Spectral triples and differential calculi related to the Kronecker foliation

27 Pages·0.27 MB·English

by  R. Matthes

Spectral triples and differential calculi
related to the Kronecker foliation
R. Matthes
Fachbereich Physik, Technische Universit¨at Clausthal,
2
0 Leibnizstraße 10, 38678 Clausthal-Zellerfeld
0
2
n O. Richter and G. Rudolph
a
J Institut fu¨r Theoretische Physik, Universit¨at Leipzig,

1 Augustusplatz 10/11, 04109 Leipzig, Germany
3

1
v
6 Abstract
6
0 Following ideasofConnesandMoscovici, wedescribetwospectraltriplesrelated
1
totheKronecker foliation, whosegeneralized Diracoperatorsarerelated tofirstand
0
second order signature operators. We also consider the corresponding differential
2
0 calculiΩ ,whicharedrasticallydifferentinthetwocases. Asaside-remark,wegive
D
/
h a description of a known calculus on the two-dimensional noncommutative torus in
p terms of generators and relations.
-
h
t 1991 MSC: 16W25; 81V15
a
m PACS numbers: 02.40.-k
:
v
i
X 1 Introduction
r
a
In Connes’approach to noncommutative differential geometry, the notion of a spec-
tral triple plays an essential role, see [1]. It encodes the differential and Riemannian
structure of the noncommutative space as well as its dimension. From the physical
point of view, spectral triples have been used to construct unified field theoretical
models, in particular the standard model (see [1], [2]), and also models including
gravitation ([3], [4], [5]). From the mathematical point of view, only a few types of
noncommutative spaces have been used in these examples: commutative algebras of
smoothfunctionsonamanifold[1],finitedimensionalalgebras(foraclassification of
spectral triples in this case see [6] and [7]) and products of both. In [8] it was shown
that it is not straightforward to define spectral triples related to covariant differen-
tial calculi on quantum groups. Explicit examples of spectral triples have also been
described for the irrational rotation algebra and higher dimensional noncommuta-
tive tori [1], [9]. For these examples, thedata of thetriple were chosen according to
physical needs or taking advantage of special structures available in the underlying
algebra. An important part of the information needed for physical purposes is the
explicit form of the differential calculus of a spectral triple. Such calculi have been
analyzed in the above-mentioned cases ([1], [2], [10], [11]). In [12] it has been shown
1
thattheextrastructureofafinitelygenerated projectivemoduleallows tointroduce
the graded algebra of differential-form-valued endomorphisms which gives a natural
mathematical language to build unified field theoretical models in the spirit of the
Mainz-Marseille approach [13]. In [14] and [15], the notion of spectral triple itself
has been modified and enriched using ideas from supersymmetric quantum theory.
One arrives at noncommutative structures generalizing classical geometrical struc-
tures (Riemannian, symplectic, Hermitian, Ka¨hler, ...structures). Physical hopes
are mainly directed to superconformal field theories (with noncommutative target
spaces).
Recently, see [16], Connes and Moscovici have described a method which makes
it possible to construct spectral triples in a systematic way for crossed product al-
gebras related to foliations. Let (M, ) be a regular foliation of a smooth manifold
F
M with Euclidean structures on both the corresponding distribution and the nor-
mal bundle. There is an associated spectral triple for the crossed product algebra
C (M) ⋊ Γ, where Γ is a group of diffeomorphisms preserving these structures.
∞
The corresponding Dirac operator is a hypoelliptic operator which is closely related
to the signature operator of the foliated manifold. This signature operator is a
modification of the standard signature operator in differential geometry, see [17].
In this paper, we construct explicitly two spectral triples related to the Kro-
necker foliation. We choose as diffeomorphism group the group which defines the
R
foliation by its action on 2 and obviously preserves natural translation invariant
Euclidean structures. ThuTs we arrive at the algebra C ( 2)⋊ , whose C version
∞ ∗
T R
is known to be Morita equivalent to the irrational rotation algebra (noncommu-
tative torus), see [18], [9]. The Dirac operator of the first spectral triple (which
has dimension 2) is closely related to the ordinary signature operator on 2. For
T
the construction of the second triple (of dimension three) we follow the strategy
proposed in [16]. The corresponding signature operators and henceforth also the
Dirac operators can be diagonalized explicitly in both cases. Then we pass to the
differential calculi associated to the spectral triples constructed before. It turns
out that for the triple related to the first order signature operator the differential
calculus can be completely determined. Restricted to C ( 2) it projects down to
∞
T
the de Rham calculus on 2. The analysis of the differential calculus for the second
T
triple turns out to be much more involved. We show that for the restriction of this
triple to the subalgebra C ( 2) (i.e. choosing the trivial diffeomorphism group)
∞
T
the corresponding one forms give just the universal calculus on C ( 2).
