On Gravity, Torsion and the Spectral Action Principle

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Einstein-Hilbert the of ion eit h orc ig oeta eesr o the for necessary potential Higgs correct the redicts n-onsseta cin eueteeutosof equations the deduce action, spectral ine-Connes eo oal nismerctrini eto .In 3. section in torsion anti-symmetric totally of se prtri ymti.Ti olw rmaresult a from follows This symmetric. is operator c iiacneto scle oso.I i seminal his In torsion. called is connection Civita ayeupe ihotooa oncin.We connections. orthogonal with equipped dary ecmoeto h oso a oefc nthe on effect no has torsion the of component pe tcePyis foecniessial twisted suitable considers one if Physics, rticle sdieadCne nrdcdtespectral the introduced Connes and mseddine oncin hc r optbewith compatible are which connections . act e In this setting many examples for commutative geometries in the sense of Connes’ spectral triples ([Co96]) can be supplied. Given the Reconstruction Theorem ([Co08]), we remark that in the even-dimensional case anti- symmetric torsion is reconstructablefrom the spectral data. In four dimensionswe calculate the purely gravitational part of the Chamseddine-Connes spectral action in some detail. We find that some terms of the action given in [HPS10] actually vanish, thus confirming the result by [ILV10]. For this action we derive the equations of motion in Theorem 5.7. One of them is a Proca equation for the torsion 3-form which suggests an interpretation of the torsion as massive vector boson. The set of critial points of the action, i.e. the solutions of the equations of motion, contains all Einstein manifolds (with zero torsion). Furthermore, in Lemmas 5.10 and 5.13 we exclude critical points which are warped products and carry special choices of non-zero torsion. We tried to keep this text elementary and accessible, and we hope that it may also serve as an introduction for non-experts. Acknoledgement: The authors appreciate funding by the Deutsche Forschungsgemeinschaft, in particular by the SFB Raum-Zeit-Materie. We would like to thank Christian Bär and Thomas Schücker for their support and helpful discussions.

2 Orthogonal connections on Riemannian manifolds

We consider an n-dimensional manifold M equipped with some Riemannian metric g. Let g denote the Levi- Civita connection on the tangent bundle. For any affine connection on the tangent bundle∇ there exists a (2, 1)- tensor field A such that ∇ Y = g Y + A(X, Y ) (1) ∇X ∇X for all vector fields X, Y . In this article we will require all connections to be orthogonal, i.e. for all vector fields X,Y,Z one has ∇ ∂ Y,Z = Y,Z + Y, Z , (2) X h i h∇X i h ∇X i where , denotes the scalar product given by the Riemannian metric g. For any tangent vector X one gets from (1) andh· (2)·i that the endomorphism A(X, ) is skew-adjoint: · A(X, Y ),Z = Y, A(X,Z) . (3) h i − h i Next, we want to express some curvature quantities for in terms of A and curvature quantities for g. To that ∇ ∇ end we fix some point p M, and we extend any tangent vectors X,Y,Z,W TpM to vector fields again denoted by X,Y,Z,W being∈ synchronous in p, which means ∈

g X = g Y = g Z = g W =0 for any tangent vector V T M. ∇V ∇V ∇V ∇V ∈ p

Furthermore, we choose a local orthogonal frame of vector ﬁelds E1,...,En on a neighbourhood of p, all being synchronous in p. Then the Lie bracket [X, Y ]= g Y g X =0 vanishes in p, and synchronicity in p implies ∇X − ∇Y Z = g g Z + ( g A) (Y,Z)+ A (X, A(Y,Z)) ∇X ∇Y ∇X ∇Y ∇X Hence, in p the Riemann tensor of reads as ∇ Riem(X, Y )Z = Z Z Z ∇X ∇Y − ∇Y ∇X − ∇[X,Y ] = Riemg(X, Y )Z + ( g A) (Y,Z) ( g A) (X,Z)+ A (X, A(Y,Z)) A (Y, A(X,Z)) (4) ∇X − ∇Y − where Riemg denotes the Riemann tensor of g. We note that Riem(X, Y )Z is anti-symmetric in X and Y . And g ∇ g by differentiation of (3) we get that ( A)(Ej , ) and ( A)(Ei, ) are skew-adjoint, and therefore we have ∇Ei · ∇Ej · Riem(E , E )E , E = Riem(E , E )E , E . (5) h i j k li − h i j l ki

2 In general, Riem does not satisfy the Bianchi identity. The Ricci curvature of is deﬁned as ∇ ric(X, Y ) = tr (V Riem(V,X)Y ) , 7→ by (4) this can be expressed as

n ric(X, Y ) = Riem(E ,X)Y, E h i ii Xi=1

n g g g = ric (X, Y )+ A (X, Y ), Ei ( A) (Ei, Y ), Ei ∇Ei − h ∇X i Xi=1 n + ( A(X, Y ), A(E , E ) + A(E , Y ), A(X, E ) ) (6) − h i i i h i i i Xi=1 where ricg is the Ricci curvature of g. We have used that A(E , ) and A(X, ) are skew-adjoint. ∇ i · · One obtains the scalar curvature R of by taking yet another trace, in p it is given as R = n ric(E , E ). For ∇ j=1 j j the following calculation we use that ( g A)(X, ) is skew-adjoint for any tangent vectors PV,X, and we get: ∇V · n g g g R = R + A (Ej , Ej ), Ei + Ej , A (Ei, Ei) ∇Ei ∇Ej i,jX=1 D E n + ( A(E , E ), A(E , E ) + A(E , E ), A(E , E ) ) − h j j i i i h i j j i i i,jX=1

n n n g g 2 = R +2 A (Ej , Ej ), Ei A(Ei, Ei) + A(Ei, Ej ), A(Ej , Ei) (7) ∇Ei − h i i,jX=1 Xi=1 i,jX=1 where Rg denotes the scalar curvature of g. ∇ The classiﬁcation of orthogonal connections with torsions traces back to [Ca25, Chap. VIII]. Here we adopt the notations of [TV83, Chap. 3] (see also [Ag06]). From (3) we know that the torsion tensor A(X, ) is skew-adjoint · on the tangent space TpM. Any torsion tensor A induces a (3, 0)-tensor by setting

A = A(X, Y ),Z for any X,Y,Z T M. XY Z h i ∈ p

We deﬁne the space of all possible torsion tensors on TpM by

3 (T M)= A T ∗M A = A X,Y,Z T M . T p ∈ p XY Z − XZY ∀ ∈ p O

This vector space carries a scalar product

n A, A′ = A A′ , (8) h i EiEj Ek EiEj Ek i,j,kX=1 and the orthogonal group O(T M) acts on (T M) via (αA) = A −1 −1 −1 . p T p XY Z α (X)α (Y )α (Z) For A (T M) and Z T M one denotes the trace over the ﬁrst two entries by ∈ T p ∈ p n

c12(A)(Z)= AEiEiZ . (9) Xi=1

3 The space of quadratic invariants on (TpM) with respect to the O(TpM)-representation is spanned by the three quadratic forms T

