Spectral Action for Scalar Perturbations of Dirac Operators

Transcription

1 Lett Math Phys 2011) 98: DOI /s Spectral Action for Scalar Perturbations of Dirac Operators ANDRZEJ SITARZ and ARTUR ZAJA C Institute of Physics, Jagiellonian University, Reymonta 4, Kraków, Poland. andrzej.sitarz@uj.edu.pl Received: 23 January 2011 / Revised: 7 April 2011 / Accepted: 26 April 2011 Published online: 21 July 2011 The Authors) This article is published with open access at Springerlink.com Abstract. We investigate the leading terms of the spectral action for odd-dimensional Riemannian spin manifolds with the Dirac operator perturbed by a scalar function. We calculate first two Gilkey de Witt coefficients and make explicit calculations for the case of n-spheres with a completely symmetric Dirac. In the special case of dimension 3, when such perturbation corresponds to the completely antisymmetric torsion, we carry out the noncommutative calculation following Chamseddine and Connes J Geom Phys 57:121, 2006) and study the case of SU q 2). Mathematics Subject Classification 2000). 58B34, 81T75. Keywords. spectral geometry, noncommutative geometry. 1. Introduction In the setup of noncommutative differential geometry developed from Alain Connes idea of spectral triples [4], the fundamental ingredient of the construction is based on the spin geometry and the Dirac operator. Although more general Diractype operators are also admitted, the fundamental theorems [3,10] give equivalence only between commutative spectral triples and usual spin geometries. Usually, the Dirac operator on spin manifold is taken to be the operator, which comes from the Levi-Civita connection on the tangent bundle. It appears, however, that a wide class of generalized operators, which come, for example, from connections with torsions also satisfies the axioms of spectral triples. Although in the classical differential geometry this might be irrelevant as we can easily pass to the language of differential geometry and select only such Diracs which come from the torsion-free connections, in the pure noncommutative situation this method is no longer available. A. Sitarz was partially supported by MNII grants 189/6.PRUE/2007/7 and N A. Zając supported by a project operated within the Foundation for Polish Science IPP Programme Geometry and Topology in Physical Models co-financed by the EU European Regional Development Fund, Operational Program Innovative Economy

2 334 ANDRZEJ SITARZ AND ARTUR ZAJA C Therefore, the class of Dirac operators with torsion is much larger than the classical Diracs and their perturbation. Classically, if one assumes that torsion is totally antisymmetric which has some natural geometric justification) the torsion Dirac operators appear only for dimension superior than 2. Whether this is the case in the noncommutative situation is not clear. In three dimensions the totally antisymmetric torsion tensor has only one component; hence, the perturbation of the Dirac operator is only by a function. We show that such perturbations are admissible in every odd dimension and calculate the correction to the spectral action in the classical case as well as in the noncommutative situation the latter in dimension 3 with the example of SU q 2)). This extends recent calculation of spectral action for compact manifolds with torsion with and without boundaries) which were carried out in [12,14] though in dimensions higher than 3 the scalar perturbation has no clear geometrical meaning SCALAR PERTURBATIONS OF SPECTRAL TRIPLES Let us begin with the definition of a real spectral triple. DEFINITION 1.1. Let be A be an algebra, H a Hilbert space and π afaithful representation of A on H. The geometric data of a real spectral triple is given by A,π,H, D, J) where D is a selfadjoint unbounded operator on H, J is antilinear unitary operator, an integer modulo 8 dimension of the spectral triple) and the following relations: a A, [D,πa)] is a bounded operator, a, b A, [J 1 πa)j,πb)]=0, CC) JD= ɛ D DJ,ɛ D =±1 depending on the dimension of the triple, a, b A, [ J 1 πa)j, [D,πb)] ] = 0, J 2 = ɛ J 1,ɛ J =±1 depending on the dimension of the spectral triple, if the dimension of the triple is even, then there exists γ = γ,suchthat: γ 2 = 1, Dγ = γ D, [γ,πa)]=0, Jγ = ɛ γ γ J, where ɛ γ =±1 depends also on the dimension of the spectral triple. In addition to the listed above conditions the spectral triple must satisfy a series of additional requirements: for their list and meaning, as well as the list of all signs, we refer to the literature [4,10]. In the case when A is an algebra of smooth functions on a manifold, we replace the commutator condition CC) by demanding that πa) = J 1 πa)j for every a A. One of the earliest results, which motivated the construction of spectral geometry, was the so-called reconstruction theorem later, under certain conditions generalized to equivalence; see [4]), which stated that for a compact spin manifold M,

