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in terms of the heat kernel expansion [37]:

1
[Λ4 f0 a0 + Λ2 f2 a2 + f4 a4 + Λ’2 f6 a6 + ...] dV, (145)
S = tr f (Dt /Λ2 ) =
2
16π 2 M

where the aj are the coe¬cients of the heat kernel expansion of the Dirac oper-
ator squared [30],

a0 = tr (14 — 1H ), (146)
6 R tr (14 — 1H ) ’ tr E,
1
a2 = (147)
72 R tr (14 — 1H ) ’ 180 Rµν R tr (14 — 1H ) + 180 Rµνρσ R — 1H )
2 µν µνρσ
1 1 1
a4 = tr (14
+ 12 tr (Rµν R ) ’ 6 R tr E + 2 tr E + surface terms.
µν 2
1 1 1
(148)

As already noted, for large Λ the positive function f is universal, only the ¬rst
three moments, f0 , f2 and f4 appear with non-negative powers of Λ. For the
322 T. Sch¨cker
u

minimax model, we get (more details can be found in [38]):
= 4 dim H = 4 — 6,
a0 (149)
= dim H R + 16|•|2 ,
tr E (150)
= 2 dim H R ’ dim H R ’ 16|•|2
a2 3
= ’ 1 dim H R ’ 16|•|2 , (151)
3
= 4 δ ad δ bc ’ δ ac δ bd ,
1a b1 c d
tr 2 [γ , γ ] 2 [γ , γ ] (152)
tr {Rµν Rµν } = ’ 1 dim H Rµνρσ Rµνρσ
2
’4 tr {[ρ(Fµν ) + Jρ(Fµν )J ’1 ]—
—[ρ(F µν ) + Jρ(F µν )J ’1 ]}
= ’ 1 dim H Rµνρσ Rµνρσ
2
’8 tr {ρ(Fµν )— ρ(F µν )}, (153)
dim H R2 + 4 tr {ρ(Fµν )— ρ(F µν )}
tr E 2 1
= 4
+16|•|4 + 16(Dµ •)— (Dµ •) + 8|•|2 R, (154)
Finally we have up to surface terms,
dim H (5 R2 ’ 8 Ricci2 ’ 7 Riemann2 ) + 4 tr ρ(Fµν )— ρ(F µν )
1
a4 = 360 3
+8|•|4 + 8(Dµ •)— (Dµ •) + 4 |•|2 R. (155)
3

We arrive at the spectral action with its conventional normalization, equation
2
(136), after a ¬nite renormalization |•|2 ’ π4 |•|2 .
f
Our ¬rst timid excursion into gravity on a noncommutative geometry pro-
duced a rather unexpected discovery. We stumbled over a , which is precisely the
electro“weak model for one family of leptons but with the U (1) of hypercharge
amputated. The sceptical reader suspecting a sleight of hand is encouraged to
try and ¬nd a simpler, noncommutative ¬nite spectral triple.

5.4 A Central Extension
We will see in the next section the technical reason for the absence of U (1)s as au-
tomorphisms: all automorphisms of ¬nite spectral triples connected to the iden-
tity are inner, i.e. conjugation by unitaries. But conjugation by central unitaries
is trivial. This explains that in the minimax example, A = H•C, the component
of the automorphism group connected to the identity was SU (2)/Z2 (±u, 1).
It is the domain of de¬nition of the lift, equation (125),
« 
±u 0 0 0
¬ 0 1 0 0·
L(±u, 1) = ρ(±u, 1)Jρ(±u, 1)J ’1 =  . (156)
0 0 ±¯ 0 u
0001
It is tempting to centrally extend the lift to all unitaries of the algebra:
« 
vw 0
¯ 0 0
¬ 0 v2 0 0·
¯
L(w, v) = ρ(w, v)Jρ(w, v)J ’1 =  ,
0 0 vw 0¯
0 v2
0 0
Forces from Connes™ Geometry 323

(w, v) ∈ SU (2) — U (1). (157)

An immediate consequence of this extension is encouraging: the extended lift is
single-valued and after tensorization with the one from Riemannian geometry,
the multi-valuedness will remain two.
Then redoing the ¬‚uctuation of the Dirac operator and recomputing the
spectral action yields gravity coupled to the complete electro“weak model of the
electron and its neutrino with a weak mixing angle of sin2 θw = 1/4.


