Summary Talk based on: The Standard Model in Noncommutative - PowerPoint PPT Presentation

Summary Talk based on: The Standard Model in Noncommutative Geometry: FD & L. Dabrowski, particles as Dirac spinors? The Standard Model in Noncommutative Geometry and Morita equivalence , preprint arXiv:1501.00156 [math-ph]; to appear in J. Noncommut. Geom. Francesco D’Andrea Keywords → Standard Model, Morita equivalence, finite-dimensional spectral triples. 08/04/2016 Summary of the talk: 1 Spectral triples (again). 2 An algebraic characterization of Dirac spinors. Conference on Noncommutative Geometry and Gauge Theories Radboud University Nijmegen, 4 – 8 april 2016 3 The finite nc space of the ν SM. 1 / 13 Introduction Examples of spectral triples Let: ( M , g ) = compact oriented Riemannian manifold without boundary, E → M herm. Definition vector bundle equipped with a unitary Clifford action c : C ∞ ( M , T ∗ C M ⊗ E ) → C ∞ ( M , E ) A unital spectral triple ( A , H , D ) is the datum of: and a connection ∇ E compatible with g . Then: (i) a (real or complex) unital C ∗ -algebra A of bounded operators on a (separable) H = L 2 ( M , E ) D = c ◦ ∇ E complex Hilbert space H , A = C ( M ) (ii) a selfadjoint operator D on H with compact resolvent, is a spectral triple. such that 1. Hodge operator 2. Signature operator ( dim M even) (iii) the unital ∗ -subalgebra E = � + T ∗ C M ⊕ � − T ∗ E = � even T ∗ C M ⊕ � odd T ∗ C M , D = d + d ∗ C M with grading � � given by the Hodge star, D = d + d ∗ . Lip D ( A ) = a ∈ A : a · Dom ( D ) ⊂ Dom D and [ D , a ] ∈ B ( H ) ⇒ Index ( D + ) = Euler char. of M ⇒ Index ( D + ) = signature of M is dense in A . 3. Dolbeault operator ( M complex m.) 4. Dirac operator ( M spin) E = � 0,even M ⊕ � 0,odd M , ∗ Example 0 (finite nc spaces) D = ∂ + ∂ E = spinor bundle, D = D / Dirac operator Take any finite-dimensional H , any A ⊂ B ( H ) and D ∈ B ( H ) . In this case Lip D ( A ) ≡ A . ⇒ Index ( D + ) = Euler char. of O M In all these examples, H carries commuting representations of A = C ( M ) and B = C ℓ ( M , g ) . 2 / 13 3 / 13

Algebraic characterization of Dirac spinors 5. The Standard Model spectral triple The underlying geometry is Definition A unital spectral triple ( A , H , D ) is called: M × F ◮ even if ∃ γ = γ ∗ on H s.t. γ 2 = 1 , γD = − Dγ and [ γ , a ] = 0 ∀ a ∈ A ; (spin manifold) (finite nc space) ◮ real if ∃ an antilinear isometry J on H s.t. J 2 = ± 1 , JD = ± DJ , Jγ = ± γJ and ∀ a , b ∈ A : with finite-dim. spectral triple ( A F , H F , D F , γ F , J F ) given by: [ a , JbJ − 1 ] = 0 [[ D , a ] , JbJ − 1 ] = 0 ◮ H F ≃ C 32 n � internal degrees of freedom of the elementary fermions. Total nr: (reality) (1st order) = 2 × 4 × 2 × 2 × n 32 n (weak isospin) (lepton + quark (L,R chirality) (particle or (generations) Theorem in 3 colors) antiparticle) 1. A closed oriented Riem. manifold M admits a spin c structure iff ∃ a Morita equivalence ◮ γ F = chirality operator C ( M ) - C ℓ ( M , g ) bimodule Σ , with C ℓ ( M , g ) the algebra of sections of the Clifford bundle. 2. Σ = C 0 sections of the spinor bundle S → M (Dirac spinors in the conventional sense). ◮ A F = C ⊕ H ⊕ M 3 ( C ) � gauge group ≈ U ( 1 ) × SU ( 2 ) × SU ( 3 ) Once we have S , we can canonically introduce the Dirac operator D of the spin c structure: J F = charge conjugation 3. M is a spin manifold iff ∃ a real structure J on L 2 ( M , S ) . ◮ D F encodes the free parameters of the theory. 4 / 13 5 / 13 What is a noncommutative spin manifold? On a property of “Hodge spinors” For simplicity, let us focus on finite-dimensional spectral triples. In the geometric examples (slide 3), [ D , f ] = c ( df ) . In the Hodge example: ( ⇒ A ≡ Lip D ( A ) and we can use the ring-theoretic Morita equivalence.) L 2 L 2 H = Ω • ( M ) ≃ C ℓ ( M , g ) B := C ℓ D ( A ) = C ℓ ( M , g ) Definition ( 1 -forms) Representation of B : by Clifford multiplication on Ω • ( M ) , or by left multiplication on itself. If ( A , H , D ) is a spectral triple, we define Ω 1 D ⊆ B ( H ) as: Real structure: J ( ω ) = ω ∗ . The algebra B ◦ = JBJ − 1 acts by right multiplication on H , � � Ω 1 D := Span a [ D , b ] : a , b ∈ A that up to completion is a self-Morita equivalence B -bimodule. Definition (Clifford algebra) [ ≈ Lord, Rennie & V´ arilly, J.Geom.Phys. 2012] Definition (2nd order condition) We call C ℓ D ( A ) ⊆ B ( H ) the algebra generated by A , Ω 1 D and possibly γ (in the even case). ( A , H , D , J ) satisfies the 2nd order condition if C ℓ D ( A ) ◦ := J C ℓ D ( A ) J − 1 ⊆ C ℓ D ( A ) ′ � � Let A ◦ := Ja ∗ J − 1 : a ∈ A ( ⋆⋆ ) . The reality and 1st order cond. are equivalent to the statement � � A ◦ ⊆ C ℓ D ( A ) ′ := b ∈ B ( H ) : [ b , ξ ] = 0 ∀ ξ ∈ C ℓ D ( A ) . ( ⋆ ) Remark: this is the old “order-two” condition by Boyle and Farnsworth (cf. also Besnard, Bizi, Brouder). Definition (Hodge condition) Definition (Dirac condition) ( dim H < ∞ ) Elements of H are “Hodge spinors” if ( ⋆⋆ ) is an equality: C ℓ D ( A ) ◦ = C ℓ D ( A ) ′ . Elements of H are “Dirac spinors” if ( ⋆ ) is an equality: A ◦ = C ℓ D ( A ) ′ . 6 / 13 7 / 13

