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Noncommutative Geometry
and Particle Physics
(draft May 28, 2011)
Walter D. van Suijlekom
Institute for Mathematics, Astrophysics and Particle Physics, Faculty of Science, Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The
Netherlands
E-mail address: [email protected]
Contents
Chapter 1.
Introduction
1
Chapter 2. Noncommutative spaces
2.1. Finite spaces
2.1.1. Commutative algebras
2.1.2. Noncommutative algebras
2.2. Noncommutative geometric finite spaces
2.2.1. Classification of finite spectral triples
2.2.2. Real algebras
2.3. Spectral triples
2.3.1. Noncommutative differential forms
2.3.2. Modules and connections
2.3.3. Unitary and Morita equivalence of spectral triples
2.3.4. Heat kernel expansion on a manifold
3
3
4
4
6
9
14
14
16
16
18
21
Chapter 3. Almost commutative (AC) manifolds and gauge theories
3.1. Background: gauge theories in physics
3.1.1. Dirac and the dawn of quantum electrodynamics
3.1.2. Non-abelian gauge theories
3.1.3. Yang–Mills gauge theory: mathematical setup
3.1.4. The Standard Model of elementary particles
3.2. Almost commutative manifolds
3.2.1. Gauge symmetries and inner automorphisms
3.2.2. Gauge fields and the spectral action
3.2.3. The Standard Model of elementary particles
23
23
23
26
27
29
29
29
30
30
Bibliography
33
iii
CHAPTER 1
Introduction
In these notes, we will give an introduction to the unified description of gauge theories
in physics as provided by noncommutative geometry. The basic idea is to push further Einstein’s unified field approach to unify in a geometrical manner the gravitational force with the
electromagnetic, weak and strong nuclear forces.
Roughly speaking, in Einstein’s general theory of relativity the main mathematical structure
is a smooth manifold equipped with a Riemannian metric. The gravitational force is then an
instance of the curvature of this metric on the background manifold. The key step made by
Connes et al. [1, 5] is to arrive at the other three fundamental forces by extending the notion of
manifold to noncommutative Riemannian manifolds. Essentially, at each point of the ordinary
manifold, there is an internal finite noncommutative space. These finite noncommutative spaces
will be the topic of most of the second Chapter, giving rise to so-called finite spectral triples.
Following Krajewski [7] we will arrive at a diagrammatic classification of such finite spectral
triples, modulo unitary equivalence. We end Chapter 2 with the notion of (general) spectral
triple and indicate how it gives rise to a gauge theory.
In Chapter 3 we will start with some background in physics, in particular the case of Yang–
Mills theories (such as the Standard Model of elementary particles) phrased in mathematical
terms. Then, we will present how direct products of manifolds with a finite noncommutative
space give rise to these gauge theories. In particular, this allows for a derivation of the full
Standard Model Lagrangian.
!!N.B. These notes are draft notes (Version: May 28, 2011)!!
1
CHAPTER 2
Noncommutative spaces
2.1. Finite spaces
Consider a space X consisting of N points:
•
1
•
······
2
•
N
The ∗-algebra C(X) of C-valued functions1 on this discrete space is rather simple, it is just CN .
A fancy way of putting this is to say that C is a functor
C : Fin → Algf
from the category of finite spaces to the category of finite-dimensional ∗-algebras. Indeed, a
map φ : X1 → X2 of finite spaces induces a map φ∗ : C(X2 ) → C(X1 ) by pullback:
φ∗ f = φ ◦ f ∈ C(X1 )
when f ∈ C(X2 ). We arrive at the following
Question: Can we ‘invert’ the functor C?
In other words, given a finite dimensional algebra A can we obtain a finite space X such
that A ' C(X)? The answer is: not quite, since C(X) is always commutative, but A not
necessarily. We suggest two ways of resolving this issue:
(1) Restrict to the category CommAlgf of (fin.dim.) commutative algebras.
(2) Change morphisms in Algf .
Before explaining both of these solutions, let us introduce some useful definitions on representations of finite-dimensional algebras, not necessarily commutative, and which moreover
extend readily to the infinite-dimensional case.
Definition 2.1. A representation of A ∈ Algf is a pair (V, π) where V is an inner product
space and π a ∗-algebra map
π : A → L(V )
The vector space V is also called a left A-module. A representation (V, π) is irreducible (or
simple) if the only subspaces in V that are invariant under the action of A are {0} or V .
Example 2.2. Consider A = Mn (C). There is a defining representation on Cn , which is
irreducile. A reducible representation is given on Cn ⊕Cn , with a ∈ Mn (C) acting in block-form:
a 0
a ∈ Mn (C) 7→
∈ L(Cn ⊕ Cn ) ' M2n (C)
0 a
which thus decomposes in the direct sum of two defining representations. See also Lemma 2.19
below.
Exercise 2.1. Show that a representation (V, π) of A is irreducible if and only if the commutant π(A)0 of π(A) consists of multiples of the identity. Here
π(A)0 = {T ∈ L(V ) : π(a)T = T π(a)∀a ∈ A}.
1The C here stands for continuous, which is automatic for discrete topological spaces
3
Definition 2.3. Two representation (V1 , π1 ) and (V2 , π2 ) of A ∈ Algf are unitarily equivalent if there exists a unitary matrix U : V1 → V2 such that
π1 (a) = U ∗ π2 (a)U.
b of A is the set of all unitary equivalence classes of
Definition 2.4. The structure space A
irreducible representations of A.
Exercise 2.2. Show that for any algebra A ∈ Algf there is a 1-1 correspondence between
unitary equivalence classes of representations of A and of Mn (A), the algebra of matrices with
entries in A. Show that this preserves irreducibility of the corresponding representations so that
\
b
M
n (A) = A.
2.1.1. Commutative algebras. We now explain how the first option above resolves the
question raised. First, any commutative (fin. dim.) algebra is of the form
A ' CN
b Of course, if A = CN then
so it should not be too hard to construct its structure space A.
b ' {1, . . . , N } where each point in A
b corresponds to the representation of the corresponding
A
copy of C on C:
πi (f ) = f (i) ∈ C;
(i = 1, . . . , N ).
We conclude that there is an equivalence between the category Fin of finite spaces and the
category CommAlgf of finite-dimensional commutative ∗-algebras.
2.1.2. Noncommutative algebras. A more interesting perspective is given by the noncommutative alternative above. We adapt the morphisms in Algf to obtain a category whose
objects are in 1-1 correspondence with finite spaces in Fin, but the notion of morphism is more
flexible.
Definition 2.5. The category KKf has as objects finite-dimensional (noncommutative) algebras. The morphisms Hom(A, B) are given by A − B-bimodules E with a B-valued inner
product. That is, E is both a left A-module and a right B-module which mutually commute, and
there is a hermitian structure h·, ·i → B satisfying
(2.1.1)
he1 , e2 bi = he1 , e2 ib;
he1 , e2 i∗ = he2 , e1 i;
he, ei ≥ 0 with equality iff e = 0.
Composition of morphisms is given by the balanced tensor product:
F ◦ E := E ⊗B F,
(E ∈ Hom(A, B), F ∈ Hom(B, C)),
with C-valued inner product given by
he1 ⊗ f1 , e2 ⊗ f2 iE⊗F = hf1 , he1 , e2 iE f2 iF
Exercise 2.3. Check that this last formula defines a C-valued inner product.
Example 2.6. The trivial A-bimodule given by A itself is the identity morphism in Hom(A, A)
(up to isomorphism). This means that if E ∈ Hom(A, B) then for the composition with
A ∈ Hom(A, A) we have E ◦ A = A ⊗A E ' E, with isomorphism as A − B bimodules.
Example 2.7. If φ : A → B is an algebra homomorphism, that is, a morphism between
objects A and B in the category Algf considered above. From it, we can construct a morphism
Eφ in KKf as follows. Let Eφ be B as a vector space with the obvious right B-module structure
and inner product, but with A acting on the left via the homomorphism A. This defines an
element in Hom(A, B) so that we have a functor:
Algf → KKf
which is the identity on objects and the above association on morphism.
Example 2.8. The vector space E = Cn is an Mn (C) − C-bimodule; with the standard
C-valued inner product it becomes a morphism in KKf from Mn (C) to C.
4
Example 2.9. Similarly, the vector space F = Cn is a C − Mn (C)-bimodule by right matrixmultiplications. A Mn (C)-valued inner product is given by
hv1 , v2 i = v 1 v2t ∈ Mn (C).
As such, it is a morphisms in KKf from C to Mn (C).
Observe in the last two examples that we can compose in two ways:
E ⊗C F ' Mn (C);
F ⊗Mn (C) E ' C.
This leads us to our first little result, which has a very powerful analogue in the infinitedimensional case.
Theorem 2.10. Two finite dimensional algebras are isomorphic in KKf if and only if they
have isomorphic structure spaces (which are finite).
Since in this case the representation theories of both algebras are equivalent, we call them
Morita equivalent.
Proof. Let A ' B in KKf . Then there exists A EB and
E ⊗B F ' A,
b
If [πB ] ∈ B then we can define πA by setting
such that
F ⊗A E ' B.
πA : A → L(E ⊗B V ),
(2.1.2)
B FA
πA (a)(e ⊗ v) = ae ⊗ v
Vice versa, we construct πB : B → L(F ⊗A W ) from πA by setting πB (b)(f ⊗ w) = bf ⊗ w and
these two maps are each other’s inverse.
b ' B.
b We will first prove the so-called Wedderburn
Now let A, B ∈ KKf such that A
Theorem in the case of finite dimensional algebras over C.
Lemma 2.11. Any finite-dimensional ∗-algebra A is isomorphic to a direct sum of complex
matrix algebras:
N
M
A'
Mni (C).
i=1
A◦
Proof. Let
denote the opposite algebra of A with product ◦ defined by a ◦ b = ba for
all a, b ∈ A. We claim that