∞
T
Intheappendixwehaveaddedtheexplicitdescriptionofthedifferentialcalculus
for the spectral triple related to the irrational rotation algebra, see [1, 9], which has
properties similar to the calculus associated to the linear signature operator.
2 The spectral triple related to a foliation
For the convenience of the reader, we recall here the definition of a spectral triple
and the differential calculus related to such a triple ([1],[9]):
Definition 1 A spectral triple (A, ,D) consists of a -algebra A, a Hilbert space
H ∗
and an unbounded operator D on , such that
H H
(i) A acts by a -representation π in the algebra B( ) of bounded operators on
∗ H
,
H
(ii) the commutators [D,π(a)], a A, are bounded and
∈
2
(iii) the operator D has discrete spectrum with finite multiplicity.
A spectral triple is said to have dimension n, if the eigenvalues (with multiplicity)
µ of D fulfil lim µk = C = 0.
k | | k→∞ k1/n 6
We will have no need to refer to gradings or real structures usually included in the
definition of a spectral triple, and also not to more general notions of dimension.
The representation π of A in B( ) can be extended to a representation π :
∗
H
Ω(A) B( ) of the universal differential calculus Ω(A) by
−→ H
πn( akdak dak)= π(ak)[D,π(a1)] [D,π(ak)].
0 1··· n 0 k ··· n
k k
X X
If J := kerπn, then J := J +dJ is a differential ideal, and one arrives at the
0 n 0 0
⊕
differential calculus Ω (A),
D
Ωn(A) := Ωn(A)/J.
D
Note that, if π is faithful, there are isomorphisms
Ω1 (A) π1(Ω1(A)) (2.1)
D ≃
and
Ω2 (A) π2(Ω2(A))/π2(dJ1). (2.2)
D ≃ 0
Now wereviewshortly theproceduregiven in [16], whichrelates aspectraltriple
to a regular foliation of a smooth manifold. Let M be a compact manifold with
a foliation given by an integrable distribution V TM. The normal bundle of
⊂
the foliation is N := TM/V, with canonical projection ρ : TM N. Assume
→
further that both V and N are equipped with Euclidean fibre metrics and with an
orientation (i.e. there are distinguished nowhere vanishing sections ω , ω of the
V N
exterior bundles vV, nN (v = dimV, n = dimN)). Furthermore, ω and ω
V N
also define a nonvanishing section of vV nN v+nT M, i.e. a volume
∗ ∗ ∗
V V ⊗ ≃
form on M. The bundle of interest for us is
V V V
E = V N .
∗ ∗
⊗
C C
^ ^
Obviously, the metrics on V and N give rise to Hermitian metrics on V and
∗
N and thus also on E. Theorientations ω and ω can bemapped by meCans of
∗ V N
V
the mCetrics to sections γ of vV and γ of nN which can be used, together
V ∗ N ∗
V
with the metrics, to define an anaClogue of the HodgCe star on the exterior bundles
V V
V and N . We choose a variant of the -operation such that 2 = 1 and
∗ ∗ ∗ ∗V
2 C= 1, i.e. C and can be considered as -grading operators (cCf. [19]).
V∗TNhCus, the Vspa∗cVeCof sec∗tNioCns of E has a naturalZin2ner product, and we denote by
= L2(M,E) the Hilbert space of square integrable sections of this bundle. From
H
now on, we always consider complexified vector bundles, but omit the subscript .
