On Spectral Triples in Quantum Gravity II
aa r X i v : . [ h e p - t h ] F e b On Spectral Triples in Quantum Gravity II
Johannes
Aastrup a , Jesper Møller Grimstrup b & Ryszard Nest c a SFB 478 “Geometrische Strukturen in der Mathematik”Hittorfstr. 27, 48149 M¨unster, Germany b The Niels Bohr InstituteBlegdamsvej 17, DK-2100 Copenhagen, Denmark c Matematisk InstitutUniversitetsparken 5, DK-2100 Copenhagen, Denmark
Abstract
A semifinite spectral triple for an algebra canonically associated tocanonical quantum gravity is constructed. The algebra is generated bybased loops in a triangulation and its barycentric subdivisions. Theunderlying space can be seen as a gauge fixing of the unconstrainedstate space of Loop Quantum Gravity. This paper is the second oftwo papers on the subject. email: [email protected] email: [email protected] email: [email protected] ontents D . . . . . . . . . . . . . . . . . . . . . . . . . . 324.5 The main theorem . . . . . . . . . . . . . . . . . . . . . . . . 33 U (1)-case . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.2 The SU (2)-case . . . . . . . . . . . . . . . . . . . . . . . . . . 39 One of the most striking application of Noncommutative Geometry [9] isConnes’ derivation of the Standard Model of high energy physics [10, 8]. Inthis derivation the Lagrangian of the full Standard Model coupled to gravityemerges from the spectral action principle [7] applied to a specific almostcommutative geometry. 2his formulation is, however, essentially classical and does, at a funda-mental level, not involve quantization. This raises the question how thequantization procedure of Quantum Field Theory fits into the frameworkon Noncommutative Geometry. Since Connes’ formulation of the StandardModel at its root is tied up to gravitation, a quantization scheme within theframework of Noncommutative Geometry must be expected to involve, atsome level, quantum gravity.In the papers [2] and [3] we started studying the question of formulatinga quantization scheme within Noncommutative Geometry using canonicalquantum gravity. Specifically, we were inspired by techniques applied inLoop Quantum Gravity. The concrete aim was to construct a spectral tripleof a noncommutative algebra naturally associated to the unconstrained statespace of Loop Quantum Gravity. Due to technical difficulties we only suc-ceeded in constructing a spectral triple on a space closely related to the statespace of Loop Quantum Gravity.This paper is one of two papers presenting a more satisfactory solutionto the problem of constructing a spectral triple on the state space of LoopQuantum Gravity as well as its physical interpretations. The paper [1] dealswith the physical background and interpretations of the construction, and wetherefore refer the reader to that paper for more thorough discussion. Thispaper deals with the concise mathematical construction.
The unconstrained state space of Loop Quantum Gravity is the space of SU (2)-connections in a trivial principal bundle on a three dimensional mani-fold. Since the construction we do works for arbitrary manifolds and arbitrarycompact Lie-groups, and since we for physical applications might need it forlater use, we will formulate it in this generality.We let G be a compact Lie group and assume we have some principal G -fiber bundle P over a manifold M . The algebra associated to the space A of connections in P we want to consider is the following:Given a representation of G in M N and given a loop L in M define amatrix valued function on A by L ( ∇ ) = Hol ( ∇ , L ) , where Hol ( ∇ , L ) denotes the holonomy of ∇ along L . Let B be the algebra3enerated by all loops based in a fixed point in M . This is the type of algebrawe want to consider.The algebra B is an algebra of matrix valued functions over A . We will fortechnical reasons explained below consider a smaller algebra than B . We willtake a triangulation of M and consider the algebra B △ which is constructedlike B but only includes loops lying in the edges of the triangulations, orone of its barycentric subdivisions. B △ is also an algebra of matrix valuedfunctions over A . Since any loop can be approximated by a loop lying in theedges of the triangulations, or one of its barycentric subdivisions, B △ canbe considered as an approximation to B . The crucial difference is that thegroups of all diffeomorphisms preserving the base point acts on B whereasonly the group differomorphisms preserving the chosen triangulation and itsbarycentric subdivisions acts on B △ .In order to construct a spectral triple over B △ we need a representation of B △ . We therefore need to construct L ( A ). A construction of L ( A ) alreadyexists within Loop Quantum Gravity, due to Ashtekar and Lewandowski. Itturns out that this construction depends on a completion of A and that thiscompletion depends on a choice of a system S of graphs. In the case of B △ the relevant system of graphs is given by finite subgraphs of the triangulationand its barycentric subdivision. The system of graphs considered in LoopQuantum Gravity is that of graphs made up of piecewise real analytic edges(of course assuming M to be real analytic).Seen from a graph Γ with edges { e i } i =1 ,...,n the space of connections A looks like G n via A ∋ ∇ → ( Hol ( ∇ , e ) , . . . , Hol ( ∇ , e n )) . We will also denote A Γ := G n . Of course A Γ tells little about A . However,by letting the complexity of the graph grow we get a more and more refinedpicture of A . This is implemented by noting that an embedding Γ ⊂ Γ naturally gives a map P Γ Γ : A Γ → A Γ , and then simply define the completion of A as the projective limit of A Γ , i.e. A S := lim Γ ∈S ←− A Γ , We will prove that under some condition on S that A is densely embeddedin A S , and hence justifies the term completion.4he construction of L ( A ) is straightforward from this completion. Since A Γ = G n we define L ( A Γ ) as square integrable functions on G n with respectto the Haar measure. The map P Γ Γ induces an embedding P Γ Γ : L ( A Γ ) → L ( A Γ ) , and L ( A ) is defined as an inductive limit L ( A ) := lim Γ ∈S −→ L ( A Γ ) . The idea for constructing the Dirac operator is that A Γ = G n is a classicalgeometry and therefore has a canonical Gauß-Bonnet-Dirac operator. Inorder to ensure that this defines an operator on the inductive limit we haveto make sure that the operator is compatible with the structure maps ofthe projective limit. It is here the technical advantages of the triangulationcompared to piecewise real analytic graphs shows up. The triangulationnarrows down the types of structure maps appearing in the projective limit.With this we can define a Dirac operator compatible with the structure maps.It turns out that there is a lot freedom in the construction. Going from onelevel in the inductive limit to the next level we add new copies of G whichcorresponds to new degrees of freedom. Each of these degrees of freedom canbe scaled. The entire construction therefore comes with a sequence of realnon zero numbers { a j,k } k ≤ j , where j, k is just a labelling of the degrees offreedoms convenient for the explicit construction.However, the constructed Dirac operator together with the algebra B △ will not fulfil the requirement of a spectral triple. The problem comes fromthe infinite dimensionality of the Clifford bundle of the space of connection.We therefore need to treat the identity operator on the Clifford bundle asa finite rank operator. The setting of semifinite spectral triples will allowexactly this. The von Neumann algebra N appearing from this constructionis a tensor product of the bounded operators on a separable Hilbert spaceand the weak closure of the CAR-algebra.In order to obtain a semifinite spectral triple we still need to perturb theoperator. At each level in the inductive limit we have to add a boundedperturbation roughly speaking of the form b j P j , where b j is a real numberand P j is the projection onto a subspace related to the kernel of a part ofthe Dirac operator at the j ’th level.The main result in the paper is 4.5.1 stating Theorem
There exist sequences { a j,k } and { b j } such that B △ together with he perturbated Dirac operator is a semifinite spectral triple with respect tothe trace on N . In the appendix we will demonstrate the case of U (1) and show that for SU (2) the perturbations are not needed. Acknowledgements
We thank the following colleagues for fruitful discussions: Christian Fleis-chhack, Victor Gayral, Gerd Grubb, Troels Harmark, Thordur Jonsson,Mario Paschke, Adam Rennie, Carlo Rovelli, Thomas Schucker, ChristianVoigh and Raimar Wulkenhaar. Furthermore, we are grateful to the fol-lowing institutes for hospitality during visits: the Institute of Mathematicsin Reykjavik, Iceland; The Max Planck Institute for Mathematics in theSciences, Leipzig, Germany; the Isaac Newton Institute for MathematicalSciences, Cambridge, UK; the Institute of Theoretical Physics in Marseilles,France.Johannes Aastrup was funded by the German Research Foundation (DFG)within the research projects
Deformation Theory for Boundary Value Prob-lems and
Geometrische Strukturen in der Mathematik (SFB 478).
