Coupling of gravity to matter, spectral action and cosmic topology
aa r X i v : . [ m a t h - ph ] O c t COUPLING OF GRAVITY TO MATTER, SPECTRALACTION AND COSMIC TOPOLOGY
BRANIMIR ´CA ´CI ´C, MATILDE MARCOLLI, KEVIN TEH
Abstract.
We consider a model of modified gravity based on the spec-tral action functional, for a cosmic topology given by a spherical spaceform, and the associated slow-roll inflation scenario. We consider thenthe coupling of gravity to matter determined by an almost-commutativegeometry over the spherical space form. We show that this produces amultiplicative shift of the amplitude of the power spectra for the densityfluctuations and the gravitational waves, by a multiplicative factor equalto the total number of fermions in the matter sector of the model. Weobtain the result by an explicit nonperturbative computation, based onthe Poisson summation formula and the spectra of twisted Dirac oper-ators on spherical space forms, as well as, for more general spacetimemanifolds, using a heat-kernel computation.
Contents
1. Introduction 21.1. Spectral triples 21.2. The spectral action 31.3. Almost-commutative geometries 31.4. Cosmic topology and inflation 41.5. Basic setup 62. Poisson summation formula 82.1. Twisted Dirac spectra of spherical space forms 92.2. Lens spaces, odd order 102.3. Lens spaces, even order 122.4. Dicyclic group 142.5. Binary tetrahedral group 152.6. Binary octahedral group 162.7. Binary icosahedral group 172.8. Sums of polynomials 183. A heat-kernel argument 183.1. Generalities 183.2. Non-perturbative results 213.3. Perturbative results 244. The inflation potential and the power spectra 264.1. Inflation potential in the heat kernel approach 28References 29 Introduction
Models of gravity coupled to matter based on Noncommutative Geometryare usually obtained (see [2], [7], [10], [11]) by considering an underlying ge-ometry given by a product X × F of an ordinary 4-dimensional (Riemanniancompact) spacetime manifold and a finite noncommutative space F .The main purpose of the paper is to show how the slow-roll inflationpotential derived in [21], [22] is affected by the presence of the matter con-tent and the almost-commutative geometry. We first consider the case ofspherical space forms using the Poisson summation formula technique andthe nonperturbative calculation of the spectral action, and then the case ofmore general spacetime manifolds using a nonperturbative heat kernel ar-gument, to show that the amplitude of the slow-roll potential is affected bya multiplicative factor N equal to the dimension of the representation, thatis, to the total number of fermions in the theory.In terms of identifying specific properties of this modified gravity modelbased on the spectral action, which set it apart from other models, thedependence of the amplitude of the inflation potential on the number offermions in the particle physics sector of the model is so far the most strikingfeature that distinguishes it from other known slow-roll inflation models.1.1. Spectral triples.
Noncommutative spaces are described, in this con-text, as a generalization of Riemannian manifolds, via the formalism of spec-tral triples introduced in [12]. An ordinary Riemannian spin manifold X isidentified with the spectral triple ( C ∞ ( X ) , L ( X, S ) , /D ), with the algebraof smooth functions acting as multiplication operators on the Hilbert spaceof square integrable spinors, and the Riemannian metric reconstructed fromthe Dirac operator /D .More generally, for a noncommutative space, a spectral triple is a similarset ( A , H , D ) consisting of a ∗ -algebra represented by bounded operators ona Hilbert space H , and a self-adjoint operator D with compact resolventacting on H with a dense domain and such that the commutators [ D, a ]extend to bounded operators on all of H . A finite noncommutative space isone for which the algebra A and Hilbert space H are finite dimensional.A recent powerful reconstruction theorem ([13], see also [30]) shows thatcommutative spectral triples that satisfy certain natural axioms, related toproperties such as orientability, are spectral triples of smooth Riemannianmanifolds in the sense mentioned above.In the models of gravity coupled to matter, the choice of the finite geom-etry F = ( A F , H F , D F ) determines the field content of the particle physicsmodel. As shown in [10], the coordinates on the moduli space of possibleDirac operators D F on the finite geometry ( A F , H F ) specify the Yukawaparameters (Dirac and Majorana masses and mixing angles) for the par-ticles. A classification of the moduli spaces of Dirac operators on generalfinite geometries was given in [4]. OUPLING TO MATTER, SPECTRAL ACTION, COSMIC TOPOLOGY 3
The spectral action.
One obtains then a theory of (modified) gravitycoupled to matter by taking as an action functional the spectral action onthe noncommutative space X × F , considered as the product of the spectraltriples ( C ∞ ( X ) , L ( X, S ) , /D ) and ( A F , H F , D F ).The spectral action functional introduced in [7] is a function of the spec-trum of the Dirac operator on a spectral triple, given by summing over thespectrum with a cutoff function. Namely, the spectral action functional isdefined as Tr( f ( D/ Λ)), where Λ is an energy scale, D is the Dirac operatorof the spectral triple, and f is a smooth approximation to a cutoff function.As shown in [7] this action functional has an asymptotic expansion at highenergies Λ of the form(1.1) Tr( f ( D/ Λ)) ∼ X k ∈ DimSp f k Λ k Z −| D | − k + f (0) ζ D (0) + o (1) , where the f k are the momenta f k = R ∞ f ( v ) v k − dv of the test function f ,for k a non-negative integer in the dimension spectrum of D (the set of polesof the zeta functions ζ a,D ( s ) = Tr( a | D | − s )) and the term R −| D | − k given bythe residue at k of the zeta function ζ D ( s ).These terms in the asymptotic expansion of the spectral action can becomputed explicitly: for a suitable choice of the finite geometry spectraltriple ( A F , H F , D F ) as in [10], they recover all the bosonic terms in theLagrangian of the Standard Model (with additional right-handed neutrinoswith Majorana mass terms) and gravitational terms including the Einstein–Hilbert action, a cosmological term, and conformal gravity terms, see alsoChapter 1 of [14]. For a different choice of the finite geometry, one canobtain supersymmetric QCD, see [2].The higher order terms in the spectral action, which appear with coeffi-cients f − k = ( − k k ! / (2 k )! f (2 k ) (0) depending on the derivatives of the testfunction, and involve higher derivative terms in the fields, were consideredexplicitly recently, in work related to renormalization of the spectral actionfor gauge theories [36], [37], and also in [8]. In cases where the underly-ing geometry is very symmetric (space forms) and the Dirac spectrum isexplicitly known, it is also possible to obtain explicit non-perturbative com-putations of the spectral action, computed directly as Tr( f ( D/ Λ)), usingPoisson summation formula techniques applied to the Dirac spectrum andits multiplicities, see [8], [21], [22], [35].1.3.
Almost-commutative geometries.
It is also natural to consider ageneralization of the product geometry X × F , where this type of almost-commutative geometry is generalized to allow for nontrivial fibrations thatare only locally, but not globally, products. This means considering almost-commutative geometries that are fibrations over an ordinary manifold X ,with fiber a finite noncommutative space F . A first instance where suchtopologically non-trivial cases were considered in the context of models ofgravity coupled to matter was the Yang–Mills case considered in [1]. BRANIMIR ´CA´CI´C, MATILDE MARCOLLI, KEVIN TEH
In the setting of [1], instead of a product geometry X × F , one consid-ers a noncommutative space obtained as an algebra bundle, namely wherethe algebra of the full space is isomorphic to sections Γ( X, E ) of a locallytrivial ∗ -algebra bundle whose fibers E x are isomorphic to a fixed finite-dimensional algebra A F . The spectral triple that replaces the product ge-ometry is then of the form ( C ∞ ( X, E ) , L ( X, E ⊗ S ) , D E ), where the Diracoperator D E = c ◦ ( ∇ E ⊗ ⊗ ∇ S ) is defined using the spin connectionand a hermitian connection on the algebra bundle E (with respect to aninner product obtained using a faithful tracial state τ x on E x ). The spectraltriple obtained in this way can be endowed with a compatible grading andreal structure and it is described in [1] in terms of an unbounded Kasparovproduct of KK-cycles. In the Yang–Mills case, where the finite-dimensionalalgebra is A F = M N ( C ), it is then shown in [1] that this type of spectraltriples describes P SU ( N )-gauge theory with a nontrivial principal bundleand the Yang–Mills action functional coupled to gravity is recovered fromthe asymptotic expansion of the spectral action.Even more generally, one can define an almost-commutative geometrywith base X , as a spectral triple of the form ( A , H , D ), where H = L ( X, V )for V → X a self-adjoint Clifford module bundle, A = C ∞ ( X, E ) for E → X a unital ∗ -algebra sub-bundle of End +Cl( X ) ( V ), and D a symmetric Dirac-type operator on V ; in this context, the compact Riemannian manifold X is no longer required to be spin. A reconstruction theorem for almost com-mutative geometries (defined in this more general topologically nontrivialsense), was recently obtained in [5], as a consequence of the reconstructiontheorem for commutative spectral triples of [13]. There, the above concretedefinition was shown to be equivalent to an abstract definition of almost-commutative geometry with base a commutative unital ∗ -algebra, analogousto the abstract definition of commutative spectral triple.1.4. Cosmic topology and inflation.
