Inner perturbations in noncommutative geometry Walter D. van Suijlekom (joint with Ali Chamseddine and Alain Connes) January 17, 2014 Overview Spectral triples as noncommutative manifolds Gauge group, semi-group of inner perturbations Relation to Morita equivalence Examples: Yang{Mills, almost-commutative manifolds, SM Noncommutative manifolds Basic device: a spectral triple (A; H; D): algebra A of bounded operators on a Hilbert space H, a self-adjoint operator D in H with compact resolvent and such that the commutator [D; a] is bounded for all a 2 A. Key example: Riemannian spin geometry Let M be an compact m-dimensional Riemannian spin manifold. A = C 1(M) H = L2(S), square integrable spinors D = @= , Dirac operator Then D has compact resolvent because @= elliptic self-adjoint. Also[ D; f ] bounded for f 2 C 1(M). Real spectral triples (A; H; D) Extend to a real spectral triple: J : H!H real structure (anti-unitary) such that J2 = ±1; JD = ±DJ Action of Aop on H: aop = Ja∗J−1 such that aopbop = Ja∗J−1Jb∗J−1 = J(ba)∗J−1 = (ba)op We demand [aop; b] = 0; a; b 2 A D is said to satisfy the first-order condition if [[D; a]; bop] = 0 Spectral invariants 1 Tr f (D=Λ)+ hJ ; D i 2 Invariant under unitaries u 2 U(A) acting as D 7! UDU∗; U = uJuJ−1 Gauge group: G(A) := fuJuJ−1 : u 2 U(A)g Compute rhs: D 7! D + u[D; u∗] +u ^[D; u^∗] +^u[u[D; u∗]; u^∗] withu ^ = JuJ−1 and blue term vanishes if D satisfies first-order condition Semi-group of inner perturbations 8 9 P <X op op j aj bj = 1 = Pert(A) := aj ⊗ bj 2 A ⊗ A P op P ∗ ∗op aj ⊗ b = b ⊗ a : j j j j j j ; with semi-group law inherited from product in A ⊗ Aop. U(A) maps to Pert(A) by sending u 7! u ⊗ u∗op. Pert(A) acts on D: X D 7! aj Dbj j For real spectral triples we use the map Pert(A) ! Pert(A ⊗ A^) sending T 7! T ⊗ T^ so that X D 7! ai ^aj Dbi b^j i;j and this extends the action of the gauge group G(A). Morita equivalence Suppose A ∼M B. Can we construct a spectral triple on B from (A; H; D)? Let B' EndA(E) with E finitely generated projective. Define 0 H = E ⊗A H Then B acts as bounded operators on H0. The self-adjoint operator (1 ⊗r D)(η ⊗ ) := rD (η) + η ⊗ D requires a universal connection on E: 1 r : E ! E ⊗A Ω (A) 1 where rD indicates that aδ(b) 2 Ω (A) is represented as a[D; b]. 0 Then( B; H ; 1 ⊗r D) is a spectral triple [Connes, 1996]. Morita equivalence with real structure Again, suppose A ∼M B. If there is a real structure J on (A; H; D), then we define 0 H := (E ⊗A H) ⊗A E with the conjugate (left A-) module E and define analogously the 0 operator (1 ⊗r D) ⊗r 1 on H , where 1 r : E! Ω (A) ⊗A E; and we also define J0 : H0 !H0; η ⊗ ⊗ ρ 7! ρ ⊗ J ⊗ η Proposition (Chamseddine{Connes{vS, 2013) We have (1 ⊗r D) ⊗r 1 = 1 ⊗r (D ⊗r 1) and the tuple 