about this mass matrix.
1) On a local trivialisation (say, around a point x E M) we can view the endo
morphism T>i as a matrix-valued function but the precise form of this matrix Di(x) depends on the choice of local trivialisation. However, since the
I00 : = ( r ° ° ( B ) , r ° ° ( E ) , O : ) , = t, [a,Jb*J*] = 0 , J1O1 = e'OiJi,
[[£>!, a], Jb*J*] — 0,
JiD = £//FiJi, V a , b E r°°(B),6.2 A L M O S T - C O M M U T A T I V E M A N I F O L D S 1C>9
transition functions are unitary, two different choices of local trivialisations yield two unitarily equivalent mass matrices, and hence the eigenvalues of the matrix Dj(x) (i.e. the masses of the particles) are independent of the choice of local trivialisation.
2) These eigenvalues of Dj(x) are (by default) allowed to vary as a function of x e M. In the standard (globally trivial) approach one can also make the (ad hoc) decision to promote the mass parameters to functions (although this is usually not done). However, this would be unnatural from the perspective that a (globally trivial) almost-commutative manifold is the (external) Kas parov product of a Riemannian spin manifold with a finite spectral triple. Instead, varying mass parameters are more naturally described by replacing the finite spectral triple by an internal space (which works equally well in the globally trivial case) and replacing the external by the internal Kasparov product. As such, the promotion of the mass parameters to functions be comes a natural attribute of our framework.
3) One could ask whether it is always possible to choose these mass paramet ers to be globally constant (as in the usual approach). We expect that this might not always be possible in the general globally non-trivial case, but it is unclear what the precise topological obstructions would be.
Proposition 6.16. An even internal space I00 = ( r ° ° ( B ) , r°°(E),Oi) yields an unbounded Kasparov B-A-module I = (S, T(E)a,Th).
Proof. By assumption, the grading V\ commutes with A and !B, and hence their C*-closures A and B are trivially graded C*-algebras. The module E = F(E) is a Z2-graded, finitely generated projective, right Hilbert A-module, with a left action of B that commutes with the (right) action of A. The properties of Fj guarantee that all conditions with respect to the grading are satisfied. For instance, the condition (El|E^) c Av+b where i, j G Z2, is satisfied, since the self-adjointness of V\ implies that (s|t) = 0 as soon as one of the arguments is odd and the other is even. The operator T>\ is a bounded, self-adjoint, odd operator by definition (and hence it is automatically regular). The boundedness of T>\ implies that [T)i, b] is also bounded for all b 6 *B.
For a compact manifold M. the compact endomorphisms of the C(M)-module F(E) are exactly the sections of the endomorphism bundle: Endc(M)(r(E)) = r(End(E)) (since F(End(E)j is already unital, the compact endomorphisms of T(E) are actually all the bounded endomorphisms, see e.g. [GVF01, Proposition 3.9]). Thus, b(l + Dj)“ 2 is compact for all b e 'S, because both (1 + and b are compact. Hence (!B, r(E)A,T>i) has all the properties mentioned in Defini
110 G L O B A L L Y N O N - T R I V I A L A L M O S T - C O M M U T A T I V E M A N I F O L D S
6.2.2 The product space
We are now ready to define almost-commutative manifolds as the product of an internal space with the canonical triple over the base manifold. This definition can be given for arbitrary dimensions, but for simplicity we will only give the explicit formula for the case of dimension 4.
Definition 6.17. Let I00 := (r°°(B), Fi, Ji) be a real even internal space over M, with M a compact 4-dimensional Riemannian spin manifold. Let V1 be a hermitian connection on E. We define a real even almost-commutative manifold to be
r x v M : = ( r ° ° ( B ) ; l2(e<£)S), $E + ^ i® r M ,r i0 rM ,Ji® jM )/
where L2(E(g)S) ~ F(E) <g>c(M) L2(S) are the L2-sections of the twisted spinor bundle E <g> S, and 0 e is the twisted Dirac operator
0E := 1 (g)V pi := 1 ® 0 + (1 0 c) O (V1 0 1).
We note that our definition of almost-commutative manifolds fits within the slightly more general definition of almost-commutative spectral triples given in [Caci2, Definition 2.3].
The order of I00 and M in the notation I00 Xy M is reversed in comparison with the order of F and M in M x F. The reason is that the order I00 x y M is more natural for its description as a Kasparov product (see Section 6.2.3), whereas the notation M x F for the globally trivial case is quite standard in the literature. The operator D := L)E + Th ® Ovi has been defined to match the existing literature. Given that we have reversed the order of the product, the more natural definition (in terms of graded tensor products) would have been (Fi0 l ) $ E + D i0 l. How ever, our definition of T> is unitarily equivalent to this more natural definition (by the unitary operator which equals —1 on F(E)1 <0c(M) ^ ( S ) 1 and 1 elsewhere), and hence there is no harm in using the operator V which matches the literature.
In the remainder of this section, we show in detail that an almost-commutative manifold I00 X y M determines an unbounded Kasparov B-C-module (i.e. a spec tral triple over *B) which represents the Kasparov product between the KK-classes of the internal space I00 and the canonical spectral triple for M.
