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Inner Perturbations in Noncommutative Geometry

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Inner Perturbations in Noncommutative Geometry 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 .
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