C
In order to construct a generalized Dirac operator, a longitudinal differential d
L
and a transversal differential operator d have to be defined. The differential d
H L
is defined canonically by means of the Bott connection ([20]) given as the partial
covariant derivative :Γ(V) Γ(N) Γ(N) defined by
∇ × −→
Y = ρ [X,Y˜] ,
X
∇
(cid:16) (cid:17)
for X Γ(V),Y Γ(N) and Y˜ Γ(TM) such that ρ(Y˜) = Y. By a standard
∈ ∈ ∈
procedure (using the Leibniz rule and duality) is extended to a differential d :
L
∇
3
Γ(E) Γ(E) defined by linear mappings Γ( kV lN ) Γ( k+1V
∗ ∗ ∗
lN −),→ ⊗ −→ ⊗
∗
V V V
V d α(X ,...,X ) = ( 1)i (α(X ,...,Xˆ ,...,X ))
L 0 k − ∇Xi 0 i k
i=0,...,k
X
+ ( 1)i+jα([X ,X ],X ,...,Xˆ ,...,Xˆ ,...,X ),
i j 0 i j k
−
i<j
X
X Γ(V). Since the Bott connection is flat, we have d2 = 0.
i ∈ L
In order to define a transversal differential operator one has to choose a sub-
bundle H TM complementary to V. This defines a bundle isomorphism j :
H
⊂
V N T M in the following way: Let us denote by pr and pr the
∗ ⊗ ∗ −→ ∗ ∗V ∗H
projections corresponding to the decomposition TM = V H , by ρ : H N
∗ ∗ ∗ H
V V V ⊕ →
the restriction of ρ to H and by ρ its transposed map. Then j is defined as the
∗H H
following composition:
id ρ∗ pr∗ pr∗
V∗ N∗ ⊗∧ H V∗ H∗ ∧ V⊗∧ H T∗M T∗M ⊗→∧T∗M ,
⊗ −→ ⊗ −→ ⊗ −→
wh^ere ^ denotes t^he replac^ement of the ten^sor produc^t by the wedge prod-
⊗ → ∧
uct. Now, the transversal operator d is obtained from the exterior differential
H
d by transporting with j and projecting to a certain homogeneous component:
H
V N has an obvious bigrading, and denoting by π(r,s) the projector to the
∗ ∗
⊗
homogeneous component of bidegree (r,s), one defines
V V
d α = π(r,s+1)(j 1 d j (α))
H H− ◦ ◦ H
for α Γ( rV sN ). Theoperator d is a graded derivation of the -graded
∗ ∗ H 2
∈ ⊗ Z
algebra Γ( V N ).
∗ ∗
V ⊗ V
In a foliation chart, d and d look as follows. Let (xi,yk), i = 1,...,v, k =
L H
V V
1,...,n be local coordinates of M such that xi are coordinates on the leaf (foliation
chart). The corresponding coordinate vector fields ( ∂ , ∂ ) form a local frame of
∂xi ∂yk
TM and ( ∂ ) a frame of V. The corresponding dual frame of T M consists of the
∂xi ∗
differentials (dxi,dyk). We define θi Γ(V ) by θi( ∂ ) = δi (i,j = 1,...,v). It is
∈ ∗ ∂xj j
immediate from the definition of N that the elements n := ∂ +V (k = 1,...,n)
k ∂yk
form a local frame of N. The elements of the corresponding dual frame of N are
∗
denoted by nk. Finally, we choose a local frame h of the transversal space H. This
k
frame is fixed by assuming ρ (h ) = n . This leads to
H k k
∂ ∂
h = hi + ,
k k∂xi ∂yk
with coefficient functions hi characterizing H. Then, the elements θi1 θir
k ∧···∧ ⊗
nj1 njs form a local frame of E, and one can show that d and d are given
L H
∧···∧
by the following local formulae:
∂α
d (α θi1 θir nj1 njs) = i1...irj1...jsθi θi1 θir nj1 njs
L i1...irj1...js ∧···∧ ⊗ ∧···∧ ∂xi ∧ ∧··· ⊗ ∧···∧
d (α θi1 θir nj1 njs) =
H i1...irj1...js ∧···∧ ⊗ ∧···∧
∂ ∂
( 1)r +hi α θi1 θir nk nj1 njs +
− ∂yk k∂xi i1...irj1...js ∧···∧ ⊗ ∧ ∧···∧
(cid:18) (cid:19)
r ∂hit
α k θi1 θl θir nk nj1 njs , (2.3)
i1...irj1...js ∂xl ∧···∧ ∧···∧ ⊗ ∧ ∧···∧
t=1
X
4
(where θl at position t replaces θit). Thelongitudinal differential d acts as a differ-
L