A 2 = A, A , (10) k k h i n A, A = A A , (11) h i EiEj Ek Ej EiEk i,j,kX=1 b n c (A) 2 = A A . (12) k 12 k EiEiEk Ej Ej Ek i,j,kX=1

Here A denotes the (3, 0)-tensor obtained from A by interchanging the ﬁrst two slots, i.e. AXY Z = AY XZ , for all tangent vectors X,Y,Z. b b

Theorem 2.1 For dim(M) 3 one has the following decomposition of (TpM) into irreducible O(TpM)- subrepresentations: ≥ T (T M) = (T M) (T M) (T M). T p T1 p ⊕ T2 p ⊕ T3 p This decomposition is orthogonal with respect to , , and it is given by h· ·i (T M) = A (T M) V s.t. X,Y,Z : A = X, Y V,Z X,Z V, Y , T1 p ∈ T p ∃ ∀ XY Z h ih i − h ih i (T M) = A (T M) X,Y,Z : A = A , T2 p ∈ T p ∀ XY Z − Y XZ (T M) = A (T M) X,Y,Z : A + A + A =0 and c (A)(Z)=0 . T3 p ∈ T p ∀ XY Z Y ZX ZXY 12

For dim(M)=2 the O(TpM)-representation

(T M) = (T M) T p T1 p is irreducible. The above theorem is just Thm. 3.1 from [TV83]. The connections whose torsion tensor is contained in (T M) = T M are called vectorial. Those whose torsion tensor is in (T M) = 3T ∗M are called to- T1 p ∼ p T2 p p tally anti-symmetric, and those with torsion tensor in 3(TpM) are called of Cartan type. V T We note that any Cartan type torsion tensor A 3(TpM) is trace-free in any pair of entries, i.e. for any Z one has ∈ T n n n

AEiEiZ =0, AEiZEi =0, AZEiEi =0. Xi=1 Xi=1 Xi=1 The second equality holds as A (T M), and the third one follows from the cyclic identity A + A + ∈ T p XY Z Y ZX AZXY =0.

Remark 2.2 The invariant quadratic form given in (12) has the null space 2(TpM) 3(TpM). More precisely, one has A (T M) (T M) if and only if c (A)(Z)=0 for any ZT T M.⊕ T ∈ T2 p ⊕ T3 p 12 ∈ p Remark 2.3 The decomposition given in Theorem 2.1 is orthogonal with respect to the bilinear form given in (11), i.e. for α, β 1, 2, 3 , α = β, and A (T M), A (T M) one gets A , A =0. ∈{ } 6 α ∈ Tα p β ∈ Tβ p h α βi Varying the base point p M, the decomposition in Theorem 2.1 is parallel with respectb to the Levi-Civita connection g (induced on∈(3, 0)-tensor ﬁelds). And from Theorem 2.1 one gets immediately: ∇

4 Corollary 2.4 For any orthogonal connection on some Riemannian manifold of dimension n 3 there exist ∇ ≥ a vector ﬁeld V , a 3-form T and a (3, 0)-tensor ﬁeld S with Sp 3(TpM) for any p M such that X Y = g Y + A(X, Y ) takes the form ∈ T ∈ ∇ ∇X A(X, Y )= X, Y V V, Y X + T (X,Y, )♯ + S(X,Y, )♯, h i − h i · · where T (X,Y, )♯ and S(X,Y, )♯ are the unique vectors with · · T (X,Y,Z)= T (X,Y, )♯,Z and S(X,Y,Z)= S(X,Y, )♯,Z for all Z. (13) · ·

For any orthogonal connection these V,T,S are unique.

Lemma 2.5 The scalar curvature of an orthogonal connection is given by

g R = Rg + 2(n 1) div∇ (V ) (n 1)(n 2) V 2 T 2 + 1 S 2 − − − − k k −k k 2 k k g with V,T,S as in Corollary 2.4, and div∇ (V ) is the divergence of the vector ﬁeld V taken with respect to the Levi-Civita connection. Proof. With the notations from (9)–(12) we rewrite (7) as

n g g 2 R = R +2 c12 A (Ei) c12(A) + A, A . (14) ∇Ei −k k h i Xi=1 b By Remark 2.2 only the vectorial part of the torsion contributes to the c12-terms, and therefore one gets n n g g g c12 A (Ei) = Ej , Ej V, Ei V, Ej Ei, Ej ∇Ei h i h∇Ei i − h∇Ei i h i Xi=1 i,jX=1

g = (n 1) div∇ (V ), (15) − n 2 2 c12(A) = c12(A)(Ej ) Ej k k Xj=1

n 2 = ( Ei, Ei V, Ej Ej V, Ei Ei, Ej Ej ) h i h i − h i h i i,jX=1

n 2 = (V V, Ei Ei) − h i Xi=1

= (n 1) V 2. (16) k − k In order to compute the last term in (14) we decompose A = A1 + A2 + A3 with Aα α(TpM). From Remark 2.3 we get ∈ T 3 A, A = A , A . h i h α αi αX=1 b b For the vectorial part we get

n A , A = δ V, E δ V, E δ V, E δ V, E h 1 1i ij h ki− ikh j i · jih ki− jkh ii i,j,kX=1 b n = δ V, E 2 δ δ V, E V, E δ δ V, E V, E + δ δ V, E V, E ij h ki − ij jk h kih ii− ik ji h j ih ki ik jk h j ih ii i,j,kX=1

5 = (n 1) V 2 (17) − k k For the totally anti-symmetric part we get

n n A , A = T T = T T = T 2. (18) h 2 2i EiEj Ek Ej EiEk − EiEj Ek EiEj Ek −k k i,j,kX=1 i,j,kX=1 b Finally, for the Cartan-type part we get

n A , A = S S h 3 3i EiEj Ek Ej EiEk i,j,kX=1 b n = S S + S S (19) − EiEj Ek EiEkEj EiEj Ek EkEj Ei i,j,kX=1

n n = S S S S (20) EiEj Ek EiEj Ek − EiEkEj EkEiEj i,j,kX=1 i,j,kX=1

= 1 S 2, (21) 2 k k where (19) is due to the cyclic identity for S, (20) follows from the anti-symmetry in the last two entries, and n S S = A , A implies (21). Plugging (15)–(21) into (14) ﬁnishes the proof. i,j,k=1 EiEkEj EkEiEj h 3 3i P b Corollary 2.6 Let M be a closed manifold of dimension n 3 with Riemannian metric g and orthogonal connection . Let dvol denote the Riemannian volume measure taken≥ with respect to g. Then the Einstein-Cartan-Hilbert functional∇ is

g 2 2 1 2 R dvol = R dvol (n 1)(n 2) V dvol T dvol + 2 S dvol . ZM ZM − − − ZM k k − ZM k k ZM k k

Considering variations over all Riemannian metrics, for which the volume volg(M) stays ﬁxed, and all orthogonal connections (i.e. over all torsion tensors), we get that (M, g, ) is a critical point of the Einstein-Cartan-Hilbert functional if and only if (M,g) is an Einstein manifold and ∇ = g is the Levi-Civita connection (i.e. V 0, T 0 and S 0). ∇ ∇ ≡ For≡ an in depth≡ treatment of the physical consequences of Einstein-Cartan-Hilbert theory in Lorentzian geometry we refer to the classical review [HHKN76] and the more recent overview [Sh02] and references therein.

