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with
«
N N
(wc ) := ρ q1,j1
(wcjM )qM,jM ,
(wcj1 ) , ..., (167)
j1 =1 jM =1

N N
(wcjM +N )qM +N ,jM +N 1nN  Jρ(...) J ’1
(wcjM +1 )qM +1,jM +1 1n1 , ...,
jM +1 =1 jM +N =1


with the (M + N ) — N matrix of charges qkj . The extension satis¬es indeed
p( (wc )) = 1 ∈ Int(A) for all wc ∈ U nc (A).
Having adjoined the harmful, continuous central unitaries, we may now stream
line our notations and write the group of inner automorphisms as
« 
N
Int(A) =  — SU (nk ) /“ [wσ ] = [(wσ1 , ..., wσN )] mod γ, (168)
k=1

where “ is the discrete group

N
“ = — Znk (z1 1n1 , ..., zN 1nN ),
k=1
zk = exp[’mk 2πi/nk ], mk = 0, ..., nk ’ 1 (169)

and the quotient is factor by factor. This way to write inner automorphisms
is convenient for complex matrices, but not available for real and quaternionic
matrices. Equation (161) remains the general characterization of inner automor-
phisms.
The lift L(wσ ) = (L —¦ i’1 )(wσ ), wσ = w mod U nc (A), is multi-valued with,
ˆ
N
depending on the representation, up to |“ | = j=1 nj values. More precisely the
multi-valuedness of L is indexed by the elements of the kernel of the projection
p restricted to the image L(Int(A)). Depending on the choice of the charge
matrix q, the central extension may reduce this multi-valuedness. Extending
harmless central unitaries is useless for any reduction. With the multi-valued
group homomorphism

(hσ , hc ) : U n (A) ’’ Int(A) — U nc (A)
’’ ((wσj , wcj )) = ((wj (det wj )’1/nj , (det wj )1/nj )),(170)
(wj )

we can write the two lifts L and together in closed form L : U n (A) ’ AutH (A):
Forces from Connes™ Geometry 327

L(w) = L(hσ (w)) (hc (w))
«
N N
= ρ q1,j1
˜
(det wjM )qM,jM ,
˜
(det wj1 ) , ...,
j1 =1 jM =1

N N
(det wjN +M )qN +M ,jN +M 
(det wjM +1 )qM +1,jM +1 , ..., wN
˜ ˜
w1
jM +1 =1 jN +M =1


— Jρ(...)J ’1 . (171)

We have set
« «  « ’1
n1
0M —N
q := q ’    
..
˜ . (172)
.
1N —N nN

Due to the phase ambiguities in the roots of the determinants, the extended
lift L is multi-valued in general. It is single-valued if the matrix q has integer
˜
0
, then q = 0 and L(w) = L(w). On the other hand, q = 0
ˆ
entries, e.g. q = ˜
1N
gives L(w) = L(i’1 (hσ (w))), not always well de¬ned as already noted. Unlike the
ˆ
extension (157), and unlike the map i, the extended lift L is not necessarily even.
We do impose this symmetry L(’w) = L(w), which translates into conditions
on the charges, conditions that depend on the details of the representation ρ.
Let us note that the lift L is simply a representation up to a phase and
as such it is not the most general lift. We could have added harmless central
unitaries if any present, and, if the representation ρ is reducible, we could have
chosen di¬erent charge matrices in di¬erent irreducible components. If you are
not happy with central extensions, then this is a sign of good taste. Indeed
commutative algebras like the calibrating example have no inner automorphisms
and a huge centre. Truly noncommutative algebras have few outer automorphism
and a small centre. We believe that almost commutative geometries with their
central extensions are only low energy approximations of a truly noncommutative
geometry where central extensions are not an issue.


6.2 Output

From the input data of a ¬nite spectral triple, the central charges and the three
moments of the spectral function, noncommutative geometry produces a coupled
to gravity. Its entire Higgs sector is computed from the input data, Fig. 6. The
Higgs representation derives from the ¬‚uctuating metric and the Higgs potential
from the spectral action.
To see how the Higgs representation derives in general from the ¬‚uctuating
Dirac operator D, we must write it as ˜¬‚at™ Dirac operator D plus internal 1-
˜
form H like we have done in equation (127) for the minimax example without
328 T. Sch¨cker
u


fj
f

f,q f




S


Fig. 6. Connes™ slot machine


extension. Take the extended lift L(w) = ρ(w)Jρ(w)J ’1 with the unitary
N N
q1j1
˜
(det wjM )qM jM ,
˜
w= (det wj1 ) , ..., (173)
j1 =1 jM =1
N N
qM +1,jM +1
˜
(det wjN +M )qN +M ,jN +M .
˜
w1 (det wjM +1 ) , ..., wN
jM +1 =1 jN +M =1

