Chiral Asymmetry and the Spectral Action

Transcription

1 Universität Potsdam Frank Pfäffle Christoph A. Stephan Chiral Asymmetry and the Spectral Action Preprints des Instituts für athematik der Universität Potsdam ) 20

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3 Preprints des Instituts für athematik der Universität Potsdam

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5 Preprints des Instituts für athematik der Universität Potsdam ) 20 Frank Pfäffle Christoph A. Stephan Chiral Asymmetry and the Spectral Action Universitätsverlag Potsdam

6 Bibliografische Information der Deutschen Nationalbibliothek Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über abrufbar. Universitätsverlag Potsdam Am Neuen Palais 10, Potsdam Tel.: +49 0) / Fax: 2292 E-ail: verlag@uni-potsdam.de Die Schriftenreihe Preprints des Instituts für athematik der Universität Potsdam wird herausgegeben vom Institut für athematik der Universität Potsdam. ISSN online) Kontakt: Institut für athematik Am Neuen Palais Potsdam Tel.: +49 0) WWW: Titelabbildungen: 1. Karla Fritze Institutsgebäude auf dem Campus Neues Palais 2. Nicolas Curien, Wendelin Werner Random hyperbolic triangulation Published at: Das anuskript ist urheberrechtlich geschützt. Online veröffentlicht auf dem Publikationsserver der Universität Potsdam URL URN urn:nbn:de:kobv:517-opus

7 Chiral Asymmetry and the Spectral Action Frank Pfäffle Christoph A. Stephan ay 7, 2012 Abstract We consider orthogonal connections with arbitrary torsion on compact Riemannian manifolds. For the induced Dirac operators, twisted Dirac operators and Dirac operators of Chamseddine-Connes type we compute the spectral action. In addition to the Einstein-Hilbert action and the bosonic part of the Standard odel Lagrangian we find the Holst term from Loop Quantum Gravity, a coupling of the Holst term to the scalar curvature and a prediction for the value of the Barbero-Immirzi parameter. 1 Introduction In [CC97] Chamseddine and Connes introduced their spectral action which is motivated by eigenvalue counting of a Dirac operator. In the high energy asymptotics the action gives the Einstein-Yang-ills-Higgs Lagrangian of the Standard odel of particle physics in its euclidean version) given the Dirac operator is suitably chosen. The geometric setup for this Dirac operator D Φ is a 4-dimensional compact Riemannian spin manifold without boundary, its spinor bundle Σ, a Hermitian vector bundle H encoding the particle content and a field Φ of endomorphisms of H containing the Higgs field and the Yukawa matrices. This operator D Φ arises naturally in the framework of noncommutative geometry see [Co94] and [C08]), where spinor fields with coefficients in H, i.e. sections of the twisted bundle Σ H, find their correspondence in almost-commutative geometries and the endomorphism field Φ is interpreted as connection of the respective finite space. On the tangent bundle and on the spinor bundle we allow more general connections than the Levi-Civita connection, namely we consider orthogonal connections with arbitrary torsion. By É. Cartan s classification [Ca23], [Ca24], [Ca25]) the torsion then consists of three components: the totally anti-symmetric one, the vectorial one and torsion of Cartan type. The Cartan type torsion does not contribute to the Dirac operator whereas the other two components do. In presence of vectorial torsion the Dirac operator is not symmetric, nevertheless its spectrum is descrete but in general not real-valued. For the physical consequences of the use of torsion connections in Lorentzian geometry we refer to the classical review [HHKN76] and the more recent overview [Sh02] and references therein. In this article we consider operators D Φ of Chamseddine-Connes type induced by orthogonal connections with torsion, and we permit that the twist bundle H may have a chiral asymmetry, i.e. the number of right-handed particles differs form the number of left-handed ones. We aim at computing the spectral action for the restriction of D Φ D Φ to sections of the physically relevant subbundle of Σ H. To this end we derive various Lichnerowicz formulas in order to determine the Seeley-deWitt coefficients which appear in the asymptotics of the spectral action. Our main result is Proposition 3.9. Here the spectral action contains more terms than in the classical situation of [CC97]. Some of them vanish if H has a chiral symmetry, the other ones are zero for the Levi-Civita connection. We find the Holst term from Loop Quantum Gravity [Ro04],[Th07]), a coupling of the Holst term to the scalar athematisches Institut, Georg-August Universität Göttingen, Bunsenstraße 3-5, Göttingen, pfaeffle@uni-math.gwdg.de Fachbereich athematik, Universität Hamburg, Bundesstraße 55, Hamburg, christophstephan@gmx.de Institut für athematik, Universität Potsdam, Am Neuen Palais 10, Potsdam 1

8 curvature and a prediction for the value of the Barbero-Immirzi parameter: it turns out that its value depends on the particle content and its absolute value is at least one. We discuss all these terms in Remarks We hope to improve the readability of this text by transfering technicalities such as curvature identies and calculations for characteristic classes into the appendices A C. Acknoledgement: The authors appreciate financial support from the SFB 647: Raum-Zeit-aterie funded by the Deutsche Forschungsgemeinschaft. We would like to thank Christian Bär, John W. Barrett and Thomas Schücker for their support and helpful discussions. 2 Heat Coefficients for Dirac Operators 2.1 Preliminaries and Definitions Let be a manifold of dimension n 3, equipped with a Riemannian metric g. Let g denote the Levi-Civita connection on the tangent bundle T. We consider an orthogonal connection on T i.e. is compatible with g in the sense that for all vector fields X, Y, Z one has X Y, Z = X Y, Z + Y, X Z, where, denotes the scalar product given by g. Such connections were first investigated by É. Cartan in [Ca23, Ca24, Ca25]. In this article we will use the notation from [PS12, Sect. 2]. It is known see [TV83, Thm. 3.1]) that orthogonal connections have a specific representation. To that end one introduces the space of Cartan type torsion tensors at p { T 3 T p ) = A 3 T p X, Y, Z : AX, Y, Z) + AY, Z, X) + AZ, X, Y ) = 0, where c 12 denotes the trace taken over the first two entries, i.e. for an orthonormal basis e 1,..., e n of T p. c 12 A)Z) = } AX, Y, Z) = AX, Z, Y ) and c 12 A)Z) = 0 n Ae i, e i, Z) i=1 Lemma 2.1 Cor. 2.4 in [PS12]) For any orthogonal connection on the tangent bundle of there exist a unique vector field V, a unique 3-form T and a unique 3, 0)-tensor field S with S p T 3 T p ) for any p such that for any vector fields X, Y one has X Y = g X Y + X, Y V V, Y X + TX, Y, ) + SX, Y, ) 1) where TX, Y, ) and SX, Y, ) are the unique vectors characterised by TX, Y, Z) = TX, Y, ), Z and SX, Y, Z) = SX, Y, ), Z for all Z. For k, 0)-tensors P, Q k Tp we will use the natural scalar product given by P, Q = i 1,...,i k Pe i1,...,e ik ) Qe i1,...,e ik ) 2) where e 1,..., e n is again an orthonormal basis of T p. The induced norm is denoted by P. The curvature tensors of an orthogonal connection are defined as RiemX, Y )Z = X Y Z Y X Z [X,Y ] Z, 3) riemx, Y, Z, W) = RiemX, Y )Z, W, 4) 2