In this section we will consider the space A of connections in a principal fiberbundle. We will address the problem of topologizing this space together withthe development of a measure theory. It turns out that the constructions,which we introduce in order to address these problems, depends on a com-pletion of A with respect to a system of graphs on the manifold. Differentchoices of graphs give different completions.The material is standard from Loop Quantum Gravity, but we have cho-sen to write a more or less selfcontained exposition, not assuming prior knowl-edge of Loop Quantum Gravity, since we want to put emphasize on severaldifferent completions and their mutual interplay. The original techniqueswere developed by Ashtekar and Lewandowski in [4]. For a survey of LoopQuantum Gravity see [5] and for a detailed account see [14].Let M be a manifold and P a G -principal bundle over M , where G is acompact connected Lie-group. We will for simplicity assume that P is the6rivial bundle, i.e. P is isomorphic to M × G . Definition 2.1.1
Let γ be a continuous piecewise smooth map γ : [0 , → M such that if γ ( t ) = γ ( t ) then t = t or t , t ∈ { , } . We call γ a simplecurve if ˙ γ ( t ) is non vanishing for all t ∈ [0 , . This requirement includes thatthe left and right derivative of γ is non vanishing in the non smooth points. We will call γ (0) the starting point and γ (1) the endpoint.If γ and γ are simple curves where the endpoint of γ coincides with thestarting point of γ , then the composition is defined by( γ ◦ γ )( t ) = (cid:26) γ (2 t ) t ∈ [0 , ] γ (2 t − t ∈ [ , . Note that the composition of two simple curves is not always a simple curve.
Definition 2.1.2
Two simple curves γ and γ are equivalent if there existan increasing piecewise smooth bijection φ : [0 , → [0 , with γ ( φ ( t )) = γ ( t ) , t ∈ [0 , , and with nowhere vanishing derivative. This requirement includes that theleft and right derivatives in the non smooth points are both non vanishing.An equivalence class of simple curves is called a simple path. Note that the relation defined above is in fact an equivalence relation.The notion of a simple path chosen here implies that a simple path hasan orientation. The inverse of a simple path represented by γ is the pathrepresented by γ − , where γ − ( t ) = γ (1 − t ) , i.e. just the path with inverse orientation.7 efinition 2.1.3 A graph Γ is the union of finite many simple paths { e i } where the only intersection points are start or end points of the simple paths.The paths are called the edges of the graph. We denote the set of edgesof Γ by E Γ . The points { e i (0) , e j (1) } are called the vertices of Γ . We denotethe set of vertices by V Γ .A graph is called connected if ∪ i,t e i ( t ) is connected. We will consider paths on the graph. These are simply compositions ofedges and inverses of edges. We will also think of each vertex as a path inthe graph. Furthermore, we would like e i ◦ ( e i ) − to be equal to the path e i (0). We fix this with the following Definition 2.1.4
Let Γ be a graph and let P (Γ) be the set of paths in Γ . Wedefine an equivalence relation ∼ on P (Γ) to be generated by ( e i ) − ◦ e i ∼ e i (1) and e i ◦ ( e i ) − ∼ e i (0) . The hoop groupoid HG (Γ) of path on Γ is defined by HG (Γ) = P (Γ) / ∼ . This equivalence relation implies that if p , p , p ∈ P (Γ) and p ∼ p then p ◦ p ∼ p ◦ p and p ◦ p ∼ p ◦ p whenever the compositions aredefined. The units for HG (Γ) are the vertices of Γ. It is easy to see that HG (Γ) is a groupoid. Definition 2.1.5
Let Γ , Γ be two graphs. We say that Γ is a subgraph of Γ , and we write Γ ⊂ Γ , if the edges of Γ are compositions of the edges ortheir inverses of Γ . Note that Γ ⊂ Γ implies that HG (Γ ) is a subgruopoid of HG (Γ ).The relation ⊂ equips the set of graphs with a partial order. We areinterested in subsystems of the set of all graphs. Definition 2.1.6
A system S of graphs is called directed if there for any Γ , Γ ∈ S exists Γ ∈ S with Γ , Γ ⊂ Γ . Definition 2.1.7
A system of graphs S is called dense if there for everypoint m ∈ M exists a coordinate chart x = ( x , . . . , x n ) around m such thatfor all open subset U containing m in this coordinate chart there exists acollection of edges e , . . . , e n ⊂ U belonging to graphs in S such that: . the e i ’s are straight lines with respect to the coordinate chart,2. the tangent vectors of the e i ’s are linearly independent. The definition of dense might appear awkward. The main purpose of thedefinition is to ensure proposition 2.2.4. Furthermore it is easy to check thedefinition in the concrete examples we have in mind. Certainly the require-ment of straight lines can be eased.We will now give three examples of dense systems of graphs, which willbe important in the rest of the paper. These systems of graphs differ partic-ularly in the size of their corresponding symmetry groups.
Example 1
Let S s be the system of all graphs. This system is clearlydense. The system is however not directed since we can have two simple paths e , e with infinitely many isolated intersection points. Hence for graphsΓ , Γ , where e ∈ Γ and e ∈ Γ , there does not exist a graphs Γ contain-ing Γ and Γ . The system S s admits a natural action of the diffeomorphismgroup Dif f ( M ). The system S s is the same as the piecewise immersed sys-tem defined in [11]. Example 2
Let M be a real analytic manifold. Let S a be the system ofgraph made up of real analytic simple curves. This system is dense and itis also directed since piecewise analytic curves have only finitely many iso-lated intersection points. The system carries a natural action of the group Dif f a ( M ) of real analytic diffeomorphism, but no action of the full diffeo-morphism group Dif f ( M ). This system was, in a base pointed version, firstconsidered in [4]. Example 3