3 Curvature calculations in case of totally anti-symmetric torsion in four dimensions

Let us collect now some equalities involving curvature tensors and the totally anti-symmetric torsion. To keep the main part of this paper as readable as possible the proofs of the following theorems and lemmata have been allocated to the appendix. We consider a 4-dimensional manifold M equipped with a Riemannian metric g. Let g denote the Levi-Civita connection on the tangent bundle. We fix some 3-form T on M and some s R,∇ and we are studying the connection which is given by ∈ ∇ Y = g Y + sT (X,Y, )♯ (22) ∇X ∇X · for any vector fields X and Y on M and T (X,Y, )♯ is defined as in (13). Hence is an orthonormal connection with totally anti-symmetric torsion. · ∇

6 3.1 Pointwise equalities In the following we want to express some curvature quantities for in terms of T and curvature quantities for g. For the Riemann curvature and the scalar curvature we will calculate∇ the norms explicitly in terms of T , the ∇Levi-Civita connection and its curvatures. The norm of the Ricci curvature is given in the appendix.

As in section 2 we ﬁx some point p M, and we extend any tangent vectors X,Y,Z,W TpM to vector ﬁelds again denoted by X,Y,Z,W being synchronous∈ in p. Hence we obtain from (4) the Riemann∈ curvature of ∇ Riem(X, Y )Z, W = Riemg(X, Y )Z, W + s (( g T ) (Y,Z,W ) ( g T ) (X,Z,W )) h i h i ∇X − ∇Y +s2 T X,T (Y,Z, )♯, W T Y,T (X,Z, )♯, W · − · = Riemg(X, Y )Z, W + s (( g T ) (Y,Z,W ) ( g T ) (X,Z,W )) h i ∇X − ∇Y +s2 T (X,Z, )♯,T (Y, W, )♯ T (Y,Z, )♯,T (X, W, )♯ . (23) · · − · · We used the identity T (X,T (Y,Z, )♯, W )= T (X, W, )♯,T (Y,Z; )♯ , which follows from (13). · − · ·

From (6) we conclude that the Ricci curvature of for any orthonormal, synchronous frame E1,...,En deﬁned on some neighbourhood of p is ∇

n n g g 2 ♯ ♯ ric(X, Y ) = ric (X, Y )+ s T (X,Y,Ei) s T (Ei,X, ) ,T (Ei,Y, ) . (24) ∇Ei − · · Xi=1 Xi=1 This formula shows that in general the Ricci curvature is not symmetric in X and Y . From Lemma 2.5 we get for the scalar curvature R of that ∇ R = Rg s2 T 2 . (25) − k k Next, we are aiming at ﬁnding an expression for the norm of the Riemann tensor. As the vector ﬁelds X,Y,Z,W are synchronous in p we get for the differential dT and the codifferential δT of the 3-form T :

dT (X,Y,Z,W ) = ( g T ) (Y,Z,W ) ( g T ) (X,Z,W ) + ( g T ) (X,Y,W ) ( g T ) (X,Y,Z), (26) ∇X − ∇Y ∇Z − ∇W n g δT (X, Y )= T (X,Y,Ei). (27) − ∇Ei Xi=1 We deﬁne the (4, 0)-tensors riem and riemg by

riem(g)(X,Y,Z,W )= Riem(g)(X, Y )Z, W . (28) D E We decompose the Riemann tensor into its symmetric and anti-symmetric component

riem(X,Y,Z,W ) = riemS(X,Y,Z,W ) + riemA(X,Y,Z,W ). (29)

The symmetric part of riem is

S 1 riem (X,Y,Z,W )= 2 (riem(X,Y,Z,W ) + riem(Z,W,X,Y )) and the anti-symmetric part of riem is then given by

riemA(X,Y,Z,W )= 1 (riem(X,Y,Z,W ) riem(Z,W,X,Y )) . 2 − Since riemS and riemA are orthogonal with respect to the scalar product of (4, 0)-tensors, as deﬁned in (57), we ﬁnd

Riem 2 = riem 2 = riemS 2 + riemA 2 (30) k k k k k k k k

7 We also decompose the Ricci curvature into its symmetric and its anti-symmetric components

ric(X, Y ) = ricS(X, Y ) + ricA(X, Y ), (31)

S 1 with ric (X, Y ) = 2 (ric(X, Y ) + ric(Y,X)) and ricA(X, Y ) = 1 (ric(X, Y ) ric(Y,X)) . 2 − Now we give an explicit formula for riem 2 in the case of M being 4-dimensional. k k Theorem 3.1 Let M be a 4-dimensional manifold with Riemannian metric g and connection as given in (22). Then the norm of the Riemann tensor of is given by ∇ ∇ 2 Riem 2 = riemg 2 + 1 s4 T 4 + 1 s2 dT 2 1 s2 Rg T 2 +4s2 B(T )+ riemA (32) k k k k 3 k k 4 k k − 3 k k with

B(T )= ricg(E , E ) T (E , E , )♯,T (E , E , )♯ + 1 Rg T 2 i k i j · j k · 4 k k i,j,kX Proof. See Appendix. 2 We notice that the term B(T ) couples the torsion to the Ricci curvature, and the term riemA is being computed in Lemma A.3.

3.2 Integral formulas In this section (M,g) will be a closed, 4-dimensional Riemannian manifold. We will exploit the topological invariance of the Euler characteristic to deduce integral formulas for 3-forms deﬁned on M.

Definition 3.2 Let be an orthogonal connection on M and let Riemij (X, Y ) := Riem(Ei, Ej )X, Y be its curvature 2-form defined∇ by equation (4). Define the 4-form h i

4 1 K = 2 ǫ Riem Riem , 32π ijkl ij ∧ kl i,j,k,lX=1 where ǫijkl is the totally anti-symmetric tensor with normalisation ǫ1234 = +1. One obtains the classical result for the interplay between the topological invariant Euler characteristic χ(M) and the curvature 2-form of : ∇ Theorem 3.3 The Euler characteristic of M is

χ(M)= K. ZM Proof. For a proof of this theorem we refer to [KN69, Vol. II, Chap. XII, Thm. 5.1]. In four dimensions the Euler characteristic can be expressed in a particularly convenient form in terms of squares of the Riemann, Ricci and scalar curvature of . ∇ Theorem 3.4 Let an orthogonal connection on M and let riem = riemS + riemA, ric = ricS + ricA and R be the Riemann curvature,∇ the Ricci curvature and the scalar curvature of decomposed into their symmetric and anti-symmetric parts according to (29) and (31). Then the Euler characteristic∇ χ(M) is

1 2 S 2 A 2 S 2 A 2 χ(M)= 2 R 4 ric +4 ric + riem riem dvol. 8π Z − k k k k k k −k k M

8 Proof. See Appendix. The classical result for the Euler characteristic in terms of the curvatures of the Levi-Civita connection is due to Berger [Ber70], and it follows immediately from the above theorem:

Corollary 3.5 Let M be a 4-dimensional manifold with Riemannian metric g and Levi-Civita connection g. Then the Euler characteristic χ(M) is given by ∇

1 g 2 g 2 g 2 χ(M)= 2 (R ) 4 ric + riem dvol 8π Z − k k k k M Proof. For g we have riemS = riemg, ricS = ricg and hence riemA 0 and ricA 0. The fact that∇ the Euler characteristic does not depend on the connection≡ allows us≡ to deduce a useful integral formula for 3-forms on closed Riemannian 4-manifolds.