9 ricx, Y ) = R = n rieme i, X, Y, e i ), 5) i=1 n rice i, e i ). 6) i=1 For = g we write Riem g, riem g, ric g and R g, respectively. The relevant calculations for norms of these curvature tensors are postponed into Appendix A. From now on, let us assume that carries a spin structure in order to have spinor fields. In particular, is then orientable. Any orthogonal connection as in 1) induces a unique connection on the spinor bundle Σ see [L89, Chap. II.4] or [Ag06, p. 17f]) which is denote by. Then the Dirac operator associated to is given by Dψ = D g ψ T ψ n 1 2 V ψ 7) where D g is the Dirac operator associated to the Levi-Civita connection and is the Clifford multiplication 1 compare equation 11) in [PS11]). As the Clifford multiplication by the vector field V is skew-adjoint we get that D is a symmetric operator in the space of L 2 -spinor fields if and only if V 0 see [FS79] and [PS12], and [GS87] for the Lorentzian setting). Note that the Cartan type torsion S does not contribute to the Dirac operator D see e.g. [PS12, Lemma 4.7]). As D D is a generalised Laplacian one has a Lichnerowicz formula. Theorem 2.2 Thm. 3.3 in [PS11]) For the Dirac operator D associated to we have D Dψ = ψ Rg ψ dt ψ 3 4 T 2 ψ + n 1 2 div g V )ψ + ) n n) V 2 ψ ) +3n 1) T V ψ + V T) ψ 8) where is the Laplacian associated to the connection X ψ = g X ψ + 3 n 1 2 X T) ψ 2 V X ψ n 1 2 V, X ψ. for any spinor field ψ. We note that the spinor connection is induced by the orthogonal connection on the tangent bundle given by X Y = g X Y + n 1) X, Y V V, Y X) + 3 TX, Y, ). 9) It is a modification of as in 1) obtained by replacing T by 3T, V by n 1)V, and S by zero. For the case of totally anti-symmetric torsion, i.e. V 0, the above Lichnerowicz formula is due to Agricola and Friedrich Theorem 6.2 in [AF04]). 2.2 Heat Coefficients Let be an compact n-dimensional Riemannian manifold without boundary, let E be Riemannian or Hermitian vector bundle. Let H be a generalised Laplacian acting on sections in E, i.e. the principal symbol of H is given by the Riemannian metric of. We assume that H is symmetric in the space of L 2 -sections in E, then the general theory [BGV96, Prop. 2.33]) states that H is essentially selfadjoint. Then there exists a unique connection 1 For the Clifford relations we use the convention X Y +Y X = 2 gx, Y ) for any tangent vectors X, Y, and any k-form θ i 1... θ i k acts on some spinor ψ by θ i 1... θ i k ψ = e i1... e ik ψ. Where θ 1,..., θ n is the dual of the orthonormal frame of tangent vectors e 1,..., e n. 3

10 H on E which is compatible with the Riemannian or Hermitian structure, and there is a unique smooth field E of symmetric endomorphisms of E such that the Bochner formula H = H E 10) holds, where H denotes the Laplacian associated to H see [BGV96, Prop. 2.5]). By elliptic regularity theory H is a smoothing operator, and therefore exp th) possesses a smooth Schwartz kernel k t x, y) E x Ey with t > 0, x, y, called the heat kernel for H. On the diagonal {x = y} the heat kernel has a complete asymptotic expansion, as t ց 0. Theorem 2.3 Lemma and Theorem in [Gi95]) For any integer l 0 there is a smooth field α 2l H) of endomorphisms of E such that the short time asymptotics k t x, x) l 0 t l n/2 α 2l H)x) as t ց 0 holds uniformly in all x. For H and E as in 10) the first of these endomorphism fields are α 0 H)x) = 4π) n/2 id Ex, α 2 H)x) = 4π) n/2 Ex) Rg x) id Ex ), α 4 H)x) = 4π) n/ H E)x) + 60 R g x)ex) Ex) n Ω ij Ω ij i,j=1 +12 g R g )x) + 5 R g x)) 2 2 ric g riem g 2 )id Ex ) where Ω ij = H e i H e j H e j H e i H [e i,e j] is the curvature endomorphism of H on the fibre E x taken with respect to an orthonormal basis e 1,..., e n of T p. Furthermore H induces a connection H on the endormorphism bundle EndE), and H denotes the associated Laplacian. From the asymptotics of the heat kernel on the diagonal one deduces the asymptotics of the heat trace. Corollary 2.4 Let be a compact Riemannian manifold without boundary, let E be Riemannian or Hermitian vector bundle. For a generalised Laplacian H acting on sections in E we choose H and E as in 10). Then one obtains the short time asymptotics of the L 2 -trace of the heat operator, as t ց 0, with a 2l H) = tr E x α 2l H)x)) dx Tr L 2 e th ) t l n/2 a 2l H). 11) l=0 The numbers a 2l H) are called the heat coefficients or the Seeley-deWitt coefficients. Theorem 2.3 gives a 0 H) = 4π) n/2 rke) vol), ) a 2 H) = 4π) n/2 tr Ex Ex)) rke) Rg x) dx, 12) a 4 H) = 4π) n/ R g x) tr Ex Ex)) tr Ex Ex) 2 ) + 30 n ) tr Ex Ωij Ω ij i,j=1 + rke) 5 R g x)) 2 2 ric g riem g 2) dx. 13) 4

11 For the evaluation of a 4 H) one uses tr Ex H E)x)) = g tr E E)x) and the fact that g f)x)dx = 0 for any smooth function f on. Now we assume that the bundle E is Z 2 -graded, i.e. there is an orthogonal decomposition E = E + E which is parallel with respect to H. This means there exists a parallel field P of endomorphisms of E such that P is pointwise the orthogonal projection of E onto E +. In all cases which occur in this article one has PE = EP for the endomorphism E from the Bochner formula 10). Then H is block diagonal with respect to the decomposition of the space of sections ΓE) = ΓE + ) ΓE ) and P commutes with H, i.e. for any section Ψ ΓE) we have PHΨ = HPΨ. We put H + = PH and notice that H + is a generalised Laplacian acting on sections of E +. Its heat kernel is given by k + t x, y) = Py)k tx, y) for t > 0, x, y and for the corresponding coefficients in the heat kernel asymptotics of Theorem 2.3 one has pointwise on and similar formulas for the Seeley-deWitt coefficients in 11). α 2l H + ) = P α 2l H) 14) 3 Connes Spectral Action Principle and Chiral Projections Now we specialise to the case of a 4-dimensional manifold, a Riemannian or Hermitian vector bundle E and a generalised Laplacian H acting on sections in E. Next, we want to consider the Chamseddine-Connes spectral action see [CC97]) for the generalised Laplacian H. For Λ > 0 it is defined as I CC H) = TrF ) H Λ 2 where Tr denotes the operator trace over L 2 E) and F : R + R + is a smooth cut-off function with support in the interval [0, +1] which is constant near the origin. Using the heat trace asymptotics one gets an asymptotic expression for I CC H) as Λ see [CC10] for details): I CC H) = TrF H Λ 2 ) = Λ 4 F 4 a 0 H) + Λ 2 F 2 a 2 H) + Λ 0 F 0 a 4 H) + OΛ ) 15) with the first three moments of the cut-off function which are given by F 4 = s Fs)ds, F 0 2 = Fs)ds 0 and F 0 = F0). These moments are independent of the geometry of the manifold. From now on we always assume that carries a spin structure so that the spinor bundle is defined and so are Dirac operator, twisted or generalised Dirac operators on. Connes spectral action principle [Co95], [Co96]) states that one can extract any action functional of interest in physics from the spectral data of a Dirac operator. In [CC97] Chamseddine and Connes have considered an operator D Φ which is the sum of a twisted Dirac operator based on the Levi-Civita connection g on ) and a zero order term. Taking H = D Φ ) 2 in 15) they recovered the Einstein-Hilbert action in a 2 H), and in the remaining terms in 15) they find the full bosonic Lagrangian of the Standard odel of particle physics. In particular I CC H) produces the correct Higgs potential for electro-weak symmetry breaking. In the following we consider various Dirac operators D induced by orthogonal connections with general torsion as in 1). We will consider H = D D since D is not selfadjoint in general) and the corresponding Seeley-deWitt coefficients in 15). If the twist bundle has a chiral asymmetry terms known from Loop Quantum Gravity [Ro04], [Th07]) arise. The situation with purely anti-symmetric torsion has been examined before in [HPS10], [ILV10], [PS12]), and there also are results on Dirac operators with scalar perturbations in [SZ11]). 3.1 The Bosonic Spectral Action First we consider the case where the vector bundle E is the spinor bundle Σ and the generalised Laplacian is H = D D where D is the Dirac operator as in 7). The Chamseddine-Connes spectral action of D D is determined if one knows the second and the fourth Seeley-deWitt coefficient. Those can be calculated. 5