The asymptotic expansion of thespectral action naturally provides an action functional for (Euclidean) mod-ified gravity, where in addition to the ordinary Einstein–Hilbert action withcosmological term one also has a topological term (the Euler characteristic)and conformal gravity terms like the Weyl curvature and a conformal cou-pling of the Higgs field to gravity. It also produces the additional bosonicterms: the action for the Higgs with quartic potential and the Yang–Millsaction for the gauge fields. Thus, it is natural to consider the spectral actionas a candidate action functional for a modified gravity model and study itsconsequences for cosmology.Cosmological implications of the spectral action, based on the asymptoticexpansion, were considered in [18], [20], [23], [24], [25], [26]. For recentdevelopments in the case of Robertson–Walker metrics see [9].In [21] and [22] the nonperturbative spectral action was computed explic-itly for the 3-dimensional spherical space forms and the flat 3-dimensionalBieberbach manifolds, via the same type of Poisson summation technique
OUPLING TO MATTER, SPECTRAL ACTION, COSMIC TOPOLOGY 5 first used in [8] for the sphere case. The spectral action for Bieberbach man-ifolds was also computed in [29]. In these computations one considers thespectral action as a pure gravity functional (that is, only on the manifold X ,without the finite geometry F ). It is shown that a perturbation D + φ ofthe Dirac operator produces in the nonperturbative spectral action a slow-roll potential V ( φ ) for the scalar field φ , which can be used as a model forcosmic inflation.While in [8] the computation of the potential V ( φ ) is performed in themodel the case of the Higgs field, we do not assume here (nor in [21], [22])that φ is necessarily related to the Higgs field and we only treat D + φ as a scalar field perturbation of the Dirac operator. In particular, in thenoncommutative geometry models of particle physics the Higgs field arisesbecause of the presence of a nontrivial noncommutative space F , and analmost commutative geometry X × F (the product of the spacetime manifold X and F , or more generally a nontrivial fibration as analyzed in [5]). TheHiggs field is described geometrically as the inner fluctuations of the Diracoperator in the noncommutative direction F . On the other hand, moregeneral fluctuations of the form D + φ are possible also in the pure manifoldcase (the pure gravity case), in the absence of a noncommutative fiber F .These are indeed the cases considered in [21], [22]. It is worth pointingout that, if one assumes that the field φ is related to the Higgs field, thenthere are very strong constraints coming from the CMB data, as recentlyanalyzed in [3], which make a Higgs-based inflation scenario, as predictedby this kind of NCG model, incompatible with the measured value of thetop quark mass. However, these constraints do not directly apply to otherscalar perturbations φ , not related to the Higgs field.In this paper we focus on the same topologies considered in [21], [35]and [22]. These include all the most significant candidates for a nontrivialcosmic topology, widely studied in the theoretical cosmology literature (seefor example [28], [31], [32]). We refer the reader to [21] and the referencestherein, for a more detailed overview of the specific physical significance ofthese various topologies.In [21], [35] and [22] the nonperturbative spectral action is computed forall the spherical space forms S / Γ and for the flat tori and for all the flatBieberbach 3-manifolds (for the latter case see also [29]). These two classes ofmanifolds provide a complete classification of all the possible homogeneouscompact 3-manifolds that are either positively curved or flat, hence theyencompass all the possible compact cases of interest to the problem of cosmictopology (see for instance [31], [32]). It is shown in [21], [35] and [22] thatthe nonperturbative spectral action for spherical space forms is, up to anoverall constant factor that depends on the order of the finite group Γ, thesame as that of the sphere S , hence so is the slow-roll potential. Similarly,the spectral action and potentials for the flat Bieberbach manifolds are amultiple of those of the flat torus T . In particular, for each such manifold,although the spectra depend explicitly on the different spin structures, the BRANIMIR ´CA´CI´C, MATILDE MARCOLLI, KEVIN TEH spectral action does not. These results show that, in a model of gravity basedon the spectral action functional, the amplitudes and slow-roll parametersin the power spectra for the scalar and tensor fluctuation would depend onthe underlying cosmic topology, and hence constraints on these quantitiesderived from cosmological data (see [19], [33], [34]) may, in principle, be ableto distinguish between different topologies.Here we discuss a natural question arising from the results of [21] and [22],namely how the presence of the finite geometry F may affect the behaviorof the slow-roll inflation potential. As a setting, we consider here the caseof the spherical space forms S / Γ as the commutative base of an almostcommutative geometry in the sense of [5], where the Clifford module bundle V on S / Γ is the spinor bundle twisted by a flat bundle corresponding toa finite-dimensional representation α : Γ → GL N ( C ) of the group Γ. Asthe Dirac operator on the almost commutative geometry we consider thecorresponding twisted Dirac operator D Γ α on S / Γ. From the point of viewof the physical model this means that we only focus on the gravity termsand we do not include the part of the Dirac operator D F that describes thematter content and which comes from a finite spectral triple in the fiberdirection.Our main result is that, for any such almost commutative geometry, thespectral action and the associated slow-roll potential only differ from those ofthe sphere S by an overall multiplicative amplitude factor equal to N/ N depending on the fiber of the almost commutative geometry. Interms of the physical model, this N represents the number of fermions in thetheory. We obtain this by computing the spectral action in its nonpertur-bative form, as in [8], [21], [22], [35], using the Poisson summation formulatechnique and the explicit form of the Dirac spectra derived in [6].More generally, one can consider a finite normal Riemannian cover Γ → f M → M together with a finite-dimensional representation α : Γ → U ( N ),and compare the spectral action and slow-roll potential on a Γ-equivariantalmost commutative geometry over f M , with the spectral action and slow-roll potential on the quotient geometry twisted by α , an almost commutativegeometry over M . Using a nonperturbative heat kernel argument, we findthat the spectral action and slow-roll potential over M differ from those over f M , up to an error of order O (Λ −∞ ) as Λ → + ∞ , by an overall multiplicativeamplitude equal to N/ Basic setup.
We recall the basic setting, following the notation of[6]. Let Γ ⊂ SU (2) be a finite group acting by isometries on S , identifiedwith the Lie group SU (2) with the round metric. The spinor bundle on OUPLING TO MATTER, SPECTRAL ACTION, COSMIC TOPOLOGY 7 the spherical form S / Γ is given by S × σ C → S / Γ, where σ is therepresentation of Γ defined by the standard representation of SU (2) on C .A unitary representation α : Γ → U ( N ) defines a flat bundle V α = S × α C N endowed with a canonical flat connection. By twisting the Diracoperator with the flat bundle, one obtains an operator D Γ α on the spheri-cal form S / Γ acting on the twisted spinors, that is, on the Γ-equivariantsections C ∞ ( S , C ⊗ C N ) Γ , where Γ acts by isometries on S and by σ ⊗ α on C ⊗ C N ; these are the sections of the twisted spinor bundle S × σ ⊗ α ( C ⊗ C N ) → S / Γ. Thus, D Γ α is the restriction of the Dirac oper-ator D ⊗ id C N to the subspace C ∞ ( S , C ⊗ C N ) Γ ⊂ C ∞ ( S , C ⊗ C N ).This setup gives rise to an almost commutative geometry in the senseof [5], where the twisted Dirac operator D Γ α represents the “pure gravity”part of the resulting model of gravity coupled to matter, while the fiber C N = H F determines the fermion content of the matter part and can bechosen according to the type of particle physics model one wishes to consider(Standard Model with right handed neutrinos, supersymmetric QCD, forexample, as in [10], [2], or other possibilities). Since we will only be focusingon the gravity terms, we do not need to specify in full the data of the almostcommutative geometry, beyond assigning the flat bundle V α and the twistedDirac operator D Γ α , as the additional data would not enter directly in ourcomputations.We later generalise this above setup as follows. Let Γ → f M → M be afinite normal Riemannian covering with f M and M compact, let e V → f M bea Γ-equivariant self-adjoint Clifford module bundle, and let Γ-equivariantsymmetric Dirac-type operator e D on e V ; let α : Γ → U ( N ) be a unitaryrepresentation. We can then define a self-adjoint Clifford module bundle V α on M by ( e V ⊗ C N ) / Γ, and a symmetric Dirac-type operator D α on V α asthe image of e D ⊗ N acting on e V ⊗ C N . Again, since we will be focusing onthe gravity terms, and since our results will allow for suitable equivariantperturbations of e D , we do not need to specify algebra bundles for e V or for V α .As it is customary when using the spectral action formalism, the compu-tation in the 4-dimensional case that determines the form of the inflationpotential V ( φ ) is obtained in Euclidean signature, on a compactificationalong an S β , of size β . For a detailed discussion of this method and of thesignificance of the parameter β , we refer the reader to § N of the representation). This paper also BRANIMIR ´CA´CI´C, MATILDE MARCOLLI, KEVIN TEH focuses on comparing two different methods for computing the nonperturba-tive spectral action, respectively based on the Poisson summation formulaand on heat-kernel techniques. As such, it is addressed primarily to anaudience of mathematicians and mathematical physicists. Ultimately, thesuccess of such models in cosmology will inevitably depend on the morestrictly cosmological aspects of this project, namely a direct comparisonbetween the model and the current CMB data. This is indeed the aim ofour ongoing collaboration with the cosmologist Elena Pierpaoli and will beaddressed elsewhere, as appropriate.2.
