, J.Geom.Phys. 58 (2008) Slogan: algebras better than Lie algebras, more constraints on reps

4 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

5 Example Composite particles (baryons) Classification in terms of elementary particles (quarks) using representations of Lie groups: SU(3)

6 Adjoint action: M bimodule over A, u U(A) unitary Ad(u)ξ = uξu ξ M Odd bimodule: M bimodule for A LR odd iff s = (1, 1, 1, 1) acts by Ad(s) = 1 Rep of B = (A LR R A op LR ) p as C-algebra p = 1 2 (1 s s0 ), with A 0 = A op Contragredient bimodule of M B = 4 times M 2 (C) M 6 (C) M 0 = { ξ ; ξ M}, a ξ b = b ξ a

7 The bimodule M F M F = sum of all inequivalent irreducible odd A LR -bimodules dim C M F = 32 M F = E E 0 E = 2 L R L R 3 0 M F = M 0 F by antilinear J F (ξ, η) = (η, ξ) for ξ, η E J 2 F = 1, ξ b = J F b J F ξ ξ M F, b A LR Sum irreducible representations of B 2 L R L R L 1 20 R 3 20 L 3 20 R Grading: γ F = c J F c J F with c = (0, 1, 1, 0) A LR J 2 F = 1, J F γ F = γ F J F Grading and KO-dimension: commutations KO-dim 6 mod 8

8 KO-dimension n Z/8Z antilinear isometry J : H H J 2 = ε, JD = ε DJ, and Jγ = ε γj n ε ε ε In particular, J 2 = 1 and J γ = γ J gives KO-dim = 6 So in this case metric dimension is zero but KO-dimension is 6

9 Interpretation as particles (Fermions) q(λ) = ( λ 0 0 λ ) q(λ) = λ, q(λ) = λ 2 L 1 0 : neutrinos ν L L 1 0 and charged leptons e L L R 1 0 : right-handed neutrinos ν R R 1 0 and charged leptons e R R L 3 0 (color indices): u/c/t quarks u L L 3 0 and d/s/b quarks d L L R 3 0 (color indices): u/c/t quarks u R R 3 0 and d/s/b quarks d R R L,R : antineutrinos ν L,R 1 0 L,R, and charged antileptons ē L,R 1 0 L,R L,R (color indices): antiquarks ū L,R 3 0 L,R and d L,R 3 0 L,R

10 Subalgebra and order one condition: N = 3 generations (input): H F = M F M F M F Left action of A LR sum of representations π Hf π H f with H f = E E E and H f = E 0 E 0 E 0 and with no equivalent subrepresentations (disjoint) If D mixes H f and H f no order one condition for A LR Problem for coupled pair: A A LR and D with off diagonal terms maximal A where order one condition holds A F = {(λ, q L, λ, m) λ C, q L H, m M 3 (C)} C H M 3 (C). unique up to Aut(A LR ) spontaneous breaking of LR symmetry

11 Subalgebras with off diagonal Dirac and order one condition Operator T : H f H f A(T ) = {b A LR π (b)t = T π(b), π (b )T = T π(b )} involutive unital subalgebra of A LR A A LR involutive unital subalgebra of A LR 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 of π(a LR ) and π (A LR ) and operator T e Te = T 0 and A A(T ) Then case by case analysis to identify max dimensional

12 Symmetries Up to a finite abelian group SU(A F ) U(1) SU(2) SU(3) Adjoint action of U(1) (in powers of λ U(1)) L R correct hypercharges of fermions (confirms identification of H F basis with fermions)

13 Classifying Dirac operators for (A F, H F, γ F, J F ) all possible D F self adjoint on H F, commuting with J F, anticommuting with γ F and [[D, a], b 0 ] = 0, a, b A F Input conditions (massless photon): commuting with subalgebra C F A F, C F = {(λ, λ, 0), λ C} ( ) S T then D(Y ) = with S = S T S 1 (S ) 0 0 Y( 1) Y S 1 = ( 1) Y ( 1) Y ( 1) 0 0 same for S 3, with Y ( 1), Y ( 1), Y ( 3), Y ( 3) GL 3 (C) and Y R symmetric: T : E R = R 1 0 J F E R

