Spinc spectral geometry and fermion doubling
Arkadiusz Bochniak
Institute of Theoretical Physics of the Jagiellonian University
joint work with Andrzej Sitarz [Phys.Rev.D 101 075038(2020)]

18 June 2020

Arkadiusz Bochniak

Spinc spectral geometry and fermion doubling

The Standard Model

One of the most successful and verified theories about the fundamental
constituents of matter and forces between them.

Arkadiusz Bochniak

Spinc spectral geometry and fermion doubling

The Standard Model

One of the most successful and verified theories about the fundamental
constituents of matter and forces between them.
Nevertheless, some puzzles still remains unsolved: neutrinos, dark matter, ...
Why the Higgs boson is different? Can all this be described geometrically?

Arkadiusz Bochniak

Spinc spectral geometry and fermion doublingNoncommutative Geometry{locc. compact Hausdorff M } ≡ {comm. C ∗ − algebra C0 (M )}{C ∗ − algebra} ≡ {ops. on a Hilbert space}differential structure (on closed, oriented Riemannian spinc manifolds): theDirac operator D acting on L2 (M, S) determines the geodesic distance on M

Arkadiusz Bochniak

Spinc spectral geometry and fermion doublingNoncommutative Geometry{locc. compact Hausdorff M } ≡ {comm. C ∗ − algebra C0 (M )}{C ∗ − algebra} ≡ {ops. on a Hilbert space}differential structure (on closed, oriented Riemannian spinc manifolds): theDirac operator D acting on L2 (M, S) determines the geodesic distance on M
[Connes, 2013]

The metric and spin structure of a closed orientable Riemannian spinc - manifold
M can be encoded into a system(C ∞ (M ), L2 (M, S), D).
For dim M even, the are two important operators in the associated Clifford
algebra Cl:
Z/2Z - grading γ 5 ,charge conjugation operator C,
that satisfy few compatibility conditions among themselves an the Dirac operator.

Arkadiusz Bochniak

Spinc spectral geometry and fermion doubling

Spectral triples(A, H, D, γ, J)A - ∗ - algebra represented (in a faithful way) on a Hilbert space H
Z/2Z - grading γ on H s.th. γ = γ ∗ and [γ, A] = 0
antilinear isometry J
selfadjoint operator D (with a compact resolvent)... see e.g. [Lizzi, 2018]

Arkadiusz Bochniak

Spinc spectral geometry and fermion doublingWhy spectral triples?

Almost-commutative geometry (C ∞ (M ) ⊗ Af ) can explain the origin of the
Higgs mass in the Standard Model. Furthermore, from the spectral action
one can derive the form of an action for this model, and find relations
between appropriate parameters. [Chamseddine-Connes-Marcoli, 2015]
There are known successes in the derivation of Hilbert-Einstein action inConnes-Lott cosmology [Ackermann, 1996][Kastler,1995][Kalau-Walze, 1995]

Arkadiusz Bochniak

Spinc spectral geometry and fermion doubling

Spectral action method

Start with a spectral triple (A, H, D, γ, J)
Define ΩD := span{a[D, b] : a, b ∈ A}
Consider DA = D + A + JAJ −1 for A ∈ ΩD
Define the bosonic spectral actionSB = Trf

DA
Λ


,

where f - smooth approximation of χ[0,1] and Λ - cut-off parameter,... and also the fermionic one: SF = hψ|DA |ψi.
Express SB as an asymptotic series in Λ−1 and then the first term should
give an effective action of the theory

Arkadiusz Bochniak

Spinc spectral geometry and fermion doubling

Few issues

Euclidean formulation

Arkadiusz Bochniak

Spinc spectral geometry and fermion doubling

Few issues

Euclidean formulation
Quadrupling of degrees of freedom

Arkadiusz Bochniak

Spinc spectral geometry and fermion doubling

Few issues

Euclidean formulation
Quadrupling of degrees of freedom
Possibility of SU (3)-symmetry breaking

Arkadiusz Bochniak

Spinc spectral geometry and fermion doubling

Few issues

Euclidean formulation
Quadrupling of degrees of freedom
Possibility of SU (3)-symmetry breaking
Disagreement in the predicted Higgs mass and its experimental value

Arkadiusz Bochniak

Spinc spectral geometry and fermion doubling

Possible partial solutions

Lorenzian formulation [Paschke – Sitarz, 2006] [Barret, 2007] [Eckstein – Franco,2014] [van den Dungen, 2015] [Brouder – Bizi – Besnard 2015] [Devastato – Farnsworth– Lizzi – Martinetti, 2018] [B. – Sitarz, 2018] [Martinetti – Singh, 2019] ...