α : A◦ → A End(A)
defined by α(x)a = ax is an isomorphism of algebras. Indeed, any f ∈ A End(A) is characterized
by its value of 1 so that f = α(f (1)). As an representation of A, we can decompose
A ' V1⊕n1 · · · ⊕ VN⊕nN .
in terms of irreducibel mutuall non-isomorphic modules Si . By Schur’s Lemma, a map f ∈
is either zero or an isomorphism given by scalar multiplication. Thus,
A Hom(Si , Sj )
A Hom(A)
' Mn1 (C) ⊕ · · · MnN (C)
so that with the above A ' Mn1 (C)◦ ⊕ · · · MnN (C)◦ . The proof is complete by noting that
matrix transposition gives an isomorphism Mn (C)◦ → Mn (C).
Continuing the proof of Theorem 2.10, we may assume
A=
N
M
Mni (C);
B=
i=1
M
M
j=1
b'B
b implies that M = N . Then, define
and A
E :=
N
M
Cni ⊗ Cmi ;
i=1
5
Mmj (C)
with A acting by block-diagonal matrices on the first tensor and B acting in a similar way by
right matrix multiplication on the second leg of the tensor product; Also, set
F :=
N
M
Cmi ⊗ Cni
i=1
with now B acting on the left and A on the right. Then, as above
E ⊗B F '
N
M
(Cni ⊗ Cmi ) ⊗Mmi (C) (Cmi ⊗ Cni ) '
i=1
N
M
Cni ⊗ Cni ' A
i=1
and similarly we obtain that F ⊗A E ' B as required.
Exercise 2.4. Fill in the gaps in the above proof:
(a) Show that the representation πA defined by Eq. (2.1.2) is irreducible if and only if πB
is.
(b) Show that the association of the class [πA ] from [πB ] through Eq. (2.1.2) is independent
of the choice of representative πB .
We conclude that the objects in the category Fin of finite spaces are in one-to-one correspondence with the objects in the category KKf of finite-dimensional noncommutative algebras.
However, KKf has a far richer structure on the leve of morphisms; it should be clear that we
prefer the category KKf over the others.
2.2. Noncommutative geometric finite spaces
Consider again a finite space X described by a noncommutative (finite dimensional) ∗algebra A. We would like to introduce some geometry on X, in particular, a notion of metric
on X.
Question: How can we describe distances between the points in x, say, as embedded in a
metric space?
Naively, this could be done by giving an array {dij }i,j∈X of real nonnegative entries, indexed
by two elements in X and requiring that dii = 0, dij = dji and dij ≤ dik + dkj .
Example 2.12. The usual discrete metric on the discrete space X is given by such an array:
1 if i 6= j
dij =
0 if i = j.
Recall the particular form of A in terms of matrix algebras:
A=
N
M
Mni (C)
i=1
This ∗-algebra can be represented (irreducibly) on the vector space
H=
N
M
Cni ,
i=1
with the canonical inner product. Moreover, an array {dij } can be implemented on H as a
symmetric linear operator on H:
D:H→H
b
In fact, we can define a metric on the structure space A
dij = sup{| Tr f (i) − Tr f (j)| : k[D, f ]k ≤ 1}
f ∈A
where i labels the defining (irreducible) representation corresponding to the matrix algebra
Mni (C) in the decomposition of A. The norm kT k of a matrix T is by definition the square
root of the largest eigenvalues of T ∗ T .
6
This suggests that the above structure of finite-dimensional algebra A, representation space
and symmetric matrix D is the data that captures a metric structure on the finite space X = Â.
Definition 2.13. A finite spectral triple is a triple (A, H, D) of a ∗-algebra represented
faithfully on a finite-dimensional Hilbert space, together with a symmetric matrix D : H → H.
We will loosely refer to D as a finite Dirac operator. In this spirit, a finite spectral triple
naturally gives rise to a notion of (discrete) differential forms.
Definition 2.14. The A-bimodule of Connes’ differential one-forms is given by
(
)
X
1
ΩD (A) :=
ak [D, bk ] : ak , bk ∈ A
k
The structure of a finite spectral triple can be further enriched by introducing a Z2 -grading
on H, denoted by γ, demanding that A is even and D is odd with respect to this grading:
γD = −Dγ;
γa = aγ;
(a ∈ A).
Also, there is a more symmetric refinement of the notion of finite spectral triple in which
H is an A-bimodule.
Definition 2.15. A real finite spectral triple is given by a spectral triple (A, H, D) and a
anti-unitary operator J : H → H 2 such that a◦ := Ja∗ J −1 furnishes a right representation of A
such that
[a, b◦ ] = 0;
(2.2.1)
[[D, a], b◦ ] = 0
for all a, b ∈ A. Moreover, we demand that J, D and (in the even case) γ satisfy commutation
relations:
J 2 = ε,
JD = ε0 DJ,
Jγ = ε00 γJ (even case).
The numbers ε, ε0 , ε00 ∈ {−1, 1} are a function of n mod 8:
n
ε
ε0
ε00
0
1
2
3
4
5
6 7
1
1 −1 −1 −1 −1
1 1
1 −1
1
1
1 −1
1 1
1
−1
1
−1
and determine the KO-dimension of (A, H, D; J, γ).
The above two conditions (2.2.1) are called commutant property and first-order or order-one
condition, respectively. They imply that left multiplication by an element in A and Ω1D (A)
commutes with the right action of A.
Remark 2.16. Recall that the opposite algebra A◦ is defined to be A as a vector space but
with opposite product ◦:
a ◦ b := ba.
◦
Thus, a defines a left representation of A◦ : (a ◦ b)◦ = a◦ b◦ .
The following exercise is inspired by Tomita-Takesaki theory of Von Neumann algebras.
Exercise 2.5. Suppose that (A, H, D = 0) is a finite spectral triple such that H posesses a
cyclic and separating vector ξ for A:
Aξ = H (cyclic);
aξ = 0 =⇒ a = 0 (separating).
Show that the operator J : H → H defined by
J(aξ) = a∗ ξ
makes (A, H, 0) a real finite spectral triple.
2An anti-unitary operator is a bijective operator J : H → H that satisfies hJξ , Jξ i = hξ , ξ i.
1
2
2 1
7
Definition 2.17. Two finite spectral triples (A1 , H1 , D1 ) and (A2 , H2 , D2 ) are called unitary
equivalent if A1 ' A2 and there exists a unitary intertwining operator U : H1 → H2 such that
U π1 (a)U ∗ = π2 (a)
(a ∈ A1 )
∗
U D1 U = D2
If there exist grading operators γ1 , γ2 then we also demand that U γ1 U ∗ = γ2 . If there exist real
structures J1 , J2 then we also demand that U J1 U ∗ = J2 .
However, following the spirit of our extended notion of morphisms between algebras, it
is more natural to have equivalences as isomorphisms in the category KKf . In fact, given a
morphism (in KKf ) from A to B, given by an A − B-bimodule E, we try to construct a finite
spectral triple on A, starting from a finite spectral triple on B. This transfer of metric structure
is accomplished as follows. Let (B, H, D) be a spectral triple; we construct (A, H 0 , D0 ). First,
we define a vector space
H 0 = E ⊗B H
which inherits a left action of A from the A-module structure of E. The naive choice of a
symmetric operator D0 by D0 (e ⊗ ξ) = e ⊗ Dv will not do, because it does not respect the ideal
defining the balanced tensor product over B, being generated by elements of the form
ea ⊗ ξ − e ⊗ aξ;
(e ∈ E, a ∈ A, ξ ∈ H).
A better definition is
D0 (e ⊗ ξ) = e ⊗ Dξ + ∇(e)ξ.
where ∇ : E → E ⊗B Ω1D (B) is a connection associated to the differential d : b 7→ [D, b] (b ∈ B).
With these definition, (A, H 0 , D0 ) is a finite spectral triple if (B, H, D) is. This is our
finite-dimension analogue of Theorem 2.37 (due to Connes [3] to be obtained below.
Analogously, we define for a real spectral triple (B, H, D, J) a real spectral triple (B, H 0 , D0 , J 0 )
by setting H 0 := E ⊗B H ⊗B E ◦ . Here E ◦ is the conjugate module to the (A − B)-bimodule
E:
E ◦ = {e : e ∈ E}
with a B − A-bimodule structure given by bea = a∗ eb∗ for any a ∈ A. It is not quite a morphism
in KKf , since we do not have a natural A-valued inner product. However, there is a B-valued
inner product on the left B-module E ◦ given by
he1 , e2 i = he2 , e1 i;
(e1 , e2 ∈ E).
As opposed to Eq. (2.1.1) this inner product is left B-linear: hbe1 , e2 i = bhe1 , e2 i for all b ∈ B,
as can be easily checked. There is a (C-valued) inner product on H 0 given by combining the
B-valued inner products on E, E ◦ with the C-valued inner product on H.
The action of A on H 0 is given by
(2.2.2)
a(e1 ⊗ ξ ⊗ e2 ) = (ae1 ) ⊗ ξ ⊗ e2
using just the A-module structure of E. In addition, there is a right action of A on H 0 by acting
on the right on the component E ◦ . In fact, it is implemented by the following anti-unitary
J 0 (e1 ⊗ ξ ⊗ e2 ) = e2 ⊗ Jξ ⊗ e1
i.e. a◦ = J 0 a∗ (J 0 )−1 understanding the above left A-action (2.2.2). Moreover, there is finite
Dirac operator given in terms of the same connection as above by
D0 (e1 ⊗ ξ ⊗ e2 ) = (∇e1 )ξ ⊗ e2 + e1 ⊗ Dξ ⊗ e2 + e1 ⊗ ξ(∇e2 )
Finally, for even spectral triples one defines a grading γ 0 = 1 ⊗ γ ⊗ 1 on E ⊗B H ⊗B E ◦ .
Theorem 2.18. If (B, H, D; J, γ) is a real (even) spectral triple of KO-dimension n, then
(in the above notation) (A, E ⊗B H ⊗B E ◦ , D0 ; J 0 ) is a real spectral triple of KO-dimension n.
8
2.2.1. Classification of finite spectral triples. In this section, we follow Krajewski’s
work [7] on the classification of all real finite spectral triples (A, H, D; J) modulo unitary equivalence. This is accomplished in a nice diagrammatic way, involving what are now called Krajewski
diagrams. They play the same role for finite spectral triple as Dynkin diagrams do for simple
Lie algebras.
The algebra: Here, there is a shortcut to the matrix-algebra decomposition of A. In fact, since
A acts faithfully on a Hilbert space, it is a subalgebra of a matrix algebra L(H) = Mdim(H) (C).
It is not hard to see that thus
N
M
A'
Mni (C).
i=1
for some n1 , . . . , nN .
The Hilbert space: Let us start with a basic result on irreducible representations of Mn (C).
Lemma 2.19. The unique irreducible representation of Mn (C) is given by left matrix multiplication on Cn .
Proof. It is clear that Cn is an irreducible representation of A ≡ Mn (C). Suppose V is
irreducible and of dimension K and define a linear map