ential in leaf direction, whereas d is a sum of a principal part, which differentiates
H
in transversal direction, and a zero order part. As examples, let us give formulae
for d acting on functions, (1,0)-, (0,1)- and (1,1)-forms:
H
d f = h (f)nk ,
H k
∂hi
d (α θi) = h (α )θi nk α kθj nk ,
H i − k i ⊗ − i∂xj ⊗
d (α nk) = h (α )nl nk ,
H k l k
∧
∂hi
d (α θi nk) = h (α )θi nl nk α lθj nl nk ,
H ik ∧ − l ik ⊗ ∧ − ik∂xj ⊗ ∧
(d (nk) = 0). For the adjoint operators d and d (in ) it is difficult to write
H ∗L ∗H H
down explicit formulae. One can show
d α = d + term of order zero,
∗L −∗V L∗V
where is the (partial) Hodge operator related to the Euclidean metric and the
V
orienta∗tion of V. Since d lowers the N -degree one has for α Γ( rV )
Γ( rV 0N ) ∗H ∗ ∈ ∗ ≡
∗ ∗
⊗ V
d α = 0 .
V V ∗H
Explicit formulae for d become rather complicated as, e.g., the case of (0,1)-forms
∗H
shows:
∂hi 1
d (α θi)= gkl(h (α ) α Γ m+α ( k + gijh (g ))) , (2.4)
∗H i − N k l − m Nkl l ∂xi 2 V k Vij
where gkl = g (nk,nl), g = g ( ∂ , ∂ ), gij = g (θi,θj) are the local compo-
N N Vij V ∂xi ∂xj V V
nents of the fibre metrics (and their duals), and Γ m are the Christoffel symbols
Nkl
corresponding to g .
Nkl
In [16], using d and d , two differential operators were introduced by
H L
Q = d d d d
L L ∗L− ∗L L
Q = d +d ,
H H ∗H
and the mixed signature operator Q for M, acting on a form with N-degree ∂ ,
N
was defined by
Q = Q ( 1)∂N +Q . (2.5)
L H
−
As noted in [16], Q is selfadjoint. Finally, a generalized Dirac operator D is defined
as the unique selfadjoint operator such that
D D =Q . (2.6)
| |
If zero is not an element of the spectrum of Q, it is given as
D = Q Q 1/2 = Q(Q2) 1/4, (2.7)
− −
| |
as shows a straightforward argument using the spectral decomposition of Q.
Onemotivationforchoosingasecondorderlongitudinalpartisthefollowing: the
index of the signature operator should not depend on the choice of the transversal
subbundle H. Usually, the index of a pseudodifferential operator only depends on
its principal symbol. However, as follows from the local formulae (2.3) and (2.4),
5
its principal part explicitly depends on H, the dependence being in the coefficients
of the partial derivatives with respect to leaf coordinates. It turns out that one can
get rid of this dependence by introducing a modified notion of pseudodifferential
operators (ψDO ) which assigns a degree 2 to transversal coordinates and a degree
′
1 to longitudinal ones. To have a contribution also from Q , one has to pass to a
L
second order operator. In [16] a homotopy argument was given to show that this
does not affect the longitudinal signature class.
Let Γ be any group of diffeomorphisms of M which preserves the distribution
V and the Euclidean metrics on both V and N. Then ψ Γ acts via the pull
∈
back as unitary operator U on , whereas functions from C (M) act there as
ψ∗ H ∞
multiplication operators. The crossed product algebra := C (M) ⋊ Γ can be
∞
A
defined as the -subalgebra of B( ) generated by these two types of operators.
∗ H
Due to U f = (f ψ)U every element of is a finite sum of elements fU . Then
ψ∗ ◦ ψ∗ A ψ∗
we have, see [16]:
Theorem 1 ( , ,D) is a spectral triple of dimension v+2n.
A H
Ourmain aim is to describeexplicitly this spectral triple for the Kronecker foliation
of the two-torus.