12 Proposition 3.1 Let C g denote the Weyl curvature of the Levi-Civita connection as in Lemma A.22) and let χ) denote the Euler characteristics of, and let R be the scalar curvature of the modified connection given in 9)). Then one has a 2 D D) = 1 48 π Rx) dx, 2 Proof. From 46) we have a 4 D D) = χ) π 2 We obtain a 2 D D) inserting 59) into 12). C g 2 dx 3 32 π 2 R = R g + 18 div g V ) 54 V 2 9 T 2. δt 2 + dv ) 2) dx We denote the Riemann curvature of by riem and recall compare Corollary 4.10 in [PS12]) that tr Σ Ω ij Ω ij ) = 1 2 riem 2. 16) i,j In order to evaluate a 4 D D) we insert the trace formulas 59) and 60) into 13) and we apply Lemma B.1 for to obtain a 4 D 1 D) = 96 π π Rg ) ricg riemg Cg 2) dx 9 δt 2 9 dv ) 2) dx. The first integral term on the right hand side equals χ) π 2 Cg 2 dx, see [CC97] and [CC10], and the claim follows. Remark 3.2 In the case of totally anti-symmetric torsion, i.e. V 0, formulas for these Seeley-deWitt coefficients have been given in [Go80], [Ob83], [Gr86], [HPS10], [ILV10], [PS12]. Next we consider the volume form ω g acting on the spinor bundle Σ. Setting P = P + = 1 2 id Σ +ω g ) we have a parallel field of orthogonal projections. If we now calculate the Seeley-deWitt coefficients of H + = P + D D we obtain relations to Loop Quantum Gravity. The Holst term 2 for the modified connection is the 4-form C H = 18 dt T, V ω g), 17) see Proposition 2.3 in [PS11] and its Erratum. Proposition 3.3 Let C g denote the Weyl curvature of the Levi-Civita connection, and let R be the scalar curvature of the modified connection. Denote the Euler characteristics of by χ) and the first Pontryagin class of by p 1 ). Then one has a 2 P + D D) = 1 96 π R ω g + C ) 2 H a 4 P + D D) = χ) 1 96 p 1) π π 2 C g 2 dx, δt 2 + dv ) 2) dx π 2 R C H. 2 Before [Ho96] this density already appeared in [HS80], a sketch of its history can be found in [BHN11, Section III.D] 6

13 Proof. For any l 0 we have a 2l P + D D) = 1 2 a 2lD D) so we only need to calculate tr Σ α 2l D D)ω g ) dx for l = 1, 2. tr Σ α 2l D D)ω g ) dx, Since tr Σ ω g ) = 0 we have 4π) 2 tr Σ α 2 ω g ) = tr Σ E ω g ). From 61) we get tr Σ E ω g )ω g = 1 C 3 H and therefore tr Σ α 2l D D)ω g ) dx = 1 48 π C 2 H. We use tr Σ ω g ) = 0 again to obtain tr Σ α 4 D D)ω g ) dx = 1 4π) 2 i,j 1 6 Rg tr Σ E ω g ) tr ΣE 2 ω g ) tr Σ Ωij Ω ij ω g)) dx. The explicit formula for Ω ij in terms of the Riemann curvature riem Theorem 4.15 in [L89, Chap. II]) gives tr Σ Ω ij Ω ij ω g ) = 1 16 riem ijkl riem ijmn tr Σ e k e l e m e n ω g ) i,j,k,l,m,n = 1 4 riem, id Λ 2 riem ) where we use the notations from Appendix A.2. By 50) we get tr Σ Ω ij Ω ij ω g ) dx = 4π 2 p 1 ). 18) i,j From 62) we obtain tr Σ E 2 ω g )ω g = Rg + R) C H, and with tr Σ E ω g )ω g = 1 C 3 H we are done. The second Seeley-deWitt coefficient a 2 P + D D) coincides with the Holst action for with critical Barbero- Immirzi parameter 1, see [PS11]. It is no surprise that one finds the first Pontryagin class and the integral of the derivatives of T and V in the fourth Seeley-deWitt coefficient a 4 P + D D). The authors did not expect that the integral of the product of the scalar curvature R and the Holst term C H occurs. The next Lemma shows that R C H is not a topological term. Lemma 3.4 For any compact oriented 4-dimensional manifold there exists a Riemannian metric g and a 3-form T such that Rg dt 0. Proof. First we fix some arbitrary smooth) 3-form T with support suppt) contained in an oriented chart. For any Riemannian metric g we find a smooth function f g such that dt = f g g dx 1 dx 2 dx 3 dx 4 in this chart. W.l.o.g. we assume that f g p) > 0 for one point p in the chart. Since g > 0 we observe that the set A = {f g > 0} does not depend on the choice of g. Next we choose a smooth function K C ) with the following properties: on some open set, which is contained in A, one has K > 0, on the set suppt) \ A one has K = 0, and there is a point q \ suppt) with Kq) < 0. By a classical result by Kazdan-Warner [KW75, Theorem 1.1] one can find a Riemannian metric g on with R g K. For this g one has Rg dt > 0. Corollary 3.5 Neither R C H nor 3 64 π δt 2 + dv ) 2) dx π R C 2 H which is the sum of the last three terms in a 4 P + D D)) are topological invariants. In particular, both are non-zero in general. Proof. If one of the above terms were topological its value would be independent of the choice of T and V. Rescaling T and V independently would imply that the term in Lemma 3.4 would always vanish. Remark 3.6 Using considerations involving the Pontryagin class see Remark B.4) it is possible to express R C H in terms of the Ricci curvature. i,j=1 7