Lemma 3.6 Let M be a closed 4-dimensional manifold with Riemannian metric g and T any 3-form on M. Let Rg denote the scalar curvature of the Levi-Civita connection g. Then ∇ 1 2 1 g 2 1 2 A 2 (33) 4 δT dvol = 3 R T 4 dT +4B(T )+ 2 riem dvol ZM k k ZM k k − k k s k k with B(T ) as deﬁned in Theorem 3.1 and riemA is the anti-symmetric component of the Riemann curvature of with sT as torsion 3-form. ∇ Proof. See Appendix.

4 Dirac operators associated to orthogonal connections

In this section we consider an n-dimensional oriented Riemannian manifold (M,g) equipped with some spin structure. Let be an orthogonal connection given as in (1)–(3). Then the connection acting on vector fields induces a connection∇ acting on spinor fields. Next, we will briefly discuss the construction∇ of this connection g (compare [Ag06, p. 17f], or see [LM89, Chap. II.4] for more details). Again, we write X Y = X Y + A(X, Y ) g ∇ ∇ with the Levi-Civita connection . For any X TpM the endomorphism A(X, ) is skew-adjoint and hence it is 2 ∇ ∈ · an element of so(TpM) ∼= TpM, we can express it as V A(X, )= α E E . (34) · ij i ∧ j Xi

Here Ei Ej is meant as the endomorphism of TpM deﬁned by Ei Ej (Z)= Ei,Z Ej Ej ,Z Ei. For any X T M∧ one determines the coefﬁcients in (34) by ∧ h i − h i ∈ p α = A(X, E ), E = A . (35) ij h i j i XEiEj Each E E lifts to 1 E E in spin(n), and the spinor connection induced by is locally given by i ∧ j 2 i · j ∇ ψ = g ψ + 1 α E E ψ = g ψ + 1 A E E ψ. (36) ∇X ∇X 2 ij i · j · ∇X 2 XEiEj i · j · Xi

Remark 4.1 The connection given by (36) is compatible with the metric on spinors and with Clifford multiplication (see e.g. [Ag06, Lemma 2.1]).

9 Remark 4.2 For totally anti-symmetric torsion, given by a 3-form T as in Corollary 2.4, one can rewrite (36) as 1 ψ = g ψ + (XyT ) ψ, ∇X ∇X 2 ·

k where XyT is the 2-form defined by XyT (X,Z)= T (X,Y,Z). We recall that a k-form ω TpM, given as ♭ ♭ ∈ ω = ω 1 E 1 ... E , acts on the spinor space as ω ψ = ω 1 E 1 V... E ψ. i ,...,ik i ∧ ∧ ik · i ,...,ik i · · ik · i1,...,iP k i1,...,iP k Remark 4.3 Not any connection on spinor fields is induced by an orthogonal connection on tangent vector fields. For example, for the connection ψ = g ψ + X ψ the endomorphism α of spinors defined by ∇X ∇X · α(ψ)= (Y ψ) Y ( ψ) (37) ∇X · − · ∇X g is given by multiplication by the Clifford element X Y + X Y Y X, which does not equal to the Clifford multiplication by any tangent vector. This consideration∇ applies· in any− dimension· n 2. ≥ Remark 4.4 If we assume that for a spinor connection for any vector fields X, Y the endomorphism α defined in (37) is the Clifford multiplication by a tangent vector∇V , i.e. α(ψ) = V ψ for all spinors ψ, then it can X,Y X,Y · be shown than the assignment X Y = VX,Y defines an orthogonal connection on tangent vector fields such that the spinor connection is compatible∇ with the Clifford multiplication. In that case, physics literature occasionally refers to (37) as the tetrad postulate. The Dirac operator associated to the spinor connection from (36) is defined as

n Dψ = E ψ i · ∇Ei Xi=1

n = Dgψ + 1 A E E E ψ 2 EiEj Ek i · j · k · Xi=1 Xj

n = Dgψ + 1 A E E E ψ, (38) 4 EiEj Ek i · j · k · i,j,kX=1 where Dg is the Dirac operator induced by the Levi-Civita connection. The next theorem tells us when the Dirac operator D is formally selfadjoint (i.e. symmetric on the space of com- pactly supported smooth spinor ﬁelds as domain), it is provided as Satz 2 in [FS79]:

Theorem 4.5 The Dirac operator D is formally selfadjoint if and only if the divergence of coincides with the divergence of g, i.e. for any vector ﬁeld Z one has ∇ ∇ n n g E Z, Ei = Z, Ei (39) h∇ i i h∇Ei i Xi=1 Xi=1 in any point p and for any orthonormal basis E , , E of T M. 1 ··· n p Taking the speciﬁc form of Y = g Y + A(X, Y ) into account, we see that (39) is equivalent to ∇X ∇X n n c (A)(Z)= A(E , E ),Z = A(E ,Z), E =0. 12 h i i i − h i ii Xi=1 Xi=1 Hence, we can conclude from Remark 2.2:

10 Corollary 4.6 The Dirac operator D associated to an orthogonal connection is formally selfadjoint if and only if the (3, 0)–torsion tensor A does not have any vectorial compontent, i.e. one has A (T M) (T M) p ∈ T2 p ⊕ T3 p in any point p M ∈ The next lemma states that for the Dirac operator the Cartan type component of the torsion is invisible:

Lemma 4.7 On a Riemannian spin manifold we consider some vector ﬁeld V , some 3-form T and some (3, 0)- tensor ﬁeld S with Sp 3(TpM) for any p M. Let 1 and 2 be the orthogonal connections determined by ∈ T ∈ ∇ ∇

A (X, Y )= X, Y V V, Y X + T (X,Y, )♯ + S(X,Y, )♯ and 1 h i − h i · · A (X, Y )= X, Y V V, Y X + T (X,Y, )♯, 2 h i − h i · respectively (compare Corollary 2.4). Denote the associated Dirac operators by D1 and D2. Then, for any spinor ﬁeld ψ one has D1ψ = D2ψ. Proof. By (38) the difference of the two Dirac operators is

n D ψ D ψ = 1 S E E E ψ. (40) 1 − 2 4 EiEj Ek i · j · k · i,j,kX=1 We use the cyclic identity for S, the fact that S is trace-free in any pair of entries and the Clifford relations E E = E E for i = j as well, in order to obtain: i · j − j · i 6 n n n S E E E = S E E E S E E E EiEj Ek i · j · k − Ej EkEi i · j · k − EkEiEj i · j · k i,j,kX=1 i,j,kX=1 i,j,kX=1

n n = S E E E S E E E − Ej EkEi j · k · i − EkEiEj k · i · j i,j,kX=1 i,j,kX=1

n = 2 S E E E . − EiEj Ek i · j · k i,j,kX=1

n Therefore we get S E E E =0, and the right hand side of (40) is zero. EiEj Ek i · j · k i,j,kP=1 One should note that the above lemma applies pointwise.