φ:A
· · ⊕ A} → V ∗ ;
| ⊕ ·{z
φ(a1 , . . . , aK ) → e1 ◦ at1 + · · · eK ◦ atK
K copies
in terms of a basis {e1 , . . . eK } of the dual vector space V ∗ . This is a map of Mn (C) representations, provided a matrix a acts on the dual vector space V ∗ by sending v 7→ v ◦ at . It
is also surjective, so that φ∗ : V → (AK )∗ is injective. Upon identifying (AK )∗ with AK as
A-representation spaces, it follows that V is a submodule of AK ' ⊕(Cn )⊕nK . By irreducibility
V ' Cn .
L
We conclude that the irreducible representations of A = N
i=1 Mni (C) are given by corresponding direct sums:
N
M
C
ni
≡
N
M
ni
i=1
i=1
on which A acts by left block-diagonal matrix multiplication. We adopt the physics notation to
indicate the irreducible representation of Mni (C) by their dimension.
Now, besides the representation of A, there should also be a representation of A◦ on H,
which is mutually commuting with that of A. In other words, we are looking for the irreducible
representations of A ⊗ A◦ . If we denote the unique irreducible representation of Mn (C)◦ by Cn◦ ,
this implies that any irreducible representation of A ⊗ A◦ is given by a subspace
H⊂
N
M
Cni ⊗ Cnj ◦ ≡
i,j=1
N
M
ni ⊗ n◦j
i=1
The integers ni and n◦j form the grid of a diagram: Whenever there is a node at the coordinates
(ni , n◦j ), the representation ni ⊗ n◦j is present in the direct sum decomposition of H.
Example 2.20. Consider the algebra A = C ⊕ M2 (C). The irreducible representations of A
are given by 1 and 2. The two diagrams
1
1◦
2◦
2
1
1◦
2◦
9
2
n1
n◦1
..
.
n◦i
..
.
n◦j
···
ni
···
nj
···
nN
..
.
n◦N
Figure 1. The nodes represent the presence of ni ⊗ n◦j in H.
correspond to H1 = C ⊕ M2 (C) and H2 = C ⊕ C2 , respectively. We used that C2 ⊗ C2◦ ' M2 (C).
The action of A on the left of H1 is given by the matrix
λ 0
0 a
with a ∈ M2 (C) acting on M2 (C) ⊂ H1 by left matrix multiplication. The right action of A on
H1 corresponds to the same matrix acting by right matrix multiplication.
For H2 the left action of A is given by matrix multiplication by the above matrix on vectors
in C ⊕ C2 . However, the right action of (λ, a) ∈ A is given by scalar multiplication with λ on
all of H2 .
The real structure: Recall that an anti-unitary operator J : H → H satisfies
hJξ1 , Jξ2 i = hξ2 , ξ1 i,
and is bijective. We can always write Jξ = Kξ in terms of a matrix K which now satisfies
hKξ1 , Kξ2 i = hξ1 , ξ2 i
and is bijective. In other words, K is a unitary operator on H.
Lemma 2.21. The presence of a real structure for the A-bimodule H requires the corresponding diagram to be symmetric along the diagonal.
Proof. The compatibility between the left and right action of A and the operator J implies
that for a = a1 ⊕ · · · ⊕ aN ∈ A according to the above direct sum decomposition, we have
K(at1 ⊕ · · · ⊕ atN ) = (a◦1 ⊕ · · · ⊕ a◦N )K.
This implies that K maps ni ⊗ n◦j bijectively to nj ⊗ n◦i . All the maps K|ni ⊗n◦j ⊕nj ⊗n◦i can be
simultaneously diagonalized so that we have a real structure on the A-bimodule H if and only
if the corresponding diagram is symmetric along the diagonal.
Remark 2.22. The condition that J 2 = implies that K t = K, on top of unitarity of
K. This means that there is little choice in the form of K (modulo base-change): on the direct
summand ni ⊗ n◦j ⊕ nj ⊗ n◦i we have
0 τ
K∼
τ 0
10
n1
n◦1
···
..
.
ni
···
n◦i
..
.
nj
···
n◦j
nN
..
.
n◦N
Figure 2. The presence of the real structure J implies a symmetry in the
diagram along the diagonal.
where τ : ni ⊗ n◦j → nj ⊗ n◦i is the flip τ (v ⊗ w◦ ) = w ⊗ v ◦ .
The situation is more subtle on the diagonals of the diagram: ni ⊗ n◦i can not be mapped
to itself by a unitary and skew-symmetric matrix. Hence, if = −1, the diagonal Cni ⊗ Cni ◦
should appear in pairs in H that can be interchanged by a K of the above form. We describe
this diagrammatically by doubling the node at a given diagonal position. If = 1 then we simply
have K : ni ⊗ n◦i → ni ⊗ n◦i given by transposition (i.e. the flip).
The finite Dirac operator: Corresponding to the above decomposition of H we can write D
as a sum of operators
Dij,kl : ni ⊗ n◦j → nk ⊗ n◦l ,
∗ . In terms of the
restricted to these subspaces. The condition D∗ = D implies that Dkl,ij = Dij,kl
above diagrammatic representation of H, we express a non-zero Dij,kl as a single line between
the nodes (ni , n◦j ) and (nk , n◦l ). In such a case, we have a non-zero Dkl,ij as well but avoid
drawing double directed lines between all pairs of nodes. Instead, we draw a single undirected
line.
Lemma 2.23. The condition JD = ±DJ and the order-one condition [[D, a], b◦ ] = 0 forces
the lines in the diagram to run only vertically or horizontally (or between the same node in the
case of degeneracies) thereby maintaining the diagonal symmetry in the diagram.
Proof. The condition JD = ±JD easily translates into a commuting diagram:
ni ⊗ n◦j
J
D
J
nj ⊗ n◦i
/ nk ⊗ n◦
l
D
/ nl ⊗ n◦
k
thus relating Dij,kl to Dji,lk : the diagonal symmetry is maintained.
If we write the order-one condition [[D, a], b◦ ] = 0 for diagonal elements a = λ1 1n1 ⊕ · · · ⊕
λN 1nN ∈ A and b = µ1 1n1 ⊕ · · · ⊕ µN 1nN ∈ A with λi , µi ∈ C, we compute that
Dij,kl (λi − λk )(λj − λl ) = 0
for all λi , µj ∈ CAs a consequence, Dij,kl = 0 whenever i 6= k or j 6= l.
11
···
..
.
n◦i
..
.
n◦j
ni
···
nj
···
..
.
◦
Figure 3. The lines between two nodes represent a non-zero Dii,ji : Cni ⊗Cni →
◦
◦
◦
Cnj ⊗ Cni , as well as its adjoint Dji,ii : Cnj ⊗ Cni → Cni ⊗ Cni . The nonzero components Dii,ij and Dij,ii are related to ±Dii,ji and ±Dji,ii , respectively,
according to JD = ∓DJ.
Grading: If, finally, there is a grading γ : H → H then each node in the diagram gets dressed
by a plus or minus sign. The rules are that
• D connects nodes with different signs,
• If the node ni ⊗ n◦j has sign ±, then the node nj ⊗ n◦i has sign ±00 , according to
Jγ = 00 γJ.
Remark 2.24. The fact that the grading is assigned to each node, even if there are degeneracies is a result of the orientation
axiom. It says that the grading γ can be implemented by
P
elements xi , yi ∈ A as γ ≡ i xi yi◦ . Hence, this is completely dictated by the operator J and
the representation of A.
Finally, this leads to Krajewski’s diagrammatic classification of real finite spectral triples,
extended to any KO-dimension.
For A a finite dimensional algebra, let Λ × Λ◦ be the lattice embedded (symmetrically) in
R2 of all irreducible representations of A ⊗ A◦ (without repetition). A symmetrical embedding
means that all points (n, n◦ ) (n ∈ Λ) lie on the diagonal of R2 and the flip in Λ×Λ◦ corresponds
to reflection with respect to the diagonal in R2 .
Definition 2.25. A Krajewski diagram Γ of KO-dimension n on a finite dimensional algebra A is a directed graph together with an embedding ι : Γ → R2 for which the embedded vertices
form a subset of the lattice Λ × Λ◦ ⊂ R2 such that
(1) the embedded graph ι(Γ) is symmetric along the diagonal in R2 , and every row or
column in Λ × Λ◦ has non-empty intersection with ι(Γ).
(2) the embedded edges connect only horizontally or vertically, or have source and target
the same point in Λ × Λ◦ .
(3) the edges are dressed by arbitrary non-zero complex matrices between the source and
target representation spaces of A ⊗ A◦ ,
(4) if n is even, the (embedded) vertices are dressed by ±1 and the edges only connect
opposite signs. Moreover, if n is 0 or 4 then the signs at the vertices are symmetric
with respect to reflection along the diagonal; if n is 2 or 4 they are anti-symmetric,
(5) if n is 2, 3, 4 or 5 then the inverse image under ι of the diagonal elements in Λ × Λ◦
contains an even and positive number of vertices of Γ.
Note that this definition allows for different vertices of Γ to be mapped to the same point
on the lattice Λ × Λ◦ ; similarly for the edges of Γ.
12
Theorem 2.26. There is a one-to-one correspondence between real finite spectral triples of
KO-dimension n and Krajewski diagrams of KO-dimension n. One associates to a Krajewski
diagram a spectral triple (A, H, D; J, γ) in the following way:
• to each vertex v of Γ with ι(v) = (ni , n◦j ) ∈ Λ × Λ◦ one associates a direct summand
Cni ⊗ Cnj ◦ in H, thus furnishing a (faithful) representation of A on H,
• to each dressed edge between (ni , n◦j ) and (nk , n◦l ), one associates a non-zero linear
operator Dij,kl : Cni ⊗ Cnj ◦ → Cnk ⊗ Cnl ◦ given by the matrix on that edge; the
operator D is the sum of all these operators and their adjoints,
• the grading γ on H is defined by setting γ to be ±1 on Cni ⊗ Cnj ◦ ⊂ H according to
the dressing by ±1 of the node (ni , n◦j ) ∈ Λ × Λ◦ ,3
• the real structure J is given by Jψ = Kψ where K is defined on the direct summand
ni ⊗ n◦j ⊕ nj ⊗ n◦i (ni 6= nj ) by
0 τ
(2.2.3)
K∼
τ 0
where τ : ni ⊗ n◦j → nj ⊗ n◦i is the flip τ (ψ ⊗ φ◦ ) = φ ⊗ ψ ◦ . If ni = nj then in
KO-dimensions 0, 1, 6 and 7 we set K ≡ τ on ni ⊗ n◦i ; in KO-dimension 2, 3, 4 and 5
we set K to be of the above form (2.2.3) on the pair ni ⊕ n◦i ⊕ ni ⊕ n◦i .
Example 2.27. Consider the case that A = C⊕C. There are at least five possible Krajewski
diagrams in KO-dimension 0; in terms of Λ = {11 , 12 }:
understanding a dressing of the diagonal vertices with a plus-sign, and the off-diagonal vertices
with a minus-sign. There are three remaining diagrams, obtained from the first three by removing
all the edges.
Let us consider the second diagram and give the corresponding spectral triple. First, the
inner product space is H = C3 . The edges indicate that there are non-zero components of D
that map between the first two copies of C in H and between the second two copies. In other
words,