3 SPECTRAL ACTION FOR SCALAR PERTURBATIONS 335 with the algebra A = C M), H being the Hilbert space of square integrable sections of the spinor bundle, J the implementation of the involution in the Clifford algebra, γ the natural Z 2 grading of the Clifford algebra and D the Dirac operator associated with the Levi-Civita connection; the data A, H, D, J,γ) form a commutative spectral triple. Now, we have: PROPOSITION 1.2. With the same data as in the definition above, in the case of odd KO dimension, all relations are satisfied if we replace D by: D = D + + ɛ D J J 1, where = πφ), for a selfadjoint element φ = φ of the algebra A. Proof. To verify our claim, it suffices to verify the algebraic relations, which depend on the Dirac operator. For the remaining part of the axioms, spectral and analytic, it suffices to observe that is a bounded operator, and hence, D is a bounded perturbation of D. First,wecalculateJD : JD = J D + + ɛ D J J 1) = ɛ D DJ+ J + ɛ J ɛ D J 1 ) = ɛ D D + ɛ D J J 1 + J. and then the order one condition: [ [D,πa)], JbJ 1] [[ = = ] D + + ɛ D J J 1,πa), JbJ 1] [ [, πa)], JbJ 1] = 0, where we have used that A and conjugation by J maps the elements of A to its commutant. Remark 1.3. Note that in the case of manifolds, when J maps elements of A to A, the perturbation + ɛ D J J 1 = 1 + ɛ D ), and it vanishes identically in the KO dimension 1 and 5 modulo The Spectral Action for Manifolds with D We shall calculate here the leading three coefficients of the spectral action for scalar perturbations of the Dirac operator on odd-dimensional compact spin manifolds. We do not assume any conditions on the reality structure; hence, the perturbation by a real function is possible in any odd dimension.

4 336 ANDRZEJ SITARZ AND ARTUR ZAJA C Therefore, D =D +, =C M, R). Let us recall the Schrödinger Lichnerowicz formula, applied to D : D ) 2 = D 2 + D + D + 2 = R +[D, ]+2 D + 2, where is the spinorial Laplacian. In order to calculate the leading term parts of the spectral action, we use the standard techniques to calculate the heat-kernel coefficients for the second-order differential operators of the Laplace type over spin manifolds. We used both the explicit formulas obtained by Barth [1], as well as the general results presented by Vassilevich [19]. In our notation, d is the dimension of the manifold, and n denoted the dimension of the fibres of the spinor bundle. The result, up to terms, which are total divergence which vanish, and since we consider manifolds without boundary) is: [a 1 ] = 4π) d 2 n 1 ) R + d 1)φ2, 12 [a 2 ] = 4π) d n R2 4R ij R ij 7 2 R ijklr ijkl 1) + 120d 1)d 3)φ d)r φ 2 ) +120d 1) i φ) i φ). The above result has been obtained earlier for [a 1 ]) by many authors [8,11,15,18] and also[20,21] in the case of dimension 3, where the scalar perturbation corresponds to torsion. The [a 2 ] coefficient in dimension 3 is again a special case of Dirac operator with an antisymmetric torsion. An interesting situation happens in dimension 3. LEMMA 2.1. The heat kernel coefficients for scalar perturbation of Dirac in dimension d = 3 read: [a 1 ] = 24π) ) 12 R + 2φ2, [a 2 ] = 4π) ) 2) 3 iφ) i φ. Proof. Indeed, observe that many terms vanish in the case of d = 3. Additionally, the Weyl tensor and so its square) as well as the Gauss Bonnet integrand must vanish: R ijkl R ijkl 2R ij R ij R2 = 0, R ijkl R ijkl 4R ij R ij + R 2 = 0. 3)