3 Spectral triples for the Kronecker foliation
3.1 The crossed product algebra for the Kronecker fo-
liation
Let us start with some conventions and notations. We consider the two-torus as the
quotient 2 = 2/2π 2. Thus, we have natural local coordinates 0 < ϑ ,ϑ < 2π.
1 2
T R Z
Consider the -manifold ( 2, ,ψ), with group action
R T R
ψ : 2 2 ,
T ×R→ T
given by
ψ((ϑ ,ϑ ),t) = (ϑ +at,ϑ +bt) ,
1 2 1 2
with a,b such that a> 0, a2+b2 = 1 and θ = b being irrational. The foliation
∈ R a
of 2 by the orbits of ψ is called the Kronecker foliation. It is well known, see [21],
T
that each leaf of this foliation is diffeomorphic to and lies dense in 2.
R T
The coordinate transformation
x = aϑ +bϑ
1 2
y = bϑ aϑ
1 2
−
is orthogonal and leads to coordinates (x,y) of a foliation chart. In these coordi-
nates, acts as follows
R
ψ((x,y),t) = (x+t,y) .
Tobemoreprecise,thisistheliftedactionof on 2,appliedtoglobalcoordinates
R R
(x,y) obtained from global coordinates (ϑ ,ϑ ) by the orthogonal transformation.
1 2
It is well known, see [22], that associated to the action of a locally compact
group K on a manifold M there is a transformation groupoid G. For the Kronecker
foliation, we have G = 2 with range and source maps r and s given by
T ×R
r(p,t) = ψ(p,t)
s(p,t) = p ,
6
p 2, the space of units being 2. The associated crossed product algebra
∈ T T
= ( 2)⋊
O O T R
is the -algebra generated by the unitary operators U , U and V acting in the
1 2 t
∗
Hilbert space L2( 2) given by
T
(U ξ)(ϑ ,ϑ ) = eiϑ1 ξ(ϑ ,ϑ )
1 1 2 1 2
·
(U ξ)(ϑ ,ϑ ) = eiϑ2 ξ(ϑ ,ϑ )
2 1 2 1 2
·
(V ξ)(ϑ ,ϑ ) = ξ(ϑ +at,ϑ +bt) , (3.1)
t 1 2 1 2
ξ L2(T2). Let e = ei(kϑ1+lϑ2) (k,l ) be the basis of trigonometric polyno-
kl
∀ ∈ ∈ Z
mials of L2( 2). Obviously from (3.1) it follows that
T
U e = e
1 kl k+1,l
U e = e
2 kl k,l+1
V e = ei(ak+bl)te . (3.2)
t kl kl
It is now immediate to show
Proposition 1 The unitary operators U , U , V satisfy
1 2 t
U U = U U , (3.3)
1 2 2 1
V U = eiatU V , (3.4)
t 1 1 t
V U = eibtU V , (3.5)
t 2 2 t
V V = V , t,s . (3.6)
t s t+s
∈ R
Remark 1 For rational a = m, m,n relative prime, there is an additional relation
b n
V = V = 1 ;
2π√m2+n2 0
2π√m2+n2 is the smallest value of t such that V = V = 1 and any other such t
t 0
is an integer multiple of it.
Proposition 2 The -algebra ( 2)⋊ is isomorphic to u ,u ,v /J, where
1 2 t
∗ O T R Ch i
u ,u ,v is the free associative unital -algebra generated by u ,u and v , t
1 2 t 1 2 t
Ch i ∗ ∈
, and J is the -ideal generated by (3.3)–(3.6) and unitarity conditions for the
R ∗
generators.
Proof: By universality of u ,u ,v /J, there exists a homomorphism π of this
1 2 t
algebraonto ( 2)⋊ senCdihngthecorirespondinggenerators intoeach other. Now,
O T R
by (3.3)–(3.6), a general element a of u ,u ,v /J is a linear combination of the
1 2 t
Ch i
monomials vtjuk1ul2, for some tj ∈ R (with j 6= j′ ⇒ tj 6= tj′), k,l ∈ Z, a =
a v ukul (finitesum). WefirstshowthatallV areindependent. V = b V
jkl tj 1 2 t t j j tj
iPs equivalent to 1 = bjei(ka+bl)(tj−t), ∀k,l. Since ab is irrational, {ak+bl|kP,l ∈ Z}
isdenseinR,andonPecanconclude1= bjeix(tj−t) ∀x∈ R. But1andeix(tj−t) are
orthogonalasalmost-periodicfunctions,see[23],andthereforelinearlyindependent.