14 3.2 Particle Lagrangians In the models of particle physics the matter content is encoded in a Hermitian vector bundle H equipped with a connection H. As in the preceeding sections we consider an orthogonal connection as in 1) inducing a connection on the spinor bundle Σ, which we denote again by. The tensor connection on the twisted bundle E = Σ H induces the Dirac operator D H which is given by D H ψ χ) = 4 ei ei ψ) χ + e i ψ) H e i χ) ) 19) i=1 for any positively oriented orthonormal frame e 1,...,e 4, any section ψ of Σ and any section χ of H. For further calculations we define the auxiliary orthogonal connection 0 by 0 X Y = g X Y + TX, Y, ). 20) Proposition 3.7 For the twisted Dirac operator D H we get D HD H ψ χ) = ψ χ) Eψ) χ 1 2 e i e j ψ) Ω H ijχ 21) i j where is the Laplacian associated to the tensor connection = id H + id Σ H and Ω H ij = H e i H e j H e j H e i H [e i,e j] is the curvature of H and E is the potential in the Lichnerowicz formula of the untwisted Dirac operator D as in 53). Proof. Given a point p, we choose the positively oriented orthonormal frame e 1,..., e 4 about p synchronous in p with respect to 0. The connection 0 has been considered in the Appendix of [HPS10] to which we refer. One has g e i e i = 0 in p. For the twisted Dirac operator we get This yields D H ψ χ) = D Hψ χ) = 4 i=1 4 i=1 D HD H ψ χ) = D Dψ) χ ei 0 e i ψ) χ + e i ψ) H e i χ) ) 3 2 V ψ) χ, ei 0 e i ψ) χ + e i ψ) H e i χ) ) V ψ) χ. 4 e j e j ψ) H e j H e i χ) 2 i,j=1 4 V e i + V, e i ) ψ H e i χ. i=1 4 0 e i ψ) H e i χ) i=1 With 9) we find which leads to X ψ = 0 X ψ X T) ψ 3 2 V X ψ 3 2 V, X ψ ψ χ) = ψ) χ ψ H e i H e i χ) 2 0 e i ψ) H e i χ) 2 e i T) ψ H e i χ i=1 i=1 i=1 8

15 + 3 4 V e i + V, e i ) ψ H e i χ. i=1 Taking the difference DH D H we see that all terms containing V explicitly drop out and we are in a situation as in the Appendix of [HPS10]. Here we recall that the curvatures of, and H are related as follows see e.g. 20) in [HPS10]): Ω ijψ χ) = Ω ij ψ) χ + ψ Ω H ijχ). 22) Let Φ be a field of selfadjoint endomorphisms of the vector bundle H. We call Φ the Higgs endomorphism. Adding a zero order term to the twisted Dirac operator we obtain the Chamseddine-Connes Dirac operator for which one has the following Lichnerowicz formula. D Φ ψ χ) = D H ψ χ) + ω g ψ) Φ χ) Proposition 3.8 For the Chamseddine-Connes Dirac operator D Φ we get where is the Laplacian associated to the tensor connection D ΦD Φ ψ χ) = ψ χ) E Φ ψ χ) 23) = id H + id Σ H and the potential E Φ is given by E Φ ψ χ) = Eψ) χ i je i e j ψ) Ω H ijχ + i ω g e i ψ) [ H e i, Φ ] χ ψ Φ 2 χ) 3 V ω g ψ) Φ χ). 24) Proof. The auxiliary connection 0 from 20) a induces connection on the spinor bundle Σ. By D 0 we denote the associated twisted Dirac operator acting on section in Σ H. From [FS79] one can conclude that D 0 is self-adjoint. As in 9) of [HPS10] we have D 0 ω g Φ) + ω g Φ)D 0) ψ χ) = i ω g e i ψ) [ H e i, Φ ] χ. 25) Furthermore we have the relations D H ψ χ) = D 0 ψ χ) 3 2 V ψ) χ and D H ψ χ) = D0 ψ χ) V ψ) χ. 26) Combining 25) and 26) we get D H ω g Φ) + ω g ) Φ)D H ψ χ) = ω g e i ψ) [ H e i, Φ ] χ + 3 V ω g ψ) Φ χ). 27) From the definition of the Chamseddine-Connes Dirac operator it follows i D ΦD Φ = D HD H + D H ω g Φ) + ω g Φ)D H + id Σ Φ 2. Into this we insert the Lichnerowicz formula 21) and equation 27) and obtain the claim. In realistic particle models there are fermions of different chirality. The mathematical description of this is a Z 2 - grading of the vector bundle H = H r H l. We may have dim H r dim H l, in that case we speak of chiral 9

16 asymmetry. Let γ denote the corresponding chirality operator, i.e. one has γ H r = id H r and γ H l = id H l. In the following we assume that the endomorphism field γ is parallel with respect to H. On the other hand the volume form ω g induces a Z 2 -grading of the spinor bundle Σ = Σ + Σ such that ω g Σ± = ± id Σ±. otivated by experiment one considers the subbundle E + = Σ + H r ) Σ H l ) of E = Σ H. Then the orthogonal projection field P : E E + can be expressed as and P is therefore parallel. P = 1 2 id Σ id H + ω g γ), 28) Finally we want to determine the asymptotic expression of the Chamseddine-Connes spectral action I CC H + ) for H + = PD Φ D Φ as in 15). To that end we just need to calculate the first three Seeley-deWitt coefficients. We denote the scalar curvature of the connection by R and the Holst term for by C H as before. Proposition 3.9 For H + = PD Φ D Φ one gets a 0 H + ) = 1 8 π rkh) vol), 2 a 2 H + ) = rkh) 96 π 2 a 4 H + ) = Rω g + tr Hγ) rkh) C H ) 1 8 π 2 11 rkh) 1440 χ) trhγ) 96 p 1 ) rkh) 640 π 2 + trhγ) 1152 π π 2 R C H π 2 tr H Φ 2 γ) C H π 2 tr H Φ 2 )dx, C g 2 dx 3 rkh) 64 π 2 trh [ H, Φ] 2 ) + tr H Φ 4 ) tr H Ω H Ω H )dx π 2 δt 2 + dv ) 2) dx R g 9 T 2) tr H Φ 2 ) ) dx tr H Ω H Ω H γ)dx with the abbreviations [ H, Φ] 2 = i [ H e i, Φ][ H e i, Φ], Ω H Ω H = i,j ΩH ij ΩH ij and Ω H Ω H = i,j ΩH ij )ΩH ij. Proof. For all following calculations we use 14). We note that tr E ω g γ) = tr Σ ω g )tr H γ) = 0 and get a 0 H + ) = 1 4 π) tr 2 E P)dx = 1 8 π rkh) vol). 2 Now we make the standard observation that for 1 k 3 and i 1,...,i k pairwise distinct one has that tr Σ e i1 e ik ) = tr Σ ω g e i1 e ik ) = 0. For the traces of the potential E Φ from 24) we then get tr E E Φ ) = tr Σ E) rkh) 4 tr H Φ 2 ), 29) tr E E Φ ω g γ) = tr Σ E ω g )tr H γ). 30) Hence the second Seeley-deWitt coefficient is given by a 2 H + ) = 1 32 π 2 tre E Φ ) Rg rke) ) dx π 2 = 1 32 π 2 rkh) trσ E) rkh)rg 4 tr H Φ 2 ) ) dx + trhγ) 32 π 2 Now we insert 59) and 61) and obtain the formula for a 2 H + ) in the Proposition. tre E Φ ω g γ) Rg tr E ω g γ) ) dx tr Σ E ω g )dx. To evaluate a 4 H + ) we use Theorem 2.3 and 14) and obtain a 4 H + ) = 1 1 4π) R g tr E E Φ ) + tr E E Φ ω g γ)) ) + 90 tr E EΦ) 2 + tr E EΦω 2 g γ)) ) 10