Let T be a triangulation of M . We let Γ be the graphconsisting of all the edges in this triangulation. Strictly speaking this is nota graph if the manifold in not compact, but in this case we can considerΓ as a system of graphs instead. Let T n be the triangulation obtained bybarycentric subdividing each of the simplices in T n times. The graph Γ n isthe graph consisting of the edges of T n . In this way we get a directed anddense system S △ = { Γ n } of graphs.The important feature of S △ which we are going to use in this paper isthe following: The step from Γ n to Γ n +1 involves: Γ M Γ ...Figure 1:1. new edges are added the edges of Γ n get subdivided into two edges. The system S △ only admits an action of the diffeomorphisms Dif f ( △ )that maps edges in ∪T n to edges in ∪T n . Hence this is a much more restrictiveclass of diffeomorphism than in the first two examples. Contrary to the firsttwo examples, the system S △ is countable. Definition 2.1.8
Let S be a directed system of graphs. We define the hoopgroupoid HG ( S ) of S to be the inductive limit HG ( S ) = lim Γ ∈S −→ HG (Γ) . Given a graph Γ define the space A Γ = Hom ( HG (Γ) , G )where G is the compact connected Lie-group. If Γ ⊂ Γ we have the em-bedding of groupoids HG (Γ ) → HG (Γ ), and we hence get a surjection P Γ Γ : A Γ → A Γ . Therefore, for a system of graphs S we have a projective system {A Γ } Γ ∈S . efinition 2.2.1 Let S be a system of graphs. The space of generalizedconnections with respect to S , denoted by A S , is defined by A S = lim Γ ∈S ←− A Γ . The projections from A S to A Γ will be denoted by P Γ . For the systems S s , S a , S △ we will denote the corresponding spaces of gener-alized connections by A s , A a , A △ .Note that when S is directed we have the equality A S = Hom ( HG ( S ) , G ) . Given an element ∇ in A Γ we can associate to it Φ Γ ( ∇ ) ∈ G n Γ where n Γ is the numbers of edges in Γ. This is done by numbering the edges in Γ as e , . . . , e n Γ , and then defining Φ Γ byΦ Γ ( ∇ ) = ( ∇ ( e ) , . . . , ∇ ( e n Γ )) . Lemma 2.2.2
The map Φ Γ : A Γ → G n Γ is a bijection.Proof. Follows since HG (Γ) is freely generated by the edges. ⊳ The bijection Φ Γ gives a topology on A Γ by requiring Φ Γ to be a home-omorphism. The topology is independent of the chosen numbering. Theprojection maps P Γ Γ : A Γ → A Γ are continuous. In factΦ Γ ◦ P Γ Γ ◦ (Φ Γ ) − : G n Γ2 → G n Γ1 is given by composition of one or more of the following operations: • Multiplying g i and g i . • Inverting g i . • Leaving out some g i in ( g , . . . , g n Γ2 ) ∈ G n Γ2 .Since A S is a projective limit of {A Γ } Γ ∈S , we define the topology on A S asthe projective limit topology. This topology is characterized by the followingproperty:Let X be a topological space and assume we have continuous maps φ Γ : X → A Γ for all Γ ∈ S such that P Γ Γ ◦ φ Γ = φ Γ for all Γ ⊂ Γ . Thenthere is a unique continuous map φ : X → A S with P Γ ◦ φ Γ = φ for all Γ ∈ S .11 .2.1 Smooth connections Let A denote the space of smooth connections in the principal bundle P .There is a natural map χ Γ : A → A Γ = Hom ( HG (Γ) , G )defined by χ Γ ( ∇ )( p ) = Hol ( p, ∇ ) , p ∈ HG (Γ) , where Hol ( p, ∇ ) denotes the holonomy of ∇ along p . Clearly P Γ Γ ◦ χ Γ = χ Γ when Γ ⊂ Γ , and hence by the property of the projective limit we get aunique map χ S : A → A S . Proposition 2.2.3
When S is directed, χ S ( A ) is dense in A S .Proof. We first prove that each χ Γ is surjective. The compositionΦ Γ ◦ χ Γ : A → G n Γ is given by Φ Γ ( χ Γ ( ∇ )) = ( Hol ( e , ∇ ) , . . . , Hol ( e n Γ , ∇ )) . Let ( g , . . . , g n Γ ) be given. Since G is connected we can for each i find aconnection ∇ i with Hol ( e i , ∇ i ) = g i . It is furthermore easy to see that since the e i ’s only intersect in the endpointswe can arrange that ∇ i = 0 on all the edges in Γ apart from e i . HenceΦ Γ (cid:16) χ Γ (cid:16)X ∇ i (cid:17)(cid:17) = ( g , . . . , g n Γ ) . Therefore χ Γ is surjective.From the directedness of S and the surjectivity of all χ Γ the density fol-lows. ⊳ The system S s is not directed. It is however possible with more elaboratemethods to prove that A is dense in A S s when G is semi-simple, see forexample the discussion in [12].We now turn to the question of injectivity of χ .12 roposition 2.2.4 When S is dense, χ S is injective.Proof. Let ∇ , ∇ be two different smooth connections. Hence thereis a point m ∈ M where ∇ ( m ) = ∇ ( m ). Let us choose a coordinatechart x = ( x , . . . , x n ) around m according to the density of S , where m corresponds to x = . We can then write ∇ k = X j g kj ( x ) dx j , k ∈ { , } , where g ij ( x ) is a smooth function of x with values in the Lie-algebra of G . Let U be a neighbourhood of such that we can assume that the functions g ij ( x )are constant in U with sufficiently good approximation. Let e , . . . , e n bethe edges in S which are straight lines with respect to the coordinate chart,which belongs to U and where the tangent vectors t , . . . , t n are linearlyindependent. We assume that the edges are parametrized by arc lengths.Since ∇ ( m ) = ∇ ( m ) there is a j such that X i g i ( ) t ji = X i g i ( ) t ji . With sufficiently good approximation we have
Hol ( e j , ∇ k ) = exp( X i g i ( ) t ji ) , k ∈ { , } . Hence, if we had chosen U small enough, we conclude Hol ( e j , ∇ ) = Hol ( e j , ∇ ) . Thus χ S ( ∇ ) = χ S ( ∇ ). ⊳ Let U be a an element in the gauge group G of M × P , i.e. U : M → G is asmooth function. Given a connection ∇ ∈ A , U induces a gauge transformedconnection ˜ ∇ . Given a path p on M with startpoint x and endpoint x theholonomy along e transforms according to Hol ( p, ˜ ∇ ) = U ( x ) Hol ( p, ∇ ) U − ( x ) . This leads to the following 13 efinition 2.2.5
Let Γ be a graphs and U : V Γ → G be a map. Define U ∗ : A Γ → A Γ by U ∗ ( ∇ )( e ) = U ( e (0)) ∇ ( e ) U ( e (1)) − for all e ∈ E Γ . Since HG (Γ) is freely generated by E Γ , this is well defined.We denote by G Γ the group of all maps U : V Γ → G . Note that via U ∗ we get a left action of G Γ on A Γ .Like with spaces of connections there are natural projections P Γ Γ : G Γ → G Γ , when Γ ⊂ Γ and P Γ Γ ( U ∗ ( ∇ )) = P Γ Γ ( U ) ∗ ( P Γ Γ ( ∇ )) , ∇ ∈ A Γ , U ∈ G Γ . (1) Definition 2.2.6
Let S be a system of graphs. Put G S = { U | U : ∪ Γ ∈S V Γ → G } . Define the left action of G S on A S by U · ∇ = ( U | V Γ ) ∗ ( ∇ ) , ∇ ∈ A Γ . Due to (1) this is well defined.A gauge transformation U ∈ G naturally gives an element in G S . When S is dense this induces an embedding G → G S . Via this embedding we getan action of G on A S , which extends the action of G on A . We will thereforecall G S the completed gauge group, or simply the gauge group. If a system S is contained in S we get a surjective continuous map P S S : A S → A S . In particular we have the commutative diagram A s P S△S s P S a S s / / A a P S△S a / / A △