0 0 (B; H ; (1 ⊗r D) ⊗r 1; J ) is a real spectral triple. Morita self-equivalence 0 0 If B = A (i.e. E = A) we have H = E ⊗A H ⊗A E'H and J ≡ J. 0 The operator D is perturbed to D ≡ D + A(1) + Ae(1) + A(2) where X X ^ −1 A(1) := aj [D; bj ]; Ae(1) := ^aj [D; bj ] = ±JA(1)J ; j j X ^ X ^ A(2):= ^aj [A(1); bj ] = ^aj ak [[D; bk ]; bj ] j j;k and blue terms vanish if D satisfies first-order condition Gauge transformations D0 7! UD0U∗ implemented by ∗ ∗ A(1) 7! uA(1)u + u[D; u ] −1 ∗ −1 −1 ∗ ∗ −1 A(2) 7! JuJ A(2)Ju J + JuJ [u[D; u ]; Ju J ] Perturbation semi-group and Morita self-equivalences Proposition (Chamseddine{Connes{vS, 2013) The linear map η : Pert(A) ! Ω1(A), η(a ⊗ bop) = aδ(b) is surjective. P op If j aj ⊗ bj 2 Pert(A) then the perturbed operator X X aj Dbj = D + aj [D; bj ] ≡ D + A(1) j j and, for real spectral triples: X ^ ai ^aj Dbi bj = D + A(1) + Ae(1) + A(2) i;j Examples: Pert(A) for matrix algebras (joint with Niels Neumann) 8 9 P <X op op j aj bj = 1 = Pert(A) := aj ⊗ bj 2 A ⊗ A P op P ∗ ∗op aj ⊗ b = b ⊗ a : j j j j j j ; N N If A = C , then in terms of basis vectors fei g of C : 8 9 N <X N2 Cii = 1 = N(N−1)=2 Pert(C ) ' Cij ei ⊗ ej 2 C ' C Cij = Cji : i;j ; Examples: Pert(A) for matrix algebras (joint with Niels Neumann) 8 9 P <X op op j aj bj = 1 = Pert(A) := aj ⊗ bj 2 A ⊗ A P op P ∗ ∗op aj ⊗ b = b ⊗ a : j j j j j j ; op If A = MN (C) 'A , then in terms of basis vectors feij g of MN (C): 8 9 P <X j Cij;jk = δik = Pert(MN (C)) ' Cij;kl eij ⊗ ekl 2 MN2 (C) Cij;kl = Clk;ji : i;j ; Ce1 = e1 ' C 2 MN2 (C) CΩ = ΩC 1 0 where Ω = N(N+1)=2 . 0 −1N(N−1)=2 Examples: Pert(A) for function algebras 8 9 P <X op op j aj bj = 1 = Pert(A) := aj ⊗ bj 2 A ⊗ A P op P ∗ ∗op aj ⊗ b = b ⊗ a : j j j j j j ; If A = C 1(M) then 1 1 f (x; x) = 1 8x 2 M Pert(C (M)) ' f 2 C (M × M) f (x; y) = f (y; x) 8x; y 2 M 1 µ Action of Pert(C (M)) on D = @= = iγ rµ is given by X µ µ aj @= bj = @= + iγ r @ f (x; y)jx=y =: @= + iγ Aµ @yµ j 1 with Aµ 2 C (M; u(1)) Non-abelian Yang{Mills theory On a 4-dimensional background: 1 A = C (M) ⊗ Mn(C) 2 H = L (S) ⊗ Mn(C) D = @= ⊗ 1 J = C ⊗ (:)∗. Proposition (Chamseddine-Connes, 1996) Tr f (D): pure gravity µ The perturbations A(1) + Ae(1) with A(1) = γ Aµ describes an su(n)-gauge field on M. Gauge group G(A) ' C 1(M; SU(n)) The spectral action of the perturbed Dirac operator is given by ! Z D + A(1) + Ae(1) f (0) µν −1 Tr f ∼ (··· ) + 2 Tr FµνF + O(Λ ) Λ 24π M Almost-commutative geometries A class of examples 1 2 (C (M) ⊗ AF ; L (S) ⊗ HF ; @= ⊗ 1 + γ5 ⊗ DF ) with real structure J = JM ⊗ JF . 