Gauge transformation for twisted spectral triples and Lorentz - PowerPoint PPT Presentation

Gauge transformation for twisted spectral triples and Lorentz signature Pierre Martinetti Universit` a di Genova Cortona , 6 th June 2018

Introduction Noncommutative geometry is an extension of Riemannian geometry, useful in mathematics to describe objects whose properties are not captured by usual geometry. Interesting for physics, for it provides a description of the standard model of elementary particles [SM] minimally coupled with (Euclidean) general relativity. Added value : the Higgs field comes out as a connection 1-form, like the other bosons, but a connection between the discrete and the continuum part of the geometry. Recent developments - e.g. twist [ D , a ] → Da − ρ ( a ) D - open the way to models beyond the SM, and might help to adress the question of the Lorentzian signature.

1. Noncommutative geometry in a nutshell 2. Gauge theory by Morita equivalence 3. The standard model: need for twist ! 4. Twisted spectral triple 5. Shadow of the Lorentz signature

1. Noncommutative geometry ina nutshell Spectral triple: algebra A acting on a Hilbert H together with selfadjoint D s.t. [ D , a ] is bounded ∀ a ∈ A . Graded spectral triple: there exists Γ = Γ ∗ , Γ 2 = I , such that { Γ , D } = 0 , [Γ , a ] = 0 ∀ a ∈ A . Real spectral triple: there exists antilinear operator J such that J 2 = ǫ I , JD = ǫ ′ DJ , J Γ = ǫ ′′ Γ J where ǫ, ǫ ′ , ǫ ′′ = ± 1 define the KO -dimension k ∈ [0 , 7].

1. Noncommutative geometry ina nutshell Spectral triple: algebra A acting on a Hilbert H together with selfadjoint D s.t. [ D , a ] is bounded ∀ a ∈ A . Graded spectral triple: there exists Γ = Γ ∗ , Γ 2 = I , such that { Γ , D } = 0 , [Γ , a ] = 0 ∀ a ∈ A . Real spectral triple: there exists antilinear operator J such that J 2 = ǫ I , JD = ǫ ′ DJ , J Γ = ǫ ′′ Γ J where ǫ, ǫ ′ , ǫ ′′ = ± 1 define the KO -dimension k ∈ [0 , 7]. J implements a map a → a ◦ := Ja ∗ J − 1 from A to the opposite algebra A ◦ . This yields a right action of A on H , ψ a := a ◦ ψ, which is asked to commute with the left action (order zero condition) [ a , Jb ∗ J − 1 ] = 0 ∀ a , b ∈ A . As well, holds the first order condition [[ D , a ] , Jb ∗ J − 1 ] = 0 ∀ a , b ∈ A .

Connes’ reconstruction theorem : with other extra-conditions one has the following spectral characterization of manifolds: ⇒ spectral triple ( C ∞ ( M ) , L 2 ( M , S ) , ∂ Compact Riemannian manifold M = / ) M such that A = C ∞ ( M ) ⇐ = ( A , H , D ) with A commutative, unital → commutative spectral triple noncommutative spectral triple � ↓ Riemannian geometry non-commutative geometry

2. Gauge theory by Morita equivalence � Fermionic fields = sections of a G -bundle E , Gauge theory with gauge group G Bosonic fields = connections on E . ⇒ a finite projective C ∞ ( M )-module. Sections of a bundle on a manifold M ⇐ Serre-Swan ◮ Bundle in noncommutative geometry ⇐ ⇒ finite projective A -module E .

2. Gauge theory by Morita equivalence � Fermionic fields = sections of a G -bundle E , Gauge theory with gauge group G Bosonic fields = connections on E . ⇒ a finite projective C ∞ ( M )-module. Sections of a bundle on a manifold M ⇐ Serre-Swan ◮ Bundle in noncommutative geometry ⇐ ⇒ finite projective A -module E . ◮ Connection on a (right) A -module E : application E → E ⊗ A Ω such that ∇ ( η a ) = ∇ ( η ) a + η ⊗ δ ( a ) ∀ η ∈ E , a ∈ A , where Ω is a A -bimodule generated by a derivation δ of A .