Remark 4.8 In the Lorentzian case it is known that torsion of Cartan type does not contribute to the Dirac action under the integral [Sh02, Chap. 2.3]. It is also known that the Dirac action is not real if the torsion has a non- vanishing vectorial component [GS87, Chap. 11.6]. Therefore only totally anti-symmetric torsion is considered to couple to fermions reasonably. The spinor connection in (36) is the connection which is induced by the tangent vector connection given in (1), ∇ hence one expects that their curvatures are related. Let (E1,...,En) be an arbitrary local orthonomal frame. For i, j the curvature endomorphism w.r.t. this frame is deﬁned as Ω ψ = ψ ψ ψ. ij ∇Ei ∇Ej − ∇Ej ∇Ei − ∇[Ei,Ej ] These curvature endomorphismsfor spinors can be naturally determined by the Riemann tensor for tangent vectors, compare with formula (4.37) in Theorem 4.15 of [LM89, Chap. II].

11 Lemma 4.9 For the spinor connection deﬁned in (36) the curvature endomorphisms in p are given by ∇ Ω ψ = 1 Riem(E , E )E , E E E ψ ij 4 h i j a bi a · b · Xa,b with Riemann tensor as deﬁned in (4).

Corollary 4.10 Let tr denote the trace over the spinor space over some footpoint p. Then one has

n tr (Ω Ω )= 1 2[n/2] Riem 2 ij ij − 8 · ·k k i,jX=1 where Riem is the Riemann tensor of the vector connection . ∇ Proof. The spinor space has dimension 2[n/2]. If a = b and c = d the Clifford relations imply 6 6 tr (E E E E )=2[n/2] (δ δ δ δ ) . a · b · c · d bc ad − bd ac From Lemma 4.9 we derive

tr(Ω Ω )= 1 Riem(E , E )E , E Riem(E , E )E , E tr (E E E E ) ij ij 16 h i j a bi h i j c di a · b · c · d Xi,j Xi,j Xa=6 b Xc6=d

= 1 2[n/2] Riem(E , E )E , E Riem(E , E )E , E 16 · h i j a bi h i j b ai Xi,j Xa=6 b Riem(E , E )E , E Riem(E , E )E , E − h i j a bi h i j a bi = 1 2[n/2] ( Riem(E , E )E , E )2 − 8 · h i j a bi Xi,j Xa=6 b

= 1 2[n/2] Riem 2 , − 8 · ·k k where we have used the anti-symmetry of Riem(Ei, Ej )Ea, Eb in the indices a and b, which holds due to (5).

5 Commutative geometries and the spectral action principle

In this section we want to discuss torsion connections within the framework of Connes’ noncommutative geometry (see [Co94]). Let M be a closed Riemannian manifold of dimension n with some fixed spin structure. We denote the algebra of smooth functions by = C∞(M), and we denotethe Hilbert space of square integrablespinor fields by . The Dirac operator Dg associatedA to the Levi-Civita connection is a selfadjoint operator in . The triple ( ,H ,Dg) forms a canonical spectral triple, and it satisfies all axioms for commutative geometry (seeH [Co96], or [GVF01]A H for more details). Now, let be an orthogonal connection on the tangent bundle of M, and let D denote the associated Dirac operator.∇ By Corollary 4.6 we know that D is symmetric if and only if the vectorial component of the torsion of is zero. In that case D is the sum of a selfadjoint operator and a bounded symmetric one, and thus selfadjoint. We∇ notice that D has the same principal symbol and the same Weyl asymptotics as Dg. Furthermore, we note that any natural algebraic structure on the spinor space such as a real structure or the Clifford multiplication with the volume element are parallel with respect to any spinor connection which is induced by an orthogonalconnection on the tangent bundle. Therefore D commutes or anti-commutes with such a structure exactly if Dg does. Following the details of the proof of [GVF01, Thm. 11.1] we see that these observations suffice to verify all axioms for commutative geometry and we conclude:

12 Lemma 5.1 If the vectorial componentof the torsion of is zero, the spectral triple ( , ,D) satisfies the axioms for commutative geometry. ∇ A H Connes’ Reconstruction Theorem (conjectured in [Co96], proved in [Co08]) states that, given a commutative geometry ( , ,D), one can construct a differentiable spin manifold such that coincides with the smooth functions onA it.H Then, from the data ( , ,D) one also gets a Riemannian metricA and one obtains that is isomorphic to the square integrable spinorA H fields (for some spin structure) (see [Co95, Théorème 6]), andH the natural Dirac operator we can always construct is the one induced by the Levi-Civita connection. In the above situation of Lemma 5.1, one can algebraically recover the totally anti-symmetric torsion component from the spectral triple ( , ,D) if the underlying manifold M has even dimension. This can be done by considering the endomorphismA ofH spinors given as difference of D and the Levi-Civita Dirac operator Dg, see (38). In even dimensions the complex Clifford algebra and the space of endomorphisms of the spinor space are identical (see [Fr00], Proposition on p. 13), and hence the endomorphism D Dg can be uniquely determined as a 3-form. In odd dimensions this argument does not apply, as one easily sees− by noticing that e.g. in the 3-dimensional case the volume form acts as multiple of the identity on the spinor space. By Lemma 4.7 the Cartan-type component of the torsion is invisible for the Dirac operator D, and therefore it cannot be recovered from ( , ,D). This can be interpreted as some sort of gauge freedom, which is schematically illustrated in Figure 1. A H

Figure 1: Cartan-type component S of torsion is invisible for Dirac operator (gauge freedom).