14f course the map P S △ S a only exists if the triangulation is real analytic.The group of all diffeomorphisms Dif f ( M ) acts on A . The question iswhat kind of groups acts on the different completions A S . In general a dif-feomorphism d preserving S induces a homeomorphism d ∗ : A S → A S . Thusin this way there is an action of Dif f ( S ), the diffeomorphisms preserving S ,on A S . We have the following diagram A sP S a S s (cid:15) (cid:15) P S△S s (cid:1) (cid:1) Dif f ( M ) o o A χ S s qqqqqqqqqqqqq χ S△ & & MMMMMMMMMMMMM χ S a / / A aP S△S a (cid:15) (cid:15) Dif f a ( M ) o o ?(cid:31) O O A △ Dif f ( △ ) o o ?(cid:31) O O -(cid:13) c c We see that the size of the completion of A is strongly related to the size ofthe symmetry group. The spaces A s , A a are non separable and the symmetrygroups are large, whereas A △ is separable and the symmetry group Dif f ( △ )is comparatively small. In this way we can think of A △ as being A s (or A a )subjected to a kind of gauge fixing of Dif f ( M ) (or Dif f a ( M )). We will here recall the construction of measures and Hilbert space structureson completions of spaces of connections. The construction first appeared in[4]. See also [13] for a different approach.Because of lemma 2.2.2 we can identify A Γ with G n Γ via Φ Γ . On G n Γ there is a canonical normalized measure, namely the Haar measure µ G n Γ .Denote by µ Γ the image measure of µ G n Γ under Φ − . Lemma 2.4.1
Let Γ ⊂ Γ . The image measure of µ Γ under P Γ Γ is µ Γ .Proof. See for example lemma I.2.9 in [14]. ⊳ Lemma 2.4.1 ensures that {A Γ , µ Γ } γ ∈S is a projective system of measurespaces. Therefore, according to Theorem I.2.10 in [14], there is a uniquemeasure µ on A S such that P Γ ( µ ) = µ Γ for all Γ ∈ S . Also lemma 2.4.1ensures that P ∗ Γ Γ : L ( A Γ ) → L ( A Γ )15s an embedding of Hilbert spaces when Γ ⊂ Γ . Proposition 2.4.2
Let L ( A ) = lim −→ L ( A Γ ) , the inductive limit of the Hilbert spaces { L ( A Γ ) } Γ ∈S . Then L ( A ) = L ( A S ) ,where the latter is with respect to the measure µ .Proof. This follows directly from the construction. See for example sec-tion I.2.4 in [14]. ⊳ In this section we will only work with the completed space A △ . We willconstruct a Dirac type operator acting on a Hilbert space H which can natu-rally be seen as square integrable functions on A △ with values in an infinitedimensional vector bundle.The idea is to construct a Dirac operator on each A Γ . Since these look likeclassical geometries we more or less only have to put a Riemannian metricon each of these. Next we need to check that the construction made on each A Γ is consistent with maps between the different graphs.In this section we have no restrictions on the manifold M . In particularwe can have infinitely many simplices in T .To simplify the notation objects indexed by the graph Γ k will be indexedsimply by k for the rest of the paper. The construction of the Dirac operator, which will be carried out in the fol-lowing subsections, looks cumbersome, but is really forced upon us by therequirement of consistency with the maps between different graphs. There-fore, in order to present the construction without the full notational weight,we will first demonstrate the case where the triangulation consists of one edgeand where we consider the step going to the first barycentric subdivision. Wethus have A = G , A = G and P , ( g , g ) = g g .16or Γ we choose a left and right invariant metric h· , ·i on G . Let ˆ e i bean orthonormal basis for T id G and let ˆ E i ( g ) = L g ˆ e i be the corresponding lefttranslated vector fields. Define the Dirac operator D ( ξ ) = X ˆ E i ∇ ˆ E i ( ξ ) , where ∇ is some SO (dim( G ))-valued connection and ξ ∈ L ( G, Cl ( T G )),where Cl ( T G ) denotes the Clifford bundle.We want to construct D acting on L ( G , Cl ( G )) on the form D ( ξ ) = X j K j ∇ K j ( ξ ) , where K j is an orthonormal frame in T G , such that P ∗ , ( D ( ξ )) = D ( P ∗ , ( ξ )) . First we have to make sense of P ∗ , . Since P , induces maps( P , ) ∗ : T G → T G , and P ∗ , : T ∗ G → T ∗ G between tangent and cotangent spaces it is also natural to let the Dirac op-erators act on Cl ( T ∗ G ), resp. Cl ( T ∗ G ) instead of Cl ( T G ), resp. Cl ( T G ).This is easily done once we have chosen a metric on T ∗ G and resp. T ∗ G . Inorder for P ∗ , : L ( G, Cl ( T ∗ G )) → L ( G , Cl ( T ∗ G ))to be an embedding of Hilbert spaces, and in fact to be defined at the level ofClifford bundles, the map induced by P , at the level of cotangent bundles,also denoted P ∗ , , must be metric.Let w ∈ T ∗ g G . It is easy to see that P ∗ , ( w ) = ( R g − w, L g − w ) , g g = g, where R g − ( w )( v ) = w ( R g v )and L g − ( w )( v ) = w ( L g v )17 We ensure that P ∗ , : T ∗ G → T ∗ G is metric by defining the inner producton T ∗ G h ( w , w ) , ( w , w ) i = 12 ( h w , w i + h w , w i ) . Denote by E i the cotangent vector field which is dual to ˆ E i via h· , ·i .Using the inner product on G we get from P ∗ , ( E i ) a vector field on G . Asmall computation shows that this vector field equals( L g L g R g − ˆ e i , L g ˆ e i ) =: ( ˆ E i , ˆ E i ) . (2)Put ˆ E + i = ( ˆ E i , ˆ E i ) and ˆ E − i = ( ˆ E i , − ˆ E i ). Since { ˆ E ± i } i is an orthonormal framefor T G it is natural to try to define D ( ξ ) = X i E + i ∇ ˆ E + i ξ + X i E − i ∇ ˆ E − i ξ. where ξ ∈ L ( G , Cl ( T ∗ G )) and where {E ± i } is the corresponding orthonor-mal frame for T ∗ G . A small computation shows thatˆ E + i ( P ∗ , ξ ) = 2 P ∗ , ( ˆ E i ( ξ )) , ξ ∈ L ( G )ˆ E − i ( P ∗ , ξ ) = 0 , ξ ∈ L ( G ) . We therefore define D ( ξ ) = 12 X i E + i ∇ ˆ E + i ξ + X i E − i ∇ ˆ E − i ξ ! . Next, write an element in ξ ∈ L ( G, T ∗ G ) as ξ = X i ξ i E i , where ξ i ∈ L ( G ). Hence P ∗ , ( ξ ) = X i P ∗ , ( ξ i ) E + i . We calculate D ( P ∗ , ( ξ ))= X i,j (cid:18) P ∗ , ( ˆ E i ( ξ j )) E + i E + j + 12 (cid:16) P ∗ , ( ξ j ) E + i ∇ ˆ E + i E + j + P ∗ , ( ξ j ) E − i ∇ ˆ E − i E + j (cid:17)(cid:19) .