5 SPECTRAL ACTION FOR SCALAR PERTURBATIONS 337 Using it, we can see that the terms, which depend only on the Riemann and Ricci tensors and the scalar curvature, add up to zero. Hence, only the kinetic term for remains. Of course, in the case of three-dimensional manifolds, the coefficient [a 2 ] will not be present in the leading terms of the perturbative expansion of the spectral action with respect to the cut-off parameter. However, when considering, for example, the spectral action on M S 1 with the product geometry, it will appear as the scale invariant term EXPLICIT SPECTRAL ACTION FOR SPHERES In the special case of the d-dimensional spheres still d being an odd number), we can independently calculate the perturbative expansion of the spectral action directly from the spectrum of the Dirac. The method, based on the Poisson summation formula, which was first presented in [2] and then used by [17] to study cosmological aspects of the spectral action will allow us to have explicit formulas for the case of φ = const. Let us recall that the spectrum of the equivariant Dirac operator on a unit sphere of an odd dimension d is given by ) d λ ± n) =± 2 + n, Nn) = 2 d 1 2 n + d 1)!, n 0, n!d 1)! where λ n is the eigenvalue and Nn) denotes the multiplicity. When we consider the scalar perturbation of D by a constant t R, the spectrum is accordingly changed to: λ ± n, t) =± d d n ± t), 2 2 n + d 1)! Nn) =, n 0. n!d 1)! Let us write an explicit formula we have for the spectral action defined by the cut-off function f : S S d t) = ) )) n + d Nn) f 2 + t n d + f 2 + t. n 0 Observe that: λ n, t) = n d 2 + t = n d) + d 2 + t = λ + n d, t). On the other hand, rewriting the formula for Nn): Nn) = 2 d 1 2 n + d 1)n + d 2) n + 1), d 1)!

6 338 ANDRZEJ SITARZ AND ARTUR ZAJA C we easily see that Nn) = N n d), as d 1isevend is odd). Therefore, we can rewrite the spectral action as S S d t) = ) n + d Nn) f 2 + t, n Z where we can sum over all integers, as the terms for n = 1,..., d + 1 do not appear as the function counting multiplicities Nn) vanishes at these points. Now, we use the Poisson summation formula to the sum over all integers. Define gx) = f x + α),α R. Wehave: ĝk) = 1 2π and f ˆ p) k) = 1 2π where ˆ gx)e ikx = 1 2π ix) p f x)e ikx = 1 2π f x + α)e ikx = ˆ f k)e ikα. ix + α)) p f x + α)e ikx e ikα, f p) k) denotes k ) p f ˆk). Using now the expansion in powers of up to terms of order o 1 ): n g = ĝ0) + o ) 1 ), n Z and for the derivatives: n in) k g = ) k+1 ĝ k) 0) + o 1 ), n Z We can now use the following lemma in order to calculate the three leading terms of the perturbative spectral action. LEMMA 2.2. For each t R we have: n + d 1)n + d 2) n + 1) = n + d ) d t d 1)t n + d2 ) d 2 + t d 1)d 2)t2 1 )n 24 dd 1)d 2) + d2 ) d 3 + t dd 1)d 2)d 3)t 1 )n 6 d 1)d 2)d 3)t3 + d2 ) d 4 + t d 1)d 2)d 3)d 4)t dd 1)d 2)d 3)d 4)t dd 1)d 2)d 3)d 4)5d+2) ) n + d ) d t + o n d 5).