Poisson summation formula
Following the method developed in [8] and [21], [35], we compute thespectral action of the quotient spaces S / Γ equipped with the twisted Diracoperator corresponding to a finite-dimensional representation α of Γ as fol-lows. We define a finite set of polynomials labeled P + m , and P − m that describethe multiplicities of, respectively, the positive and negative eigenvalues of thetwisted Dirac operator. More precisely, the index m takes values in the setof residue classes of the integers modulo c Γ , where c Γ is the exponent of thegroup Γ, the least common multiple of the orders of the elements in Γ. For k ≡ m mod c Γ , k ≥
1, and(2.1) λ = − / ± ( k + 1) ,P ± m ( λ ) equals the multiplicity of λ .The main technical result we will prove is the following relation betweenthese polynomials:(2.2) c Γ X m =1 P + m ( u ) = c Γ − X m =0 P − m ( u ) = N c Γ (cid:18) u − (cid:19) . Since the polynomial on the right-hand side is a multiple of the polynomialfor the spectral multiplicities of the Dirac spectrum of the sphere S (see [8]),we will obtain from this the relation between the non-perturbative spectralaction of the twisted Dirac operator D Γ α on S / Γ and the spectral action onthe sphere, see Theorem 2.1 below.Furthermore, we shall show that the polynomials P + m ( u ) match up per-fectly with the polynomials P − m ( u ), so that the polynomials P + m ( u ) alonedescribe the entire spectrum by allowing the parameter k in equation 2.1 torun through all of Z . Namely, what we need to show is that(2.3) P + m ( u ) = P − m ′ ( u ) , where for each m , m ′ is the unique number between 0 and c Γ − m + m ′ + 2 is a multiple of c Γ , or more precisely, OUPLING TO MATTER, SPECTRAL ACTION, COSMIC TOPOLOGY 9 (2.4) m ′ = c Γ − − m, if 1 ≤ m ≤ c Γ − ,c Γ − , if m = c Γ − ,c Γ − m = c Γ . Define(2.5) g m ( u ) = P + m ( u ) f ( u/ Λ) . Now, we apply the Poisson summation formula to obtainTr( f ( D/ Λ)) = X m X l ∈ Z g m (1 / c Γ l + m + 1)= N X m c g m (0) + O (Λ −∞ )= N (cid:18)Z R u f ( u/ Λ) − Z R f ( u/ Λ) (cid:19) + O (Λ −∞ )= N (cid:18) Λ b f (2) (0) −
14 Λ b f (0) (cid:19) + O (Λ −∞ ) , and so we have the main result. Theorem 2.1.
Let Γ be a finite subgroup of S , and let α be a N -dimensionalrepresentation of Γ . Then the spectral action of S / Γ equipped with thetwisted Dirac operator is (2.6) Tr f ( D/ Λ) = N (cid:18) Λ b f (2) (0) −
14 Λ b f (0) (cid:19) + O (Λ −∞ ) , where b f (2) denotes the Fourier transform of u f ( u ) . Similar computations of the spectral action have also been performed in[21], [22], and [35]. In the sequel we describe how to obtain equation (2.2),by explicitly analyzing the cases of the various spherical space forms: lensspaces, dicyclic group, and binary tetrahedral, octahedral, and icosahedralgroups. In all cases we compute explicitly the polynomials of the spectralmultiplicities and check that (2.2) is satisfied. Our calculations are basedon a result of Cisneros-Molina, [6], on the explicit form of the Dirac spectraof the twisted Dirac operators D Γ α , which we recall here below.2.1. Twisted Dirac spectra of spherical space forms.
The spectra ofthe twisted Dirac operators on the quotient spaces are derived in [6]. Let usrecall the notation and the main results.Let E k denote the k + 1-dimensional irreducible representation of SU (2)on the space of homogeneous complex polynomials in two variables of degree k . By the Peter–Weyl theorem, one can decompose C ∞ ( S , C ) = ⊕ k E k ⊗ E ∗ k as a sum of irreducible representations of SU (2). This gives that, on C ∞ ( S , C ⊗ C N ) = ⊕ k E k ⊗ E ∗ k ⊗ C ⊗ C N , the Dirac operator D ⊗ id C N decomposes as ⊕ k id E k ⊗ D k ⊗ id C N , with D k : E ∗ k ⊗ C → E ∗ k ⊗ C . Uponidentifying C ∞ ( S , C ⊗ C N ) Γ = ⊕ k E k ⊗ Hom Γ ( E k , C ⊗ C N ), one seesthat, as shown in [6], the multiplicities of the spectrum of the twisted Diracoperator D Γ α are given by the dimensions dim C Hom Γ ( E k , C ⊗ C N ), whichin turn can be expressed in terms of the pairing of the characters of thecorresponding Γ-representation, that is, as h χ E k , χ σ ⊗ α i Γ . One then obtainsthe following: Theorem 2.2. (Cisneros-Molina, [6])
Let α : Γ → GL N ( C ) be a represen-tation of Γ . Then the eigenvalues of the twisted Dirac operator D Γ α on S / Γ are − − ( k + 1) with multiplicity h χ E k +1 , χ α i Γ ( k + 1) , if k ≥ , −
12 + ( k + 1) with multiplicity h χ E k − , χ α i Γ ( k + 1) , if k ≥ . Proposition 2.3. (Cisneros-Molina, [6])
Let k = c Γ l + m with ≤ m < c Γ . (1) If − ∈ Γ , then h χ E k , χ α i Γ = ( c Γ l ( χ α (1) + χ α ( − h χ E m , χ α i Γ if k is even, c Γ l ( χ α (1) − χ α ( − h χ E m , χ α i Γ if k is odd. (2) If − / ∈ Γ , then h χ E k , χ α i Γ = N c Γ l h χ E m , χ α i Γ . Lens spaces, odd order.
In this section we consider Γ = Z n , where n is odd. When n is odd, − / ∈ Γ, which affects the expression for thecharacter inner products in Proposition 2.3.For m ∈ { , . . . , n } , we introduce the polynomials, P + m ( u ) = Nn u + ( β αm − mNn ) u + β αm − mN n − N n , where β αm = h χ E m − , χ α i Γ , and m takes on values in { , , . . . , n } Using Theorem 2.2 and Proposition 2.3, it is easy to see that the polyno-mials P + m ( u ) describe the spectrum on the positive side of the real line, i.e., P + m ( λ ) equals the multiplicity of λ = − / k + 1) , k ≥ k ≡ m mod n.For the negative eigenvalues, the multiplicities are described by the poly-nomials P − m ( u ) = Nn u + (cid:18) Nn + mNn − γ αm (cid:19) u + 3 N n + mN n − γ αm , OUPLING TO MATTER, SPECTRAL ACTION, COSMIC TOPOLOGY 11 m ∈ { , , . . . n − } , i.e., P − m ( λ ) equals the multiplicity of the eigenvalue λ = − / − ( k + 1) , k ≥ , whenever k ≡ m mod n; here γ αm is defined by γ αm = h χ E m +1 , χ α i Γ . Let us denote the irreducible representations of Z n by χ t , which sends thegenerator to exp( πitN ), for t a residue class of integers modulo n .For the sake of computation, we take Z n to be the group generated by B = " e πin e − πin . Then in the representation E k , B acts on the basis polynomials P j ( z , z ), j ∈ { , , . . . k } as follows. B · P j ( z , z ) = P j (( z , z ) B )= P j ( e πin z , e − πin z )= ( e πin z ) k − j ( e − πin z ) j = e πin ( k − j ) P j ( z , z ) . Hence, B is represented by a diagonal matrix with respect to this basis, andwe have Proposition 2.4.
The irreducible characters χ E k of the irreducible repre-sentations of SU (2) restricted to Z n , n odd, are decomposed into the irre-ducible characters χ [ t ] of Z n by the equation (2.7) χ E k = j = k X j =0 χ [ k − j ] . Here, [ t ] denotes the number from to n − to which t is equivalent mod n . In the case where − / ∈ Γ, that is to say, when Γ = Z n where n is odd, byequating coefficients of the quadratic polynomials P + m and P − m ′ , the condition2.3 is replaced by one that may be simply checked. Lemma 2.5.
Let Γ be any finite subgroup of SU (2) such that − / ∈ Γ . Thencondition 2.3 is equivalent to the condition (2.8) β αm + γ αm ′ = ( χ α (1) , if ≤ m ≤ c Γ − χ α (1) , if m = c Γ − , c Γ , where α is an irreducible representation of Γ . Furthermore this conditionholds in all cases. Using Proposition 2.4, it is a simple combinatorial matter to see that(2.9) n X m =1 h χ E m − , χ α i Γ = N n + 12 , for any representation α of Z n For the argument to go through, one also needs to check the special case P + c Γ (1 /
2) = 0 . By direct evaluation one can check that this indeed holds.For the negative side, we see that(2.10) n X m =1 h χ E m +1 , χ α i Γ = N n + 32 , for any representation α of Z n , and so Proposition 2.6.