14 Moduli space C 3 C 1 C 3 = pairs (Y ( 3), Y ( 3) ) modulo W j unitary matrices Y ( 3) = W 1 Y ( 3) W 3, Y ( 3) = W 2 Y ( 3) W 3 C 3 = (K K)\(G G)/K G = GL 3 (C) and K = U(3) dim R C 3 = 10 = (3 + 3 eigenvalues, 3 angles, 1 phase) C 1 = triplets (Y ( 1), Y ( 1), Y R ) with Y R symmetric modulo Y ( 1) = V 1 Y ( 1) V 3, Y ( 1) = V 2 Y ( 1) V 3, Y R = V 2 Y R V 2 π : C 1 C 3 surjection forgets Y R fiber symmetric matrices mod Y R λ 2 Y R dim R (C 3 C 1 ) = 31 (dim fiber 12-1=11)

15 Physical interpretation: Yukawa parameters and Majorana masses Representatives in C 3 C 1 : Y ( 3) = δ ( 3) Y ( 3) = U CKM δ ( 3) U CKM Y ( 1) = U PMNS δ ( 1) U PMNS Y ( 1) = δ ( 1) δ, δ diagonal: Dirac masses c 1 s 1 c 3 s 1 s 3 U = s 1 c 2 c 1 c 2 c 3 s 2 s 3 e δ c 1 c 2 s 3 + s 2 c 3 e δ s 1 s 2 c 1 s 2 c 3 + c 2 s 3 e δ c 1 s 2 s 3 c 2 c 3 e δ angles and phase c i = cos θ i, s i = sin θ i, e δ = exp(iδ) U CKM = Cabibbo Kobayashi Maskawa U PMNS = Pontecorvo Maki Nakagawa Sakata neutrino mixing Y R = Majorana mass terms for right-handed neutrinos

16 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?

17 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...?)

18 Symmetries and NCG Symmetries of gravity coupled to matter: G = U(1) SU(2) SU(3) G = Map(M, G) Diff(M) Is it G = Diff(X )? Not for a manifold, yes for an NC space Example: A = C (M, M n (C)) G = PSU(n) 1 Inn(A) Aut(A) Out(A) 1 1 Map(M, G) G Diff(M) 1.

19 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

20 Product geometry M F Two spectral triples (A i, H i, D i, γ i, J i ) of KO-dim 4 and 6: A = A 1 A 2 H = H 1 H 2 D = D γ 1 D 2 γ = γ 1 γ 2 J = J 1 J 2 Case of 4-dimensional spin manifold M and finite NC geometry F : A = C (M) A F = C (M, A F ) H = L 2 (M, S) H F = L 2 (M, S H F ) D = / M 1 + γ 5 D F D F chosen in the moduli space described last time

21 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 ζ a,d (s) = Tr(a D s ) 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 = (equal 2 mod 8); DimSp k Z 0 with k 4

22 Variant: almost commutative geometries (C (M, E), L 2 (M, E S), D E ) M smooth manifold, E algebra bundle: fiber E x finite dimensional algebra A F C (M, E) smooth sections of a algebra bundle E Dirac operator D E = c ( E S ) with spin connection S and hermitian connection on bundle Compatible grading and real structure An equivalent intrinsic (abstract) characterization in: Branimir Ćaćić, A reconstruction theorem for almost-commutative spectral triples, arxiv: Here we will assume for simplicity just a product M F

23 Inner fluctuations and gauge fields Setup: Right A-module structure on H ξ b = b 0 ξ, ξ H, b A Unitary group, adjoint representation: ξ H Ad(u) ξ = u ξ u ξ H Inner fluctuations: D D A = D + A + ε J A J 1 with A = A self-adjoint operator of the form A = a j [D, b j ], a j, b j A Note: not an equivalence relation (finite geometry, can fluctuate D to zero) but like self Morita equivalences