Then

D = LDL’1
˜
’1
= ρ(w) Jρ(w)J ’1 D ρ(w) Jρ(w)J ’1
˜
= ρ(w) Jρ(w)J ’1 D ρ(w’1 ) Jρ(w’1 )J ’1
˜
= ρ(w)Jρ(w)J (ρ(w’1 )D + [D, ρ(w’1 )])Jρ(w’1 )J ’1
’1 ˜ ˜
= Jρ(w)J DJρ(w )J + ρ(w)[D, ρ(w’1 )]
’1 ˜ ’1 ’1 ˜
= Jρ(w)Dρ(w’1 )J ’1 + ρ(w)[D, ρ(w’1 )]
˜ ˜
= J(ρ(w)[D, ρ(w’1 )] + D)J ’1 + ρ(w)[D, ρ(w’1 )]
˜ ˜ ˜
= D + H + JHJ ’1 ,
˜ (174)

with the internal 1-form, the Higgs scalar, H = ρ(w)[D, ρ(w’1 )]. In the chain
˜
(174) we have used successively the following three axioms of spectral triples,
[ρ(a), Jρ(˜)J ’1 ] = 0, the ¬rst order condition [[D, ρ(a)], Jρ(˜)J ’1 ] = 0 and
˜
a a
[D,
˜ J] = 0. Note that the unitaries, whose representation commutes with the
internal Dirac operator, drop out from the Higgs, it transforms as a singlet
under their subgroup.
The constraints from the axioms of noncommutative geometry are so tight
that only very few s can be derived from noncommutative geometry as pseudo
forces. No left-right symmetric model can [41], no Grand Uni¬ed Theory can [42],
Forces from Connes™ Geometry 329

Yang-Mills-Higgs

left-right symm.
NCG




GUT
standard model supersymm.

Fig. 7. Pseudo forces from noncommutative geometry


for instance the SU (5) model needs 10-dimensional fermion representations,
SO(10) 16-dimensional ones, E6 is not the group of an associative algebra.
Moreover the last two models are left-right symmetric. Much e¬ort has gone
into the construction of a supersymmetric model from noncommutative geome-
try, in vain [43]. The standard model on the other hand ¬ts perfectly into Connes™
picture, Fig. 7.


6.3 The Standard Model

The ¬rst noncommutative formulation of the standard model was published by
Connes & Lott [33] in 1990. Since then it has evolved into its present form [18“
20,28] and triggered quite an amount of literature [44].


Spectral Triple. The internal algebra A is chosen as to reproduce SU (2) —
U (1) — SU (3) as subgroup of U (A),

A = H • C • M3 (C) (a, b, c). (175)

The internal Hilbert space is copied from the Particle Physics Booklet [13],

HL = C2 — CN — C3 • C 2 — CN — C , (176)
HR = C — CN — C3 • C — C N — C3 • C — CN — C . (177)

In each summand, the ¬rst factor denotes weak isospin doublets or singlets, the
second denotes N generations, N = 3, and the third denotes colour triplets or
singlets. Let us choose the following basis of the internal Hilbert space, counting
330 T. Sch¨cker
u

fermions and antifermions (indicated by the superscript ·c for ˜charge conju-
gated™) independently, H = HL • HR • HL • HR = C90 :
c c


u c t νe νµ ν„
, , , , , ;
d sL b e µ „
L L L L L
uR , cR , tR ,
eR , µR , „R ;
dR , sR , bR ,
c c c c c c
u c t νe νµ ν„
, , , , , ;
d sL b e µ „
L L L L L
uc , cc , tc ,
ec , µc , c
R R R „R .
dR , sc ,
c
bc , R R
R R

This is the current eigenstate basis, the representation ρ acting on H by
« 
ρL 0 0 0
¬ 0 ρR 0 0·
ρ(a, b, c) :=   (178)
c
0 0 ρL 0
¯
0 ρc
0 0 ¯R
with
« 
b1N — 13 0 0
a — 1N — 13 0  0 ,

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