Remark 5.2 For some even-dimensional compact Riemannian manifold M with a given spin structure we fix = C∞(M) and the space of square integrable spinor fields. The above considerations show that one has a familyA of Dirac operatorsH parametrized by the 3-forms T Ω3(M) such that the associated spectral triples are pairwise disctinct commutative geometries. Notice that all these∈ Dirac operatorsare in the same K-homology class since they all have the same principal symbol. We leave it open how big the class of first order operators D in is for which ( , ,D) forms a commutative geometry. H A H Remark 5.3 For spectral triples of odd KO-dimension it has recently been shown in [SZ10, Prop. 1.2] that one can modify the Dirac operator by adding a term induced by a selfadjoint element of the algebra and still finds the axioms of spectral triples satisfied. In the case of = C∞(M) and KO-dimensions 3 or 7 moduloA 8, i.e. M is of dimension 3 or 7 modulo 8, this modification is realisedA by adding a real-valued function Φ to the Dirac operator, see [SZ10, Rem. 1.3]. ∈ A In the following we will only consider orthogonal connections with zero vectorial component to ensure selfad- jointness of the induced Dirac operator D. For the computation∇ of the Chamseddine-Connes spectral action (see 2 [CC97]) we need the Seeley-deWitt coefficients a2k(D ) of the heat trace asymptotics [Gi95]

2 Tr e−t D tk−n/2a (D2) as t 0. ∼ 2k → kX≥0

13 Proposition 5.4 The ﬁrst two Seeley-deWitt coefﬁcients are

2 1 [n/2] a0(D )= n/2 2 dvol, (4π) ZM

2 1 [n/2] 3 2 1 g a2(D )= n/2 2 T R dvol. (4π) ZM 4k k − 12 Proof. By Lemma 4.7 we can assume without loss of generality that the Cartan-type component of the torsion g ♯ vanishes. The orthogonal connection is given by X Y = X Y + T (X,Y, ) . Adapting [AF04, Thm. 6.2] into our notation we get the Bochner formula∇ ∇ ∇ · 3 1 3 D2 =∆+ dT + Rg T 2 (41) 2 4 − 4k k where ∆ is the Laplacian associated to the spin connection 3 ψ = g ψ + (XyT ) ψ, (42) ∇X ∇X 2 · e which is induced (as in Remark 4.2) by the orthogonal connection

Y = g Y +3T (X,Y, )♯. (43) ∇X ∇X · We notice that the trace of dT taken overe the spinor space is zero due to Clifford relations. Inserting this into the general formulas for the Seeley-deWitt coefﬁcients (see [Gi95, Theorem 4.1.6]) the claim follows. 2 If we consider the spectral action given by the a2(D ) and variations with respect to the torsion 3-form T we ﬁnd that T = 0 is the only possibility for critical points. Therefore this spectral action detects the Dirac operator induced by the Levi-Civita connection within the class of Dirac operators induced by orthogonal connections without vectorial torsion. We note that this holds in any dimension. This is in complete accordance with [CM08, Section 18.2]. 2 The computation of a4(D ) is more involved, we will give it only for 4-dimensional manifolds. In [ILV10] it has been noted that some terms given in [HPS10] vanish. Similar results have been found before (compare [Go80], [Ob83], [Gr86]). The calculation given below is elementary, it takes place essentially in the tangent bundle and should therefore be easily accessible.

Proposition 5.5 If M is 4-dimensional, the third Seeley-deWitt coefﬁcient is

2 11 1 2 3 2 a4(D )= 720 χ(M) 320π2 C dvol 32π2 δT dvol, (44) − ZM k k − ZM k k where C is the Weyl curvature of M (computed from the Levi-Civita connection).

2 3 1 g 3 2 Proof. We read (41) as D = ∆ E with potential E = 2 dT 4 R + 4 T . From [Gi95, Theorem 4.1.6,c)] we get − − − k k

2 1 g 2 g 2 g 2 g 2 a4(D )= 2 tr 60R E + 180E + 30 Ωij Ωij +20(R ) 8 ric +8 Riem dvol , 5760π Z − k k k k M Xi,j where we have omitted the terms that integrate to zero over the closed manifold M (Laplacians of functions). The term Ω is the curvature endomorphism for the spinor connection . The traces over the spinor space are ij ∇ tr(E)= Rg +3 T 2 , e − k k

14 1 3 9 9 tr(E2)= (Rg)2 Rg T 2 + T 4 + dT 2, 4 − 2 k k 4 k k 24k k where we note that tr(ω2)= 1 ω 2 for any ω 4. For the orthogonal connection on the tangent bundle we 6 k k ∈ ∇ denote the Riemannian curvature by Riem and applyV Corollary 4.10. Then we get e

2 1 1 g 2 g 2 g 2 2 a4(D ) = 2 5 (R ) 8 Ric +8 Riem 15 Riem 16π 360 Z − k k k k − k k M 90 Rg T 2 + 405 T 4 + 135 dT 2 dvol − k k k k 2 k k 1 1 g 2 g 2 g 2 = 16π2 360 5 (R ) 8 Ric 7 Riem dvol ZM − k k − k k

1 1 g 2 3 2 1 A 2 2 R T dT +12 B(T )+ riem dvol − 16π 8 Z k k − 4 k k 3 M by means of Proposition 3.1. With Lemma 3.5 we identify the ﬁrst integral as the Euler characteristic plus the square of the Weyl curvature. Lemma 3.6 shows that the second integral equals

1 1 g 2 3 2 1 A 2 1 3 2 2 R T dT +12 B(T )+ riem dvol = 2 δT dvol. 16π 8 Z k k − 4 k k 3 16π 2 Z k k M M

This ﬁnishes the proof. Next, we want to consider the Chamseddine-Connes spectral action (see [CC97]) for the Dirac operator D. For Λ > 0 it is deﬁned as D2 I = Tr F CC Λ2 where Tr denotes the operator trace over as before, and F : R+ R+ is a cut-off function with support in the interval [0, +1] which is constant nearH the origin. Using the heat→ trace asymptotics one gets an asymptotic expression for I as Λ (see [CC10] for details): CC → ∞ D2 I = Tr F =Λ4 F a (D2)+Λ2 F a (D2)+Λ0 F a (D2)+O(Λ−∞) (45) CC Λ2 4 0 2 2 0 4

∞ ∞ with the ﬁrst three moments of the cut-off function which are given by F = s F (s) ds, F = F (s) ds 4 0 · 2 0 and F0 = F (0). Note that these moments are independent of the geometry of tRhe manifold. R

Now we want to deduce the equation of motion for ICC . In analogy to the Riemannian Einstein-Hilbert case we consider variations with respect to the metric and the torsion 3-form while keeping the volume ﬁxed. Then 2 a0(D ) and χ(M) are constant and their variation vanishes. In order to avoid further complications we neglect the contributions of the O(Λ−∞)-term. Therefore we will consider the following action functional

g 2 2 2 ICC = α R dvol β C dvol + γ1 T dvol γ2 δT dvol, (46) − ZM − ZM k k ZM k k − ZM k k e where α,β,γ1,γ2 > 0. Before we proceed let us brieﬂy recall the standard scalar product on k-forms induced by a Riemannian metric g (compare e.g. with [Bl81]). Let E1,...,En be an orthonormal basis of some tangent space TpM. Then the scalar product on k T ∗M is uniquely determined by the requirement that E∗ ... E∗ , i <...