17 + 15 i,j By means of 22) we identify the terms containing Ω ij as tr E tr E Ω ij Ω ) ij + 15 tr E Ω ij Ω ij ωg γ) ) i,j + tr E P) 5R g ) 2 2 ric g riem g 2) ) dx 31) i,j Ω ij Ω ij ) ) = rkh) trσ Ω ij Ω ij + 4 trh Ω H ) ij ΩH ij i,j tr E Ω ij Ω ij ωg γ) ) = tr H γ)tr Σ Ω ij Ω ij ω g) 33) i,j where we have used 22) and tr Σ ω g ) = 0. In order to evaluate the traces that involve EΦ 2 one has to write down more than two dozen of terms. Then one has to check that in the trace most of these terms vanish due to the Clifford relations and the cyclicity of the trace, similarly as in [HPS10, equations 11) and 12)]. The traces are tr E EΦ) 2 = rkh) tr Σ E 2 ) ) + tr H + trh [ H e i, Φ] 2) + 4 tr H Φ 4 ) i,j i,j Ω H ijω H ij 2 tr Σ E) tr H Φ 2 ) + 36 V 2 tr H Φ 2 ) + 12 V tr H Φ 2 ), 34) tr E EΦ 2 ωg γ)) = tr H γ) tr Σ E 2 ω g ) + tr H Ω H ij )ΩH ij γ) 2 tr Σ E ω g ) tr H Φ 2 γ). 35) Integration by parts gives V tr H Φ 2 )dx = divg V ) tr H Φ 2 )dx. We integrate the last three summands in 34) and find with 59) that ) 2 tr Σ E) + 36 V 2 12 div g V ) tr H Φ 2 ) dx = 2 R g 3 T 2) tr H Φ 2 ) dx. 36) To conclude we insert 29), 30) and equations 32) 36) into 31) and observe that the occuring terms can be arranged in following way: a 4 H + ) = rkh) 2 tr Σ α4 D D) ) dx + trhγ) 2 tr Σ α4 D D)ω g) dx π π 2 trh [ H, Φ] 2 ) + tr H Φ 4 ) tr H Ω H Ω H )dx π 2 i,j i i,j 32) R g 9 T 2) tr H Φ 2 ) 1 2 tr ΣE ω g ) tr H Φ 2 γ) ) dx tr H Ω H Ω H γ)dx. The terms tr Σ α4 D D) ) dx and tr Σ α4 D D)ω g) dx have already been computed in Propositions 3.1 and 3.3. To simplify the term tr Σ E ω g ) we take equation 61) into account. In the following remarks we want to discuss the terms in a 2 H + ) and a 4 H + ) which seem interesting to us. Remark 3.10 a) In a 4 H + ) one recovers the Yang-ills functional tr HΩ H Ω H )dx. For the Standard odel of particle physics the details have already been given by Chamseddine and Connes in [CC97], see also [IKS97]. b) If H is an associated bundle the term tr H Ω H Ω H γ)dx is topolgical and can be expressed by Chern classes compare 4.66) in [LS97]). Details for the Standard odel can be found in [CC10], equation 7.5) and the discussion thereafter. Remark 3.11 a) The roles of V and T seem dual to each other in the dynamical term δt 2 + dv ) 2) dx. But by the term R g 9 T 2) tr H Φ 2 )dx only the anti-symmetric torsion component T couples to the Higgs 11

18 endomorphism. b) If the relation γφ = Φγ holds, which is the case in models based on the axioms of noncommutative geometry [Co94], [CC97], [CC10]) the trace tr H Φ 2 γ) is zero and the term tr HΦ 2 γ) C H vanishes. Otherwise the vectorial torsion component V couples to Φ via the Holst term C H ). Rω g + trhγ) rkh) Remark 3.12 Up to a multiplicative constant the term the modified connection with Barbero-Immirzi parameter rkh) tr Hγ) C H ) coincides with the Holst action for whose modulus is larger or equal to 1). A different derivation for the numerical value of the Barbero-Immirzi parameter in a similar situation can be found in [BS10]. A Norms and Decomposions of Curvature Tensors A.1 The Kulkarni-Nomizu Product In this section let U denote an n-dimensional real vector space, equipped with a scalar product g. Definition A.1 For two 2, 0)-tensors h and k on U the Kulkarni-Nomizu product is the 4, 0)-tensor h k given by h k)x, Y, Z, W) = hx, Z)kY, W) + hy, W)kX, Z) hx, W)kY, Z) hy, Z)kX, W) for any X, Y, Z, W U. We would like to point out that we do not require h and k to be symmetric. This deviates from the usual definition, see e.g. [Bes87, Def ]. In any case we have h k = k h. Following the usual conventions S 2 U denotes the space of symmetric 2, 0)-tensors on U, and Λ 2 U is the space of anti-symmetric 2, 0)-tensors. We recall that for any 2, 0)-tensor k on U the trace or g-trace) is given by tr g k) = n ke i, E i ) i=1 for any basis E 1,..., E n of U which is orthonormal w.r.t. g. For a 4, 0)-tensor Q the Ricci contraction cq) is the 2, 0)-tensor n cq)x, Y ) = QE i, X, Y, E i ) for any g-orthonormal basis E 1,...,E n of U and any X, Y U. Elementary calculations show: Lemma A.2 Let k and h be a 2, 0)-tensors on U. Then one has a) The Ricci contraction of the Kulkarni-Nomizu product is ck g) = 2 n)k tr g k)g. i=1 b) k g, h g = 4n 2) k, h + 4 tr g k) tr g h), where, denotes the natural scalar product for k, 0)-tensors as in 2). In particular one obtains for the norms k g 2 = 4n 2) k 2 + 4tr g k)) 2 and g g 2 = 8nn 1). 12