7 SPECTRAL ACTION FOR SCALAR PERTURBATIONS 339 We skip the proof based on explicit calculations. For an even function f, the terms of the order d+1 2k shall vanish as the value of the Fourier transform at 0 is just the integral of an odd function over R). Therefore, the first three nontrivial terms are up to the overall sign 1) d 1 2, which arises from the Fourier transform): S S d t)= d ˆ f d 1) 0) d 2 d 1)d 2) 2 t 2 d ) 12 d 4 d 1)d 2)d 3)d 4) + 24 f ˆ d 3) 0) t 4 d 2 t2 + d5d+2) 240 ) ˆ f d 5) 0). 4) We can easily compare this result with the one obtained previously, which could be treated as the independent test that the results are correct. Let us recall that for the unit d-dimensional sphere the following identities are true: R ijkl ) 2 = 2dd 1), R ij ) 2 = dd 1) 2, R = dd 1). Therefore, using the formula 1) we calculate: [a 1 ] 1 ) dd 1) + d 1)t2 = d 1) 12 [a 2 ] 120d 1)d 3)t 4 60dd 1)d 3)t ) 2 dd 1) 7dd 1) 4dd 1)2 = 120d 1)d 3) t 2 d 12 t 4 d 2 t2 + d5d+2) ). 240 ), As we can see, it is exactly the result obtained above in 4). 5) 2.2. NONCOMMUTATIVE SPECTRAL ACTION IN DIMENSION 3 Assume that we have a regular real spectral triple A, H, D) with a simple dimension spectrum {1, 2, 3} which we shall later abbreviate by saying that its dimension is 3). By analogy with the commutative example, we propose to call a perturbation of the Dirac operator by a selfadjoint element of the algebra, A =,a torsion-perturbed Dirac, D = D + + J J 1 recall that JD= DJ in dimension 3). It is clear that D is still a good Dirac operator for the real spectral triple. 1 Let us recall the following theorem of Connes and Chamseddine [2], which we state here in a more general version. 1 The only possible exception is the axiom of the existence of the Hochschild cycle. However, this fails also in the case of the usual perturbation of the Dirac by a one-form in some noncommutative examples.

8 340 ANDRZEJ SITARZ AND ARTUR ZAJA C THEOREM 2.3 see Theorems 1.145, of [6]). If A, D, H) is a regular spectral triple with a simple dimension spectrum consisting of positive integers {1, 2,...,d}, then the spectral action has the following expansion in powers of the scale coefficient : S D = d j=1 f d j d j D d+ j + f 0)ζ D 0) +, 6) where the coefficients f k depend on the function f through: f k = 0 f u)u k 1 du, and the symbol denotes the noncommutative integral: T := Res ζ D T s) = Res Tr T D s). s=0 s=0 The difference in the scale-invariant part of the action between D and a perturbation of D is: ζ D+X 0) ζ D 0) = d 1) k XD 1) k. 7) k k=1 Although the last part of the theorem was originally proved for d = 4 and for X being a one-form, it is evident from the proof in [2] that it generalizes for any bounded X and any dimension spectrum of the assumed form. We apply now the above theorem to our specific case, when the dimension is 3 and the perturbation of the Dirac is of the assumed scalar type. We have: PROPOSITION 2.4. The components of the full perturbative spectral action on a real regular spectral triple A, H, D ) of dimension 3, with a Dirac operator perturbed by a positive torsion term, are: a) D 3 = D 3. b) D 2 = D 2 4 F D 3. c) D 1 = D 1 2 F D D J J 1 D 3. d) ζ D 0) ζ D 0) = 2 D J J 1) D 2 + [D, ] + J J 1) D where F denotes the sign of D, F = D D 1. 3 D J J 1 D 3.

9 SPECTRAL ACTION FOR SCALAR PERTURBATIONS 341 Proof. We use the noncommutative pseudodifferential calculus developed by Connes and Moscovici in [5]. For simplicity, we assume that the kernels of D and D + + J J 1 are empty if it is not the case, then one can correct D and D by a finite rank operator, which is a projection on the kernel, for details of such procedure see [9]). To facilitate computations, we use φ = + J J 1. Let us briefly recall main definitions and properties. We say that an operator T is smooth if the map: t e it D Te it D, is smooth, and we denote by OP 0 the algebra of smooth operators. We let OP s be the space of operators P such that D s P is in OP 0. We consider the algebra A) of operators, which have an expansion: P b r D r + b r 1 D r 1 +, where the operators b j are from the algebra generated by A, JAJ and all commutators with D and D we assume that A and JAJ 1 are OP 0 ). The algebra structure of operators of arbitrary order is established by the following relation: D α b = k=0 Ɣα + 1) Ɣα k + 1)Ɣk + 1) δk b) D α k, 8) where δb) =[ D, b]. Using this rule, we see for example that [F, a] is in OP 1 for a A. Toshowit, observe that D 1 [D, a] is in OP 1. Expanding it further with the use of 8), we see that it is [F, a] modulo OP 2. 2 We shall compute the leading terms of the expansion for the operators, which appear in the spectral action. Let us begin with D ) 2 : D ) 2 = D Dφ D 2 + φ D 1 + φ 2 D 2) 1. Using the series expansion of the last term and the rule 8) we obtain, modulo operators of order OP 5 : D ) 2 = D 2 2φ D 3 [D,φ]+φ 2) D 4 + 4φ 2 D 4 + 2[D 2,φ]D 5 mod OP 5. Observe that our spectral triple has metric dimension 3, and therefore, it is sufficient to compute the expansion of all operators modulo OP 4. Assume that D 1 = D 1 + A D 2 + B D 3 mod OP 4, 2 In some cases, [F, a] could be lower order, like in the SU q 2) case discussed later, where it is actually a smoothing operator see [13]).