Morita equivalence A , B Morita equivalent ⇐ ⇒ B = End A ( E ), E a Hermitian finite proj. A -module. How to export a spectral triple ( A , H , D ) to a Morita equivalent algebra B ?

Morita equivalence A , B Morita equivalent ⇐ ⇒ B = End A ( E ), E a Hermitian finite proj. A -module. How to export a spectral triple ( A , H , D ) to a Morita equivalent algebra B ? Assume E = E R is a right A -module. The algebra B acts on H R := E R ⊗ A H as b ( η ⊗ ψ ) = b η ⊗ ψ ∀ b ∈ B , η ∈ E R , ψ ∈ H . The “natural” action of D on H R , D R ( η ⊗ ψ ) := η ⊗ D ψ, is ill-defined since ∀ a ∈ A one gets D R ( η a ⊗ ψ ) − D R ( η ⊗ a ψ ) = − η ⊗ [ D , a ] ψ.

Morita equivalence A , B Morita equivalent ⇐ ⇒ B = End A ( E ), E a Hermitian finite proj. A -module. How to export a spectral triple ( A , H , D ) to a Morita equivalent algebra B ? Assume E = E R is a right A -module. The algebra B acts on H R := E R ⊗ A H as b ( η ⊗ ψ ) = b η ⊗ ψ ∀ b ∈ B , η ∈ E R , ψ ∈ H . The “natural” action of D on H R , D R ( η ⊗ ψ ) := η ⊗ D ψ, is ill-defined since ∀ a ∈ A one gets D R ( η a ⊗ ψ ) − D R ( η ⊗ a ψ ) = − η ⊗ [ D , a ] ψ. Take a connection ∇ on E R with value in the A -bimodule generated by the derivation δ ( . ) = [ D , . ]: Ω 1 D ( A ) := { a i [ D , b i ] , a i , b i ∈ A} . ◮ The covariant derivative D R ( η ⊗ ψ ) := η ⊗ D ψ + ( ∇ η ) ψ is well defined.

Morita equivalence A , B Morita equivalent ⇐ ⇒ B = End A ( E ), E a Hermitian finite proj. A -module. How to export a spectral triple ( A , H , D ) to a Morita equivalent algebra B ? Assume E = E R is a right A -module. The algebra B acts on H R := E R ⊗ A H as b ( η ⊗ ψ ) = b η ⊗ ψ ∀ b ∈ B , η ∈ E R , ψ ∈ H . The “natural” action of D on H R , D R ( η ⊗ ψ ) := η ⊗ D ψ, is ill-defined since ∀ a ∈ A one gets D R ( η a ⊗ ψ ) − D R ( η ⊗ a ψ ) = − η ⊗ [ D , a ] ψ. Take a connection ∇ on E R with value in the A -bimodule generated by the derivation δ ( . ) = [ D , . ]: Ω 1 D ( A ) := { a i [ D , b i ] , a i , b i ∈ A} . ◮ The covariant derivative D R ( η ⊗ ψ ) := η ⊗ D ψ + ( ∇ η ) ψ is well defined. Morita self-equivalence: B = E R = A , then D R = D + A R with A R ∈ Ω 1 D ( A ) .

Same construction with left module E L , H L = H ⊗ A E L , ∇ ◦ a connection on E L with value in the bimodule �� � Ω 1 D ( A ◦ ) = a ◦ i [ D , b ◦ a ◦ i , b ◦ i ∈ A ◦ i ] , . i The covariant derivative D L ( ψ ⊗ η ) := D ψ ⊗ η + ( ∇ ◦ η ) ψ is well defined, Morita self-eq.: D L = D + A ◦ = D + ǫ ′ J A L J − 1 for A ◦ , A L ∈ Ω 1 D ( A ◦ ) , Ω 1 D ( A ).