Let Γ be cyclic with odd, and let α be a N -dimensionalrepresentation of Γ . Then n X m =1 P + m ( u ) = n − X m =0 P − m ( u ) = N u − N . Note that in the statement of Theorem 2.2, the first line holds even ifwe take k = −
1, since the multiplicity for this value evaluates to zero.Therefore, we automatically have P − c Γ − ( − /
2) = 0 , which we still needed to check.2.3. Lens spaces, even order.
When n is even, we have − ∈ Z n . When − ∈ Γ, from Theorems 2.2 and 2.3 it follows that the multiplicity of theeigenvalue λ = 1 / lc Γ + m, l ∈ N is given by, P + m , m ∈ { , , . . . , c Γ } , P + m ( u ) = 1 χ α (1) + ( − m +1 χ α ( − u + (cid:18) β αm − m ( χ α (1) + ( − m +1 χ α ( − (cid:19) u + 12 β αm − χ α (1) + ( − m +1 χ α ( − − m ( χ α (1) + ( − m +1 χ α ( − . The one case that is not clear is that of λ = 1 /
2, which is not an eigenvalueof the twisted Dirac operator. It is not clear from Theorems 2.2 and 2.3 that(2.11) P + c Γ (1 /
2) = 0 , OUPLING TO MATTER, SPECTRAL ACTION, COSMIC TOPOLOGY 13 and this needs to hold in order for the argument using the Poisson summa-tion formula to go through. However, by evaluating equation (2.11), we seethat one needs to check that(2.12) h χ E c Γ − , χ α i = c Γ χ α (1) + ( − c Γ +1 χ α ( − , so that Equation 2.11 indeed holds for each subgroup Γ and irreduciblerepresentation α . Proposition 2.7.
For any subgroup Γ ⊂ S of even order, the sum of thepolynomials P + m is c Γ X m =1 P + m ( u ) = c Γ χ α (1) u + − c χ α (1)2 − c Γ ( χ α (1) − χ α ( − c Γ X m =1 β αm ! u − c Γ χ α (1)2 − c χ α (1)4 c Γ χ α ( −
1) + 12 c Γ X m =1 β αm Since the coefficients of the polynomial are additive with respect to directsum, it suffices to consider only irreducible representations.In the case of lens spaces, c Γ = χ t ( −
1) = ( − t . As a matter ofcounting, one can see that Proposition 2.8. c Γ X m =1 β tm = ( n +22 if t is even, n if t is odd. Putting this all into the expression of Proposition 2.7, we have, for an N -dimensional representation α ,(2.13) c Γ X m =1 P + m ( u ) = N (cid:18) u − (cid:19) . The negative eigenvalues are described by the polynomials P − m ( u ) = 1 χ α (1) + ( − m +1 χ α ( − u + (cid:18) m χ α (1) + ( − m +1 χ α ( − − γ αm (cid:19) u m χ α (1) + ( − m +1 χ α ( − − γ αm , for m ∈ { , , . . . c Γ − } , so that we have the following proposition. Proposition 2.9.
For any subgroup Γ ⊂ S of even order, the sum of thepolynomials P − m is c Γ X m =1 P − m ( u ) = c Γ χ α (1) u + χ α (1) c χ α (1) c Γ χ α ( − c Γ − c Γ − X m =0 γ αm ! u + χ α (1) c Γ χ α (1) c χ α ( − c Γ − c Γ − X m =0 γ αm . By counting, one can see that(2.14) c Γ − X m =0 γ tm = ( n +42 if t is even, n +22 if t is odd.To complete the computation of the spectral action one still needs to verifythe condition (2.3). We have the following lemma, obtained by equating thecoefficients of P + m and P − m ′ , that covers the cases of the binary tetrahedral,octahedral and icosahedral groups as well. Lemma 2.10.
Let Γ be any finite subgroup of SU (2) such that − ∈ Γ thecondition (2.3) is equivalent to the condition (2.15) β αm + γ αm ′ = χ α (1)( χ α (1) + ( − m +1 χ α ( − , if ≤ m ≤ c Γ − , χ α (1)( χ α (1) + χ α ( − , if m = c Γ − , χ α (1)( χ α (1) − χ α ( − , if m = c Γ ,where α is an irreducible representation of Γ . Furthermore this conditionholds in all cases. Dicyclic group.
The character table for the dicyclic group of order4 r is, for r odd, Class 1 + − l r r ψ t − t ζ lt r + ζ − lt r χ χ − − l i − iχ − − χ − − l − i i ,and for r even,Class 1 + − l r r ψ t − t ζ lt r + ζ − lt r χ χ − − l i − iχ − − χ − − l − i i OUPLING TO MATTER, SPECTRAL ACTION, COSMIC TOPOLOGY 15
Here ζ r = e πir , 1 ≤ t ≤ r − , ≤ l ≤ r −
1. The notation for the differentconjugacy classes can be understood as follows. The number indicates theorder of the conjugacy class, while a sign in the subscript indicates the signof the traces of the elements in the conjugacy class as elements of SU (2).For the dicyclic group of order 4 r , the exponent of the group is c Γ = ( r if r is even,4 r if r is odd.One can decompose the characters χ E k into the irreducible characters byinspection, and with some counting obtain the following propositions. Proposition 2.11.
Let Γ be the dicyclic group of order r , where r is even. c Γ X m =1 β αm = r if χ α ∈ { χ , χ , χ , χ } , r if χ α = ψ t , t is even, r + 1 if χ α = ψ t , t is odd; c Γ − X m =0 γ αm = r + 1 if χ α ∈ { χ , χ , χ , χ } , r + 2 if χ α = ψ t , t is even, r + 1 if χ α = ψ t , t is odd. Proposition 2.12.
Let Γ be the dicyclic group of order r , where r is odd. c Γ X m =1 β αm = r if χ α ∈ { χ , χ } , r + 1 if χ α ∈ { χ , χ } , r if χ α = ψ t , t is even, r + 2 if χ α = ψ t , t is odd; c Γ − X m =0 γ αm = r + 2 if χ α ∈ { χ , χ } , r + 1 if χ α ∈ { χ , χ } , r + 4 if χ α = ψ t , t is even, r + 2 if χ α = ψ t , t is odd. Binary tetrahedral group.
The binary tetrahedral group has order24 and exponent 12. The character table of the binary tetrahedral group isClass 1 + − a + b + a − b − χ χ ω ω ω ω χ ω ω ω ω χ − − − χ − ω ω − ω − ω χ − ω ω − ω − ω χ − Here, ω = e πi .For the remaining three groups, we can use matrix algebra to decomposethe characters χ E k .Let χ j , x j , j = 1 , . . . , d denote the irreducible characters and represen-tatives of the conjugacy classes of the group Γ, respectively. Then, sinceevery character decomposes uniquely into the irreducible ones, we have aunique expression for χ E k as the linear combination χ E k = d X j =0 c kj χ j . If we let b = ( b j ) j = 1 , . . . , d be the column with b j = χ E k ( x j ), and let A = ( a ij ) be the d × d matrix where a ij = χ j ( x i ) and let c = ( c kj ) j = 1 , . . . d be another column, then we have b = Ac ;but A is necessarily invertible by the uniqueness of the coefficient column c ,and so c is given by c = A − b. By this method, we obtain the following proposition.
Proposition 2.13.
Let Γ be the binary tetrahedral group. Then, c Γ X m =1 β αm = , if χ α ∈ { χ , χ , χ } , , if χ α ∈ { χ , χ , χ } , , if χ α = χ ; c Γ − X m =0 γ αm = , if χ α ∈ { χ , χ , χ } , , if χ α ∈ { χ , χ , χ } , , if χ α = χ . Binary octahedral group.
The binary octahedral group has order48 and exponent 24. The character table of the binary octahedral group isClass 1 + − + − + − χ χ − − − χ − − χ − √ −√ − χ − −√ √ − χ − − − χ − − χ − − OUPLING TO MATTER, SPECTRAL ACTION, COSMIC TOPOLOGY 17
Proposition 2.14.
Let Γ be the binary octahedral group. Then, c Γ X m =1 β αm = , if χ α ∈ { χ , χ } , , if χ α = χ , , if χ α ∈ { χ , χ } , , if χ α ∈ { χ , χ } , , if χ α = χ ; c Γ − X m =0 γ αm = , if χ α ∈ { χ , χ } , , if χ α = χ , , if χ α ∈ { χ , χ } , , if χ α ∈ { χ , χ } , , if χ α = χ . Binary icosahedral group.
The binary icosahedral group has order120 and exponent 60. The character table of the binary icosahedral groupis Class 1 + −
30 20 + − a + b + a − b − Order 1 2 4 6 3 10 5 5 10 χ χ − − µ ν − µ − νχ − − − ν − µ ν µχ − − ν µ − ν µχ − µ − ν µ − νχ − − − − χ − − − − χ − − χ − − − µ = √ , and ν = √ − . Proposition 2.15.