24 Properties of inner fluctuations (A, H, D, J) Gauge potential A Ω 1 D, A = A Unitary u A, then Ad(u)(D + A + ε J A J 1 )Ad(u ) = D + γ u (A) + ε J γ u (A) J 1 where γ u (A) = u [D, u ] + u A u D = D + A (with A Ω 1 D, A = A ) then D + B = D + A, A = A + B Ω 1 D B Ω 1 D B = B D = D + A + ε J A J 1 then D + B + ε J B J 1 = D + A + ε J A J 1 A = A + B Ω 1 D B Ω 1 D B = B

25 Gauge bosons and Higgs boson Unitary U(A) = {u A uu = u u = 1} Special unitary SU(A F ) = {u U(A F ) det(u) = 1} det of action of u on H F Up to a finite abelian group SU(A F ) 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

26 More on Gauge bosons Inner fluctuations A (1,0) = i a i[/ M 1, a i ] with with a i = (λ i, q i, m i ), a i = (λ i, q i, m i ) in A = C (M, A F ) U(1) gauge field Λ = i λ i dλ i = i λ i[/ M 1, λ i ] SU(2) gauge field Q = i q i dq i, with q = f 0 + α if ασ α and Q = α f α[/ M 1, if ασ α ] U(3) gauge field V = i m i dm i = i m i[/ M 1, m i ] reduce the gauge field V to SU(3) passing to unimodular subgroup SU(A F ) and unimodular gauge potential Tr(A) = 0 V = V 1 3 Λ Λ Λ = V 1 3 Λ1 3

27 Gauge bosons and hypercharges The (1, 0) part of A + JAJ 1 acts on quarks and leptons by 4 3 Λ + V Λ + V Q Λ + V Q Q 21 Q Λ + V correct hypercharges! Λ Q 11 Λ Q Q 21 Q 22 Λ

28 More on Higgs boson Inner fluctuations A (0,1) in the F -space direction i a i [γ 5 D F, a i](x) Hf = γ 5 (A (0,1) q + A (0,1) l ) ( A (0,1) 0 X q = X 0 ( Υ X = u ϕ 1 Υ ) uϕ 2 Υ d ϕ 2 Υ d ϕ 1 ( Υ Y = ν ϕ 1 Υ ) νϕ 2 Υ e ϕ 2 Υ e ϕ 1 ) 1 3 A (0,1) 1 = ( 0 Y Y 0 ) and X = ( Υu ϕ 1 Υ d ϕ 2 Υ u ϕ 2 Υ d ϕ 1 and Y = ( Υν ϕ 1 Υ e ϕ 2 Υ ν ϕ 2 Υ e ϕ 1 ϕ 1 = λ i (α i λ i ), ϕ 2 = λ i β i ϕ 1 = α i (λ i α i ) + β i β i and ϕ 2 = ( α i β i + β i( λ i ᾱ i ( )), for a i(x) ) = (λ i, q i, m i ) and α β a i (x) = (λ i, q i, m i ) and q = β ᾱ ) )

29 More on Higgs boson Discrete part of inner fluctuations: quaternion valued function H = ϕ 1 + ϕ 2 j or ϕ = (ϕ 1, ϕ 2 ) D 2 A = (D1,0 ) (D 0,1 ) 2 γ 5 [D 1,0, 1 4 D 0,1 ] [D 1,0, 1 4 D 0,1 ] = 1 γ µ [( s µ + A µ ), 1 4 D 0,1 ] This gives D 2 A = E where Laplacian of = s + A E = 1 4 s id + µ<ν γ µ γ ν F µν i γ 5 γ µ M(D µ ϕ) (D 0,1 ) 2 with s = R scalar curvature and F µν curvature of A D µ ϕ = µ ϕ + i 2 g 2W α µ ϕ σ α i 2 g 1B µ ϕ SU(2) and U(1) gauge potentials