P
VTth′Uen1k′aUl2sl′o(V(tkU,1lk)U6=2l a(nkd′,Vl′t)′)Us1khUif2lt(at 6=difft′e)raernetlninuemarbleyrinodfespteepnsdeinnt.thSeinacbeoVvteUb1kaUs2ilsaenkdl
they are obviously independent. In other words, the monomials V UkUl constitute
t 1 2
7
a basis in ( 2)⋊ . If π(a) = a V UkUl = 0 we have a = 0 and π is a
bijection. O T R jkl tj 1 2 jkl (cid:3)
P
Inanalogy withthe definitiongiven beforeTheorem1, puttingM = 2 andΓ = ,
T R
we define the crossed product
:= C ( 2)⋊ (3.7)
∞
A T R
as a -subalgebra of B(L2( 2)).
∗ T
Remark 2 We can introduce a set of seminorms on = ( 2)⋊ as follows.
O O T R
Let ( 2) be the -subalgebra generated by U and U . We define a family
1 2
O T ⊂ O ∗
(p ) of seminorms on this subalgebra by
n n
∈N
p ( a UjUk) = sup(1+ j + k )n a .
n jk 1 2 | | | | | jk|
j,k
Xjk ∈Z
It is well-known ([24, 22.19.2.-4.]) that the completion of ( 2) in the correspond-
O T
ing Fr´echet topology is C ( 2). Now we define seminorms on by
∞
T O
p⋊( a V UkUl) = p ( a UkUl).
n jkl tj 1 2 n jkl 1 2
jkl jkl
X X
Then it is easy to show that is the completion with respect to the Fr´echet
A ⋊ O
topology defined by the family of seminorms p . To this end, one first notes that
n
every element of is a finite sum of products fV , f C ( 2). By the above, f
t ∞
is a limit of elemeAnts f of ( 2) with respect to p , a∈nd by tThe definition of p⋊ it
k O T n n
is obvious that fV is the limit of f V .
t k t
Let us now describe the Hilbert space of the spectral triple of Theorem 1 for the
Kronecker foliation.
Here, both V and N are one-dimensional, with local frames consisting each of
one vector ∂ and n = ∂ +V respectively. Let τ and ν denote the corresponding
∂x ∂y
elementsofthedualframes. ThenE = V N consistsoffourone-dimensional
∗ ∗
⊗
subspaces of elements of degrees (0,0), (1,0), (0,1), (1,1), with local frames 1, τ,
V V
ν, τ ν, respectively. Thenatural choice of translation invariant (under the natural
⊗
action of 2 on 2) Euclidean fibre metrics makes these frame elements mutually
R T
orthogonal unit vectors in L2( 2,E). We may identify
T
L2( 2,E) = L2(T2) L2(T2) L2(T2) L2(T2) ,
T ⊕ ⊕ ⊕
with e 1 (e ,0,0,0),...,e τ ν (0,0,0,e ).
kl kl kl kl
→ ⊗ →
Remark 3 Sincethegeneratorsact, according to(3.1), componentwise inL2( 2,E)
T
the crossed product algebra of Theorem 1 coincides with (3.7).
We choose the transversal subspace H in the simplest way, i.e. we put hi = 0,
k
i.e. H is generated by the coordinate vector field ∂ . Then the general formulae
∂y
of the foregoing section lead (with some easy computations for the adjoints) to the
8
following expressions:
f fτ fν fτ ν
⊗
∂f ∂f
d 0 τ ν 0
L
∂x ∂x ⊗
∂f ∂f
d ν τ ν 0 0 ,
H
∂y −∂y ⊗
∂f ∂f
d 0 0 ν
∗L −∂x −∂x
∂f ∂f
d 0 0 τ
∗H −∂y ∂y
with f C ( 2). To prove, e.g.,
∞
∈ T
∂f
d (fτ ν) = τ,
∗H ⊗ ∂y
we denote by ( ) the scalar product in L2( 2,E) and observe that
·|· T
∂g
(gτ d (fτ ν)) (d (gτ)fτ ν) = ( τ ν fτ ν)
| ∗H ⊗ ≡ H | ⊗ −∂y ⊗ | ⊗
∂g(x,y) ∂f(x,y)
= f(x,y)dxdy = g(x,y) dxdy
− ∂y ∂y
Z Z
∂f
= (gτ τ) .
|∂y
Note that all the above operators can also be written as matrix differential
operators.