Let Γ be the binary icosahedral group. c Γ X m =1 β αm = , if χ α = χ , , if χ α ∈ { χ , χ } , , if χ α ∈ { χ , χ } , , if χ α = χ , , if χ α = χ , , if χ α = χ , , if χ α = χ ; c Γ − X m =0 γ αm = , if χ α = χ , , if χ α ∈ { χ , χ } , , if χ α ∈ { χ , χ } , , if χ α = χ , , if χ α = χ , , if χ α = χ , , if χ α = χ . Sums of polynomials.
If we input the results of Propositions 2.8,2.11, 2.12, 2.13, 2.14, 2.15 into Propositions 2.7, 2.9, and also recall Propo-sition 2.6, we obtain the following.
Proposition 2.16.
Let Γ be any finite subgroup of SU (2) and let α be an N -dimensional representation of Γ . Then the sums of the polynomials P + m and P − m are given by c Γ X m =1 P + m ( u ) = c Γ − X m =0 P − m ( u ) = N c Γ (cid:18) u − (cid:19) . A heat-kernel argument
It may at first seem surprising that, in the above calculation, using thePoisson summation formula and the explicit Dirac spectra, although thespectra themselves depend in a subtle way upon the representation theoreticdata of the unitary representation α : Γ → U ( N ), through the pairing ofthe characters of representations, the resulting spectral action only dependsupon the dimension N of the representation, the order of Γ, and the spectralaction on S .This phenomenon is parallel to the similar observation in the Poissonformula computation of the spectral action for the spherical space forms andthe flat Bieberbach manifolds in the untwisted case [21], [22], [35], whereone finds that, although the Dirac spectra are different for different spinstructures, the resulting spectral action depends only on the order S or T .In this section, we give a justification for this phenomenon based on aheat-kernel computation that recovers the result of Theorem 2.1 and justifiesthe presence of the factor N/ Generalities.
We begin with some background on the spectral actionfor almost commutative spectral triples. In what follows, let L denote theLaplace transform, and let S (0 , ∞ ) = { φ ∈ S ( R ) | φ ( x ) = 0 , x ≤ } . Theorem 3.1.
Let ( A, H, D ) be a spectral triple of metric dimension p , andlet f : R → C be even. If | f ( x ) | = O ( | x | − α ) as x → ∞ , for some α > p ,then for any Λ > , f ( D/ Λ) = f ( | D | / Λ) is trace-class. If, in addition, OUPLING TO MATTER, SPECTRAL ACTION, COSMIC TOPOLOGY 19 f ( x ) = L [ φ ]( x ) for some measurable φ : R + → C , then (3.1) Tr ( f ( D/ Λ)) = Z ∞ Tr (cid:16) e − sD / Λ (cid:17) φ ( s ) ds. Proof.
Fix Λ >
0. Let µ k denote the k -th eigenvalue of D in increasingorder, counted with multiplicity; since ( A, H, D ) has metric dimension p ,( µ k + 1) − p/ = O ( k − ) as k → ∞ , and hence for k > dim ker D , µ − k = O ( k − /p ) as k → ∞ . By our hypothesis on f , then, for k > dim ker D , | f ( µ / k / Λ) | = O ( k − α/p ) , k → ∞ ;since 2 α/p >
1, this implies that P ∞ k =1 f ( µ / k / Λ) is absolutely convergent,as required.Now, suppose, in addition, that f ( x ) = L [ φ ]( x ) for some measurable φ : [0 , ∞ ) → C . ThenTr ( f ( D/ Λ)) = ∞ X k =1 L [ φ ]( µ k / Λ )= ∞ X k =1 Z ∞ e − sµ k / Λ φ ( s ) ds = Z ∞ " ∞ X k =1 e − sµ k / Λ φ ( s ) ds = Z ∞ Tr (cid:16) e − sD / Λ (cid:17) φ ( s ) ds, as was claimed. (cid:3) The above result raises the question of when a function φ : R + → C definesa function f ( x ) = L [ φ ]( x ) such that f ( D/ Λ) is trace-class; a sufficientcondition is given by the following lemma.
Lemma 3.2. If φ ∈ S (0 , ∞ ) , then L [ φ ]( s ) = O ( s − k ) as s → + ∞ , for all k ∈ N .Proof. Since φ ∈ S (0 , ∞ ), φ ( k ) is a bounded function with φ ( k ) (0) = 0 forall k ∈ N , and hence s n L [ φ ]( s ) = L [ φ ( n ) ]( s ) too is bounded, as required. (cid:3) As a corollary of the above results, together with the asymptotic expan-sion of the heat kernel of a generalised Laplacian, we obtain the followingbasic result on the spectral action for almost commutative spectral triples:
Corollary 3.3 (cf. [27, Theorem 1]) . Let V be a self-adjoint Clifford modulebundle on a compact oriented Riemannian manifold M , and let D be asymmetric Dirac-type operator on V . Let f : R → C be of the form f ( x ) = L [ φ ]( x ) for φ ∈ S (0 , ∞ ) . Then for Λ > , f ( D/ Λ) is trace-class with (3.2) Tr ( f ( D/ Λ)) = Z ∞ (cid:20)Z M tr (cid:0) K ( s/ Λ , x, x ) (cid:1) d Vol( x ) (cid:21) φ ( s ) ds, where K ( t, x, y ) denotes the heat kernel of D , and Tr f ( D/ Λ) admits theasymptotic expansion (3.3) Tr ( f ( D/ Λ)) ∼ ∞ X k = − dim M Λ − k φ k Z M a k +dim M ( x, D ) d Vol( x ) , as Λ → + ∞ , where a n ( x, D ) is the n -th Seeley-DeWitt coefficient of thegeneralised Laplacian D , and the constants φ n are given by φ n = Z ∞ φ ( s ) s n/ ds. Proof.
The first part of the claim follows immediately from the fact thatTr( e − tD ) = Z M tr ( K ( t, x, x )) d Vol( x ) , t > , while the second part of the claim follows immediately from the asymptoticexpansion tr ( K ( t, x, x )) ∼ t − dim M/ ∞ X k =0 t k/ a k ( x, D ) , t → , together with the assumption that φ has rapid decay, so that, in particular,the φ n are all finite. (cid:3) In fact, since a n ( · , D ) = 0 for n odd [16, Lemma 1.7.4], one has that theasymptotic form of Tr ( f ( D/ Λ)), as Λ → + ∞ , is given by (P ∞ n =0 Λ m − n ) φ m − n ) R M a n ( x, D ) d Vol( x ) , if dim M = 2 m , P ∞ n =0 Λ m − n )+1 φ m − n )+1 R M a n ( x, D ) d Vol( x ) , if dim M = 2 m + 1.Note also that for n > φ − n = Z ∞ φ ( s ) s − n/ ds = 1Γ( n/ Z ∞ f ( u ) u n − du. The following result guarantees that the φ k can be chosen at will: Proposition 3.4.
For any ( a n ) ∈ C Z there exists some φ ∈ S (0 , ∞ ) suchthat a n = Z ∞ s n/ φ ( s ) ds, n ∈ Z . In fact, this turns out to be a simple consequence of the following resultby Dur´an and Estrada, solving the Hamburger moment problem for smoothfunctions of rapid decay:
Theorem 3.5 (Dur´an–Estrada [15]) . For any ( a n ) ∈ C Z there exists some φ ∈ S (0 , ∞ ) such that a n = Z ∞ s n φ ( s ) ds, n ∈ Z . OUPLING TO MATTER, SPECTRAL ACTION, COSMIC TOPOLOGY 21
Proof of Proposition 3.4.
By Theorem 3.5, let ψ ∈ S (0 , ∞ ) be such that a n = 2 Z ∞ s n +1 ψ ( s ) ds, n ∈ Z . Then for φ ( s ) = ψ ( √ s ) ∈ S (0 , ∞ ), Z ∞ s n/ φ ( s ) ds = 2 Z ∞ t n +1 ψ ( t ) dt = a n , n ∈ Z , as required. (cid:3) Non-perturbative results.
We now give a non-perturbative heat-kernel-theoretic analysis of the phenomenon mentioned above.Let f M → M be a finite normal Riemannian covering with f M and M compact, connected and oriented, and let Γ be the deck group of the cov-ering. Let e V → f M be a Γ-equivariant self-adjoint Clifford module bun-dle, and let e D be a Γ-equivariant symmetric Dirac-type operator on e V .We can therefore form the quotient self-adjoint Clifford module bundle V := e V / Γ → M = f M /
Γ, with e D descending to a symmetric Dirac-typeoperator D on V ; under the identification L ( M, V ) ∼ = L ( f M , e V ) Γ , we canidentify D with the restriction of e D to C ∞ ( f M , e V ) Γ , where the unitary action U : Γ → U ( L ( f M , e V )) is given by U ( γ ) ξ ( e x ) := ξ ( e xγ − ) γ .Our first goal is to prove the following result, relating the spectral actionof D to the spectral action of e D in the high energy limit: Theorem 3.6.