6 Connes™ Do-It-Yourself Kit

Our ¬rst example of gravity on an almost commutative space leaves us wondering
what other examples will look like. To play on the Yang“Mills“Higgs machine,
one must know the classi¬cation of all real, compact Lie groups and their unitary
representations. To play on the new machine, we must know all ¬nite spectral
triples. The ¬rst good news is that the list of algebras and their representations is
in¬nitely shorter than the one for groups. The other good news is that the rules of
Connes™ machine are not made up opportunistically to suit the phenomenology of
electro“weak and strong forces as in the case of the Yang“Mills“Higgs machine.
On the contrary, as developed in the last section, these rules derive naturally
from geometry.


6.1 Input

Our ¬rst input item is a ¬nite dimensional, real, associative involution algebra
with unit and that admits a ¬nite dimensional faithful representation. Any such
algebra is a direct sum of simple algebras with the same properties. Every such
simple algebra is an algebra of n — n matrices with real, complex or quaternionic
entries, A = Mn (R), Mn (C) or Mn (H). Their unitary groups U (A) := {u ∈
A, uu— = u— u = 1} are O(n), U (n) and U Sp(n). Note that U Sp(1) = SU (2).
The centre Z of an algebra A is the set of elements z ∈ A that commute with all
elements a ∈ A. The central unitaries form an abelian subgroup of U (A). Let us
denote this subgroup by U c (A) := U (A) © Z. We have U c (Mn (R)) = Z2 ±1n ,
exp(iθ)1n , θ ∈ [0, 2π), U c (Mn (H)) = Z2 ±12n . All
U c (Mn (C)) = U (1)
automorphisms of the real, complex and quaternionic matrix algebras are in-
ner with one exception, Mn (C) has one outer automorphism, complex conju-
gation, which is disconnected from the identity automorphism. An inner au-
tomorphism σ is of the form σ(a) = uau’1 for some u ∈ U (A) and for all
a ∈ A. We will denote this inner automorphism by σ = iu and we will write
Int(A) for the group of inner automorphisms. Of course a commutative algebra,
e.g. A = C, has no inner automorphism. We have Int(A) = U (A)/U c (A), in
particular Int(Mn (R)) = O(n)/Z2 , n = 2, 3, ..., Int(Mn (C)) = U (n)/U (1) =
SU (n)/Zn , n = 2, 3, ..., Int(Mn (H)) = U Sp(n)/Z2 , n = 1, 2, ... Note the ap-
parent injustice: the commutative algebra C ∞ (M ) has the nonAbelian automor-
phism group Di¬(M ) while the noncommutative algebra M2 (R) has the Abelian
324 T. Sch¨cker
u