10 342 ANDRZEJ SITARZ AND ARTUR ZAJA C where A, B are in OP 0. Using the algebra rule of the pseudodifferential calculus, we easily obtain: D 2 = D 2 + 2A D 3 + A 2 δa) + 2B) D 4 mod OP 5, and if we compare it with the earlier result we see that we must have: A = φf, B =[D,φ]+φ 2. Therefore, in the end, we have: D 3 = D 3 mod OP 4, D 2 = D 2 2φF D 3 mod OP 4, D 1 = D 1 φf D 2 + [D,φ]+φ 2) D 3 mod OP 4, where we have also used that [F, a] is in OP 1 for a A. If we substitute now φ = + J J 1, D = F D and use both trace property of the noncommutative integral as well as its invariance with respect to conjugation by J we obtain the results a) c). To see d) we apply directly 7), where we just need to substitute X = + J J 1. Using that the noncommutative integral is invariant under conjugation by J and the simple relation: D 1 = D 1 D 1 [D, ]D 1 we recover d) in a straightforward way. It is interesting to see the application of the above calculations in the commutative case of a three-dimensional manifold M. Then = = J J 1 is a function on M. To see that the spectral action simplifies significantly in this case we first observe: LEMMA 2.5. Let D be a Dirac operator on M and F = signd). Then ψ F D 3 = 0, for any function ψ C M). Proof. Indeed, since we are dealing with pseudodifferential operators over a manifold both f and F are of order 0, D is of order 1), we can use the symbols and the relation of the noncommutative integral with the Wodzicki residue. Infact,whenwerewriteψ F D 3 as ψ DD 4, we see that the principal symbol of this expression is scalar multiple of the symbol of D as the leading symbol of D 4 is scalar). Hence, its Clifford trace vanishes.

11 SPECTRAL ACTION FOR SCALAR PERTURBATIONS 343 Finally, we can show: PROPOSITION 2.6. The leading terms of the perturbative expansion of the spectral action for the Dirac operator with torsion on the three-dimensional manifold are: 3 term: VolM) 1 term: 1 2π M ) R M In particular, there are no 2 terms and scale-invariant contributions. 9) Proof. First, observe that using Lemma 2.5 and the arguments there we can at once say that these terms must vanish: F D 3, F D 2. Furthermore, since D 1 = F D 1, 3 D 3 = 3 F D 3, the above integrals must vanish as well. A bit more work is necessary to show that 2 D 2 vanishes. Here, the best way is to use the result of direct calculations by Kastler [16]. Using the explicit form of the paramatrix for D 2, which he derived, we see that its component of order 3,σ 3 ξ, x) is an odd polynomial in ξ. Therefore, the above integral vanishes as well. Finally, from the trace property of the noncommutative integral, we immediately have [D, ]D 3 = 0 in dimension 3. For the remaining components of the spectral action, using the identities: D 3 = 1 π 2 VolM), ψ D 3 = 1 π 2 ψ, M D 1 = 1 24π 2 Rg), M where ψ C M) and Rg) is the scalar curvature of M, we obtain the result from the proposition recall that in the case of a spectral triple over a manifold of dimension 3 one has J J 1 = = ). 3. The Quantum Sphere SU q 2) We shall show here that the spectral action terms for the quantum deformation of the three-sphere are not much different from the undeformed case. We start with the undeformed invariant Dirac operator and the spectral triples as described in [7] we omit here details of the notation and construction of spectral geometry).