0 λ µ
D = λ 0 0 
µ 0 0
for some λ, µ ∈ C, and in this basis:


1 0
0
γ = 0 −1 0 
0 0 −1
Finally, J is given by the matrix K composed with

1 0
K = 0 0
0 1
complex conjugation on H where

0
1
0
From this it is clear that
Dγ = −γD;
DJ = JD;
Jγ = γJ.
Exercise 2.6. Use the five diagrams of the previous example to show that on A = C ⊕ C a
finite spectral triple of KO-dimension 6 must have vanishing finite Dirac operator.
3That the grading is the same for all vertices in the inverse image of a given node in Λ × Λ◦ corresponds to
orientability, cf. Remark 2.24.
13
Example 2.28. Consider A = Mn (C) so that Λ = {n}. We then have a Krajewski diagram
n
n◦
The node can be labeled only by either plus or minus one, the choice being irrelevant. This
means that H = Cn ⊗ Cn◦ ' Mn (C) with trivial grading γ. The operator J is a combination
of complex conjugation and the flip on n ⊗ n◦ : this translates on Mn (C) as taking the matrix
adjoint. Moreover, since the single node has dressing ±1, there are no non-zero Dirac operators:
D = 0. Hence, the finite spectral triple of this diagram corresponds to
(A = Mn (C), H = Mn (C), D = 0; J = (·)∗ , γ = 1).
2.2.2. Real algebras. In the previous sections, we have considered finite spectral triples
on complex algebras. In practice, it is useful to allow for real algebras as well. We must be
careful here, because the Hilbert space is a complex space. Thus, a real algebra A is represented
on H through a R-linear algebra map π : A → L(H).
A particularly interesting example in this context is given by H, the real algebra of quaternions. We define it as a real subalgebra of M2 (C):
α
β
H=
: α, β ∈ C ,
−β ∗ α∗
which is indeed closed under multiplication. Its unique irreducible representation is given by
left multiplication on H ⊗R C which, as before, we denote by its complex dimension 2.
2.3. Spectral triples
Definition 2.29. A spectral triple (A, H, D) is given by an unital ∗- algebra A represented
as operators in a Hilbert space H and a self-adjoint operator D such that (1 + D2 )−1/2 is a
compact operator and [D, a] bounded for a ∈ A.
A spectral triple is even if the Hilbert space H is endowed with a Z/2Z-grading γ such that
[γ, a] = 0 and {γ, D} = 0.
A real structure of KO-dimension n ∈ Z/8Z on a spectral triple is an antilinear isometry
J : H → H such that
JD = ε0 DJ,
J 2 = ε,
Jγ = ε00 γJ
(even case).
The numbers ε, ε0 , ε00 ∈ {−1, 1} are a function of n mod 8:
n
ε
ε0
ε00
0
1
2
3
4
5
6 7
1
1 −1 −1 −1 −1
1 1
1 −1
1
1
1 −1
1 1
1
−1
1
−1
Moreover, with b0 = Jb∗ J −1 we impose that
[a, b0 ] = 0,
[[D, a], b0 ] = 0,
∀ a, b ∈ A,
A spectral triple with a real structure is called a real spectral triple.
The basic example is the commutative spin geometry of a Riemannian spin manifold given
by the triple
• A = C ∞ (M ), the algebra of smooth functions on M .
• H = L2 (M, S), the Hilbert space of square integrable sections of a spinor bundle
S → M.
• D, the Dirac operator associated with the Levi-Civita connection.
14
If the manifold is even dimensional, there is a grading defined by Γ := −γ 1 γ 2 · · · γ dim M , where
γ µ are the Dirac gamma matrices satisfying {γ µ , γ ν } = 2g µν . The real structure is given by the
charge conjugation.
In general, given a real spectral triple (A, H, D; J) we can construct a spectral triple on a
commutative algebra, derived from this data. Define
AJ := {a ∈ A : aJ = Ja∗ } .
This is a complex subalgebra, contained in the center of A.
Remark 2.30. This definition is very similar to the definition of a subalgebra of A defined
in [1, Prop. 3.3] (cf. [5, Prop. 1.125]), which is the real commutative subalgebra in the center
of A consisting of elements for which aJ = Ja. We have provided a similar but different
definition, since this subalgebra will turn out to be very useful for the description of the gauge
group associated to any spectral triple.
Just as in [1, Prop. 3.3] we have
Proposition 2.31. Let (A, H, D; J) be a real spectral triple. Then
(1) AJ = {a ∈ A : aJ = Ja∗ } defines an involutive commutative complex subalgebra of the
center of A.
(2) (AJ , H, D; J) is a real spectral triple,
P
(3) Any a ∈ AJ commutes with the algebra generated by the sums j aj [D, bj ] with aj , bj ∈
A.
Proof. (1) If a ∈ AJ then also Ja∗ J −1 = (JaJ −1 )∗ = a since J is isometric. Hence, AJ
is involutive. Moreover, for all a ∈ A and b ∈ AJ we have that [a, b] = [a, Jb∗ J −1 ] = 0 by the
commutant property (2.2.1). Thus, AJ is in the center of A.
(2) AJ being a subalgebra of A, all conditions for a spectral triple are automatically satisfied.
(3) This follows from the order-one condition (2.2.1): [a, [D, b]] = [JaJ −1 , [D, b]] = 0 for
a ∈ AJ and b ∈ A.
Under suitable condition on the triple (A, H, D; J) the spectral triple (AJ , H, D) is a commutative spin geometry [3]. Then, Connes’ Reconstruction Theorem [4] establishes the existence of
a compact Riemannian spin manifold M such that (AJ , H, D) ' (C ∞ (M ), L2 (M, S ⊗ E), DE ).
The spinor bundle S → M is twisted by a vector bundle E → M and the twisted Dirac operator is of the form DE = ∂/ +ρ with ρ ∈ Γ End(S ⊗ E). Moreover, the Dirac operator DA
obtained from the inner fluctuations of the noncommutative manifold (A, H, D) can also be
written as DA = ∂/ +ρA with ρA again a section of the endomorphism bundle End(S ⊗ E), but
now parametrized by A ∈ Ω1D (A).
A spectral triple is called regular
(or smooth) if the algebra generated by A and [D, A]
T∞
lies within the smooth domain n=0 Dom δ n of the operator derivation δ(T ) := |D|T − T |D|.
This condition permits to introduce the analogue of Sobolev spaces Hs := Dom(1 + D2 )s/2 for
s ∈ R. One can develop this theory to an abstract differential calculus, cf. the notes by Higson.
Corresponding to the direct product of manifolds, one can take the product of spectral
triples as follows. Suppose (A1 , H1 , D1 , γ1 , J1 ) and (A2 , H2 , D2 , γ2 , J2 ) are even real spectral
triples, then we define the (exterior) product spectral triple by
A = A1 ⊗ A2
H = H1 ⊗ H 2
D = D1 ⊗ 1 + γ1 ⊗ D2
γ = γ1 ⊗ γ2
J = J1 ⊗ J2
Note that D2 = D12 ⊗ 1 + 1 ⊗ D22 since the cross-terms vanish due to γ1 D1 = −D1 γ1 .
15
Example 2.32. We are mainly interested in almost commutative manifolds, which are by
definition products of Riemannian spin manifolds M with finite noncommutative spaces F . More
precisely, we will consider
M × F := (C ∞ (M ), L2 (S), ∂/; γM , JM ) ⊗ (AF , HF , DF ; γF , JF )
with (AF , HF , DF ; γF , JF ) as in Definition 2.13. Note that this can be identified with:
M × F = (C ∞ (M, AF ), L2 (S ⊗ (M × HF )), ∂/ ⊗1 + γM ⊗ DF ; γM ⊗ γF , JM ⊗ JF ).
2.3.1. Noncommutative differential forms. Let A be an algebra with unit over C. The
universal differential algebra Ωun (A) is the graded algebra generated by a ∈ A of degree 0 and
symbols δa, a ∈ A of degree 1, such that
δ(ab) = (δa)b + aδb
δ(αa + βb) = αδa + βδb;
(a, b ∈ A, α, β ∈ C).
We can write Ωun (A) as a direct sum of subspaces Ωpun (A) generated by linear combinations of
a0 δa1 · · · δap . Furthermore, there is the isomorphism of vector spaces
A⊗A
(2.3.1)
⊗p
' Ωpun (A),
where A := A/CI. The operator δ is defined on Ωun (A) by
δ(a0 δa1 · · · δap ) = δa0 δa1 · · · δap ,
δ(δa1 · · · δap ) = 0.
By construction, the algebra Ωun (A) is also a A-bimodule. As the name suggests, the universal
differential algebra satisfies the following universal property.
Proposition 2.33. Let (Ω, d) be a graded differential algebra and let ρ be a morphism of
unital algebras. Then, there exists a unique extension of ρ to a morphism of graded differential
algebras ρe : Ωun (A) → Ω such that ρe ◦ δ = d ◦ ρe.
An example of a frequently used differential calculus in the text and more generally, in
noncommutative geometry, is Connes’ differential calculus [2]. Let (A, H, D) be a spectral
triple. The A-bimodule ΩpD (A) of Connes’ differential p-forms is made of classes of operators
of the form
X j
ω=
a0 [D, aj1 ] · · · [D, ajp ], aji ∈ A,
j
modulo the sub-bimodule of operators
X
[D, bj0 ][D, bj1 ] · · · [D, bjp−1 ] : bji ∈ A, bj0 [D, bj1 ] · · · [D, bjp−1 ] = 0 .
j
The exterior differential dD is given by
X
X
j
j
j
dD
a0 [D, a1 ] · · · [D, ap ] =
[D, aj0 ][D, aj1 ] · · · [D, ajp ].
j
j
In the case of the canonical triple (C ∞ (M ), H, D) of a Riemannian spin manifold M , this
differential calculus is isomorphic to the de Rham differential calculus.
2.3.2. Modules and connections. We recall some basic definitions on modules and connections thereon. We derive a general Bianchi identity for the curvature of such connections
and link with gauge theory.
16
2.3.2.1. Modules. Let A be an algebra over the complex numbers C.
Definition 2.34. A right module E is a vector space over C that carries a right representation of A, i.e. there is a map E × A 3 (η, a) → ηa such that
η(ab) = (ηa)b,
η(a + b) = ηa + ηb,
(η + ξ)a = ηa + ξa,
for any η, ξ ∈ E and a, b ∈ A.
There is the natural notion of a morphism of (right) A-modules as linear maps that respect
this structure. Thus, a morphism between two (right) A-modules E and F is a linear map
ρ : E → F that is also right A-linear:
∀η ∈ E, a ∈ A.
ρ(ηa) = ρ(η)a;
Left modules and morphisms of left modules are defined similarly. A bimodule over an algebra
A is both a left and a right A-module such that the left and right action of A commute:
∀η ∈ E, a, b ∈ A.
(aη)b = a(ηb);
Given a right A-module, we define its dual module E 0 as the collection of all morphisms from E
into A, where A is seen as the trivial right A-module; in other words:
E 0 := φ : E → A | φ(ηa) = φ(η)a, η ∈ E, a ∈ A .
Definition 2.35. A right A-module E is said to be finite projective if there exists an idempotent p = p2 ∈ MN (A) such that E ' pAN as right A-modules.
Here MN (A) ' MN (C) ⊗C A denotes the algebra of N × N matrices with entries in A
whereas AN := Cn ⊗C A which can be thought of as the set of N -dimensional vectors with
entries in A, and is clearly a right A-module.
2.3.2.2. Connections. Let us suppose we have an algebra A with a differential calculus
(Ω(A) = ⊕p Ωp (A), d). We now review the notion of a (gauge) connection on a (finite projective)
module E over A with respect to the given calculus; we take a right module structure.
A connection on the right A-module E is a C-linear map
∇ : E ⊗A Ωp (A) 7→ E ⊗A Ωp+1 (A) ,
defined for any p ≥ 0, and satisfying the Leibniz rule
∇(ωρ) = (∇ω)ρ + (−1)p ωdρ , ∀ ω ∈ Ωp (A) , ρ ∈ Ω(A) .
A connection is completely determined by its restriction
∇ : E → E ⊗A Ω1 (A) ,
which satisfies
∇(ηa) = (∇η)a + η ⊗A da , ∀ η ∈ E , a ∈ A ,
and which is extended by the Leibniz rule. It is the latter property that implies that the
composition,