15 where the coeffients ωi1...ik , ηi1...ik are anti-symmetric in the indices i1 ...ik, the scalar product is then given as 1 ω, η = gi1j1 gik jk ω η . (47) h ig k! ··· i1...ik j1...jk i1X,...,ik j1,...,jk Referring to the norm of 2-forms and 3-forms as used above we note that S,S = 1 S 2, T,T = 1 T 2 for h ig 2 k k h ig 6 k k S 2 T ∗M and T 3 T ∗M, compare (58). With respect to this scalar product the Hodge -operator is an ∈ p ∈ p ∗ isometry,V and on a 4-manifoldV the L2-adjoint of d is δ = d independently of the degree k of the form. For 3-forms δ is given as in (27). −∗ ∗ Let g be a Riemannian metric on M. For any k-form η on M we define a symmetric (2, 0)-tensor gη for k 2 by ≥ gη(X, Y )= Xyη, Y yη h ig and for k =1 by gη(X, Y )= Xyη Y yη = η(X) η(Y ) for any tangent vectors X,Y. · For (2, 0)-tensors a and h the natural scalar product defined in (57) reads as

ir js a,h = aij g g hrs, (48) h i Xr,s i,j in local coordinates, as above.

Lemma 5.6 Let (g(t))t be a smooth family of Riemannian metrics on M with g(0) = g and g˙(0) = h, and let k 1. Then for any k-form η on M we get ≥ d η, η = gη,h . dt t=0h ig(t) −h i

Proof. In coordinates we write η = η dxi1 ... dxik with η anti-symmetric in the indices. i1...ik ∧ ∧ i1...ik i1<...

1 i2j2 ik jk i1r j1s = ηi1...ik ηj1...jk g g g g hrs −(k 1)! ··· − Xr,s i1X,...,ik j1,...,jk

η i1r j1s = (g )i1j1 g g hrs − Xr,s i1,j1

= gη,h , −h i where we have used the total anti-symmetry of ηi1...ik in the indices for the second equality.

Theorem 5.7 Any critical point (M,g,T ) of ICC satisﬁes 0=3γ T γ dδT e (49) 1 − 2 1 1 0= αGg βBg + γ 6g∗T + T 2g γ 2gd∗T + δT 2g (50) − 1 − 2k k − 2 − 2k k g g 1 g g where G = ric 2 R g is the Einstein tensor of the metric g and B denotes its Bach tensor (for a deﬁnition see [Bes87, (4.77.)]).−

16 Proof. Let (M,g,T ) be a critial point. We consider an arbitrary variation T (t) of 3-forms with T (0) = T and T˙ (0) = τ. Then we have

d 2 2 0= γ1 T (t) γ2 δT (t) dvol dt t=0 Z k k − k k M

d = 6γ1 T (t),T (t) g 2γ2 δT (t),δT (t) g dvol dt t=0 Z h i − h i M

= 3γ1 T γ2 dδT, 4τ gdvol ZM h − i since d is the adjoint of δ. As τ can be chosen arbitrarily we have established (49). Now we ﬁx T and consider an arbitrary variation g(t) of Riemannian metrics with g(0) = g and g˙(0) = h. In the d following we label any object which depends on g(t). First we note that dt t=0dvolg(t) = g,h dvolg and by help of Lemma 5.6 we compute | h i

d d T 2 dvol = 6 T,T dvol dt t=0 Z k kg(t) g(t) dt t=0 Z h ig(t) g(t) M M

d = 6 T, T dvol dt t=0 Z ∗g(t) ∗g(t) g(t) g(t) M

∗gT d =3 2g + gT, gT g,h +4 g T, T dvolg Z − h∗ ∗ ig ∗ dt t=0 ∗g(t) g M D E

∗gT 1 2 d = 6g + T g,h + 12 g T, T dvolg (51) Z − 2k k ∗ dt t=0 ∗g(t) g M

Now we calculate d d δ T 2 dvol = 2 δ T,δ T dvol dt t=0 Z k g(t) kg(t) g(t) dt t=0 Z g(t) g(t) g(t) g(t) M M

d = 2 d T, d T dvol dt t=0 Z ∗g(t) ∗g(t) g(t) g(t) M

d∗gT = 2g + d g T, d g T g g,h ZM D− h ∗ ∗ i E d +4 δ d T, T dvol g ∗g dt t=0 ∗g(t) g g 

d∗gT 1 = 2g + δgT g,h ZM − 2k k γ d + 12 1 T, T dvol (52) γ ∗g dt t=0 ∗g(t) g g 2 where we have inserted 3γ T = γ δ d T which we obtained from (49). 1 ∗g 2 g ∗g Finally [Bes87, Proposition 4.17] and [Bes87, (4.77)] tell us that

d g(t) g(t) 2 g g αR β C dvol = αG βB ,h dvolg. (53) dt t=0 Z − − k kg(t) g(t) Z h − i M M

Combining (51), (52) and (53) gives the assertion (50). 

17 Remark 5.8 a) Equation (49) is a Proca equation for a 3-form. This suggests a physical interpretation of torsion as a massive vector boson. This feature has been observed earlier in the Lorentzian context, see [OVEH97], and appears to be natural for dynamical Lagrangians of the torsion. b) Equivalently, the Proca equation (49) can be expressed as γ2 under the condition that . ∆T = 3 γ1 T dT =0

Remark 5.9 Ricci ﬂat manifolds (M,g) with T = 0 are critical points of ICC . This follows from the fact that Ricci ﬂat manifolds have vanishing Bach tensors (see [Bes87, Prop. 4.78]). e Finding solutions for the equations of motions (49) and (50) with T = 0 is a challenge. The following lemmas show that classes of warped products with special choices of T can be6 excluded.

Lemma 5.10 Let (N,h) be a compact oriented 3-dimensional Riemannian manifold with constant curvature, and let f : S1 (0, ) be some smooth function on the circle S1 = R/Z. Consider M = S1 N equipped with the warped→ product∞ metric g = dt2 f(t)2 h. Let τ : M R be a smooth function and× set the 3-form ∗ ⊕ → 1 T = τ π dvol(N,h) where dvol(N,h) is the Riemannian volume form of (N,h) and π : S N N is the canonical· projection, and let this triple (M,g,T ) solve the equations of motions (49) and (50). Then× the→ torsion is zero: T =0. Proof. As (N,h) is locally conformally ﬂat, so is the warped product (M,g). Therefore the Bach tensor of (M,g) vanishes, Bg = 0. In order to get the Einstein tensor Gg we need to calculate the curvatures of M. By X, Y , ∂ Z we denote vectors tangent to the leaves t N, ∂t is the unit normal vector ﬁeld. For the Levi-Connection connection we obain { }×

g ∂ f˙(t) g ∂ g h ˙ ∂ X ∂t = f(t) X, ∂ ∂t =0, X Y = X Y f(t) f(t) h(X, Y ) ∂t , ∇ ∇ ∂t ∇ ∇ − where X, Y are also considered as tangent vectors ﬁelds of (N,h) and h is the corresponding Levi-Connection connection. The Riemann tensor of g is given by ∇ ∇ 2 g h f˙(t) Riem (X, Y )Z = Riem (X, Y )Z + f(t) g(X,Z)Y g(Y,Z)X ·{ − } g ∂ f¨(t) ∂ Riem (X, ∂t )Y = f(t) g(X, Y ) ∂t