Let f : R → C be of the form f ( x ) = L [ φ ]( x ) for φ ∈S (0 , ∞ ) . Then for Λ > , (3.4) Tr ( f ( D/ Λ)) = 1 (cid:16) f ( e D/ Λ) (cid:17) + O (Λ −∞ ) , as Λ → + ∞ . Remark 3.7.
Theorem 3.6 continues to hold even when inner fluctuationsof the metric are introduced, since for A ∈ C ∞ ( M, End( V )) symmetric, D + A on V lifts to e D + e A on e V , where e A is the lift of A to e V . To prove this result, we will need a couple of lemmas. First, we have thefollowing well-known general fact:
Lemma 3.8.
Let G be a finite group acting unitarily on a Hilbert space H , and let A be a G -equivariant self-adjoint trace-class operator on H . Let H G denote the subspace of H consisting of G -invariant vectors. Then therestriction A | H G of A to H G is also trace-class, and Tr (cid:0) A | H G (cid:1) = 1 G X g ∈ G Tr ( gA ) . Proof.
This immediately follows from the observation that G P g ∈ G g is theorthogonal projection onto H G . (cid:3) Now, we can compute the heat kernel trace of D using the heat kernel for e D : Lemma 3.9.
For t > , (3.5)Tr (cid:16) e − tD (cid:17) = 1 (cid:16) e − t e D (cid:17) + 1 X γ ∈ Γ \{ e } Z f M tr (cid:16) ρ ( γ )( e xγ − ) e K ( t, e xγ − , e x ) (cid:17) d Vol( e x ) , where e K ( t, e x, e y ) denotes the heat kernel of e D , and ρ denotes the right actionof Γ on the total space e V .Proof. Let γ ∈ Γ. Then for any ξ ∈ C ∞ ( f M , e V ), (cid:16) U ( γ ) e − t e D (cid:17) ξ ( e x ) = U ( γ ) (cid:18)Z f M e K ( t, e x, e y ) ξ ( e y ) d Vol( e y ) (cid:19) = ρ ( γ )( e xγ − ) (cid:18)Z f M ( e xγ − ) e K ( t, e xγ − , e y ) ξ ( e y ) d Vol( e y ) (cid:19) = Z f M ρ ( γ )( e xγ − ) e K ( t, e xγ − , e y ) ξ ( e y ) d Vol( e y )so that the operator U ( γ ) e − t e D has the integral kernel( t, e x, e y ) ρ ( γ )( e xγ − ) e K ( t, e xγ − , e y ) . Since L ( M, V ) ∼ = L ( f M , e V ) Γ , we can therefore apply Lemma 3.8 to obtainthe desired result. (cid:3) Finally, we can proceed with our proof:
Proof of Theorem 3.6.
By Corollary 3.3 and Lemma 3.9, it suffices to showthat for γ ∈ G \ { e } , Z ∞ (cid:20)Z f M tr (cid:16) ρ ( γ )( e xγ − ) e K ( s/ Λ , e xγ − , e x ) (cid:17) d Vol( e x ) (cid:21) φ ( s ) ds = O (Λ −∞ ) , as Λ → ∞ .Now, since f M is compact and since the finite group Γ acts freely andproperly, inf ( e x,γ ) ∈ f M × Γ d ( e xγ − , e x ) = min ( e x,γ ) ∈ f M × Γ d ( e xγ − , e x ) > . Hence, by [17, Proposition 3.24], there exist constants
C > c > e x ∈ f M k e K ( t, e xγ − , e x ) k ≤ Ce − c/t , t > , for k · k the fibre-wise Hilbert-Schmidt norm, implying, in turn, that forevery n ∈ N there exists a constant C n > e x ∈ f M k e K ( t, e xγ − , e x ) k ≤ C n t n , t > . OUPLING TO MATTER, SPECTRAL ACTION, COSMIC TOPOLOGY 23
Hence, for each n ∈ N , (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ (cid:20)Z f M tr (cid:16) ρ ( γ )( e xγ − ) e K ( s/ Λ , e xγ − , e x ) (cid:17) d Vol( e x ) (cid:21) φ ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z ∞ Vol( M ) sup e x ∈ f M k ρ ( γ )( e x ) k ! sup e x ∈ f M k e K ( s/ Λ , e xγ − , e x ) k ! | φ ( s ) | ds ≤ Vol( M ) · sup e x ∈ f M k ρ ( γ )( e x ) k ! · C n Z ∞ ( s/ Λ ) n | φ ( s ) | ds = Vol( M ) · sup e x ∈ f M k ρ ( γ )( e x ) k ! · C n · Z ∞ s n | φ ( s ) | ds ! Λ − n , yielding the desired result. (cid:3) Now, let α : Γ → GL N ( C ) be a representation of Γ; by endowing C N with a Γ-equivariant inner product, we take α : Γ → U ( N ). Since f M → M is a principal Γ-bundle, we form the associated Hermitian vector bundle F := f M × α C N → M ; since Γ is finite, we endow F with the trivial flatconnection d . We can therefore form the self-adjoint Clifford module bundle V ⊗ F → M , which admits the symmetric Dirac-type operator D α obtainedfrom D by twisting by d , that is, D α = D ⊗ c (1 ⊗ d ) , where c denotes the Clifford action on V ⊗ F .We now obtain the following generalisation of Theorem 3.6, which explainsthe factor of N/ Theorem 3.10.
Let f : R → C be of the form f ( x ) = L [ φ ]( x ) for φ ∈S (0 , ∞ ) . Then for Λ > , (3.6) Tr ( f ( D α / Λ)) = N (cid:16) f ( e D/ Λ) (cid:17) + O (Λ −∞ ) , as Λ → + ∞ . Remark 3.11.
This result is again compatible with inner fluctuations of themetric, insofar as if A ∈ C ∞ ( M, End( V )) is symmetric, then D α + A ⊗ on V ⊗ F is induced from e D + e A on e V , where e A is A viewed as a Γ -equivariantelement of C ∞ ( f M ,
End( e V )) .Proof of Theorem 3.10. On the one hand, consider the trivial bundle e F := f M × C N over f M , together with the trivial flat connection d . Then for the ac-tion ( e x, v ) γ := ( e xγ, α ( γ − ) v ), e F is a Γ-equivariant Hermitian vector bundle,and d is a Γ-equivariant Hermitian connection on e F . Then, by taking thetensor product of Γ-actions, we can endow e V ⊗ e F with the structure of a Γ-equivariant self-adjoint Clifford module bundle, admitting the Γ-equivariantsymmetric Dirac-type operator f D α = e D ⊗ c (1 ⊗ d ). As a vector bundle,however, we may simply identify e V ⊗ e F with e F ⊕ N , in which case we mayidentify f D α with e D ⊗ N . On the other hand, by construction, the bundle F defined above is thequotient of e F by the action of Γ. Hence, under the action of Γ, the quotientof e V ⊗ e F is the the self-adjoint Clifford module bundle V ⊗ F , with f D α descending to the operator D ⊗ c (1 ⊗ d ) = D α .Finally, by Theorem 3.6 and our observations above,Tr ( f ( D α / Λ)) = 1 (cid:16) f ( f D α / Λ) (cid:17) + O (Λ −∞ )= 1 (cid:16) f ( e D/ Λ) ⊗ N (cid:17) + O (Λ −∞ )= N (cid:16) f ( e D/ Λ) (cid:17) + O (Λ −∞ ) , as Λ → + ∞ ,as was claimed. (cid:3) One can apply these results to give a quick second proof of Theorem 2.1.
Second proof of Theorem 2.1.
Recall that Γ ⊂ SU (2) is a finite group actingby isometries on S , identified with SU (2) endowed with the round metric,and that α : Γ → U ( N ) is a representation. Since S is parallelizable and Γacts by isometries, the spinor bundle C → S S → S and the Dirac oper-ator /D S are trivially Γ-equivariant. Then, by construction, the Dirac-typeoperator D Γ α on S S ⊗ V α is precisely the induced operator D α correspondingto e D = /D S , so that by Theorem 3.10,Tr ( f ( D α / Λ)) = N f ( /D S / Λ)) + O (Λ −∞ ) , as Λ → + ∞ .However, by [8, § f ( /D S / Λ)) = Λ d f (2) (0) −
14 Λ b f (0) + O (Λ −∞ ) , where d f (2) denotes the Fourier transform of u f ( u ). Hence,Tr ( f ( D α / Λ)) = N (cid:18) Λ d f (2) (0) −
14 Λ b f (0) (cid:19) + O (Λ −∞ ) , as required. (cid:3) Perturbative results.
Let us now turn to the perturbative picture.In light of Corollary 3.3, it suffices to compare the Seeley-DeWitt coefficientsof e D with those of D and D α . Proposition 3.12.
Let e D and D be as above. Let π : f M → M denote thequotient map. Then for all n ∈ N , a n ( π ( e x ) , D ) = a n ( e x, e D ) , e x ∈ f M , and hence Z M a n ( x, D ) d Vol( x ) = 1 Z f M a n ( e x, e D ) d Vol( e x ) . OUPLING TO MATTER, SPECTRAL ACTION, COSMIC TOPOLOGY 25
Proof.