∇2 = ∇ ◦ ∇ : E → E ⊗ Ω2 (A) ,
is Ω(A)-linear. Indeed, for any ω ∈ Ωp (A), ρ ∈ Ω(A),
∇2 (ωρ) = ∇ ((∇ω)ρ + (−1)p ωdρ)
= (∇2 ω)ρ + (−1)p+1 (∇ω)dρ + (−1)p (∇ω)dρ + ωd2 ρ
= (∇2 ω)ρ .
The restriction of ∇2 to E is the curvature
(2.3.2)
F : E → E ⊗A Ω2 (A) ,
17
of the connection. It is A-linear, F (ηa) = F (η)a for any η ∈ E, a ∈ A, and satisfies
∇2 (η ⊗A ρ) = F (η)ρ , ∀ η ∈ E , ρ ∈ Ω(A) .
Thus, F ∈ EndA (E, E ⊗A Ω2 (A)), the collection of (right) A-linear endomorphisms of E, taking
values in the two-forms Ω2 A.
Connections always exist on a projective module. On the free module E = CN ⊗C A ' AN ,
a connection is given by the operator
∇0 = 1 ⊗ d : CN ⊗C Ωp (A) → CN ⊗C Ωp+1 (A) .
With the canonical identification CN ⊗C ⊗A = (CN ⊗C A) ⊗A Ω(A) ' (Ω(A))N , one thinks of
∇0 as acting on (Ω(A))N as the operator ∇0 = (d, d, · · · , d) (N -times).
For a generic projective module E one has a canonical inclusion map, λ : E → AN , which
identifies E as a direct summand of the free module AN and a canonical idempotent p : AN → E
which allows to identify E = pAN . Using these maps as well as their natural extension to Evalued forms, on E a connection ∇0 is given by the composition
λ
p
1⊗d
E ⊗A Ωp (A) −→ CN ⊗C Ωp (A) −→ CN ⊗C Ωp+1 (A) −→ Ωp+1 (A)
(we have also used canonical identifications for the free module). This connection is called the
Levi-Civita or Grassmann connection and is explicitly given by
∇0 = p ◦ (1 ⊗ d) ◦ λ
although one simply indicates it by
∇0 = pd.
(2.3.3)
2.3.3. Unitary and Morita equivalence of spectral triples. In the previous chapter
we have seen the prominent role that is played by symmetries in physics. We now consider
symmetries of (A, H, D, γ, J), an even real spectral triple. The first candidate is unitary equivalence.
Definition 2.36. Two spectral triples (A1 , H1 , D1 ) and (A2 , H2 , D2 ) are called unitary
equivalent if A1 ' A2 and there exists a unitary intertwining operator U : H1 → H2 such that
U π1 (a)U ∗ = π2 (a)
(a ∈ A1 )
∗
U D1 U = D2
If there exist grading operators γ1 , γ2 then we also demand that U γ1 U ∗ = γ2 . If there exist real
structures J1 , J2 then we also demand that U J1 U ∗ = J2 .
As a special case, we consider the gauge group U(A), defined as the unitary elements in
the algebra A of a spectral triple (A, H, D). It implements unitary transformations from the
spectral triple to itself, transforming
D 7→ D + u[D, u∗ ].
If the spectral triple is real, the unitary intertwiner is given by U = uJu∗ J −1 ≡ u(u∗ )◦ for
u ∈ U(A), thus transforming
D 7→ D + u[D, u∗ ] + 0 Ju[D, u∗ ]J −1 .
Effectively, the unitary group acts as automorphisms on A by conjugation, a 7→ uau∗ . Such
automorphisms are called inner, in contrast to the outer automorphisms which are defined as
the quotient Aut(A)/Inn(A). This is nicely summarized by
1 → Inn(A) → Aut(A) → Out(A) → 1.
18
Note that if A = C ∞ (M ) is commutative, there are no non-trivial inner automorphisms and
Out(A) = Diff(M ). Moreover, unitaries in the subgroup U(AJ ) of U(A) act trivially on both A
and H. In fact, U(AJ ) ⊂ U(Z(A)) so that we can define a group
G(A, H, D; J) =
(2.3.4)
U(A)
U(AJ )
which we will call the gauge group for (A, H, D; J). On the infinitesimal level we can define
the corresponding gauge Lie algebra as
u(A)
g(A, H, D; J) =
u(AJ )
with u(A) the Lie algebra of skew-hermitian elements in A.
We have seen that a non-abelian gauge group appears naturally when A is noncommutative.
Even more, noncommutative algebras allow for a more general – and more natural – notion of
equivalence than automorphisms, namely, Morita equivalence. Let us imitate the construction
in Theorem 2.18 and see if we can lift Morita equivalence to the level of spectral triples in this
more general setting.
Recall that given an algebra A, a Morita equivalent algebra B is the algebra of endomorphisms of a finite projective (right) module E over A,
B = EndA (E).
Let (A, H, D) be a given spectral triple and try to construct a spectral triple (B, H0 , D0 ). Naturally, H0 := E ⊗A H carries an action of φ ∈ B:
φ(η ⊗ ψ) = φ(η) ⊗ ψ
(η ∈ E, ψ ∈ H).
The naive choice of an operator D0 by D0 (η ⊗ ψ) = η ⊗ Dψ will not do, because it does not
respect the ideal defining the tensor product over A, being generated by elements of the form
ηa ⊗ ψ − η ⊗ aψ;
(η ∈ E, a ∈ A, ψ ∈ H).
A better definition is
D0 (η ⊗ ψ) = η ⊗ Dψ + ∇(η)ψ.
where ∇ : E → E ⊗A Ω1D (A) is a connection associated to the differential d : a 7→ [D, a] (a ∈ A).
Theorem 2.37. If (A, H, D) is a spectral triple and ∇ is a connection on a finite projective
right A-module E, then (B, E ⊗A H, ∇ ⊗ 1 + 1 ⊗ D) is a spectral triple.
Analogously, we define for a real spectral triple (A, H, D, J) a real spectral triple (B, H0 , D0 , J 0 )
by setting H0 := E ⊗A H ⊗A E ◦ . Here E ◦ is the conjugate module to E:
E ◦ = {ξ : ξ ∈ E}
with a left A action defined by aξ = ξa∗ for any a ∈ A. Still φ ∈ B acts on H0 by
φ(η ⊗ ψ ⊗ ξ) = φ(η) ⊗ ψ ⊗ ξ
and
D0 (η ⊗ ψ ⊗ ξ) = (∇η)ψ ⊗ ξ + η ⊗ Dψ ⊗ ξ + η ⊗ ψ(∇ξ)
J 0 (η ⊗ ψ ⊗ ξ) = ξ ⊗ Jψ ⊗ η
Theorem 2.38. If (A, H, D, J) is a real spectral triple, then (B, E ⊗A H ⊗A E ◦ , ∇ ⊗ 1 ⊗ 1 +
1 ⊗ D ⊗ 1 + 1 ⊗ 1 ⊗ ∇) is a real spectral triple.
Finally, for even spectral triples one defines a grading γ 0 = 1 ⊗ γ ⊗ 1 on E ⊗A H ⊗A E ◦ .
We now focus on Morita self-equivalences, for which B = A and consequently E = A.
Let us look at connections
∇ : A → Ω1D (A).
19
P
Clearly, by the Leibniz rule ∇ = d + A where A ≡ ∇(1) = j aj [D, bj ] is a generic element in
Ω1D (A). Similarly, ψ∇a = (0 JdaJ −1 + 0 JAaJ −1 )ψ. Since H0 ' H we have
D0 (ψ) ≡ D0 (1 ⊗ ψ ⊗ 1) = ∇(1)ψ + ψ∇(1) + Dψ = Dψ + Aψ + 0 JAJ −1 ψ.
In other words, D is ‘innerly perturbed’ to DA := D + A + 0 JAJ −1 where A∗ = A ∈ Ω1D (A) is
called the gauge field. Another name used for these fields is inner fluctuations of the Dirac
operator, since it is the algebra A that generates – through Morita self-equivalences – the fields
A.
On the new spectral triple (A, H, DA ) there is an action of the unitary group U(A) by
unitary equivalences. Recall that U = uJu∗ J −1 so that
DA 7→ U DA U ∗
or, equivalently,
A 7→ uAu∗ + u[D, u∗ ]
which is the usual rule for a gauge transformation on a gauge field.
2.3.3.1. Spectral action functional. Having identified the gauge group canonically associated
to a spectral triple, and derived the gauge fields, we are ready to find action functionals of these
fields that are invariant under the gauge group.
Definition 2.39. Let f be a positive and even function from R to R. The spectral action
is defined by
Sb [A] := Tr f (DA /Λ)
where Λ is a real cutoff parameter. The fermionic action is defined as the inner product
Sf [A, ψ] := (ψ, DA ψ).
We will assume that f is given by a Laplace–Stieltjes transform:
Z
2
f (x) =
e−tx dµ(t).
t>0
with µ a measure on R, and further that there exists the following heat kernel expansion:
X
2
Tr e−tD =
tα/2 cα
α
as t → 0. Note that this is defined for the unperturbed operator D, but similar expression hold
for any bounded perturbation such as DA of D.
Using the Laplace–Stieltjes transform, we now derive an asymptotic expansion of the spectral
in terms of the heat coefficients cα .
Proposition 2.40. The spectral action is given, asymptotically (as Λ → ∞) by
X
Tr f (D/Λ) =
Λ−α f−α cα
α
where fα :=
R
t−α/2 dµ(t).
Proof. This follows directly after inserting the heat expansion in the Laplace–Stieltjes
transform:
Z
Tr f (D/Λ) =
tα/2 Λα cα dµ(t).
t>0
20
2.3.4. Heat kernel expansion on a manifold. We end this Chapter by a discussion on
the motivating case of a heat kernel on a manifold M , and provide a preliminary result on the
spectral action in the commutative case.
Let M be a compact Riemannian spin manifold of dimension m and let S → M be the
spinor bundle with the spinor connection ∇S . Let V be a hermitian vector bundle over M with
a compatible connection ∇V . First, we state a more general result (cf. [6] for more details).
Theorem 2.41 (Generalized Lichnerowicz formula). Let ∂/V be the twisted Dirac operator
on S ⊗ V . Then
1
∂/2V = ∆S⊗V − R + RV
4
∗ S⊗V
S⊗V
S⊗V
where ∆
=∇
∇
is the Laplacian of S ⊗ V and RV is the bundle endomorphism
X
RV =
γµ γν ⊗ [∇Vµ , ∇Vν ]
µ<ν
Theorem 2.42 (Gilkey). Let M be an m-dimensional compact Riemannian spin manifold.
If P = ∇∗ ∇ − E for a connection ∇ and an endomorphism E on some vector bundle, then there
is an asymptotic expansion (at t → 0):
Z
X
ak (x, P )dµ(x)
Tr e−tP =
t(k−m)/2
M
k≥0
with respect to the Riemannian volume form dµ. The so-called Seeley–DeWitt coefficients
ak (x, P ) vanish for k odd and the first three non-zero are
a0 (x, P ) = (4π)−m/2 Tr(1)
R
−m/2
a2 (x, P ) = (4π)
Tr − + E
6
1
a4 (x, P ) = (4π)−m/2
Tr − 60R E + 180E 2 + 60E;µ µ + 30Ωµν Ωµν
360
µ
2
− 12R;µ + 5R − 2Rµν R
µν
+ 2Rµνρσ R
µνρσ
Corollary 2.43. Let ∂/ be a Dirac operator on a 4-dimensional Riemannian spin manifold
(M, g). The first three non-zero residues can be expressed in terms of the curvature Rµνρσ of g:
Z
Z
1
1
a0 (x, ∂/2 ) = 2
dµ(x)
a2 (x, ∂/2 ) =
R dµ(x)
4π M
48π 2 M
Z
1
2
a4 (x, ∂/ ) = −
Cµνρσ C µνρσ dµ(x)
160π 2 M
modulo boundary and topological terms. Here, Cµνρσ is the Weyl curvature, which is by definition
the trace-free part of Rµνρσ .
As a consequence, the spectral action on a commutative manifold becomes:
Z
Z
Z
f4
f2
f0
Sb [g] = 2
dµ(x) +
R dµ(x) −
Cµνρσ C µνρσ dµ(x)
4π M
48π 2 M
160π 2 M
Thus, we obtain a cosmological constant, the Einstein–Hilbert action of General Relativity, and
an action of conformal gravity.
21
CHAPTER 3
Almost commutative (AC) manifolds and gauge theories
3.1. Background: gauge theories in physics
In this section we will give a crash course to gauge theories, put into a historical context.
3.1.1. Dirac and the dawn of quantum electrodynamics. In 1928, Dirac asked the
question whether there exists a differential operator D such that its square is equal to the
Laplace (d’Alembert) operator:
X
η µν ∂µ ∂ν
D2 =
µ,ν
where η = diag(+1, −1, −1, −1) The motivation for this was to find a relativistic version of the
Schrödinger equation:
3
X
∂ 2
∂
1
(3.1.1)
−i~
ψ(x, t) = i~ ψ(x, t)
2m
∂xj
∂t
j=1
Indeed, the fact that the left-hand-side involves a second-order differential operator and the
right-hand-side a first-order, breaks the special relastivistic symmetry between space and time.
Two solutions can be found to this question, based on replacing the non-relativistic relation
E = p2 /2m by Einstein’s relations:
p
(3.1.2)
E = p2 c2 + m2 c4
keeping in mind the quantum mechanical identification pj = −i~∂/∂xj .
3.1.1.1. The Klein–Gordon equation. The first solution is to square the right-hand-side involving the time-derivative, leading to the Klein–Gordon equation:


2
2
3 X
∂
∂
2
4

−i~
ψ(x, t)
+ m c  ψ(x, t) = i~
∂xj
∂t
j=1
or, equivalently, ( + m2 c2 /~2 )ψ(x, t) = 0 with the d’Alembert operator:
3 X
1 ∂2
∂ 2
.
= 2 2 −
c ∂t
∂xj
j=1
η µν ∂
More compactly, =
µ ∂ν after writing x0 = ct. The Klein–Gordon equation describes the
relativistic motion of a free scalar particle with mass m.
3.1.1.2. The Dirac equation. Another solution to the above problem is to try to take the
square root at the left-hand-side of (3.1.1). Dirac postulated a first-order differential operator
H by setting
3
X
∂
2
H = α0 mc + c
αi −i~
∂xj
j=1
P
2
2
2 4
and then demanding that H 2 =
i c (−i~∂/∂xi ) + m c , according to Einstein’s relation
(3.1.2) for the energy. One computes:
X
X
X
H 2 = α02 m2 c4 + (α0 αi +αi α0 )mc3 (−i~∂i )+
αi2 c2 (−i~∂i )2 + (αi αj +αj αi )c2 (−i~∂i )(−i~∂j )
i
i
23
i>j
which implies that the αµ satisfy
α02 = 1
α0 αi + αi α0 = 0
αi2 = 1
αi αj + αj αi = 0
(i 6= j)
Clearly, these relations cannot be satisfied by ordinary numbers, and the smallest representation
of this (Clifford) algebra such that H is a symmetric operator is four dimensional.
The Dirac equation is given by
∂
ψ
∂t
Introducing the so-called Dirac gamma matrices γ µ = (α0 , −α0 α1 , −α0 α2 , −α0 α3 ) this is
equivalent to
mc (3.1.3)
iγ µ ∂µ −
ψ(t, x) = 0.
~
It describes the relativistic motion of a free electron, or, more generally, of a free fermion. The
Dirac matrices satisfy
{γµ , γν } = 2η µν 14
Hψ = i~
Of course, on a (pseudo)-Riemannian spin manifold one has the analogous Dirac equation
written concisely as (D − m)ψ = 0 in terms of the Dirac operator D = iγ ◦ ∇S with ∇S a
spin connection, and a section ψ of a spinor bundle S. In this case, the square of the Dirac
operator is not precisely the d’Alembertian, but we have
Theorem 3.1 (Weitzenböck). If ∆ = gµν ∇Sµ ∇Sν is the Laplace–Beltrami operator, then
1
D2 = ∆ − R
4
3.1.1.3. Principal of extremal action and electrodynamics. In physics, it is convenient to
work with action functionals and obtain equation of motions – such as the Dirac equation
– as extremas of this functional. Let us illustrate this in the equation of interest, that is, the
Dirac equation. A fermionic action functional is given as the inner product on Γ(M, S):
Sf [ψ] = (ψ, (D − m)ψ);
(ψ ∈ Γ(M, S)).
R
In terms of the hermitian structure, we have Sf [ψ] = M hψ, (D − m)ψix dµ(x). Now, ψ extremizes the action Sf means that the directional derivative
Sf0 [ψ][χ] = lim(Sf [ψ + tχ] − Sf [ψ])/t,
t→0
vanishes for all χ ∈ Γ(M, S). One computes that this happens if and only if (D − m)ψ = 0.
We conclude that the fermionic action Sf describes the physical system of a relativistic particle
moving in spacetime M . More generally, the vanishing of the directional derivative of an action
functional gives the equation of motion for the corresponding physical system.
Remark 3.2. Note that the Klein–Gordon equation can be obtained as the equation of motion
of the action functional
Z
mc2 2
1
φ .
η µν (∂µ φ)(∂ν φ) −
Skg [φ] =
2 M
~
An interesting observation is that the above action Sf is invariant under the following global
U (1) symmetry:
ψ → eiθ ψ;
(θ ∈ [0, 2π]).
Indeed, with h·, ·ix being anti-linear and linear in the first and second entry, respectively, we
find that Sf [eiθ ψ] = Sf [ψ].
24
Next, suppose that this symmetry transformation on ψ is position dependent, θ = θ(x).
Clearly, Sf is not invariant under this local U (1)-symetry unless we make the following minimal
replacement:
∇S
∇S + ieA
Here we introduce a new field A ∈ Ω1 (M ) that transforms under a U (1)-transformation as
A 7→ A − e−1 dθ
The fermionic action now depends on two fields A and ψ:
Z
hψ, (iγ µ ∇Sµ − eγ µ Aµ − m)ψix dµ(x)
Sf [A, ψ] =
M
Invariance of this functional under the U (1)-action follows:
Z
Z
−1
iθ
µ
Sf [A + ie dθ, e ψ] = Sf [A, ψ] −
hψ, γ (∂µ θ)ψix dµ(x) +
hψ, γ µ (∂µ θ)ψix dµ(x)
M
M
The second term on the right-hand-side comes from the Leibniz rule for ∇S on eiθ ψ, and the
last term comes from the transformation of the A-field.
If we extremize the new action Sf0 with respect to ψ we obtain the equation of motion
(iγ µ ∂µ − eγ µ Aµ − m) ψ = 0
This describes the relativistic motion of an electron in the presence of an electromagnetic field
Aµ . As such, Aµ satisfies Maxwell’s equations; let us derive them here from the principle of
extremal action.
The curvature of A is defined to be F = dA ∈ Ω2 (M ). In local coordinates, we write:
F = 21 Fµν dxµ ∧ dxν .
where Fµν = ∂µ Av − ∂ν Aµ . Maxwell’s theory is described by the following action
Z
Z
1
Fµν F µν .
Sem [A] :=
F ∧ (∗F ) ≡
4 M
M
The so-called Hodge star operator ∗ : Ωr (M ) → Ωn−r (M ) is given locally by
p
|g| µ1 ···µr
µ1
µr
ν1
νn−r
∗(dx ∧ · · · ∧ dx ) =
ν1 ···νn−r dx ∧ · · · ∧ dx
(n − r)!
with µ1 ···µn = ±1 depending on whether µ1 · · · µn is an even or odd permutation of 12 · · · n.
The equation of motion for S[A] is
d(∗F ) = 0
which together with the Bianchi identity dF = 0, which is always satisfied, forms Maxwell’s
equation for electromagnetism. More explicitly, we identify the components of Fµν with the
electric and magnetic field as