12 344 ANDRZEJ SITARZ AND ARTUR ZAJA C We take a torsion term being a finite sum of homogeneous polynomials in a, a, b: = C α,β,γ a α b β b ) γ, 10) where α Z if α<0 then we take a ) α )andβ,γ N. For simplicity, we do not take into account the condition of J-reality; thus, we restrict ourselves only to D+. If is selfadjoint, then: C α,β,γ = q αγ+β) C α,γ,β. PROPOSITION 3.1. The coefficients of the leading terms of the perturbative spectral action on a real spectral triple with arbitrary torsion over SU q 2) are: a) b) c) D 3 = 2, D 2 = 0, D 1 = D 3, where we have used that for the standard Dirac operator on S 3 : D 3 = 2, D 1 = 1 2. For the form of as assumed in 10) we have: 2 D 3 = C α,0,0 2. α Proof. Using the results for the classical Dirac operator and the noncommutative expansion from Proposition 2.4, we obtain a) and the part of b). The vanishing of the terms linear in in b) and c) is a consequence of Theorem 3.4 of [13]. To calculate the noncommutative integral of 2 D 3, we use the explicit form of it from [13]. Remark 3.2. Observe that unlike in the classical case the second term of the spectral action has no single minimum as the moduli space of scalar perturbations, for which it reaches the minimum value is infinite dimensional as it contains all functions, which have nontrivial dependence on b or b ). Therefore, the proposed spectral condition for the vanishing of torsion makes no sense in the considered noncommutative geometry of SU q 2).

13 SPECTRAL ACTION FOR SCALAR PERTURBATIONS 345 Let us turn our attention to the terms, which vanish identically in the classical situation. PROPOSITION 3.3. The scale invariant part of the action does not vanish, and we have: ζ D 0) ζ D 0) = 1 α C α,l n,m C α,l m,n [ ] 1) k q kk 1) α 4 2 k q 2 1 q 2k+l), where [ ] n k q 2 = α Z l 0 0 m,n l is the quantum binomial. ni=1 1 q 2i ) ki=1 ) n k ), 1 q 2i ) i=1 1 q2i ) k=0 Proof. First, let us see that using the same expansion as from the proof of proposition 2.4 the scale invariant terms are: D D 2 1 [D, ]D D Let us begin with the first term, D 1. It is clear that only the terms with bb ) n can possibly contribute. The shortest way to show that the integral vanishes, bb ) n D 1 = 0, is to use the property of the following algebra automorphism of SU q 2): ρa) := a, ρa ) := a,ρb) := b,ρb ) := b. 11) Using the Lemma 5.17 of [13], we see that for any homogeneous polynomial in a, a, b, b the noncommutative integral: pa, a, b, b ) D k = 1) k pρa), ρa ), ρb), ρb )D k, is up to sign) invariant with respect to ρ. Since in our case k =1 andρbb )=bb the integral must vanish. In fact, this proves also that the last term, which proportional to D 3, vanishes as well. For the second term of the scale invariant part, we again use Lemma 5.14 of [13]. First of all, we calculate 2 : 2 = α,β,γ,α,β,γ C α,β,γ C α,β,γ aα b β b ) γ a α b β b ) γ.

14 346 ANDRZEJ SITARZ AND ARTUR ZAJA C Then the diagonal part is the sum of elements, where α = α and β + β =γ +γ : 2 diag C α,m,l n C α,l m,n a α b m b ) l n a α b l m b ) n = α Z α Z l 0 0 m,n l l 0 0 m,n l C α,l n,m C α,l m,n a α a α bb ) l. Next, the commutation relations for SU q 2) yields: a α a α = 1 q 2α 2 bb ) 1 q 2α 4 bb ) 1 bb ) α [ ] = 1) k q kk 1) α bb ) k. k q 2 k=0 Next, using it and Lemma 5.14 we obtain: 2 D 2 = α [ ] ) C α,l n,m C α,l m,n 1) k q kk 1) α 4 k α Z l 0 0 m,n l k=0 q 2 1 q 2k+l). The next component of the scalar invariant part vanishes: [D, ] D 3 = 0. This is an easy consequence of the fact that one can rewrite the above expression as δ ) D 3 = 1 δ 2) D 3, 2 and then, using expansion of diagonal terms of 2 we obtain the desired result. 4. Conclusions We have seen that in the case of three-dimensional manifolds the torsion term contributes only to the spectral action through a quadratic term. Therefore, the postulate to restrict to torsion-free geometries by minimizing at a fixed metric the term, which comes from the noncommutative integral of D n+2, appeared quite plausible. Indeed, it works in the classical commutative case and might be extended to other torsion-type perturbation of the Dirac operators. The scalar-type perturbations considered in this paper only in dimension 3 have the interpretation of torsion and appear to have no direct geometrical meaning in higher dimensions. We have shown that in the case of lowest possible dimension for which they are nontrivial the next term spectral action includes coupling of the scalar field to the scalar curvature and the standard kinetic term, thus making the field dynamical. In higher dimensions, an extra selfinteraction quartic term