Proposition 6.18. Let I00 = ( r ° ° ( B ) , r°°(E),Dj, Fi, Ji) be a real even internal space of even KO-dimension k over a compact Riemannian spin manifold M. Let V1 be a hermitian connection on E that commutes with the grading V\, satisfies V jji = JiV^, and is such that the induced connection [V1, •] on End E restricts to a connection on B. Then the real even almost-commutative manifold I00 x v M. is a real even spectral triple of KO-dimension
6 . 2 A L M O S T - C O M M U T A T I V E M A N I F O L D S
Proof. Let us write D : = 0 E + Di®r/vi- We need to show that [T>, a] is bounded for all a € r°°(B). Since T)\ is bounded itself, we need only check this for the twisted Dirac operator $ E/ and we find
[0 E,a] = c ([V ‘,a]),
where, w ith some abuse of notation, we set c(T <8> a.) = T 0 c(a) for T E r°°(End E) and a E 0 ] (M). Hence for smooth a the commutator [0 E, a] indeed acts as a bounded operator on L2(E (g> S). Furthermore we need to show that D has compact resolvent, and (as M is compact) for this it is sufficient to show that D 2 (and hence T>) is elliptic. The Lichnerowicz-Weitzenböck formula shows that the square of the twisted Dirac operator $ E is a generalised Laplacian, and hence is elliptic. The bounded (zeroth-order) perturbation 0 E —> 0 E-|-Di<g>rM does not affect this ellipticity. Hence I°° x y M is indeed a spectral triple.
Given the grading operators V\ and Tm, it is straightforward to check that we have D(Fi ® FM) = — [V\ <g> Fm)TL provided that [V1, V\] = 0.
Given the real structures Ji and Jm/ the operator Ji ® Jm is anti-unitary and satisfies
(Ji ® Jm)2 = — £/ T>(JI <8>Jm) = (Ji® Jm)£ ,
(J i® jM )(ri® rM) = £ " ( r i® r M)(ji® j M)/ (6-2) where the signs £, e" are determined by the KO-dimension k of Ji. The first equal ity in Eq. (6.2) is immediate from = — 1 and J2 = £. Using the relations
Jm$ = $ Jm/ YhJm = = Jm^M/ Ji^Di = 2)iJi, V [ j i = JiV^, the second equality in Eq. (6.2) is checked by a local calculation (where we write
(1 <g> c) o (V 1 ® 1) = <g) y^):
T)(Ji ® Jm)(s ®ip) = (Jis) 0 ( $ Jm^ ) + (V [jis) <g> + (DiJis) ® (FmJm4>)
= (Jis) ® (Jm0 ^ ) + (JiV^s) <g> (JmY ^ J + (JiDis) <g> = (Ji ®
The third equality in Eq. (6.2) immediately follows from [Jm/TmI — 0 and JiH =
e'TiJi. From the values of —£ and 1 " it is immediate that the KO-dimension of I00 x M should be 4 + k (mod 8) (see Table 1).
The zeroth-order condition on I00 x y M. is immediate from the zeroth-order condition on I00. Moreover,
[[0 E,a],JbJ*] = [c([V‘, a]). JbJ*] = c([[V ,< a], JbJ*]) = 0,
because, by assumption, [Vx,a] E F°°(B) Oc°°(M) (M), which commutes with JbJ*. Together w ith the first-order condition o n D i , this implies that V satisfies
G L O B A L L Y N O N - T R I V I A L A L M O S T - C O M M U T A T I V E M A N I F O L D S
For a unital real spectral triple T = (A,TC, D, J), the gauge group is defined in [DSi2, Definition 2.5] as
S(T) := {ujuj* I u G U(A)} ~ UlAVUlAi), (6.3) where the central subalgebra A] is defined as A] := {a G A | aj = Ja*}. For an almost-commutative manifold (constructed from a compact manifold M), we therefore obtain the gauge group
S(I°° x v M) = lX(3)/lt(3j ),
for the real structure J = Ji <8> Jm- Flowever, since 3j ~ 3j,, we find that the gauge group of the almost-commutative manifold is completely determined by the internal space, and we write
S(I°° x v M) ~ S(I°°) : = {ujiuj; I u G U(3)}. (6.4)
6.2.3 The Kasparov product
We now show that the product I00 xy M is an unbounded representative for the Kasparov product of the KK-classes of I00 and the canonical spectral triple for M. We first prove this for the cases where T)\ = 0, and then show that the presence of
T)\ is irrelevant at the level of KK-classes.
Let I00 be an internal space over M, where T)\ = 0, and consider the unbounded Kasparov module I := (3, EA,0), where E = T(E). We know from Proposition 6.18 that I°°XvM = [3, L2(E 0 S), D) is a spectral triple, which thus yields an unboun ded Kasparov module I x y M = [3, L2(E 0 S)c, 3) G ^(ELC) (Definition 2.25).
Proposition 6.19. The unbounded Kasparov module I x v M represents the Kasparov