Fermion doubling problem [Lizzi – Mangano – Miele – Sparano, 1997][Gracia-Bondia – Iochum – Schucker, 1998] [D’Andrea – Kurkov – Lizzi, 2016] ...

Classification of finite Dirac operators and the problem of leptoquarks[Krajewski, 1998] [Paschke – Sitarz, 1998] [Paschke – Scheck – Sitarz, 1999] [Farnsworth– Boyle, 2014] [Dąbrowski – D’Andrea – Sitarz, 2018] [B. – Sitarz, 2018] ...

New fermions approach [Stephan, 2006] [Stephan, 2007]
σ field [Stephan, 2009] [Chamseddine – Connes, 2012] ...
Grand Symmetry [Devastato – Lizzi – Martinetti, 2014] ...
Twisted spectral triples [Landi – Martinetti, 2016] [Devastato – Martinetti, 2017] ...

Arkadiusz Bochniak

Spinc spectral geometry and fermion doubling

New point of view

All of the mentioned extensions are still based on the concept of (real) spectral
triples with the product geometry. Furthermore, most of them similarly require
cutting down the quadrupled number of the degrees of freedom.

Arkadiusz Bochniak

Spinc spectral geometry and fermion doubling

New point of view

All of the mentioned extensions are still based on the concept of (real) spectral
triples with the product geometry. Furthermore, most of them similarly require
cutting down the quadrupled number of the degrees of freedom.
Our approach
We do not demand the almost-commutative structure, but rather a specificbehaviour of the Lorentzian structure.

Arkadiusz Bochniak

Spinc spectral geometry and fermion doublingKrein-shifted geometry

Dirac operator on the 4-dim Minkowski (1, 3) space: D = iγ µ ∂µ , where{γ µ , γ ν } = 2η µν .
RRe where ψ̄ = ψ † γ 0
Lorentz-invariant fermion action: M ψ̄Dψ = M ψ † Dψ,0e
and D = γ D is the Krein shift of D.
e is symmetric, while D is Krein self-adjoint: D = γ 0 Dγ 0 with theDfundamental symmetry γ 0 . [Paschke – Sitarz, 2006] [B. – Sitarz, 2018]
12
0
grading γ =, real structure J = iγ 2 ◦ cc.
0−12Dγ = −γD, DJ = J D, J 2 = 1, J γ = −γJ .
e = γ D,e DJ
e = −J D,
e J 2 = 1, J γ = −γJ .
DγArkadiusz Bochniak

Spinc spectral geometry and fermion doublingRiemannian and pseudo-Riemannian spectral triples

A Riemannian finite spectral triple built over a finite-dimensional algebra A is acollection of data (A, H, D, πL , πR ), where πL is a representation of A on H and
πR is a representation of Aop on H such that
[πL (a), πR (b)] = 0, (0th order)[[D, πL (a)], πR (b)] = 0 (1st order)
for all a, b.

Arkadiusz Bochniak

Spinc spectral geometry and fermion doublingRiemannian and pseudo-Riemannian spectral triples

A Riemannian finite spectral triple built over a finite-dimensional algebra A is acollection of data (A, H, D, πL , πR ), where πL is a representation of A on H and
πR is a representation of Aop on H such that
[πL (a), πR (b)] = 0, (0th order)[[D, πL (a)], πR (b)] = 0 (1st order)
for all a, b.
We say that the triple if ofspinc type if (ClD (πL (A)))0 = πR (A),
Hodge type if (ClD (πL (A)))0 = ClD (πR (A)),
where ClD (πL (A)) is the algebra generated by πL (a) and [D, πL (a)] for all a ∈ A.
Similarly we define ClD (πR (A)).
This notion can be extended into genuine Riemannian geometries (requiring
compact resolvent for D etc.) and also into the pseudo-Riemannian cases following
the path of [B. – Sitarz, 2018].