19 The Bianchi map b : 4 U 4 U is an endomorphism of the space of 4, 0)-tensors, mapping Q bq) with bq)x, Y, Z, W) = 1 3 QX, Y, Z, W) + QY, Z, X, W) + QZ, X, Y, W) ) for any X, Y, Z, W U. It is known [Bes87, 1.107]) that b is an idempotent, and one easily checks that b is symmetric w.r.t.,. Therefore b is an orthogonal projection. Let S 2 Λ 2 U ) denote the space of 4, 0)-tensors on U that are anti-symmetric in the first and the last pair of their entries and with the symmetry QX, Y, Z, W) = QZ, W, X, Y ) for any X, Y, Z, W U. Note that b leaves S 2 Λ 2 U ) invariant, thus S 2 Λ 2 U ) orthogonally decomposes into the kernel and the image of b. One calls RU ) = S 2 Λ 2 U ) kerb) 37) the space of algebraic curvature tensors, compare [Bes87, 1.108], furthermore one has S 2 Λ 2 U ) imb) = Λ 4 U. Remark A.3 In general for an orthogonal connections the associated curvature tensor riem is not an algebraic curvature tensor, it is not even contained in S 2 Λ 2 U ). Remark A.4 For k, h S 2 U we have k h RU ). According to the conventions of [Bes87] one defines S 2 0U = { h S 2 U trg h) = 0 }. The following classical result is called the Ricci decomposition of an algebraic curvature tensor, see 1.116) in [Bes87]. For Q RU) the algebraic scalar curvature is sq) = tr g cq) ) R and the algebraic tracefree Ricci curvature is hq) = cq) 1 n sq)g S2 0 U. The algebraic Weyl tensor weylq) is defined as Then the three summands in the decomposition weylq) = Q + 1 n 2 hq) g + 1 2nn 1) sq) g g. Q = 1 1 2nn 1) sq) g g n 2 hq) g + weylq) 38) are orthogonal with respect to the natural scalar product for 4, 0)-tensors given as in 2). Remark A.5 An important property of the algebraic Weyl tensor weylq) is that it is tracefree in any pair of entries. In particular, for any 2, 0)-tensor k it follows that the 4, 0)-tensors weylq) and k g are perpendicular. Remark A.6 The algebraic Weyl tensor vanishes, weylq) = 0, if n 3, and the algebraic tracefree Ricci curvature hq) is zero if n 2. Finally, let Λ 2 Λ 2 U ) denote the space of 4, 0)-tensors on U that are anti-symmetric in the first and the last pair of their entries and with the symmetry QX, Y, Z, W) = QZ, W, X, Y ) for any X, Y, Z, W U. For any k Λ 2 U we note that k g Λ 2 Λ 2 U ). If we take k = cq) for Q Λ 2 Λ 2 U ) we can give an orthogonal decomposition similar to 38): Lemma A.7 For any Q Λ 2 Λ 2 U ) the two summands in the decomposition Q = 1 n 2 cq) g + Q + 1 n 2 cq) g) 39) are orthogonal with respect to the natural scalar product for 4, 0)-tensors. 13

20 Proof. For Q, P Λ 2 Λ 2 U ) one verifies c cq) g ) = 2 n)cq) and P, cq) g = cp) g, Q. Hence the map π : Λ 2 Λ 2 U ) Λ 2 Λ 2 U ), Q 1 2 ncq) g is a symmetric idempotent and therefore an orthogonal projection. Then 39) is just the orthogonal decomposition into image and kernel of π. We summerise our consideration so far and a decomposition of 2 Λ 2 U ) = S 2 Λ 2 U ) Λ 2 Λ 2 U ). We introduce the following vector spaces of tensors: SU ) = Rg g, H S U ) = S 2 0U ) g, H A U ) = Λ 2 U ) g, W S U ) = kerc RU ) ), W A U ) = kerc Λ2 Λ 2 U ) ), Λ 4 U ) = b S 2 Λ 2 U ) ). Proposition A.8 The decomposition 2 Λ 2 U ) = SU ) H S U ) W S U ) Λ 4 U ) H A U ) W A U ) is orthogonal with respect to the natural scalar product for 4, 0)-tensors given as in 2). Proof. This is a direct consequence of the above considerations. Remark A.9 If U is 4-dimensional we have dim SU ) = dimλ 4 U ) = 1, dim H S U ) = dim W A U ) = 9, dim W S U ) = 10 and dim H A U ) = 6. A.2 The Hodge -Operator Now let U denote a 4-dimensional real vector space, equipped with a scalar product g and an orientation. We denote the induced volume form by ω g. Throughout this section we fix an oriented orthonormal basis E 1,..., E 4 of U and for any 4, 0)-tensor Q and any 2, 0)-tensor h we denote Q ijkl = QE i, E j, E k, E l ) and h ij = he i, E j ). Furthermore ω g E i, E j, E k, E l ) = ǫ ijkl and ge i, E j ) = δ ij. The Hodge -operator is an endomorphism of Λ 2 U with = id Λ 2. It naturally induces an endomorphism id Λ 2 : 2 Λ 2 U ) 2 Λ 2 U ). Lemma A.10 For any P, Q 2 Λ 2 U ) we have P, id Λ 2 Q) = id Λ 2 P), Q. In other words id Λ 2 is a selfadjoint endomorphism of 2 Λ 2 U ) 4 U. Proof. Note that P, id Λ 2 Q) = P ijkl Q ijab ǫ abkl = id Λ 2 P), Q. Corollary A.11 id Λ 2 is an isometry of 2 Λ 2 U ) as id Λ 2 ) 2 = id Λ 2 id Λ 2. Lemma A.12 Let h and k be a 2, 0)-tensors on U. Then we have: a) id Λ 2 g g) = 4 ω g. b) If either of h and k is symmetric, then h g, id Λ 2 k g) = 0. c) If h and k are both anti-symmetric, then h g, id Λ 2 k g) = 8 h, k. 14

21 Proof. For a) we just note that id Λ 2 g g) ijlm = 2 δ ia δ jb δ ja δ ib )ǫ ablm = 4 ǫ ijlm. For arbitrary 2, 0)-tensors h and k we calculate h g, id Λ 2 k g) = 8 h ij k lm ǫ ijlm, from which b) and c) follow. i,j,l,m In the following proposition we study to what extent the map id Λ 2 respects the decomposition from Proposition A.8. The relations which we prove have been found before by Hehl et al. in [HN95, Formula B.4.35)]. Proposition A.13 If we restrict the isometry id Λ 2 : 2 Λ 2 U ) 2 Λ 2 U ) to the components of the decomposition of 2 Λ 2 U ) given in Proposition A.8, we the following isomorphisms: id Λ 2 : SU ) id Λ 2 : H S U ) id Λ 2 : W S U ) = Λ 4 U ), id Λ 2 : Λ 4 U ) = W A U ), id Λ 2 : W A U ) = W S U ), id Λ 2 : H A U ) = SU ), 40) = H S U ), 41) = H A U ). 42) Proof. First we note that id Λ 2 is its own inverse since id Λ 2 ) 2 = id Λ 2 id Λ 2. Lemma A.12a) states that id Λ 2 maps a basis of SU ) to a basis of Λ 4 U ). This shows that the two maps in 40) have the given range and are isomorphisms. It is a classical fact [Bes87, Chap. 13B] that, if dimu) = 4, the map id Λ 2 preserves the space W S U ) of algebraic Weyl tensors and is an isomorphism. Next, we show that id Λ 2 maps H S U ) into W A U ). Let h g H S U ), i.e. h S 2 0U ). We will verify that id Λ 2 h g) is perpendicular to any component of the decomposition from Proposition A.8 other than W A U ). For Q W S U ) we have id Λ 2 Q) W S U ) and, therefore, Q, id Λ 2 h g) = id Λ 2 Q), h g = 0 as W S U ) and H S U ) are orthogonal. For ω g Λ 4 U ) one obtains ω g, id Λ 2 h g) = id Λ 2 ω g ), h g = 1 4 g g, h g = 0 as SU ) and H S U ) are perpendicular. For k g H S U ) H A U ) Lemma A.12b) gives k g, id Λ 2 h g) = 0. Hence id Λ 2 H S U )) W A U ). Since dim H S U ) = dim W A U ) = 9 the first map in 41) is an isomorphism and the second one as well. Finally, since the decomposition from Proposition A.8 is orthogonal and id Λ 2 is an isometry, the second map in 42) is also an isomorphism. Corollary A.14 For Q H A U ) W A U ) we consider the 2, 0)-tensor q = 1 2 c id Λ 2 Q)). Then one has tr g q) = 0 and id Λ 2 Q) = q g. Furthermore we find that q S 2 0 U ), if Q W A U ), and that q Λ 2 U ), if Q H A U ). Proof. Proposition A.13 provides the existence and uniqueness of a tracefree q with id Λ 2 Q)) = q g. With Lemma A.2 a) one identifies this q as in the corollary. 15