Spin + 2nd order Back to the Standard Model. . . Observation 1. Recall that in the ncg approach to the Standard Model,one has: Dirac condition + 2nd order condition ⇒ Hodge condition. M × F (spin manifold) (finite nc space) In fact: For the continuous part, elements of H M are Dirac spinors. What about the finite part? C ℓ D ( A ) ◦ ⊆ C ℓ D ( A ) ′ = A ◦ C ℓ D ( A ) = A = ⇒ We have the following dictionary: Therefore: Geometry ← → Algebra Observation 2. Spin c A - C ℓ D ( A ) Morita equivalence Dirac condition + 2nd order ⇒ H is a self-Morita equivalence A -bimodule (a “line bundle”). Spin A - C ℓ D ( A ) Morita equivalence with J C ℓ D ( A ) self-Morita equivalence ∗ Hodge An example of spectral triple satisfying both conditions (Einstein-Yang Mills): ∗ in progress with L. Dabrowski & A. Sitarz J ( a ) = a ∗ A = M N ( C ) H = A D = 0 What kind of nc space is F ? 8 / 13 9 / 13 Postdictions on D F The 1st order condition Let ( A , H , J ) be finite dim. One can completely characterize D ’s of 1st order: Not every D F is allowed! ⇒ Restrictions on the free parameters/on the interactions. Constraints of the 1st kind: Theorem ( ≈ Krajewski) • D ∈ End C ( H ) satisfies the 1st order condition iff it is of the form 1 The parity ( γ F D F = − D F γ F ) and 1st (or 2nd) order condition put constraints on D F : some matrix entries must be zero. D = D 0 + D 1 ( † ) For example, the 1st order cond. does not allow a vertex with D 0 ∈ ( A ◦ ) ′ and D 1 ∈ A ′ . e − • D selfadjoint resp. odd ⇒ one can always choose D 0 and D 1 selfadjoint resp. odd. ? • JD = DJ ⇒ one can choose D 1 = JD 0 J − 1 . e − Proof. Lemma: Let H be finite-dimensional and V ⊂ End ( H ) a ∗ -subalgebra. Then, there exists a direct complement W of V in End ( H ) such that [ V , W ] ⊂ W . ♣ Nothing forbids taking D F = 0 (all conditions are satisfied). For V = A ′ let W be the complement above. Write D = D 0 + D 1 with D 0 ∈ V and D 1 ∈ W . Constraints of the 2nd kind: From the 1st order condition we deduce that in fact D 1 ∈ ( A ◦ ) ′ . � 2 The request that elements of H F are Dirac spinors (or Hodge spinors) on F implies, in Remark: In [Krajewski, J.Geom.Phys. 1998] uniqueness of the decomposition ( † ) follows from the general, that some matrix entries cannot be zero. orientability condition. In the ν SM orientability is not satisfied, and the decomposition is not unique. 10 / 13 11 / 13

The Dirac condition On the Higgs mass Several modifications of the original model have been proposed. One can: Theorem 1. enlarge the Hilbert space thus introducing new fermions [Stephan, 2009] ; If γ F = χ is the chirality operator, there is no compatible D F satisfying the Dirac condition. 2. turn one element of D F into a field by hand, rather than getting it as a fluctuation of On the other hand, consider the following grading, given on particles by the metric [Chamseddine & Connes, 2012] ; γ F := ( B − L ) χ 3. break (relax) the 1st order condition, thus allowing more terms in the Dirac operator (or in the algebra) [Chamseddine, Connes & van Suijlekom, 2013] ; with B , L = barion/lepton nr. Then it is possible to find D F satisfying the Dirac condition (we 4. Grand Symmetry + twisted spectral triples [Devastato, Lizzi & Martinetti, 2014] . have theorems both with necessary conditions and sufficient conditions). In 2,3,4: the Majorana mass term of the neutrino is replaced by a new scalar field Φ . Remarks: Theorem ◮ 16 free parameters or 25 with the non-standard γ F (for a toy model with 1 generation). In order to satisfy the Dirac condition, we must add two terms to Chamseddine-Connes D F . ◮ In the Standard Model: 19 parameters, whose numerical values are established by We get: experiments. One of these is the Higgs mass: m H ≈ 126 GeV. → a new scalar field close to the Φ above (but doesn’t break the 1st order condition); ◮ In Chamseddine-Connes’ original spectral triple, m H is not a free parameter. It was → a field coupling leptons with quarks. predicted m H ≈ 170 GeV, a value ruled out by Tevatron in 2008. Physical implications are under investigation (see the talk at this conference by F . Lizzi). 12 / 13 13 / 13