Arkadiusz Bochniak

Spinc spectral geometry and fermion doublingFermions in the Standard Model

Parametrization of the particle content in the one-generation Standard Model:
νR

u1R

u2R

u3R



 eR

Ψ=

 νL


d1R

d2R

u1L

u2L



d3R 

 ∈ M4 (HW )

3
uL 


d1L

d2L



eL



d3L

As an algebra we take A = C ⊕ H ⊕ M3 (C) with representations:πL (λ, q, m)Ψ = 



λ
λ̄

 Ψ,
q

Arkadiusz Bochniak

πR (λ, q, m)Ψ = Ψ


λ


mT

.

Spinc spectral geometry and fermion doublingDirac Operator

At every point of the Minkowski space, we can encode any linear operator on thespace of particles as an operator in M4 (C) ⊗ M2 (C) ⊗ M4 (C).
Lorentzian Dirac Operator
DSM Ψ = iσ µ ∂µie
σ µ ∂µieσµ ∂iσ µ ∂µ Ψ + DF Ψ,

µ

|

{z
D

}

where DF is a finite endomorphism of the Hilbert space M4 (HW ).

Arkadiusz Bochniak

Spinc spectral geometry and fermion doublingDirac Operator

At every point of the Minkowski space, we can encode any linear operator on thespace of particles as an operator in M4 (C) ⊗ M2 (C) ⊗ M4 (C).
Lorentzian Dirac Operator
DSM Ψ = iσ µ ∂µie
σ µ ∂µieσµ ∂iσ µ ∂µ Ψ + DF Ψ,

µ

|

{z
D

}

where DF is a finite endomorphism of the Hilbert space M4 (HW ).
D is Lorentz-invariant, while DF will transform covariantly only if is anelement of M4 (C) ⊗ id ⊗ M4 (C),
If it is the case, DF commutes with the chirality Γ = πL (1, −1, 1), so wehave Lorentzian part D that anticommutes with Γ and a finite part that
commutes with Γ. The Krein-shifted operators have opposite behaviour.

Arkadiusz Bochniak

Spinc spectral geometry and fermion doubling

First-order and spinc conditions]
When DSM satisfies the first-order condition?

Arkadiusz Bochniak

Spinc spectral geometry and fermion doubling

First-order and spinc conditions]
When DSM satisfies the first-order condition?
e
D alone obviously satisfies the order-one conditiongself-adjoint D
F are resctricted to

g
D
F =

Ml

Ml†
|

{z

Dl




⊗e11 +}Mq

⊗(14 − e11 ),

Mq†
|

{z

Dq

}

where Ml , Mq ∈ M2 (C).

Arkadiusz Bochniak

Spinc spectral geometry and fermion doubling

First-order and spinc conditions]
When DSM satisfies the first-order condition?
e
D alone obviously satisfies the order-one conditiongself-adjoint D
F are resctricted to

g
D
F =

Ml

Ml†
|

{z

Dl




⊗e11 +}Mq

⊗(14 − e11 ),

Mq†
|

{z

Dq

}

where Ml , Mq ∈ M2 (C).
Furthermore, analyzing the commutant of the generalized Clifford algebra]
generated by πL (a) and [DSM , πL (a)] one can check that for one-generation model
the spinc condition is satisfied.

Arkadiusz Bochniak

Spinc spectral geometry and fermion doublingThree generations

The Hilbert space is now M4 (HW ) ⊗ C3 with diagonal representation, so that Ml
and Mq are now in M2 (C) ⊗ M3 (C).
The spinc condition will hold if algebras generated by πL (A) and Dl , Dq ,respectively, will be (M4 (C) ⊗ id ⊗ id) ⊗ M3 (C) independently for the leptonand for quarks.
Using similar argument as in [Dąbrowski – Sitarz, 2019] we infer the same
condition for the Hodge property to be satisfied:

Arkadiusz Bochniak

Spinc spectral geometry and fermion doublingThree generations

Mass matrices:

Ml =



Υν
Υe


,

Mq =



Υu
Υd

,

where Υe and Υu are chosen diagonal with the masses of electron, muon,and tau and the up, charm, and top quarks, respectively, andfν U † ,Υν = U Υ

fd V †
Υd = V Υfν , Υ
fd diagonal, and U - PMNS, V - CKM.
with Υ

Arkadiusz Bochniak

Spinc spectral geometry and fermion doublingThree generations

Mass matrices:

Ml =



Υν
Υe


,

Mq =



Υu
Υd

,

where Υe and Υu are chosen diagonal with the masses of electron, muon,and tau and the up, charm, and top quarks, respectively, andfν U † ,Υν = U Υ

fd V †
Υd = V Υfν , Υ
fd diagonal, and U - PMNS, V - CKM.
with Υ
The sufficient condition to fulfill the Hodge property: for both pairs ofmatrices (Υν , Υe ) and (Υu , Υd ) their eigenvalues are pairwise different.
This requirement is satisfied in the case of the physical standard model,
provided that there is no massless neutrino [Dąbrowski – Sitarz, 2019]

Arkadiusz Bochniak

Spinc spectral geometry and fermion doublingFrom spinc to Hodge dualityg(HSM = M4 (C), D
F , Γ) is an Euclidean even spectral triple.
g
Suppose DF is such that the spinc condition holds.
Idea: Doubling of the triple ⇒ The resulting real spectral triple will satisfy
the Hodge duality and is the finite spectral triple of the SM studied so far as
the finite component of the product geometry. :2Take HSM= HSM ⊕ HSM with πL ⊕ πR .

Define J(M1 ⊕ M2 ) = M2∗ ⊕ M1∗ .
Extend Γ so that JΓ = ΓJ.
−1 .
gg
D0 := DF ⊕ 0 + J(DF ⊕ 0)J
Then D0 Γ = −ΓD0 and D0 J = JD0 .
ClD0 ((πL ⊕ πR )(A)) = ClD
g (πL (A)) ⊕ πR (A).
F

Arkadiusz Bochniak

Spinc spectral geometry and fermion doubling

Reality and CP violation

Consider M4 (HW ) with the real structure id ⊗ J ⊗ id (it means that J acts on
finite part by complex conjugation). It does not implement the usual zero-ordercondition, but we still have a milder version:
πR (A) ⊆ JπL (A)J −1 .

Imposing the same KO-dimension for the Euclidean finite spectral triple as
for the Lorentzian spatial part, the mass matrices Ml and Mq have to bereal.
One generation: reality of fermion masses: OK
Three generations: reality of PMNS and CKM matrices, CP symmetry
preservation (vanishing of the Wolfenstein parameter η̄ and CP-violating
+0.52
ν
ν
phase δCP
; exp.: η̄ = 0.355+0.012−0.011 and δCP = 1.38−0.38 [PDG, 2018])
The existence of CP-violation may be interpreted as a shadow ofJ-symmetry violation in the nondoubled spectral triple.

Arkadiusz Bochniak

Spinc spectral geometry and fermion doublingTwisted triples^
The Krein-shifted Dirac operator D
SM satisfies 1st order condition^
The Lorentzian Dirac operator DSM = β D
SM satisfied twisted condition:[[DSM , πL (a)]β , πR (b)]β = 0,
where β = id ⊗ γ 0 ⊗ id and [x, y]β = xy − βyβ −1 x.

Arkadiusz Bochniak

Spinc spectral geometry and fermion doublingSummary

We proposed a geometry of the SM which is not a product of spectral triples.
When restricted to the commutative algebra of real-valued functions, we geteven real Lorentzian spectral triple of KO-dimension 6 and with the Dirac
operator satisfying the 1st order condition
The restriction of the spectral triple to the constant functions over the
Minkowski space gives a Euclidean even spectral triple, which fails to be real.
The lack of real structure can be related to the appearance of the violationof CP symmetry in the SM
The full spectral triple satisfies the spinc condition: the Clifford algebra
generated by the commutators of the Krein-shifted Dirac operator with therepresentation πL of the algebra has, as the commutant, the right
representation of the algebra πR .

Arkadiusz Bochniak

Spinc spectral geometry and fermion doublingOutlook

What is the best description of this geometry? There is no product here but
rather a quotient spectral geometry.
Classify all such possible geometries as extensions of the SM.
Compute the spectral action for such geometries [B.– Sitarz – Zalecki, in
preparation] and compare values of parameters with experimental data.
Beyond SM: application to e.g. Pati-Salam models with Lorentzian
structures [B. – Williams – Zalecki, 2020] [B. – Williams – Zalecki, in preparation]...

Arkadiusz Bochniak

Spinc spectral geometry and fermion doublingThank you for your attention!

Arkadiusz Bochniak

Spinc spectral geometry and fermion doubling