3.2 The first order signature operator as Dirac operator
We will first show that (C ( 2)⋊ ,L2( 2,E),Q˜), with Q˜ being the linear signa-
∞
T R T
ture operator
Q˜ = d +d +d +d ,
L ∗L H ∗H
is a spectral triple of dimension 2. Using the foliation chart (x,y) and the local
frame 1,τ,ν,τ ν , this operator can be written as
{ ⊗ }
0 ∂ ∂ 0
−∂x −∂y
∂ 0 0 ∂
Q˜ =  ∂x ∂y .
∂ 0 0 ∂
 ∂y −∂x 
 0 ∂ ∂ 0 
 −∂y ∂x 
 
The eigenvalue problem for Q˜ is most easily solved by considering its square
∂2 ∂2 0 0 0
−∂x2 − ∂y2
Q˜2 =  0 −∂∂x22 − ∂∂y22 0 0 .
0 0 ∂2 ∂2 0
 −∂x2 − ∂y2 
 0 0 0 ∂2 ∂2 
 −∂x2 − ∂y2 
 
9
In coordinates (ϑ ,ϑ ) one is immediately led to the eigenvalue equations
1 2
2 2
∂ ∂ ∂ ∂
a +b b a f =λ2f ,
j j
− ∂ϑ ∂ϑ − ∂ϑ − ∂ϑ
(cid:18) 1 2(cid:19) (cid:18) 1 2(cid:19) !
for the four components of an eigenvector f = f1+f2τ +f3ν+f4τ ν of Q˜2. It is
⊗
now straightforward to see that
e 0 0 0
kl
0 e 0 0
e1 =  ,e2 =  kl ,e3 =  ,e4 =  
kl 0 kl 0 kl e kl 0
kl
 0   0   0   e 
       kl 
       
are eigenvectors of Q˜2 to the eigenvalue
λ2 = (ak+bl)2+(al bk)2.
kl −
The operator Q˜ itself acts in this basis of L2( 2,E) as follows:
T
Q˜(e1 ) = (ak+bl)e2 +(al bk)e3
kl kl − kl
Q˜(e2 ) = ( ak bl)e1 +( al+bk)e4
kl − − kl − kl
Q˜(e3 ) = ( al+bk)e1 +(ak+bl)e4
kl − kl kl
Q˜(e4 ) = (al bk)e2 (ak+bl)e3 .
kl − kl− kl
Thus, we have
λ = (ak+bl)2+(al bk)2 (3.8)
±kl ± −
and p
bk al ak+bl
e+1 = i − e1 +e3 +i e4 ,
kl − λ+ kl kl λ+ kl
kl kl
ak+bl al bk
e+2 = i e1 +e2 +i − e4 ,
kl − λ+ kl kl λ+ kl
kl kl
ak+bl al bk
e−kl1 = e1kl−i λ+ e2kl+i λ−+ e3kl,
kl kl
al bk ak+bl
e−kl2 = i λ−+ e2kl+i λ+ e3kl+e4kl
kl kl
form a complete set of eigenvectors of Q˜.
Proposition 3 (C ( 2)⋊ ,L2( 2,E),Q˜) is a spectral triple of dimension 2.
∞
T R T
Proof: Theeigenvalues (3.8) ofQ˜ have finitemultiplicity, tendtoinfinityfork,l
→
andhavenofiniteaccumulationpoint. Thus,theresolventofQ˜ iscompact. Since
∞
[Q˜,V ]= 0, (3.9)
t
boundednessof the commutators of Q˜ with elements of the algebra follows from the
fact that
0 ∂f ∂f 0
−∂x −∂y
∂f 0 0 ∂f
[Q˜,fV ]= f[Q˜,V ]+[Q˜,f]V =  ∂x ∂y V ,
t t t ∂f 0 0 ∂f t
 ∂y −∂x 
 0 ∂f ∂f 0 
 −∂y ∂x 
 
10