automorphism group O(2)/Z2 . All exceptional groups are missing from our list
of groups. Indeed they are automorphism groups of non-associative algebras, e.g.
G2 is the automorphism group of the octonions.
The second input item is a faithful representation ρ of the algebra A on
a ¬nite dimensional, complex Hilbert space H. Any such representation is a
direct sum of irreducible representations. Mn (R) has only one irreducible rep-
resentation, the fundamental one on Rn , Mn (C) has two, the fundamental one
and its complex conjugate. Both are de¬ned on H = Cn ψ by ρ(a)ψ = aψ
and by ρ(a)ψ = aψ. Mn (H) has only one irreducible representation, the fun-
¯
damental one de¬ned on C2n . For example, while U (1) has an in¬nite number
of inequivalent irreducible representations, characterized by an integer ˜charge™,
its algebra C has only two with charge plus and minus one. While SU (2) has
an in¬nite number of inequivalent irreducible representations characterized by
its spin, 0, 1 , 1, ..., its algebra H has only one, spin 1 . The main reason behind
2 2
this multitude of group representation is that the tensor product of two repre-
sentations of one group is another representation of this group, characterized by
the sum of charges for U (1) and by the sum of spins for SU (2). The same is
not true for two representations of one associative algebra whose tensor product
fails to be linear. (Attention, the tensor product of two representations of two
algebras does de¬ne a representation of the tensor product of the two algebras.
We have used this tensor product of Hilbert spaces to de¬ne almost commutative
geometries.)
The third input item is the ¬nite Dirac operator D or equivalently the
fermionic mass matrix, a matrix of size dimHL — dimHR .
These three items can however not be chosen freely, they must still satisfy
all axioms of the spectral triple [39]. I do hope you have convinced yourself of
the nontriviality of this requirement for the case of the minimax example.
The minimax example has taught us something else. If we want abelian
gauge ¬elds from the ¬‚uctuating metric, we must centrally extend the spin lift,
an operation, that at the same time may reduce the multivaluedness of the
original lift. Central extensions are by no means unique, its choice is our last
input item [40].
To simplify notations, we concentrate on complex matrix algebras Mn (C) in
the following part. Indeed the others, Mn (R) and Mn (H), do not have central
unitaries close to the identity. We have already seen that it is important to
separate the commutative and noncommutative parts of the algebra:

N
A=C • nk ≥ 2.
M
Mnk (C) a = (b1 , ...bM , c1 , ..., cN ), (158)
k=1


Its group of unitaries is

N
— — U (nk )
M
U (A) = U (1) u = (v1 , ..., vM , w1 , ..., wN ) (159)
k=1
Forces from Connes™ Geometry 325

and its group of central unitaries

U c (A) = U (1)M +N uc = (vc1 , ..., vcM , wc1 1n1 , ..., wcN 1nN ). (160)

All automorphisms connected to the identity are inner, there are outer automor-
phisms, the complex conjugation and, if there are identical summands in A, their
permutations. In compliance with the minimax principle, we disregard the dis-
crete automorphisms. Multiplying a unitary u with a central unitary uc of course
does not a¬ect its inner automorphism iuc u = iu . This ambiguity distinguishes
between ˜harmless™ central unitaries vc1 , ..., vcM and the others, wc1 , ..., wcN , in
the sense that

Int(A) = U n (A)/U nc (A), (161)

where we have de¬ned the group of noncommutative unitaries

N
U (A) := — U (nk )
n
w (162)
k=1

and U nc (A) := U n (A) © U c (A) wc . The map

i : U n (A) ’’ Int(A)
’’ iw
w (163)

has kernel Ker i = U nc (A).
The lift of an inner automorphism to the Hilbert space has a simple closed
form [19], L = L —¦ i’1 with
ˆ

L(w) = ρ(1, w)Jρ(1, w)J ’1 .
ˆ (164)

ˆ ˆ
It satis¬es p(L(w)) = i(w). If the kernel of i is contained in the kernel of L, then
the lift is well de¬ned, as e.g. for A = H, U nc (H) = Z2 .

AutH (A) H Y
CC 6 A HH
OL K A
A L HHH
p (165)
C ˆ
?C A H
Int(A) ←’ U (A) ← U nc (A)
-
i n
det

For more complicated real or quaternionic algebras, U nc (A) is ¬nite and the lift
L is multi-valued with a ¬nite number of values. For noncommutative, complex
algebras, their continuous family of central unitaries cannot be eliminated except
for very special representations and we face a continuous in¬nity of values. The
solution of this problem follows an old strategy: ˜If you can™t beat them, adjoin
them™. Who is them? The harmful central unitaries wc ∈ U nc (A) and adjoining
means central extending. The central extension (157), only concerned a discrete
326 T. Sch¨cker
u

group and a harmless U (1). Nevertheless it generalizes naturally to the present
setting:

L : Int(A) — U nc (A) ’’ AutH (A)
ˆ —¦ i’1 )(wσ ) (wc )
’’ (L
(wσ , wc ) (166)

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. 70
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