1 1 Gauge group G(C (M) ⊗ AF ) = C (M; G(AF )) Inner perturbations: 0 µ D 7! D = @= ⊗ 1 + γ ⊗ adAµ + γ5 ⊗ Φ −1 with adAµ a g(AF )-gauge potential and Φ = DF + φ + JF φJF a map HF !HF Explicitly, X X Aµ = −i aj @µ(bj ); φ = aj [DF ; bj ] j j As G(AF )-representations: ∗ ∗ ∗ Aµ 7! uAµu − iu@µu ; Φ 7! UΦU Almost-commutative geometries Spectral action Proposition (Van den Dungen{vS, 2012) In the above setting, D0 f (0) 2f Λ2 f (0) Tr f ∼ (··· ) + Tr (F F µν) − 2 Tr (Φ2) + Tr (Φ4) Λ 24π2 µν 4π2 8π2 f (0) f (0) + sTr (Φ2) + Tr (D Φ)(DµΦ) + O(Λ−1): 48π2 8π2 µ with f2 the first moment of f . The noncommutative Standard Model 1 2 (C (M) ⊗ (C ⊕ H ⊕ M3(C)); L (S) ⊗ HF ; @= ⊗ 1 + γ5 ⊗ DF ) Fermions are given by: ⊕3 HF := Hl ⊕ Hl ⊕ Hq ⊕ Hq : Algebra acts as: 0λ 0 0 0 1 0λ 0 0 0 1 Hl B 0 λ 0 0 C Hq B 0 λ 0 0 C (λ, q; m) −! B C ; (λ, q; m) −−! B C ⊗ 13; @ 0 0 α β A @ 0 0 α β A 0 0 −β α 0 0 −β α Hl Hq (λ, q; m) −! λ14; (λ, q; m) −−! 14 ⊗ m Real structure JF interchanges fermions and anti-fermions. The noncommutative Standard Model The finite Dirac operator ST ∗ D := F T S The operator S is given by 0 1 0 1 0 0 Yν 0 0 0 Yu 0 0 0 0 Y 0 0 0 Y S := Sj = B e C ; S ⊗ 1 := Sj = B d C ⊗ 1 ; l Hl B ∗ C q 3 Hq B ∗ C 3 @Yν 0 0 0 A @Yu 0 0 0 A ∗ ∗ 0 Ye 0 0 0 Yd 0 0 where Yν, Ye , Yu and Yd are 3 × 3 mass matrices acting on the three generations. The symmetric operator T only acts on the right-handed (anti)neutrinos, T νR = YR νR for a 3 × 3 symmetric Majorana mass matrix YR , and Tf = 0 for all other fermions f 6= νR . The noncommutative Standard Model The spectral action Proposition (Chamseddine{Connes{Marcolli, 2007) In the above setting, The unimodular gauge group SG(C ⊕ H ⊕ M3(C)) = U(1) × SU(2) × SU(3) The inner perturbations of @= ⊗ 1 + γ5 ⊗ DF are parametrized by U(1); SU(2) and SU(3) gauge fields Λµ; Qµ; Vµ and a Higgs doublet H The spectral action is given by D0 f (0)10 Tr f ∼ (··· ) + Λ Λµν + Tr (Q Qµν) + Tr (V V µν) Λ π2 3 µν µν µν bf (0) −2af Λ2 + ef (0) af (0) + jHj4 + 2 jHj2 + jD Hj2 + O(Λ−1): 2π2 π2 2π2 µ Example beyond first-order [Chamseddine{Connes{vS, 2013] 0 AF = CR ⊕ CL ⊕ M2(C); 2 ◦ 2 ◦ ◦ HF = (CR ⊕ CL) ⊗ (C ) ⊕ C ⊗ (CR ⊕ CL); 0 1 J = ◦ C (C : complex conjugation); F 1 0 0 1 0 k ⊗ 1 ky 0 0 x 2 0 0 Bkx ⊗ 12 0 0 C DF = B C B ky 0 C @ 0 0 0 0 12 ⊗ kx A 0 0 12 ⊗ kx 0 The algebra action of (λR ; λL; m) 2 A on H is given explicitly by t ! λR 12 m op mt π(λ ; λ ; m) = λL12 ; π (λ ; λ ; m) = : R L m R L λR 12 m λL12 Proposition 0 The largest (even) subalgebra AF ⊂ AF ≡ CR ⊕ CL ⊕ M2(C) for which the first-order condition holds (for the above HF ; DF and JF ) is given by λ 0 A = λ ; λ ; R :(λ ; λ ; µ) 2 ⊕ ⊕ F R L 0 µ R L CR CL C Proposition The inner perturbed Dirac operatorD 0 is parametrized by three complex scalar fields φ, σ1; σ2 entering in A(1) and A(2): t 0 0 kx (1+φ)⊗12 ky vv 0 1 ^ kx (1+φ)⊗12 0 0 DF + A(1) + A(1) + A(2) = @ t A ky v v 0 0 12⊗kx (1+φ) 0 0 12⊗kx (1+φ) 0 1 + σ with v = 1 .