22 A.3 The Norm of Curvature Tensors We consider an n-dimensional Riemannian manifold equipped with an orthogonal connection without Cartan type torsion, thus there is a vector field V and a 3-form T such that 1) takes the form for any vector fields X, Y. X Y = g X Y + X, Y V V, Y X + TX, Y, ) 43) For a fixed point p we will apply the considerations of section A.1 to the tangent space T p. For convenience we introduce the symmetric 2, 0)-tensors g T, g V, g V on T p, for X, Y T p they are given by g T X, Y ) = X T, Y T, g V X, Y ) = V, X V, Y, Lemma A.15 The traces of the above 2, 0)-tensors are g V X, Y ) = g X V, Y + g Y V, X. tr g g T ) = T 2, tr g g V ) = V 2, tr g g V ) = 2 div g V ). Proof. Direct calculation. We note that the anti-symmetric 2, 0)-tensors dv ) on T p can be written as dv )X, Y ) = g X V, Y g Y V, X. Furthermore, we define the 4, 0)-tensors G T, K V and K V T on T p by G T X, Y, Z, W) = TY, Z, ), TX, W, ) TX, Z, ), TY, W, ), K T X, Y, Z, W) = g X T)Y, Z, W) g Y T)X, Z, W) g Z T)X, Y, W) + g WT)X, Y, Z), K V T X, Y, Z, W) = V, X TY, Z, W)) V, Y TX, Z, W) V, Z TX, Y, W) + V, W TX, Y, Z). Remark A.16 For the Ricci contraction we get cg T ) = g T. Furthermore we see that G T S 2 Λ 2 T p ), and therefore id b)g T ) RT p ) Remark A.17 We note that V T is a 4-form, and hence it is in the image of the Bianchi map and its Ricci contraction vanishes. For, given as in 43), the Riemann curvature tensor riem is defined as in 4). We note that riem is anti-symmetric in the first and the last pair of its entries. Yet it is not an algebraic curvature tensor in the sense of 37) since in general it does not coincide with its symmetrisation riem S X, Y, Z, W) = 1 2 riemx, Y, Z, W) + riemz, W, X, Y )). The anti-symmetric part of the Riemann curvature tensor is accordingly defined as riem A X, Y, Z, W) = 1 2 riemx, Y, Z, W) riemz, W, X, Y )), which yields the decomposition riem = riem S + riem A which is orthogonal with respect to the scalar product of 4, 0)-tensors. 16

23 Lemma A.18 The components of the Riemann curvature tensor of are riem S = riem g 1 2 g V g V 2 g g g V g dt V T G T, 44) riem A = 1 2 K T 1 2 dv ) g + V T) g + K V T. 45) Proof. This is verified by a lengthy but still elementary calculation. Corollary A.19 As riem S S 2 Λ 2 T p ), it has a component riem S ker = id b)riems ) in the kernel of the Bianchi map and a component riem S im = briem S ) in the image: riem S ker = riem g 1 2 g V g V 2 g g g V g id b)g T ) RT p ), riem S im = 1 2 dt V T bg T ) Λ 4 T p. Proof. We use Λ 4 T p = S 2 Λ 2 T p ) imb) and Remarks A.3, A.16 and A.17. We can rewrite the definition of the Ricci curvature 5) as ric = criem). Its symmetric and anti-symmetric component are defined as ric S X, Y ) = 1 2 ricx, Y ) + ricy, X)) and rica X, Y ) = 1 2 ricx, Y ) ricy, X)). Remark A.20 One notices that ric S = criem S ) and ric A = criem A ). Lemma A.21 We have ric S = ric g + div g V ) n 2) V 2) g + n 2)g V + n 2 2 g V g T, ric A = δt + n 2 2 dv ) + 4 n)v T), where δt is the codifferential of T, which is given as δt = n i=1 E i g E i T) for any orthonormal basis E 1,..., E n or equivalently δt = d T, if carries an orientation). Proof. The Ricci contractions of 44) and 45) are evaluated using Lemma A.15. Applying Lemma A.15 again, we evaluate the scalar curvature as R = tr g ric S ) = R g + 2n 1)div g V ) n 1)n 2) V 2 T 2, 46) which agrees with the formula given in [PS12, Lemma 2.5]. For the remainder of this section we assume that is 4-dimensional. Since riem S ker is an algebraic curvature tensor it possesses a Ricci decomposition as 38). Its algebraic Weyl tensor can be identified with the Weyl curvature of the Levi-Civita connection: Lemma A.22 Let dim) = 4 and let C g = weylriem g ) denote the Weyl curvature of the Levi-Civita connection. Then we have C g = weylriem S ker). Proof. In this situation we have weylriem S ker ) = 1 12 R g g rics g + riem S ker = C g T 2 g g 1 2 gt g id b)g T ) ). Setting C T = 1 12 T 2 g g 1 2 gt g id b)g T ) we need to show that C T = 0. We note that C T is an algebraic Weyl tensor, C T = weyl id b)g T )). This can easily be verified for the situation riem g 0 and V 0 where riem S ker = id b)gt ).) As the Ricci decomposition is orthogonal we have C T 2 = C T, id b)g T ) 17

24 = id b)c T, G T = C T, G T, the last equality holds as C T kerb). Finally, we note that [PS12, Lemma A.1] still holds true if one replaces the Weyl curvature of the Levi-Civita connection by any algebraic Weyl tensor such as C T. It states that C T, G T = 0 if dim) = 4. Corollary A.23 If dim) = 4 the 4, 0)-tensor norm of riem S ker is given by riem S ker 2 = C g ric S R2. Proof. Since hriem S ker ) is tracefree Lemma A.2b) shows hriem S ker ) g, hriems ker ) g = 8 hriems ker ), hriems ker ) = 8 ric S 1 4 R g, rics 1 4 R g = 8 ric S 2 4 R ric S, g R2 g, g = 8 ric S 2 2 R 2. We use the orthogonality of the Ricci decomposition and Lemma A.22 to finish the proof. Lemma A.24 If dim) = 4 the 4, 0)-tensor norm of riem S im is given by riem S im 2 = 1 4 dt V 2 T 2 12 V T 2 4 V dt, T. Proof. For any 4-form ω the argument which leads to 62) in [PS12] shows that 0 = ω g, G T = bω g ), G T = ω g, bg T ). 47) Hence dt, bg T ) = V T, bg T ) = bg T ), bg T ) = 0. From this we get riem S im 2 = 1 4 dt 2 + V T, V T dt, V T. Using the definition of the scalar product of 4, 0)-tensors we find V T, V T = 4 V 2 T 2 12 V T 2, dt, V T = 4 V dt, T, from which the Lemma follows. B Calculations with Topological Classes B.1 Euler Characteristics We consider a 4-dimensional compact Riemannian manifold without boundary. Let be an orthogonal connection which takes the form of 43). Lemma B.1 The integral over the square of the norm of the curvature tensor of is given by riem 2 1 dx = 3 R2 2 3 Rg ) ric g 2 + C g δt dv ) 2 ) dt V 2 T 2 24 V T 2 8 V dt, T dx 18