18f we therefore require ∇ to be a SO (2dim( G ))-valued connection in T ∗ G satisfying ∇ ˆ E + i ( E + j ) = P ∗ , ( ∇ ˆ E i ( E j )) ∇ ˆ E − i ( E + j ) = 0we see that P ∗ , ( D ( ξ )) = D ( P ∗ , ( ξ )) for all ξ ∈ L ( G, Cl ( T ∗ G )). Due to lemma 2.2.2 we have at each level the identification A k = G n k , where n k is the number of edges in Γ k . We want to construct a Riemannian metricon T A k = T G n k . However, we require the metric to be consistent with theembeddings of graphs. Embeddings of graphs Γ k ⊂ Γ k +1 induces a surjectivesmooth map P k,k +1 : A k +1 → A k , and therefore a map of tangent bundles( P k,k +1 ) ∗ : T A k +1 → T A k . Dualizing this map we get an embedding P ∗ k,k +1 : T ∗ A k → T ∗ A k +1 of cotangent bundles.We want to construct the Hilbert space on which the Dirac operator actsas a inductive limit of Hilbert spaces. It is hence natural to construct themetric on the cotangent bundle of A k , since we have canonical maps P ∗ k,k +1 : L ( A k , T ∗ A k ) → L ( A k +1 , T ∗ A k +1 ) . Definition 3.2.1
Let h· , ·i be a left and right invariant Riemannian metricon T ∗ G . Let Γ k be the graph consisting of the edges in T k , the k ’s barycentricsubdivision of T . On T ∗ A k = T ∗ G n k define h· , ·i k = 12 k h· , ·i n k , (3) where h· , ·i n k is the product metric on T ∗ G n k . roposition 3.2.2 The map P ∗ k,k +1 : T ∗ A k → T ∗ A k +1 preserves the metric (3).Proof. We subdivide G n k +1 as G n k × G n , where G n k corresponds tosubdivision of the edges in Γ k and G n corresponds to the new edges addedwhen going from k to k + 1. Write T ∗ ( g ,...,g nk +1) A k +1 = T ∗ g G ⊕ · · · ⊕ T ∗ g nk +1 G. We choose orientations of the edges in such a way that P k,k +1 ( g , . . . , g n k +1 ) = ( g g , g g , . . . , g n k − g n k ) . For a tangent vector v = ( v , . . . , v n k +1 ) in T ∗ A k +1 we have( P k,k +1 ) ∗ ( v ) = ( L g v + R g v , L g v + R g v , . . . , L g nk − v n k + R g nk v n k − ) , where L g means left translation of tangent vectors and R g right translation.Hence for a cotangent vector w = ( w , . . . , w n k ) in T ∗ ( g g ,...,g nk − g nk ) A k wehave P ∗ k,k +1 ( w ) = ( R g − w , L g − w , . . . , R g − nk w n k , L n k − w n k , , . . . , , where by definition L g ( w )( v ) = w ( L g − ( v )) , R g ( w )( v ) = w ( R g − ( v )) , v ∈ T G, w ∈ T ∗ G. From this and the left and right invariance of h· , ·i the proposition follows. ⊳ By proposition 3.2.2 the map P ∗ k,k +1 induces a map, also denoted P ∗ k,k +1 ,from Cl ( T ∗ A k ) to Cl ( T ∗ A k +1 ), where Cl ( T ∗ A k ) denotes the Clifford bundleof T ∗ A k with respect to h· , ·i k . Furthermore this map is isometric. We thusget embeddings of Hilbert spaces P ∗ k,k +1 : L ( A k , Cl ( T ∗ A k )) → L ( A k +1 , Cl ( T ∗ A k +1 ) . Definition 3.2.3
Put H k = L ( A k , Cl ( T ∗ A k )) . Define H = lim −→ H k , the inductive limit of of the system {H k , P ∗ k,k +1 } k . Due to proposition 2.4.2 we can consider H as L ( A △ , Cl ( A △ )) or morefreely written L ( A , Cl ( A )). 20 .3 Some special covector fields For the general construction of the Dirac operator we will need some specialcovector fields generalizing duals of the vector fields (2). For notationalsimplicity we will only present the construction on a single edge which isthen subdivided infinitely many times. Thus A n = G n and the structuremaps are given by P n − ,n ( g , g , . . . , g n − , g n ) = ( g g , . . . , g n − g n ) . Let { e i } be an orthonormal basis of T ∗ id G . Define covector fields on A = G by E , i ( g ) = L g ( e i ) . The construction of the covector fields on A n will be by induction. Assumethat covector fields {E j,ki } j ≤ n − ,k ≤ j − on A n − has been defined (For j = 0 the set k ≤ j − is { } ). We adopt thenotation E i = L g g ··· g n R ( g g ··· g n ) − e i E i = L g g ··· g n R ( g g ··· g n ) − e i ...for covector fields on G depending on g , . . . , g n . Define covector fields on A n by { P ∗ n − ,n ( E j,ki ) } j ≤ n − ,k ≤ j − and E n, i = 2 n − ( E i , −E i , , . . . , E n, i = 2 n − (0 , , E i , −E i , , . . . , E j,ki instead of P ∗ n − ,n ( E j,ki ). Lemma 3.3.1 {E j,ki } j ≤ n,k ≤ j − is an orthonormal frame for T ∗ A n with re-spect to the inner product h· , ·i n . roof. Since P ∗ n − ,n preserves the inner product by proposition 3.2.2 itis enough to check that {E n,ki } k ≤ j − are orthonormal and are orthogonal to {E j,ki } j ≤ n − ,k ≤ j − .It is clear that {E n,ki } k ≤ j − are orthonormal. By induction and thecomputation in the proof of proposition 3.2.2 we see that the vectors in {E j,ki } j ≤ n − ,k ≤ j − are of the form( c E i , c E i , c E i , c E i , . . . , c n − E n − i , c n − E n i ) . From this the lemma follows. ⊳ We want to construct a Dirac type operator acting on H . For this we con-struct Dirac type operators acting on H n which are consistent with P ∗ n,n +1 .We first need to determine sufficient conditions on the connections whichpermit the existence of the Dirac operator.Let ˆ E j,ki be the vector field obtained from E j,ki by using h· , ·i n to identify T ∗ A n with T A n . Also let ˆ e i denote the basis in T id G obtained from e i byusing h· , ·i to identify T id G with T ∗ id G . Note also under this identification thecovectorfield (0 , . . . , , E ji , , . . . )gets mapped to 12 n (0 , . . . , , L g j ··· g n R ( g j +1 ··· g n ) − ˆ e i , , . . . , . Definition 3.4.1
A system of connections {∇ n } n , where ∇ n is a connectionin T ∗ A n is called admissible if ∇ n is a SO (2 n · dim ( G )) -valued connectionand if ∇ n ˆ E j,ki ( E m,pl ) = P ∗ n − ,n ( ∇ n − E j,ki ( E m,pl )) j, m < n ∇ n ˆ E n,ki ( E m,pl ) = 0 m < n We want to define the Dirac operator in the usual fashion using an admissiblefamily of connection, i.e. at each level it should be on the form D n = X E j,ki · ∇ n ˆ E j,ki . roposition 3.4.2 Let {∇ n } be an admissible system of connections and let { a j,k } k ≤ j − be a sequence of complex numbers. For ξ ∈ H n define D n ( ξ ) = X j ≤ n,k,i a j,k E j,ki ∇ n ˆ E j,ki ξ. Then P ∗ n,n +1 ( D n ( ξ )) = D n +1 ( P ∗ n,n +1 ( ξ )) and hence the system of operators { D n } defines a densely defined operator D on H .Proof. We first check the identity on functions, i.e. ξ ∈ L ( A n ): P ∗ n,n +1 ( D n ( ξ ))( g , g , . . . , g n +1 )= P ∗ n,n +1 X j ≤ n,k,i a j,k E j,ki · ˆ E j,ki ( ξ ) ! ( g , . . . , g n +1 )= X j ≤ n,k,i a j,k E j,ki · P ∗ n,n +1 ( ˆ E j,ki ( ξ ))( g , . . . , g n +1 )= X j ≤ n,k,i a j,k E j,ki · ( ˆ E j,ki ( ξ ))( g g , . . . , g n +1 − g n +1 ) . Let γ i be a curve in G with ˙ γ i (0) = ˆ e i . Write E j,ki = ( c E i , c E i , . . . , c n E n i ) . Thus ˆ E j,ki ( ξ )( g , . . . , g n )= 12 n X l c l ddt ξ ( g , . . . , g l − , g l · · · g n γ i ( g l +1 · · · g n ) − ,g l +1 , . . . , g n ) , and ˆ E j,ki ( ξ )( g g , . . . , g n +1 − g n +1 )= 12 n ddt X l c l ξ ( g g , . . . , g l − − g l − , g l − · · · g n +1 γ i ( g l +1 · · · g n +1 ) − , g l +1 , . . . g n +1 ) .