0 −Ex −Ey −Ez
Ex
0
Bz −By 

(Fµν ) = 
Ey −Bz
0
Bx 
Ez By −Bx
0
so that d(∗F ) = 0 = dF become
∇·B=0
∂B
= −∇ × E
∂t
∇·E=0
∂E
∇×B=
∂t
25
3.1.2. Non-abelian gauge theories. The above U (1)-symmetry principle led Yang and
Mills to consider theories with a non-abelian symmetry group. At first, there seemed to be no
direct physical application of such theories. However, at the beginning of the 1960s, Glashow,
later joined by Weinberg and Salam, used a U (1) × SU (2)-symmetry as underlying their electroweak theory. The Standard Model of elementary particles was finally completed by
adding a SU (3) quark color symmetry. This model has been tested up to previously unencountered precision, the only missing piece being the Higgs particle.
Remark 3.3. As said, the weak interactions correspond to a SU (2) gauge group. Matter
is supposed to be in a representation of this group. For example, the neutron and proton are
supposed to be organized in the definining representation:
p
a
b
p
g·
7→
n
−b∗ a∗
n
Historically, this occurence of SU (2)-symmetry was first motivated by the similarity in mass for
the neutron and proton (940 and 938 MeVc−2 , respectively). The slight difference in mass was
later explained by interpreting the proton as a combination of two up and one down quark, and
the neutron as one up and two down quarks. Then, the up and down quark are combined in a
defining representation of SU (2), which in addition both constitute a representation of SU (3):
the three colors of the quarks.
More generally, one considers matter as representations of a Lie group G, typically a matrix
group such as SU (N ). One might consider N -vectors in the defining representation of SU (N ):
 
ψ1
 .. 
Ψ= . 
ψN
with each ψi ∈ Γ(S) a spinor. The action functional that describes the dynamics of these N
free massless particles is given by
Z
N Z
X
µ S
Sf [ψ1 , . . . , ψN ] =
hΨ, iγ ∇µ Ψix dµ(x) =
hψj , iγ µ ∇Sµ ψj ix dµ(x)
M
j=1
M
Indeed, the extremal points of this action are precisely sections ψi that satisfy the Dirac equation. This action has a global SU (N )-invariance, since U ∈ SU (N ) acting as
Ψ 7→ U Ψ.
leaves Sf invariant. Again, by promoting this to a local SU (N )-symmetry – i.e. U = U (x),
requires replacing
∇S
∇S + A
The gauge field A is an element in Ω1 (M )⊗su(N ) that transforms under a SU (N )-transformation
as
(3.1.4)
U : A 7→ U AU ∗ + U dU ∗
The curvature of A is defined to be
FA = dA + A ∧ A
and is an element in Ω2 (M )⊗su(N ). It transform as F 7→ U F U ∗ under a SU (N )-transformation.
Yang and Mills then introduce an action functional for such a field, now carrying their name.
For A ∈ Ω1 (M ) ⊗ su(N ) the Yang–Mills action functional is given by
Z
Sym [A] = −
Tr FA ∧ ∗FA
M
26
One checks that it is invariant under the action of U (x) ∈ G, as in (3.1.4). Observe, however,
that a mass term of the form
1
− M 2 Aµ Aµ
2
is not gauge invariant, forcing the gauge fields to be massless.
3.1.3. Yang–Mills gauge theory: mathematical setup. As the previous example
should indicate, the proper mathematical setting for gauge theories is vector bundles and connections thereon.
Definition 3.4. Let E → M be a vector bundle. A connection ∇E on E is a map
∇E : ΓE → Ω1 (M ) ⊗C ∞ (M ) ΓE
such that the Leibniz rule is satisfied, i.e.
∇E (f η) = f ∇E (η) + df ⊗C ∞ (M ) η;
(f ∈ C ∞ (M ), η ∈ ΓE).
The curvature F of ∇E is given by
F = (∇E )2 : ΓE → Ω2 (M ) ⊗C ∞ (M ) ΓE
In other words, F ∈ Ω2 (M ) ⊗C ∞ (M ) Γ End(E).
Locally, we can always write ∇E = d + A with A ∈ Ω1 (M ) ⊗C ∞ (M ) Γ End(E). Similarly,
F ∈ Ω2 (M ) ⊗C ∞ (M ) Γ End(E).
Suppose that there is a (smooth) action of a Lie group G on the fibers of E → M . For
instance, if E is an associated vector bundle to a G-principal bundle P → M it naturally comes
with such an action. Indeed, one considers the associated fiber bundle
Ad P = P ×G G
so that the gauge group G := Γ Ad P acts fiberwise on ΓE. The Lie algebra of G is the vector
space of section Γ ad P where
ad P = P ×G g
In this case, it is also natural to assume that ∇E comes from a connection on the principal
bundle G, so that it is given by a g-valued one-form ω on P , satisfying:
ω(X ∗ ) = X;
Rg∗ ω = Adg−1 ω
(X ∈ g).
Of course, we can also write locally ∇E = d + A with A ∈ Ω1 (M ) ⊗ g. There is an action of the
gauge group G on A:
A 7→ uAu∗ + udu∗ ;
(u ∈ G)
identifying G locally with maps from M to G.
Next, we consider the tensor product of the bundle E with the spinor bundle S → M .
Essentially, one takes the fiberwise tensor product, and on S ⊗ E one can define the tensor
product connection:
∇S⊗E = ∇S ⊗ 1 + 1 ⊗ ∇E
In local coordinates, one writes
∇S⊗E = d + ω + A.
Definition 3.5. The Dirac operator with coefficients in E is given (locally) as
DE = iγ µ ∇S⊗E
= iγ µ ∇Sµ + iγ µ Aµ
µ
The corresponding fermionic action is
Sf [A, ψ] = (ψ, (DE − m)ψ)
with ψ a section of S ⊗ E.
The dynamics of the field A is described by the Yang–Mills action functional, now in its full
form.
27
Definition 3.6. The Yang–Mills action functional is defined for a connection ∇E locally
of the form d + A with A ∈ Ω1 (M ) ⊗ g:
Z
Sym [A] = −
Tr F ∧ ∗F.
M
(∇E )2
where F :=
is the curvature of
on the abelian part of g).
∇E
and Tr the Killing form on g (and minus the identity
The Yang–Mills action functional transforms under the action of G as
F 7→ uF u∗
Thus, Sym [A] is invariant under the Γ Ad P -action.
The equations of motion of the above action reads
[∇E , ∗F ] = 0
This is called the Yang–Mills equation. Note its similarity with the Bianchi identity, which
is simply [∇E , ∗F ] = 0 and is always satisfied. This is the starting point of instantons, i.e.
connections with a selfdual curvature F = ∗F . For these connections, the Bianchi identity
implies the Yang–Mills equation so that instantons are minima of the Yang–Mills action. It was
realized later, through the work of Donaldson, that the moduli space of instantons plays a key
role in the classification of smooth structures on four-dimensional manifolds.
3.1.3.1. Higgs mechanism. Although the above is intriguing from a mathematical viewpoint,
nature is (as usual) slightly more complicated. In fact, the U (1) × SU (2)-symmetry discussed
above is not observed in nature, only a residual U (1)-symmery (namely, electrodynamics). Let
us describe the mathematical structure underlying this symmetry breaking.
Suppose that H ⊂ G, and that Φ is a scalar vector, that is, a section of P ×G V . Then, one
considers the action functional:
Z
1
E
g µν ∇E
Sh [Φ, A] =
µ Φ · ∇ν Φ − V (Φ).
2 M
Here V (Φ) is a potential: a polynomial in the components Φi . The minima of this potential
are supposed to be only invariant under a subgroup H, rather than under the full group G.
Physically, this means that when the field Φ ‘rolls down’ to such a minimum, the symmetry
group G is spontaneously broken to H.
Let us illustrate this in an example of great physical interest, namely the Weinberg–Salam
electroweak theory. In this case, G = U (1) × SU (2) which will be broken to H = U (1) as
follows. The field Φ has two components, Φ1 and Φ2 on which (eiθ , U ) ∈ U (1) × SU (2) acts by
matrix multiplication:
Φ1
Φ1
iθ
7→ e U
Φ2
Φ2
The potential in the above action Sh is taken of the form
V (Φ) = µ2 |Φ|2 + λ|Φ|4
where, conventionally, µ2 < 0. This potential has the form of a mexican hat, with minima at
|Φ|2 = −µ2 /2λ =: v. After choosing a basis of V , we can assume that a minimum is of the form
v
Φ0 =
0
Clearly, this is not invariant any more under the U (1) × SU (2) action, however Φ0 is invariant
under the subgroup
−iθ
e
0
iθ
H = (e ,
' U (1)
0
eiθ
This is the celebrated Higgs spontaneous symmetry breaking mechanism, with the Higgs
field h appearing as quantum fluctuations v
v + h. Note that when we put Φ = Φ0 in the
action Sh , one generates in this way terms of the form v 2 g µν Aµ Aν , which are interpreted as
28
mass terms for the gauge fields. More precisely, the physical gauge fields are the photon A, the
Z-boson and the W ± -bosons. They are a linear combination (a rotation) of the gauge fields
(B, W ) ∈ Ω1 (M ) ⊗ u(1) ⊕ su(2) that would arise from the previous discussion:
3
B
A
cos θw sin θw
W
W+
=
;
W =
Z
− sin θw cos θw
W3
W − −W 3
where θw is the so-called Weinberg angle.
3.1.4. The Standard Model of elementary particles. The Standard Model is based
on Yang–Mills gauge theory with group U (1) × SU (2) × SU (3). The word ‘based’ is essential
because, as already mentioned, this full symmetry group is spontaneously broken to U (1) ×
SU (3) via the Higgs mechanism. In any case, there are three corresponding gauge fields Bµ , Wµ
and Vµ describing the photon, the Z and W -bosons, and the gluons. After symmetry breaking,
only the photon and gluons remain massless.
The particle content of the Standard Model is as follows. There are left-handed leptons
νL , eL forming a defining representation of U (1) × SU (2), the right handed νR is in the trivial
representation of the full group, and eR is in the defining U (1)-representation. Similarly, the
quarks uL , dL form a defining representation of SU (2) × SU (3) with some more complicated
(fractional) representations of U (1). Also, the right-handed quarks uR , dR are trivial in the
SU (2)-representation but defining in SU (3).
This adds to a total of 16 degrees of freedom, which together with anti-particles doubles to
32. Finally, there are three generations of particles (muon, tauon and their neutrinos, strange,
charm, top and bottom quark). Thus, we have in total 96 fermionic degrees of freedom. As
before, all these fermionic fields are gathered in a vector Ψ.
The action of the Standard Model is the sum of the Yang–Mills actions SYM for the three
gauge fields, the Higgs action Sh and fermionic action of the form
W
V
Sf [ψi ] = hΨ, iγ µ ∇B
µ + ∇µ + ∇µ Ψi
where ∇B , ∇W and ∇V are the covariant derivatives on the spin fields, with the gauge fields
acting according to the above representations.
3.2. Almost commutative manifolds
Already introduced in Example 2.32 we now focus on almost commutative manifolds:
M × F = (C ∞ (M, AF ), L2 (S ⊗ (M × HF )), ∂/ ⊗1 + γM ⊗ DF ; γM ⊗ γF , JM ⊗ JF ).
3.2.1. Gauge symmetries and inner automorphisms. Let us determine the gauge
group G(M × F ) for M × F , as in Eq. (2.3.4). First, note that AJ = C ∞ (M, (AF )JF ). Then,
since also U(A) = C ∞ (M, U(AF )) we find that
G(M × F ) = C ∞ (M, G(F )) where G(F ) =
U(AF )
.
U((AF )JF )
Thus, the gauge group is completely determined by the Lie group G(F ) which can be determined
in the examples at hand. Similarly,
g(M × F ) = C ∞ (M, g(F )) where g(F ) =
u(AF )
.
u((AF )JF )
It follows easily that g(F ) = Lie G(F ).
Example 3.7. Consider once more the finite spectral triple
FYM := (Mn (C), Mn (C), D = 0; J = (·)∗ , γ = 1).
The algebra (AF )JF consists of matrices for which left and right multiplication on any other
matrix coincide. This happens only for multiples of the identity, so (AF )JF = C1n . Also,
U(AF ) = U (n) so that G(FYM ) = P SU (n) and, correspondingly, g(FYM ) = su(n).
29
3.2.2. Gauge fields and the spectral action. Let us apply the discussion in Section
2.3.3 on Morita self-equivalences to the almost commutative manifold M × F and see what the
corresponding gauge fields look like.
Proposition 3.8. Let A ∈ ΩA
1 (A). Then
D + A + JAJ −1 = ∂/ +iγ µ Aµ + γ 5 Φ
with Aµ (x) ∈ g(F ) and Φ(x) ∈ End(HF ).
Corollary 3.9 (Weitzenböck). For an AC manifold M × F we have DA 2 = ∇∗ ∇ − E for
a G(F )-connection ∇ on S ⊗ (M × HF ), and
1
1
E = R ⊗ 1 − 1 ⊗ Φ2 + γ µ γ ν ⊗ Fµν − iγ µ γ5 ⊗ Dµ Φ,
4
2
where Dµ = [∇µ , Φ] = ∂µ Φ + [Aµ , Φ] and Fµν = [∇µ , ∇ν ].
Consequently, we can apply Gilkey’s Theorem 2.42 to arrive after a long calculation at
Proposition 3.10. The spectral action on an AC manifold is
Z
DA
∼
LEH (gµν ) + LYM (Aµ ) + LH (gµν , Aµ , Φ) + Ø(Λ−1 )
Tr f
Λ
M
where
N f4 Λ 4 N f2 Λ 2
N f (0)
+
R−
Cµνρσ C µνρσ
2
2
2π
24π
320π 2
f (0)
:= −
Tr(Fµν Fµν )
24π 2
2f2 Λ2
f (0)
1
2
4
2
2
:= −
Tr(Φ ) +
Tr(Φ ) − R Tr(Φ ) + Tr (Dµ Φ)
4π 2
8π 2
6
LEH :=
LYM
LH
and N is the dimension of the finite Hilbert space HF .
This means that besides the Lagrangian LEH which contains the Einstein–Hilbert Lagrangian (given by R), we find a Yang–Mills action for the G(F )-gauge field A and a Higgs-like
action for the endomorphisms Φ.
3.2.3. The Standard Model of elementary particles. The Standard Model is the
almost commutative manifold based on the real algebra
AF = C ⊕ H ⊕ M3 (C).
We use a Krajewski diagram (Figure 1) to summarize the additional structure of the finite
spectral triple (AF , HF , DF ; γF , JF ). Thus, the spectral triple M × F is now given by
(C ∞ (M ) ⊗ (C ⊕ H ⊕ M3 (C), L2 (M, S) ⊗ C96 , ∂/ ⊗1 + γ5 ⊗ DF ).
Here 96 is 2 (particles and anti-particles) times 3 (families) times 4 leptons times 4 quarks with
3 colors each. We write the representation of A in terms of the suggestive basis of C96 :
( νL
eL νR eR uL dL uR dR ν L eL ν R eR uL dL uR dR )
t
.
The relation to the nodes in the Krajewski diagram is that first eight basis vectors correspond
(from left to right) to the horizontally aligned nodes: top row corresponds to leptons, bottom
row to quarks. Similarly for the anti-particles, obtained via reflection in the diagonal.
Explicitly, the representation on HF for an element (λ, q, m) ∈ C ⊕ H ⊕ M3 (C) is given by
q h i