By [16, Lemma 4.8.1], there exist a unique connection ∇ and en-domorphism E on V such that D = ∇ ∗ ∇ − E , and similarly a uniqueconnection e ∇ and endomorphism e E on e V such that e D = e ∇ ∗ e ∇ . Since e D isthe lift of D to e V , it follows by uniqueness that e ∇ and e E are the lifts of ∇ and E , respectively, to e V as well.Now, since the finite group Γ acts freely and properly on f M , let { ( U α , Ψ α ) } be an atlas for f M such that for each α , π | U α : U α → π ( U α ) is an isometry.Hence, the local data defining a n ( · , D ) on U α lifts to the local data defining a n ( · , e D ) on π − ( U α ); since for P = ∇ ∗ ∇ − E a generalised Laplacian on aHermitian vector bundle E → X , each a n ( · , P ) is given by a universal poly-nomial in the Riemannian curvature of X , the curvature of ∇ , E , and theirrespective covariant derivatives [16, § a n ( · , e D )is indeed the lift to f M of a n ( · , D ), as was claimed. (cid:3) Proposition 3.13.
Let e D and D α be as above. Then for all n ∈ N , a n ( π ( e x ) , D α ) = N a n ( e x, e D ) , e x ∈ f M , and hence Z M a n ( x, D α ) d Vol( x ) = N Z M a n ( e x, e D ) d Vol( e x ) . Proof.
On the one hand, by [16, Lemma 1.7.5], a n ( · , e D ⊗ N ) = N a n ( · , e D ),for e D ⊗ N on e V ⊕ N . On the other hand, e D ⊗ N is the lift to e V ⊕ N of D α on V ⊗ F , so that by Proposition 3.12, a n ( · , e D ⊗ N ) is the lift to f M of a n ( · , D α ). Hence, N a n ( · , e D ) is the lift to f M of a n ( · , D α ), as required. (cid:3) Let us now apply these results to the Dirac operator /D S on the round3-sphere S , together with a finite subgroup Γ of SU (2) acting freely andproperly on S ∼ = SU (2), and a representation α : Γ → U ( N ). Since /D S =( ∇ S ) ∗ ∇ S + by the Lichnerowicz formula, it follows from [16, Theorem4.8.16] that Z S a ( x, /D S ) d Vol( x ) = Z S (4 π ) − / tr(id) d Vol( x ) = √ π , Z S a ( x, /D S ) d Vol( x ) = Z S (4 π ) − / tr (cid:18)
66 id −
32 id (cid:19) d Vol( x ) = − √ π . Since the operator D Γ α is precisely D α as induced by e D = /D S , it thereforefollows by Proposition 3.13 that Z S / Γ a ( y, ( D Γ α ) ) d Vol( y ) = N Z S a ( x, /D S ) d Vol( x ) = N √ π ( , Z S / Γ a ( y, ( D Γ α ) ) d Vol( y ) = N Z S a ( x, /D S ) d Vol( x ) = − N √ π ( . Finally, one has that φ − = 2Γ(3 / Z ∞ f ( u ) u du = 2 √ π Z ∞−∞ f ( u ) u du = 2 √ π d f (2) (0) ,φ − = 2Γ(1 / Z ∞ f ( u ) du = 1 √ π Z ∞−∞ f ( u ) du = 1 √ π b f (0) , where d f (2) is the Fourier transform of f ( u ) u . Hence,Tr ( f ( /D S / Λ)) ∼ Λ φ − Z S a ( x, /D S ) d Vol( x )+ Λ φ − Z S a ( x, /D S ) d Vol( x ) + O (Λ − )= Λ d f (2) (0) −
14 Λ b f (0) + O (Λ − ) , and Tr (cid:0) f ( D Γ α ) (cid:1) ∼ Λ φ − Z S / Γ a ( y, ( D Γ α ) ) d Vol( y )+ Λ φ − Z S / Γ a ( y, ( D Γ α ) ) d Vol( y ) + O (Λ − )= N (cid:18) Λ d f (2) (0) −
14 Λ b f (0) (cid:19) + O (Λ − ) , which is indeed consistent with Theorem 2.1.4. The inflation potential and the power spectra
It was shown in [21], [22] that for a 3-manifold Y that is a spherical spaceform S / Γ or a flat Bieberbach manifold (a quotient of the flat torus T by a finite group action), the non-perturbative spectral action determines aslow-roll potential for a scalar field φ by settingTr( h (( D Y × S + φ ) / Λ )) − Tr( h ( D Y × S / Λ )) = V Y ( φ ) , up to terms of order O (Λ −∞ ), where, in the spherical space form case thepotential is of the form V Y ( φ ) = π Λ βa V Y ( φ Λ ) + π βa W Y ( φ Λ ) , for h the test function for the computation of the spectral action on the4-manifold Y × S , a > β > S . The functions V Y and W Y are of the form(4.1) V Y ( x ) = λ Y V S ( x ) and W Y ( x ) = λ Y W S ( x ) , where, for Y = S / Γ, the factor λ Y = ( − , and(4.2) V S ( x ) = Z ∞ u ( h ( u + x ) − h ( u )) du and W S ( x ) = Z x h ( u ) du. OUPLING TO MATTER, SPECTRAL ACTION, COSMIC TOPOLOGY 27
Thus, the potential satisfies(4.3) V Y ( φ ) = λ Y V S ( φ ) = V S ( φ ) . The slow-roll potential V Y ( φ ) can be used as a model for cosmological in-flation. As such, it determines the behavior of the power spectra P s,Y ( k ) and P t,Y ( k ) for the density fluctuations and the gravitational waves, respectivelygiven in the form(4.4) P s ( k ) ∼ M P l V ( V ′ ) and P t ( k ) ∼ VM P l , with M P l the Planck mass, see [33] and [22] for more details. Includingsecond order terms, these can be written also as power laws as in [33],(4.5) P s ( k ) ∼ P s ( k ) (cid:18) kk (cid:19) − n s + αs log( k/k ) P t ( k ) ∼ P t ( k ) (cid:18) kk (cid:19) n t + αt log( k/k ) , where the exponents also depend on the slow roll potentials through certainslow-roll parameters. Since, as already observed in [21], [22], the slow-rollparameters are not sensitive to an overall multiplicative scaling factor in thepotential, we focus here only on the amplitude only, which, as shown in [22],correspondingly changes by a multiplicative factor. Namely, in the case ofa spherical space form with the spectral action computed for the untwistedDirac operator, one has(4.6) P s,Y ( k ) ∼ λ Y P s ( k ) (cid:18) kk (cid:19) − n s,S + αs,S log( k/k ) P t,Y ( k ) ∼ λ Y P t ( k ) (cid:18) kk (cid:19) n t,S + αt,S log( k/k ) , where, as above, λ Y = 1 / a , and β is included in [22], while here we focuson how the coupling of gravity to matter affects these parameters.By directly comparing the argument given in [21] proving (4.6) with theresult of Theorem 2.1 above, we see that, in our case, we obtain then thefollowing version of (4.6), modified by an overall multiplicative factor N ,the total number of fermions in the model of gravity coupled to matter. Proposition 4.1.
For a spherical space form Y = S / Γ , consider the slow-roll potential V Y,α ( φ ) determined by the nonperturbative spectral action Tr( h (( D α,Y × S + φ ) / Λ )) − Tr( h ( D α,Y × S / Λ )) = V Y,α ( φ ) , where D α,Y × S is the Dirac operator induces on the product geometry Y × S by the twisted Dirac operator D Γ α on Y . Then the associated power spectraas in (4.4) , (4.5) satisfy (4.6) , with λ Y = N/ . Inflation potential in the heat kernel approach.
Let us now con-sider inflation potentials on space-times of the form M × S β for M compactoriented Riemannian and odd-dimensional, arising from general almost com-mutative triples over M .Let D is a symmetric Dirac-type operator on a self-adjoint Clifford modulebundle V → M , with M compact oriented Riemannian and odd-dimensional,and let /D β be the Dirac operator with simple spectrum β ( Z + ) on thetrivial spinor bundle C → S S β → S β . We may immediately generalise theconstruction of [8, § D M × S β on the self-adjoint Clifford module bundle (cid:16) V ⊠ S S β (cid:17) ⊕ → M × S β .Hence, we may define an inflation potential V M : C ∞ (cid:16) M × S β (cid:17) → R by V M ( φ ) := Tr (cid:16) h (( D M × S β + φ ) / Λ ) (cid:17) − Tr (cid:16) h ( D M × S β / Λ ) (cid:17) , where h = L [ ψ ] for ψ ∈ S (0 , ∞ ); note that D M × S β + φ has heat traceTr e − t D M × S β + φ ! = 2Tr (cid:16) e − t /D β (cid:17) Tr (cid:16) e − tD (cid:17) e − φ t for φ locally constant.Let Γ → f M → M , e V → f M , V → M , e D , D , α , F → M and D α bedefined as in Subsection 3.2, with M and f M odd-dimensional, generalisingthe discussion above of Γ → S → Y . On the one hand, we may form oddDirac-type operators e D ˜ M × S β , D M × S β , and D α,M × S β from e D , D and D α ,respectively, as above. On the other hand, if one trivially extends the actionof Γ on f M to f M × S β and the action on e V → f M to (cid:16) V ⊠ S S β (cid:17) ⊕ → M × S β ,then D f M × S β becomes a Γ-equivariant Dirac-type operator on (cid:16) V ⊠ S S β (cid:17) ⊕ ,and the constructions of Subsection ?? applied to the Γ-equivariant Dirac-type operator D f M × S β reproduce precisely the Dirac-type operators D M × S β ,and D α,M × S β .Now, let V f M , V M , and V M,α denote the inflation potentials correspondingto ˜ D , D , and D α , respectively, which we all view as nonlinear functionalson C ∞ ( f M × S β , R ) Γ ∼ = C ∞ ( M × S β , R ). Then, since we also have that OUPLING TO MATTER, SPECTRAL ACTION, COSMIC TOPOLOGY 29 D f M × S β + φ is the lift of D M × S β + φ and (cid:18) D f M × S β + φ (cid:19) ⊗ N is the liftof D M × S β ,α + φ , Theorems 3.6 and 3.10, mutatis mutandis , therefore implythe following result: Proposition 4.2.