23n the other hand D n +1 ( P ∗ n,n +1 ( ξ ))( g , . . . , g n +1 )= X j ≤ n +1 ,k,i a j,k E j,ki · ˆ E j,ki ( P ∗ n,n +1 ( ξ ))( g , . . . , g n +1 )= X j ≤ n,k,i a j,k E j,ki · ˆ E j,ki ( P ∗ n,n +1 ( ξ ))( g , . . . , g n +1 )+ X j = n +1 ,k,i a j,k E j,ki · ˆ E j,ki ( P ∗ n,n +1 ( ξ ))( g , . . . , g n +1 )The terms in the last sum areˆ E n +1 ,ki ( P ∗ n,n +1 ( ξ ))( g , . . . , g n +1 )= 12 ddt (cid:0) ξ ( g g , . . . , g k − · · · g n +1 γ i ( g k · · · g n +1 ) − g k , . . . , g n +1 − g n +1 ) − ξ ( g g , . . . , g k − g k · · · g n +1 γ i ( g k +1 · · · g n +1 ) − ,g k +1 , . . . , g n +1 − g n +1 ) (cid:1) = 0The terms in the first sum areˆ E j,ki ( P ∗ n,n +1 ( ξ ))( g , . . . , g n +1 )= 12 n +1 ddt X l c l (cid:0) ξ ( g g , . . . ,g l − · · · g n +1 γ i ( g l · · · g n +1 ) − g l , . . . , g n +1 − g n +1 )+ ξ ( g g , . . . , g l − g l · · · g n +1 γ i ( g l +1 · · · g n +1 ) − , . . . , g n +1 − g n +1 ) (cid:1) = 12 n ddt X l c l ξ ( g g , . . . , g l − − g l − , g l − · · · g n +1 γ i ( g l +1 · · · g n +1 ) − , g l +1 , . . . g n +1 ) . This proves the compatibility for functions.Because of the derivation property of the connection on the Clifford bun-dle it only remains to prove compatibility on vectors of the form E j,ki , j ≤ n .24his follows from P ∗ n,n +1 ( D n ( E m,pl ))= P ∗ n,n +1 ( X j ≤ n,k,i a j,k E j,ki ∇ n ˆ E j,ki E m,pl )= X j ≤ n +1 ,k,i a j,k E j,ki ∇ n +1ˆ E j,ki ( E m,pl )= D n +1 ( E m,pl )where we have used the admissibility condition for the connections. ⊳ In section 2.2.2 we have constructed a left action of G △ on A △ . It follows fromthe construction that this action preserves the inner product on L ( A △ ), andhence is a unitary action of G △ on L ( A △ ). In order to talk about gaugeinvariance of of the Dirac operator we first need to extend this action to aunitary action on L ( A △ , Cl ( A △ )).Let U be a gauge transformation written as U n ( g , . . . , g n ) = ( u g u − , . . . , u n − g n u − n ) , on A n , where u , . . . , u n ∈ G . Since the metric is left and right invariant ineach copy of G , we see that ( U n ) ∗ : T ∗ A n → T ∗ A n preserves the metric andtherefore U extends to a unitary on L ( A △ , Cl ( A △ )). This unitary will alsobe denoted by U .A vectorfield of the form( g , . . . , g n ) → (0 , . . . , , L g j ··· g n R ( g j +1 ··· g n ) − ˆ e i , , . . . , u g u − , . . . , u n − g n u − n ) → (0 , . . . , , L u j − g j ··· g n R ( u − j g j +1 ··· g n ) − ˆ e i , , . . . , , under ( U n ) ∗ . Written differently( g , . . . , g n ) → (0 , . . . , , L g j ··· g n R ( g j +1 ··· g n ) − u n ˆ e i u − n , , . . . , . U n ) ∗ is, on a covector field of the form( g , . . . , g n ) → (0 , . . . , , L g j ··· g n R ( g j +1 ··· g n ) − e i , , . . . , , given by( g , . . . , g n ) → (0 , . . . , , L g j ··· g n R ( g j +1 ··· g n ) − u − n e i u n , , . . . , . Put f i = u n e i u n . Since { f i } be another orthonormal basis for the dual ofthe Lie algebra of G , there exist a matrix O ij ∈ O ( dim ( G )) with f i = X j O ij e j . From the construction of the covector fields {E j,ki } it follows that U ( E j,ki ) = X l O il E j,kl . It turns out that the operator D is not always gauge invariant. Wetherefore need Definition 3.5.1
An admissible system of connections is gauge admissibleif the conditions X r,i,q O ql O ir E j,kr U ( ∇ n ˆ E j,ki E m,pq ) = X i E j,ki ∇ n ˆ E j,ki E m,pl hold for all n, j, k and all gauge transformations U ∈ G △ . Note that the system of trivial connections with respect to the trivializa-tions given by {E j,ki } is a gauge admissible system of connections. Proposition 3.5.2
When D is constructed from a gauge admissible connec-tion, D is gauge invariant, i.e. D = U DU ∗ for all U ∈ G △ .Proof. Let ξ ∈ L ( A n ). We compute U DU ∗ ( ξ )= X j ≤ n,k,i a j,k ( U n ) ∗ ( E j,ki ) d ( U − n ) ∗ ( ˆ E j,ki ) ξ = X j ≤ n,k a j,k X i,l,m O il E j,kl d O im ˆ E j,km ξ = X j ≤ n,k,i a j,k E j,ki d ˆ E j,ki ξ = D ( ξ ) 26ue to the derivation and product structures we now only need to checkgauge invariance on the covectorfields {E j,ki } . Here we get U DU ∗ ( E m,pl )= U ( X j ≤ n,k,i,q a j,k E j,ki O ql ∇ n ˆ E j,ki E m,pq )= X j ≤ n,k,i,q a j,k U ( E j,ki O ql ) U ( ∇ n ˆ E j,ki E m,pq )= X j ≤ n,k a j,k X r,i,q O ql O ir E j,kr U ( ∇ n ˆ E j,ki E m,pq ) ! = X j ≤ n,k,i a j,k E j,ki ∇ n ˆ E j,ki E m,pl = D ( E m,pl ) , where we have used gauge admissibility of ∇ n . ⊳ In this section we will construct a semifinite spectral associated to A △ orrather to the algebra of holonomy loops, see definition 4.1.1. The followingdefinition first appeared [6]. Definition 4.0.3
Let N be a semifinite von Neumann algebra with a semifi-nite trace τ . Let K τ be the τ - compact operators. A semifinite spectral triple ( B , H , D ) is a ∗ -subalgebra B of N , a representation of N on the Hilbert space H and an unbounded densely defined self adjoint operator D on H affiliatedwith N satisfying1. b ( λ − D ) − ∈ K τ for all b ∈ B and λ / ∈ R ..2. [ b, D ] is densely defined and extends to a bounded operator. Let v be a vertex in S △ . Denote by HG v ( S △ ) the subgroupoid of HG ( S △ )of loops based in v , i.e. paths starting and ending in v . Let G → M N be a27nitary matrix representation of G and let H ′ = H ⊗ M N . Hence H ′ = lim −→ ( H n ⊗ M N ) . Definition 4.1.1
Let L be a loop in HG v ( S △ ) . Let ξ ∈ H n ⊗ M N , where n is such that L ∈ HG ( n ) . Define L ( ξ )( ∇ ) = ∇ ( L ) ξ ( ∇ ) , ∇ ∈ A n = Hom ( HG ( n ) , G ) . Since P ∗ n,m ( L ( ξ )) = L ( P ∗ n,m ( ξ )) we get a densely defined operator on H ′ , also denoted L . Clearly L is bounded.Denote by B v the ∗ -algebra generated by { L } L ∈HG v ( S △ ) . We will call B v the algebra of holonomy loops. Proposition 4.1.2
For L ∈ HG v ( S △ ) define the function HL : A → M N by HL ( ∇ ) = Hol ( L, ∇ ) . The ∗ -algebra generated by { HL } L ∈HG v ( S △ ) equipped with the sup norm isisomorphic to B v as normed ∗ -algebra.Proof. Follows from the dense embedding