15 SPECTRAL ACTION FOR SCALAR PERTURBATIONS 347 appears, thus making it possible that in the first three terms one might have a nonzero vacuum value of the scalar perturbation. The analysis of the spectral action for the genuine noncommutative example yields even more interesting result. Clearly, for the SU q 2), it is no longer possible to eliminate the torsion by minimizing the D 1 term of the spectral action. In addition, unlike in the classical case, the scale invariant terms do not vanish. It appears that there is no reason to eliminate the scalar perturbation from the considerations in the setup of noncommutative geometry. Its geometrical meaning and consequences for the model building in physics are still to be discussed. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original authors) and source are credited. References 1. Barth, N.H.: Heat kernel expansion coefficient. I. An extension. J. Phys. A Math. Gen. 20, ) 2. Chamseddine, A., Connes, A.: Inner fluctuations of the spectral action. J. Geom. Phys. 57, ) 3. Connes, A.: On the spectral characterization of manifolds. arxiv: Connes, A.: Noncommutative Geometry and Reality. J. Math. Phys. 36, ) 5. Connes, A., Moscovici, H.: The local index formula in noncommutative geometry. Geom. Funct. Anal. 52) 1995) 6. Connes, A., Marcolli, M.: Noncommutative Geometry, Quantum Fields and Motives. American Mathematical Society, New York 2007) 7. Da browski, L., Landi, G., Sitarz, A., van Suijlekom, W., Várilly, J.C.: The Dirac operator on SU q 2). Commun. Math. Phys. 259, ) 8. Donnelly, H.: Heat equation asymptotics with torsion. Indiana Univ. Math. J. 341) 1985) 9. Essouabri, D., Iochum, B., Levy, C., Sitarz, A.: Spectral action on noncommutative torus. J. Noncommut. Geom. 2, ) 10. Gracia-Bondía, J.M., Várilly, J.C., Figueroa, H.: Elements of Noncommutative Geometry. Birkhäuser, Boston 2001) 11. Gusynin, V.P., Gorbar, E.V., Romankov, V.V.: Heat kernel expansion for nonminimal differential operations and manifolds with torsion. Nucl. Phys. B 362, ) 12. Hanisch, F., Pfffle, F., Stephan, C.: The Spectral Action for Dirac Operators with Skew-Symmetric Torsion. Commun. Math. Phys. 3003), ) 13. Iochum, B., Levy, C., Sitarz, A.: Spectral action on SU q 2). Commun. Math. Phys. 289, ) 14. Iochum, B., Levy, C., Vassilevich, D.: Spectral action for torsion with and without boundaries. arxiv: Kalau, W., Walze, M.: Gravity, noncommutative geometry and the Wodzicki residue. J. Geom. Phys. 16, ) 16. Kastler, D.: The Dirac operator and gravitation. Commun. Math. Phys. 1663), ) 17. Marcolli, M., Pierpaoli, E., Teh, K.: The spectral action and cosmic topology. arxiv:

16 348 ANDRZEJ SITARZ AND ARTUR ZAJA C 18. Nieh, H.T., Yan, M.L.: Quantized Dirac field in curved Riemann-Cartan background. Ann. Phys. 138, ) 19. Vassilevich, D.V.: Heat kernel expansion: user s manual. Phys. Rep. 388C, ) 20. Yajima, S.: Mixed anomalies in 4 and 6 dimensional space with torsion. Prog. Theor. Phys. 792), ) 21. Yajima, S.: Evaluation of the heat kernel in Riemann-Cartan space. Class. Quantum Gravity 139), )