Let Γ → ˜ M → M , ˜ V → X , ˜ D and α : Γ → U ( N ) beas above, and define V , D , V α and D α as above. Let V ˜ M , V M and V M,α bedefined as above. Then (4.7) V M ( φ ) = 1 V f M ( φ ) + O (Λ −∞ ) , as Λ → + ∞ ,and (4.8) V M,α ( φ ) = N V f M ( φ ) + O (Λ −∞ ) , as Λ → + ∞ . This, therefore, explains the factor λ Y in Equations 4.3 and 4.5 and Propo-sition 4.1. Acknowledgment.
This work is partially supported by NSF grantsDMS-0901221, DMS-1007207, DMS-1201512, PHY-1205440. The secondauthor thanks Jos´e Luis Cisneros-Molina for a useful conversation and forpointing out to us the results of [6], and the first author thanks MathaiVarghese for useful conversations.
References [1] J. Boeijink, W.D. van Suijlekom,
The noncommutative geometry of Yang-Mills fields ,J. Geom. Phys., Vol. 61 (2011) 1122–1134.[2] T. van den Broek, W.D. van Suijlekom,
Supersymmetric QCD and NoncommutativeGeometry , Comm. Math. Phys., Vol. 303 (2011), no. 1, 149–173.[3] M. Buck, M. Fairbairn, M. Sakellariadou,
Inflation in models with conformally coupledscalar fields: An application to the noncommutative spectral action , Phys. Rev. D 82(2010) 043509 [14 pages][4] B. ´Ca´ci´c,
Moduli spaces of Dirac operators for finite spectral triples , in “QuantumGroups and Noncommutative Spaces: Perspectives on Quantum Geometry” (eds.M. Marcolli, D. Parashar), Vieweg Verlag, 2010.[5] B. ´Ca´ci´c,
A reconstruction theorem for almost-commutative spectral triples ,Lett. Math. Phys., Vol. 100 (2012), no. 2, 181–202.[6] J. Cisneros-Molina,
The η -Invariant of Twisted Dirac Operators of S / Γ, GeometriaeDedicata 84, (2001) 207–228.[7] A. Chamseddine, A. Connes,
The spectral action principle . Comm. Math. Phys. 186(1997), no. 3, 731–750.[8] A. Chamseddine, A. Connes,
The uncanny precision of the spectral action , Com-mun. Math. Phys. 293 (2010) 867–897.[9] A. Chamseddine, A. Connes,
Spectral Action for Robertson-Walker metrics ,arXiv:1105.4637.[10] A.H. Chamseddine, A. Connes, M. Marcolli,
Gravity and the standard model withneutrino mixing . Adv. Theor. Math. Phys. 11 (2007), no. 6, 991–1089. [11] A. Connes,
Gravity coupled with matter and the foundation of non-commutative ge-ometry . Comm. Math. Phys. 182 (1996), no. 1, 155–176.[12] A. Connes,
Geometry from the spectral point of view . Lett. Math. Phys. 34 (1995),no. 3, 203–238.[13] A. Connes,
On the spectral characterization of manifolds , arXiv:0810.2088.[14] A. Connes, M. Marcolli,
Noncommutative geometry, quantum fields and motives , Col-loquium Publications, Vol. 55, American Mathematical Society, 2008.[15] A. L. Dur´an, R. Estrada,
Strong moment problems for rapidly decreasing smoothfunctions , Proc. Am. Math. Soc. 120 (1994), no. 1, 529–534.[16] P. B. Gilkey,
Invariance theory, the heat equation, and the Atiyah-Singer index theo-rem , 2nd ed., CRC Press, 1995.[17] A. Kahle,
Superconnections and index theory , Ph.D. thesis, University of Texas atAustin, 2008.[18] D. Kolodrubetz, M. Marcolli,
Boundary conditions of the RGE flow in the noncom-mutative geometry approach to particle physics and cosmology , Phys. Lett. B, Vol.693(2010) 166–174.[19] J.E. Lidsey, A.R. Liddle, E.W. Kolb, E.J. Copeland, T. Barreiro, M. Abney,
Re-constructing the Inflaton Potential – an Overview , Rev. Mod. Phys (1997) Vol. 69,373–410.[20] M. Marcolli, E. Pierpaoli,
Early universe models from noncommutative geometry .Adv. Theor. Math. Phys., Vol. 14 (2010), no. 5, 1373–1432.[21] M. Marcolli, E. Pierpaoli, K. Teh,
The spectral action and cosmic topology .Comm. Math. Phys. 304 (2011), no. 1, 125–174.[22] M. Marcolli, E. Pierpaoli, K. Teh,
The coupling of topology and inflation in noncom-mutative cosmology , Comm. Math. Phys., Vol. 309 (2012), no. 2, 341–369.[23] W. Nelson, J. Ochoa, M. Sakellariadou,
Constraining the noncommutative SpectralAction via astrophysical observations
Phys. Rev. Lett., Vol. 105 (2010) 101602 [5pages][24] W. Nelson, J. Ochoa, M. Sakellariadou,
Gravitational waves in the Spectral Action ofnoncommutative geometry , Phys. Rev. D, Vol.82 (2010) 085021 [15 pages][25] W. Nelson, M. Sakellariadou,
Cosmology and the noncommutative approach to theStandard Model , Phys. Rev. D, Vol. 81 (2010) 085038 [7 pages][26] W. Nelson, M. Sakellariadou,
Natural inflation mechanism in asymptotic noncommu-tative geometry
Phys. Lett. B, Vol. 680 (2009) 263–266.[27] R. Nest, E. Vogt, W. Werner,
Spectral action and the Connes-Chamseddine model ,in “Noncommutative Geometry and the Standard Model of Elementary ParticlePhysics,” (eds. F. Scheck, H. Upmeier, W. Werner), Springer, 2002.[28] A. Niarchou, A. Jaffe,
Imprints of spherical nontrivial topologies on the cosmic mi-crowave background , Phys. Rev. Lett. 99, 081302 (2007)[29] P. Olczykowski, A. Sitarz,
On spectral action over Bieberbach manifolds , Acta Phys.Polon. B42 (2011) 1189–1198.[30] A. Rennie, J.C. Varilly,
Reconstruction of manifolds in noncommutative geometry ,arXiv:math/0610418.[31] A. Riazuelo, J.P. Uzan, R. Lehoucq, J. Weeks,
Simulating Cosmic Microwave Back-ground maps in multi-connected spaces , Phys. Rev. D 69, 103514 (2004)[32] A. Riazuelo, J. Weeks, J.P. Uzan, R. Lehoucq, J.P. Luminet,
Cosmic microwavebackground anisotropies in multiconnected flat spaces , Phys. Rev. D 69, 103518 (2004)[33] T.L. Smith, M. Kamionkowski, A. Cooray,
Direct detection of the inflationary gravi-tational wave background , Phys. Rev. D. (2006), Vol. 73, N.2, 023504 [14 pages].[34] E.D. Stewart, D.H. Lyth,
A more accurate analytic calculation of the spectrum ofcosmological perturbations produced during inflation , Phys. Lett. B 302 (1993) 171–175.
OUPLING TO MATTER, SPECTRAL ACTION, COSMIC TOPOLOGY 31 [35] K. Teh,
Nonperturbative Spectral Action of Round Coset Spaces of SU (2),arXiv:1010.1827, to appear in Journal of Noncommutative Geometry.[36] W.D. van Suijlekom, Renormalization of the spectral action for the Yang-Mills system ,JHEP 1103 (2011) 146–152.[37] W.D. van Suijlekom,
Renormalization of the asymptotically expanded Yang-Mills spec-tral action , arXiv:1104.5199, to appear in Communications in Mathematical Physics.
Department of Mathematics, California Institute of Technology, Pasadena,CA 91125, USA
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