Same construction with left module E L , H L = H ⊗ A E L , ∇ ◦ a connection on E L with value in the bimodule �� � Ω 1 D ( A ◦ ) = a ◦ i [ D , b ◦ a ◦ i , b ◦ i ∈ A ◦ i ] , . i The covariant derivative D L ( ψ ⊗ η ) := D ψ ⊗ η + ( ∇ ◦ η ) ψ is well defined, Morita self-eq.: D L = D + A ◦ = D + ǫ ′ J A L J − 1 for A ◦ , A L ∈ Ω 1 D ( A ◦ ) , Ω 1 D ( A ). Assume E is a A -bimodule. By combining the two constructions one gets D ′ = D + A R + ǫ ′ J A L J − 1 . One has that D ′ J = ǫ ′ JD ′ if and only if there exists A ∈ Ω 1 D ( A ) such that D ′ = D A := D + A + J AJ − 1

Same construction with left module E L , H L = H ⊗ A E L , ∇ ◦ a connection on E L with value in the bimodule �� � Ω 1 D ( A ◦ ) = a ◦ i [ D , b ◦ a ◦ i , b ◦ i ∈ A ◦ i ] , . i The covariant derivative D L ( ψ ⊗ η ) := D ψ ⊗ η + ( ∇ ◦ η ) ψ is well defined, Morita self-eq.: D L = D + A ◦ = D + ǫ ′ J A L J − 1 for A ◦ , A L ∈ Ω 1 D ( A ◦ ) , Ω 1 D ( A ). Assume E is a A -bimodule. By combining the two constructions one gets D ′ = D + A R + ǫ ′ J A L J − 1 . One has that D ′ J = ǫ ′ JD ′ if and only if there exists A ∈ Ω 1 D ( A ) such that D ′ = D A := D + A + J AJ − 1 ◮ The substitution of D with a D A is called a fluctuation of the metric. ◮ When D A is selfadjoint, it is called a covariant Dirac operator.

Gauge transformation Group U ( E ) of unitary endomorphisms of E : u ∈ End A ( E ) such that u ∗ u = I . Its adjoint action on Ω-value connections on E , ∇ u := u ∇ u ∗ ∀ u ∈ U ( E ) , yields a new Ω-value connection ∇ u .

Gauge transformation Group U ( E ) of unitary endomorphisms of E : u ∈ End A ( E ) such that u ∗ u = I . Its adjoint action on Ω-value connections on E , ∇ u := u ∇ u ∗ ∀ u ∈ U ( E ) , yields a new Ω-value connection ∇ u . Both connections decompose as ∇ u = ∇ Gr + A u , ∇ = ∇ Gr + A , where ∇ Gr is the Grassmann connection, while the gauge potentials A and A u are A -linear maps E → E ⊗ A Ω (right module case) or E → Ω ⊗ A E (left module). ◮ A gauge transformation is the map A → A u .

Consider a covariant operator obtained by Morita self-equivalence: D A = D + A + J A J − 1 with A ∈ Ω 1 D ( A ) . Substituting the connections ∇ , ∇ ◦ with ∇ u , ( ∇ ◦ ) u for some u ∈ U ( A ) yields D A u = D + A u + J A u J − 1 with A u := u [ D , u ∗ ] + uAu ∗ .

Consider a covariant operator obtained by Morita self-equivalence: D A = D + A + J A J − 1 with A ∈ Ω 1 D ( A ) . Substituting the connections ∇ , ∇ ◦ with ∇ u , ( ∇ ◦ ) u for some u ∈ U ( A ) yields D A u = D + A u + J A u J − 1 with A u := u [ D , u ∗ ] + uAu ∗ . Furthermore, D A u is also obtained by the conjugate action of Ad( u ) : ψ → u ψ u ∗ = u ( u ∗ ) ◦ ψ = uJuJ − 1 ψ, namely Ad( u ) D A Ad( u ) − 1 = D A u . ◮ Gauge transformations are implemented by inner automorphisms of A . They preserve the selfadjointness of the covariant Dirac operator.