25 Proof. The classical formula of the Euler characteristics of see [Ber70]) states 8π 2 χ) = R g ) 2 4 ric g 2 + riem g 2) dx, 48) and one can show see [PS12, Thm. 3.4]) that 8π 2 χ) = R 2 4 ric S ric A 2 + riem S 2 riem A 2) dx. 49) Hence the difference of the right hand sides of 49) and 48) is zero. By adding this difference and using riem 2 = riem S 2 + riem A 2 we obtain riem 2 dx = R 2 R g ) 2 4 ric S ric A ric g riem S 2 riem g 2) dx. The Ricci decomposition yields riem g 2 = C g ric g Rg ) 2. Furthermore we know riem S 2 = riem S ker 2 + riem S im 2. Since δt, dv dx = T, ddv dx = 0 we have ) ric A 2 dx = δt, δt + dv, dv 2 δt, dv dx = δt 2 + dv 2) dx Then we are able to impose the explicit formulas for all occuring terms Lemma A.21, Corollary A.23 and Lemma A.24), which finishes the proof. B.2 First Pontryagin Class We consider a 4-dimensional compact oriented Riemannian manifold without boundary. Let be an orthogonal connection which takes the form of 43). Let E 1,..., E 4 be a local orthonormal frame of the tangent bundle T. The curvature 2-forms w.r.t. this frame are defined as Riem ij = k,l rieme 1, E j, E k, E l )E k E l for i, j = 1,...,4. By Chern-Weil theory see e.g. [KN69, Chap. XII.4] and [Bes87, Chap. 13B]) the 4-form i,j Riem ij Riem ij is independent of the choice of the frame and integrates to the first Pontryagin class of up to a prefactor: p 1 ) = 1 8π Riem 2 ij Riem ij = 1 16π 2 = 1 16π 2 i,j Riem ij, Riem ij dx Now we decompose the curvature tensor of according Proposition A.8: i,j riem, id Λ 2 riem) dx. 50) riem = 1 24 R g g 1 2 hs g + C g + briem S ) 1 2 rica g + riem A rica g) 51) where we have inserted Lemma A.22 and have denoted h S = hriem S ker ) = rics 1 4 R g. We use Proposition A.13 and the orthogonality of the decomposition in Proposition A.8 and obtain riem, id Λ 2 riem) = C g, id Λ 2 C g ) rica g, id Λ 2 ric A g) R 12 riems im, id Λ 2 g g) riema rica g, id Λ 2 h S g). 52) To the authors knowledge Baekler and Hehl were the first to notice that the density of the first Pontryagin class can be put in such a form, see [BH11, Section 3]. 19

26 Remark B.2 The integral over the second summand of the right hand side of 52) is ric A g, id Λ 2 ric A g) dx = 4 d T, dv dx. 1 4 Proof. By Lemma A.12c) we have ric A g, id Λ 2 ric A g) = 8 ric A, ric A. We observe that δα = d α for any k-form α, which gives that δt, δt dx = 0 and dv, dv dx = 0. This leads to ric A, ric A dx = δt dv, δt dv dx = 2 d T, dv dx, from which the claim follows. Remark B.3 For the third summand of the right hand side of 52) we get R 12 riems im, id Λ 2 g g) = R 6 dt, ωg + 4R 3 T, V. Proof. From Lemma A.12a) and Corollary A.19 we obtain riem S im, id Λ 2 g g) = 4 riem S im, ω g = 2 dt, ω g 4 V T, ω g 4 bg T ), ω g. The last term is zero due to 47), and from V T, ω g = 4 T, V the claim follows. Remark B.4 The integral over the fourth summand of the right hand side of 52) is ) riem A rica g, id Λ 2 h S g) dx = 4 d T, dv + 4R 3 T, V dx Proof. Since the Pontryagin class is a topological invariant we get from 50) and 52) that 16π 2 p 1 ) = riem, id Λ 2 riem) dx = C g, id Λ 2 C g ) dx. 4R dt Therefore the last three terms on the right hand side of 50) integrate to zero. We note that η, ω g ω g = 24 η for any 4-form η, and with Remarks B.2 and B.3 we obtain the claim. C Some Traces of the Potential In order to evaluate the Seeley-deWitt coefficients for D D we need to compute several traces involving the potential that comes from the Lichnerowicz formula 8). We only consider the case of a 4-dimensional Riemannian spin manifold. Comparing 8) with the Bochner formula 10) gives us the potential E = 1 4 Rg T divg V ) V 2) id Σ 3 2dT 9 T V 9 V T), 53) which acts as an endomorphism of the spinor bundle Σ. We also need the square of the potential: E 2 = 1 4 Rg T divg V ) V 2) 2 idσ Rg 3 4 T divg V ) 9 2 V 2) 3 2dT + 9 T V + 9 V T)) This leads us to the following lemma: dt dt + 81 T V + V T)) dt T V + T V dt) dt V T) + V T) dt). 54) 20

27 Lemma C.1 The traces of E and E 2 taken over the spinor bundle are tr Σ E) = R g + 3 T 2 6 div g V ) + 18 V 2 55) tr Σ E 2 ) = Rg T divg V ) V 2) dt V T V 2 T 2 18 V dt, T 56) Proof. As Σ has rank 4 we get tr Σ id Σ ) = 4 and 55) follows from 53) immediately. In order to conclude 56) from 54) we take into account that in the Clifford algebra only elements of degree zero i.e. elements proportional to the identity) contribute in the trace. Furthermore we only use cyclicity of the trace and the relations for any tangent vector V and any k-forms α, β: V α + 1) k+1 α V = 2 V α), tr Σ α β) = 4 k! 1)kk+1)/2 α, β, and V α = V α) if α is a 4-form. We denote the Riemannian volume form on by ω g and we evaluate tr Σ E ω g ) and tr Σ E 2 ω g ). Lemma C.2 The traces of E ω g and E 2 ω g taken over the spinor bundle are tr Σ E ω g ) = 1 4 dt, ωg + 6 T, V 57) tr Σ E 2 ω g ) = ) 1 4 Rg 3 4 T divg V ) 9 2 V 2) 1 2 dt, ωg 12 T, V 58) Proof. Apart from the ingredients mentioned in the proof of Lemma C.1 we only need that V ω g = V for any tangent vector V in order to prove 57). For the proof of 58) we use again that Clifford elements of non-zero degree have trace zero. Therefore, we have 0 = tr Σ 1 4 Rg T divg V ) V 2) 2 idσ ω g), 0 = tr Σ 9 4 dt dt ωg), 0 = tr Σ 27 2 dt T V + T V dt)ωg dt V T) + V T) dt)ωg). Since T V + V T) = 1 2 T V V T) as endomorphisms of spinors we get tr Σ T V + V T)) 2 ω g) = 1 4 tr Σ T V V T) 2 ω g) = 0 when we take into account that V ω g = ω g V and that the trace is cyclic. Therefore the only term in E 2 ω g that contributes to the trace is Rg 3 4 T divg V ) 9 2 V 2) 3 2 dt + 9 T V + 9 V T)) ω g. Its trace is evaluated in the same way as it is done in the proof of 57). The scalar curvature of the modified connection given in 9) is R = R g + 18 div g V ) 54 V 2 9 T 2. We recall that the Holst term C H computed from the connection defined originally in [Ho96] or as in 7) in [PS11]) is given by C H = 6 dt 2 T, V ω g 1 2 S+ 2 S 2) ω g, 21