A → A △ . ⊳ Let A n = K ( L ( A n )) ⊗ Cl ( T ∗ id A n ) ⊗ End ( M N ) . where K ( L ( A n )) denotes the compact operators on L ( A n ). We define maps,with an abuse of notation also denoted P ∗ n,n +1 , from A n to A n +1 in the fol-lowing way • On End ( M N ), P ∗ n,n +1 is the identity. • On K ( L ( A n )), P ∗ n,n +1 is the map induced by the embedding P ∗ n,n +1 : L ( A n ) → L ( A n +1 ) , i.e. if P n,n +1 denotes the projection in L ( A n +1 ) onto P ∗ n,n +1 ( L ( A n ))we have K ( L ( A n )) ∋ a → aP n,n +1 ∈ K ( L ( A n +1 )) . We use the map P n,n +1 : Cl ( T ∗ A n ) → Cl ( T ∗ A n +1 ) , which has already been defined in subsection 3.2 and restrict it to theidentity. If we write T ∗ id A n +1 = T ∗ id A n ⊕ V n,n +1 , and hence Cl ( T ∗ id A n +1 ) = Cl ( T ∗ id A n ) ˆ ⊗ Cl ( V n,n +1 ) , then P ∗ n,n +1 : Cl ( T ∗ id A n ) → Cl ( T ∗ id A n +1 )is given by P ∗ n,n +1 ( E ) = E ⊗ Cl ( V n,n +1 ) . Note that P ∗ n,n +1 is a C ∗ -algebra homomorphism. Therefore { A n } is an in-ductive system of C ∗ -algebras. Let A = lim A n be the inductive limit. By P ∗ n, ∞ : A n → A, we denote the induced embeddings.From the construction of A it follows that A = K ( L ( A )) ⊗ B ⊗ M N , (4)where B is a UHF-algebra. Since the dimension of the Clifford algebra is apower of 2 when n ≥ B is the CAR-algebra.There is a trace T r on A k defined by T r = T r o ⊗ T r n ⊗ T r o , where T r o are the operator traces on K ( L ( A k )), resp. End ( M N ) and T r n is the normalized trace on Cl ( T ∗ id A k ). By construction T r ( a ) = T r ( P ∗ k,k +1 ( a )) , A .Using the trivialization {E j,ki } of T ∗ id A n we see that A acts on H ′ . In fact H ′ factors like (4) into H ′ = L ( A ) ⊗ Cl ( T ∗ id A ) ⊗ M N . Note that the action of B on Cl ( T ∗ id A ) is just the GNS-representation of B with respect to the normalized trace on B . Definition 4.2.1
Let N be the weak closure of A in B ( H ′ ) . The trace τ : N → C is defined as the extension of the trace T r on A to N . Note that τ is a semifinite trace, since it is the tensor product of the usualsemifinite trace on B ( H ), H separable, and the finite trace on the hyperfiniteII factor B w . We define the coordinate transformationΘ n : A n = G n → G n by Θ n ( g , . . . , g n ) = ( g g · · · g n , g g · · · g n , . . . , g n − g n , g n ) . It is easy to see that Θ n preserves the Haar measure on G n .The inverse of θ n is given byΘ − n ( g , . . . , g n ) = ( g g − , g g − , . . . , g n − g − n , g n ) , The main purpose of Θ n is the following: A n is a trivial G n − -principal fiber bundle over A n − , where the actionof G n − on A n is given by( g , . . . , g n − )( g ′ , . . . , g ′ n ) = ( g ′ g − , g g ′ , . . . , g ′ n − g − n − , g n − g ′ n ) . n we get the following commutative diagram... ... G n → A n Θ n → G n ↓ ↓ pr o G n − → A n − n − → G n − ... ... G → A → G ↓ ↓ pr o A = G where pr o means projection onto the odd coordinates, i.e. pr o ( g , g , . . . , g n − , g n ) = ( g , g , . . . , g n − ) . In other words { Θ n } is just a consistent way of trivializing the principalbundles. Lemma 4.3.1
Let ˆ E i ( g ) = L g (ˆ e i ) , the left translate of ˆ e i . On A n we have ((Θ n ) ∗ ( ˆ E n,ki ))( g , . . . , g n ) = −
12 (0 , . . . , , ˆ E i ( g k ) , , . . . , , and if we write ˆ E j,ki = ( c ˆ E i , c ˆ E i , c ˆ E i , c ˆ E i , . . . , c n − ˆ E n − i , c n − ˆ E n i ) , j < n, then ((Θ n ) ∗ ( ˆ E j,ki ))( g , . . . , g n ) = (cid:16) X j =1 c j ˆ E i ( g ) , ( c + 2 X j =2 c j ) ˆ E i ( g ) , X j =2 c j ˆ E i ( g ) , . . . , c n − ˆ E i ( g n − ) , c n − ˆ E i ( g n ) (cid:17) Proof.
Straightforward computation. ⊳ We will also write ˆ E ji ( g , . . . , g n ) = (0 , . . . , ˆ E i ( g j ) , . . . , roposition 4.3.2 Let {∇ n } be an admissible system of connections. As-sume that D is self adjoint, ∇ n ˆ E j,ki ( E n,ml ) = 0 for all i, j, k, m, l, and all n ≥ and that a j,k are real and non zero for all j, k . Then D is self adjoint.Proof. We will prove that D n is formally self adjoint for each n . From thisthe statement follows because D n is an elliptic pseudo differential operatoron a compact manifold and therefore by elliptic regularity D n is self adjoint.We can even find an orthonormal basis for H ′ n which diagonalizes D n withreal eigenvalues. We can therefore find an orthonormal basis for H ′ whichdiagonalizes D with real eigenvalues. Hence D is self adjoint.Write D n = X i a , E , i ∇ n ˆ E , i + X ≤ j ≤ n,k,i a j,k E j,ki ∇ n ˆ E j,ki . By the assumptions on the connection and lemma 4.3.1 the first summandon the right hand side is a linear combination of D ’s acting on the differentcopies of G , and thus by assumption self adjoint. The second summand is,in the trivialization induced by {E j,ki } and transported by Θ n on the form X ≤ j ≤ n,k,i a j,k E j,ki d (Θ n ) ∗ ( ˆ E j,ki ) , where d is the exterior derivative. Since (Θ n ) ∗ ( ˆ E j,ki ) is a left invariant vectorfield according to lemma 4.3.1, d (Θ n ) ∗ ( ˆ E j,ki ) is formally skew self adjoint. Theformal selfadjointness now follows since Clifford multiplication with E j,ki isalso skew self adjoint and commutes with d (Θ n ) ∗ ( ˆ E j,ki ) . ⊳ D The spectral projections of D n will by construction belong to K ( L ( A n )) ⊗ End ( Cl ( T ∗ id A n )) ⊗ End ( M N ) . Since we can split Cl ( T ∗ id A n ) into irreducible representations of Cl ( T ∗ id A n ),and D n acts on each of these, the spectral projections of D n is in A n = K ( L ( A n )) ⊗ Cl ( T ∗ id A n ) ⊗ End ( M N ) . T ∗ id A n +1 = T ∗ id A n ⊕ V n,n +1 . More generally we can write T ∗ id A m = T ∗ id A n ⊕ V n,m , and we thus have Cl ( T ∗ id G m ) = Cl ( V n,m ) ˆ ⊗ Cl ( T ∗ id A n ) . (5)We assume that the system of connections defining the Dirac operatorsatisfies the properties in 4.3.2. These properties ensure the following equa-tion: D m ( P ∗ n,m ( ξ ) ⊗ v ) = P ∗ n,m ( D n ( ξ )) ⊗ v, ξ ∈ H ′ n , v ∈ Cl ( V n,m ) . If ξ is an eigenvector with eigenvalue λ , then P ∗ n,m ( ξ ) ⊗ v is an eigenvector witheigenvalue λ if v ∈ Cl ( V n,m ). Therefore, if P λ,n is the spectral of eigenvalue λ of D n the projection P ∗ n, ∞ ( P λ,n ) is a subprojection of the spectral projectionof eigenvalue λ of D . By construction P ∗ n, ∞ ( P λ,n ) ∈ N ⊳ Proposition 4.4.1
The spectral projections of the Dirac operator is con-tained in N .Proof. For a spectral projection P λ of eigenvalue λ of D we have P ∗ n, ∞ ( P λ,n ) ր P λ . Hence P λ is in the weak closure of A , i.e. P λ ∈ N . ⊳ Corollary 4.4.2