λ


π(λ, q, m) = 

λ
q⊗13 h
λ
i
λ
⊗13
λ14
30




14 ⊗m
2L
1
1◦








2L ◦
1R ◦
1R
3
1R






1R

3
◦
◦
Figure 1. The Krajewski diagram of the Standard Model
Here, the quaternion
q is considered as a 2×2-matrix. The 96×96-matrix DF is of the following
∗
form: DF = TS TS where
h

 Υ∗v


S=


Υv
i

Υe
Υ∗e
h
Υ∗u ⊗13
Υu ⊗13



i;

Υd ⊗13 
 0
T =
0

ΥR
0

04
Υ∗d ⊗13
in terms of the 3 × 3 Yukawa-mixing-matrices Υν , Υe , Υu , Υd and a real constant ΥR responsible
for neutrino mass terms (it corresponds to the diagonal dotted line in Figure 1).
The grading γF is +1 on all L-particles, and −1 on all R-particles; the total grading is
then γ5 ⊗ γF . The anti-linear operator
J is a combination of charge conjugation on S and the
(anti-linear) matrix JF = 148 148 , reflecting with respect to the diagonal in Figure 1.
The rest then follows from a long calculation; the inner fluctuations are DA = ∂/ +iγµ Aµ +
γ5 Φ with
g1
Aµ =
2
Bµ −
0
0
g2
Wµ
2
0 0
0 0
0 g1 B µ
Φ = DF +
−
⊕
g
g2
Wµ ⊗13 − 61 Bµ ⊗3
2
0
0
Υν φ1 Υν φ2
−Υe φ2 Υe φ1
(h.c.)
31
0
0
2g
− 31 Bµ ⊗13
0
!
⊕
− 14 ⊗
0
g1
Bµ ⊗13
3
Υu φ1 Υu φ2
−Υd φ2 Υd φ1
(h.c.)
!
!
g3
Vµ
2
Here Bµ , Wµ , Vµ are U (1), SU (2) and SU (3)-gauge fields, resp. and (φ1 φ2 )t two scalar (Higgs)
fields. The spectral action is modulo gravitational terms:
Z
Z
Z
−2af2 Λ2 + ef0
f0
f0
2
2
SΛ =
|φ|
+
a|D
φ|
−
aR|φ|2
µ
π2
2π 2
12π 2
Z Z
f0
5 2
f0
2 i µi
2 a µνa
µ
+ 2 b|φ|4 + Ø(Λ−2 )
g3 G µ G + g2 Fµ F
+ g1 Bµ B
− 2
2π
3
2π
with a, b, c, d, e constants depending on the Yukawa parameters. For example,
a = Tr (Υ∗ν Υν + Υ∗e Υe + 3 (Υ∗u Υu + Υ∗d Υd ))
b = Tr (Υ∗ν Υν )2 + (Υ∗e Υe )2 + 3 (Υ∗u Υu )2 + (Υ∗d Υd )2
When we add the fermionic term hJψ, DA ψi to SΛ , we obtain the Standard Model Lagrangian, including the Higgs boson and neutrino mass terms, provided we have
1
g32 f0
5
=
g32 = g22 = g12 .
2
2π
4
3
These GUT-type relations between the coupling
constants
allows for predictions. For example,
p
one identifies the mass of the W as 2MW = a/2 so that the Higgs vacuum reads 2M/g2 . The
above relation for a then gives a postdiction for the mass of the top quark as mt ≤ 180 GeV.
Moreover, the mass of the Higgs is mH = 8λM 2 /g22 with λ = g32 b/a2 resulting in a prediction
of mH ∼ 168 GeV. More details on this computation can be found in [5].
32
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Theor. Math. Phys. 11 (2007) 991–1089.
[2] A. Connes. Noncommutative Geometry. Academic Press, San Diego, 1994.
[3] A. Connes. Gravity coupled with matter and the foundation of non-commutative geometry. Commun. Math.
Phys. 182 (1996) 155–176.
[4] A. Connes. On the spectral characterization of manifolds. arXiv:0810.2088.
[5] A. Connes and M. Marcolli. Noncommutative Geometry, Quantum Fields and Motives. AMS, Providence,
2008.
[6] P. B. Gilkey. Invariance theory, the heat equation, and the Atiyah-Singer index theorem, volume 11 of Mathematics Lecture Series. Publish or Perish Inc., Wilmington, DE, 1984.
[7] T. Krajewski. Classification of finite spectral triples. J.Geom.Phys. 28 (1998) 1–30.
33