The operator D is affiliated with N . We begin by enlarging our Hilbert space slightly. Let H = H ′ ⊗ Cl (1). Wealso enlarge N by tensoring with Cl (1). By abuse of notation we will alsocall the enlargement N . The trace τ is also enlarged by tensoring with thenormalized trace on Cl (1) and denoted by τ .33he Dirac operator extends to an operator on H . We write D n = X j There exist sequences { a j,k } and { b k } such that ( B v , H , D p ) is a semifinite spectral triple with respect to ( N , τ ) .Proof. The selfadjointness and affiliation to N of D p are already takencare of. What remains is to prove that [ b, D p ] ∈ B ( H ) and that b ( λ − D p ) − ∈K τ for all b ∈ B v .The boundedness of the commutator: Let L be a loop in Γ n . If we write H n = L ( G n ) ⊗ Cl ( T ∗ A n ) ⊗ M N ⊗ Cl (1)34he loop operator L acts by point wise matrix multiplication over G n inthe M N factor, i.e. the matrix entries is of the form f ( g , . . . , g n ). Theaction of L on H n +1 is then matrix multiplication with entries of the form f ( g g , . . . , g n +1 − g n +1 ). Conjugating the operator with Θ n +1 we see, sinceΘ − n +1 ( g , . . . , g n +1 ) = ( g g − , g g − , . . . , g n − g − n +1 , g n +1 ) , that the result is independent of g , g , . . . , g n +1 , and thus [ L, D − n +1 ] = 0.It therefore follows that [ L, P n +1 ] = 0. Hence [ L, D n +1 ,p ] = [ L, D n,p ] andtherefore [ L, D p ] is bounded.To prove that there exist sequences { a j,k } and b j such D p has τ -compactresolvent we will prove that for any real sequence c n converging to ∞ we canchoose a n,k and b n such that the new eigenvalues, modulo the extra multi-plicity of the existing eigenvalue due to the growth of the Clifford bundle,introduced by going from ( D n − ,p ) to ( D n,p ) are bigger than c n .In the following we will omit the M N part. This will play no role, since theDirac operator does not act on the M N part. First we rewrite the operatorin the following way D n,p = D + n + X j ≤ n − b j eP j ! + (cid:0) D − n + b n eP n (cid:1) = D + n,p + (cid:0) D − n + b n eP n (cid:1) , and (cid:0) D − n + b n eP n (cid:1) = X k,i a n,k E n,ki ∇ n ˆ E n,ki + b n eP n = a X k,i a n,k a E n,ki ∇ n ˆ E n,ki + b n a eP n ! =: aD − n,p . Using the coordinate change Θ n we factorize H n = L ( G n − ) ⊗ L ( G n − ) ⊗ Cl ( T ∗ id A n ) ⊗ Cl (1)= H ⊗ H ⊗ Cl ( T ∗ id A n ) ⊗ Cl (1) , where H = L ( G n − ) corresponds to the coordinates ( g , g , . . . , g n − ) un-der Θ n and H corresponds to the coordinates ( g , g , . . . , g n ). In particularby lemma 4.3.1, D − n,p acts trivially on H . Taking the square of D n,p we get( D n,p ) = ( D + n,p ) + a { D − n,p , D + n,p } + a ( D − n,p ) . (7)35sing lemma 4.3.1 it is easy to see that { D − n,p , D + n,p } does not act on H .Let ξ belong to the orthogonal complement of H ⊗ ⊗ Cl ( T ∗ id A n ) ⊗ Cl (1).We know, due to the fact that D p is self adjoint and commutes with the maps P ∗ k,k +1 that this complement is an invariant subspace for D p . Decompose ξ with respect to the above decomposition of H n into ξ = X k ξ k ⊗ ξ k , (8)where { ξ k } is an orthonormal basis for H and ξ k belongs to the orthogonalcomplement of 1 ⊗ Cl ( T ∗ id A n ) ⊗ Cl (1)in H ⊗ Cl ( T ∗ id A n ) ⊗ Cl (1) . Combining (7) and (8) we get h ( D n,p ) ξ, ξ i = h ( D + n,p ) ξ, ξ i + a h{ D + n,p , D − n,p } ξ, ξ i + a h ( D − n,p ) ξ, ξ i≥ h{ D + n,p , D − n,p } ξ, ξ i + a h ( D − n,p ) ξ, ξ i = X k h ξ k , ξ k i (cid:0) a h{ D + n,p , D − n,p } ξ k , ξ k i + a h ( D − n,p ) ξ k , ξ k i (cid:1) Since ( D − n,p ) is an elliptic second order operator on H ⊗ Cl ( T ∗ id A n ) ⊗ Cl (1)and the operator { D + n,p , D − n,p } is a first order operator on H ⊗ Cl ( T ∗ id A n ) ⊗ Cl (1), the operator A = ( D − n,p ) − { D + n,p , D − n,p } ( D − n,p ) − is bounded. We thus get h (( D n,p ) ξ, ξ i≥ X k h ξ k , ξ k i (cid:0) a h{ D + n,p , D − n,p } ξ k , ξ k i + a h ( D − n,p ) ξ k , ξ k i (cid:1) = X k h ξ k , ξ k i ( a h AD − n,p ξ k , D − n,p ξ k i + a h ( D − n,p ) ξ k , ξ k i ) ≥ X k h ξ k , ξ k i (cid:0) − a k A kh D − n,p ξ k , D − n,p ξ k i + a h ( D − n,p ) ξ k , ξ k i (cid:1) ≥ X k h ξ k , ξ k i λ ( a − a k A k ) h ξ k , ξ k i = λ ( a − a k A k ) k ξ k λ is the lowest eigenvalue of ( D − n,p ) on the complement of H ⊗ ⊗ Cl ( T ∗ id A n ) ⊗ Cl (1) . Hence by choosing a big enough we have h ( D n,p ) ξ, ξ i ≥ c n k ξ k . ⊳ The sequence { b n } is needed in the case when the operator D e = X i E i d ˆ E i , where ˆ E i is an orthonormal frame of left translated vectorfields on one copyof G , has a non trivial kernel in the sense that the kernel is given by the spanof { E i } i . This is clearly not the case for U (1). In the appendix we will showthat this is also not the case for SU (2). Proposition 4.5.2 When D b is constructed from a gauge admissible sys-tem of connections satisfying the demands in proposition 4.3.2, D b is gaugeinvariant, i.e. D b = U D b U ∗ for all U ∈ G △ . ( U acts trivially on the Cl (1) -part.)Proof. From the proof of proposition 3.5.2 it follows that D − n is gaugeinvariant, i.e. invariant under U n for all U n ∈ G n . In particular the kernel of D − n is gauge invariant. Also the space (6) is invariant under G n and thereforethe projection P n onto the orthogonal complement of (6) in the kernel of D − n is gauge invariant under G n . From this the invariance follows. ⊳ Note that the system of trivial connections with respect to the trivializa-tions given by {E j,ki } fulfills the the demands of proposition 4.5.2. In this appendix we will first demonstrate the case of U (1) to show what kindof growth conditions are needed on a j,k . Secondly we show that for SU (2)the perturbation with the P n ’s is not needed to obtain a semifinite spectraltriple. 37 .1 The U (1) -case We write U (1) = { e πiθ | θ ∈ [0 , } and choose the metric such that h ddθ , ddθ i = 1 . The system of connections we will use is the system of trivial connections.The operator D n has the form X j ≤ n,k,i a j,k E j,ki d ˆ E j,ki Since i can be only one in this formula, we will simply omit it. Also we willassume that a j,k = a j,k for all k , k , and simply denote a j := a j,k . All ˆ E j,k commute. Hence D n = − X j ≤ n,k a j ( d ˆ E j,k ) . Since D n acts trivially in the Clifford bundle, and since the identity on theClifford bundle is normalized to have trace 1, we will omit the Clifford bundlein the rest of this computation.We will use the coordinate change Θ n to rewrite D n . The rewrittenoperator will be denoted ˜ D n . 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