¨ Riemg(X, ∂ ) ∂ = f(t) X. ∂t ∂t − f(t) From that we get the Ricci curvature and the scalar curvature

g f¨(t) 2 h ¨ ˙ 2 ric = 3 f(t) dt ric (f(t) f(t)+2(f(t)) ) h − ⊕ − g 1 h ¨ ˙ 2 R = f(t)2 R 6f(t) f(t) 6(f(t)) . − − As (N,h) is assumed to have constant curvature there is a κ R such that rich =2κh and Rh =6κ. Hence the Einstein tensor of (M,g) is ∈

g 3 ˙ 2 2 ¨ ˙ 2 G = f(t)2 (f(t)) κ dt 2f(t) f(t) + (f(t)) κ h. (54) − ⊕ − 1 2 3 ∂ ∂ ∂ Next, we consider normal coordinates (x , x , x ) of N about p such that ∂x1 , ∂x2 , ∂x3 forms an orthonormal basis of TpN with respect to h. For the volume distortion √h, given by √h(x)= det(hij (x), we get in p: p 3 2 ∂ ∂ 1 h 1 √h =1, i √h =0, i 2 √h = R (p)= κ. ∂x (∂x ) − 3 − 3 Xi=1

18 Now we take the product chart (t, x1, x2, x3) of M. In these coordinates the torsion 3-form T reads as τ(t, x1, x2, x3)√h(x1, x2, x3)dx1 dx2 dx3. From (49) we get dT =0 which implies that ∂τ 0. ∧ ∧ ∂t ≡ 1 2 3 ∗ 1 2 3 As dx , dx , dx is an orthonormalbasis of Tp N w.r.t. h, we getthat dt, f dx , f dx , f dx forms an orthonormal ∗ basis of T(t,p)M w.r.t. g. Hence, we get

2 2 6τ T =6 T,T = 6 . (55) k kg h i f 1 2 3 1 In (t,p) we get T = τ(x , x , x ) 3 dt and thus ∗ f(t)

2 ∗T τ 2 g = f(t)6 dt (56)

3 1 ∂τ i Furthermore we obtain d T = 3 i dx dt, and therefore ∗ f(t) ∂x ∧ iP=1 d∗T ∂ ∂ 1 g 2 g ( , ) = 6 grad τ ∂t ∂t f(t) k kg d∗T ∂ ∂ g ( ∂t , ∂xi ) = 0

d∗T ∂ ∂ 1 ∂ ∂ g ( i , j ) = 6 dτ( i ) dτ( j ). ∂x ∂x f(t) ∂x · ∂x If we now assume that equation (50) holds, we observe that after restricting the occuring (2, 0)-tensors to TN every ingredient is a multiple of h except gd∗T . So (50) can only hold if dτ = 0. Therefore the function τ is constant, and d T =0 and so gd∗T =0. Then we decompose (50) into its dt2-component and its h-component: ∗ 2 2 3α ˙ 2 6γ1τ 6γ1τ f(t)2 f(t)) κ = f(t)6 2f(t)6 − − 2 ¨ ˙ 2 6γ1τ α 2f(t) f(t) + (f(t)) κ = 2f(t)4 − − We divide the ﬁrst equation by 3, the second one by (f(t)2, and substract the ﬁrst from the second. We get:

2 γ1τ 1 f¨(t)= 5 . − α · f(t) Since f > 0 and α > 0 we conclude that f¨ 0 and therefore f˙ increases monotonically. On the other hand f˙ being a function on S1 = R/Z is periodic. Therefore≥ f˙ is constant and hence τ is zero.

Remark 5.11 It should be noted that with the ansatz in Lemma 5.10 one can obtain restrictions on the geometry of (N,h) pointwise from equation (49). Namely, in the normal coordinates from above one ﬁnds

3 2 ∂ 1 dδT = τ i 2 √h = κT. − (∂x ) 3 Xi=1

By (49) we have 1 3γ1 , therefore is a spherical space form. 3 κ = γ2 > 0 (N,h)

Remark 5.12 If we now consider formally the same equations of motions for Lorentzian manifolds, and if we admit also non-compact manifolds as solutions, the argument from the proof of Lemma 5.10 cannot discard the R 2 2 ∗ Robertson-Walker ansatz M = N with g = dt f(t) h and T = τ π dvol(N,h) because in the above proof the compactness of S1 is essential.× − ⊕ · One could argue that in the above examples the torsion T is not dynamical. In the last example we consider a situation where torsion may be dynamical, and we show that the torsion vanishes by other reasons.

19 Lemma 5.13 Let (N,h) be the ﬂat torus T 3 = R3/Z3, let f : S1 (0, ) be some smooth function. Consider M = S1 N equipped with the warped product metric g = dt2 →f(t)∞2 h. Let τ : S1 (0, ) be a smooth function, and× let ⊕ → ∞ T = τ(t)(dx1 dx2 + dx2 dx3 + dx3 dx1) dt. ∧ ∧ ∧ ∧ Assume that the triple (M,g,T ) solves the equations of motions (49) and (50). Then the torsion is zero: T =0. Proof. For the T as above we ﬁnd

T = τ(t) dx3 + dx1 + dx2 , ∗ f(t) f(t)˙τ(t)−τ(t)f˙(t) 3 1 2 d T = 2 dt dx + dx + dx . ∗ f(t) ∧ 2 2 τ(t) 2 2 2 ˙ Furthermore we get T = 18 ( f(t)2 ) and δT = f(t)6 f(t)τ ˙(t) f(t)τ(t) and k k · k k · − ∗T ∂ ∂ g ( ∂t , ∂t ) = 0,

2 d∗T ∂ ∂ (f(t)˙τ(t)−τ(t)f˙(t)) g ( , ) = 3 6 . ∂t ∂t · f(t) ∂ Now we insert twice ∂t into the (2, 0)-tensors of (50) and use the speciﬁc form of the Einstein tensor (54). This yields 2 2 2 (f˙(t)) (τ(t)) (f(t)˙τ(t)−τ(t)f˙(t)) 0=3α 2 +9γ 4 +5γ 6 . · (f(t)) 1 · (f(t)) 2 · (f(t)) Since α, γ1,γ2 > 0 each summand is nonnegative, therefore each term vanishes individually, in particular the second one. From this we conclude τ =0.

Remark 5.14 In the Lorentzian setting actions similar to ICC have already been consider and some cosmological consequences for possible critical points with non-vanishing torsion have been discussed (see e.g. [HHKN76] or e [Sh02] and the references therein).

A Appendix: Proofs of Proposition 3.1, Theorem 3.4 and Lemma 3.6

We consider a manifold M equipped with a Riemannian metric g. For (k, 0)-tensor ﬁelds S, T on M one has pointwise the natural scalar product

S,T = S(E ,...,E ) T (E ,...,E ), (57) h i i1 ik · i1 ik i1X,...,ik where (Ei)i is an orthonormal basis of the tangent space. This coincides with the deﬁnition given in (8) for (3, 0)-tensors. If S and T are k-forms there is another natural scalar product, as deﬁned in (47), given by

S,T = S(E ,...,E ) T (E ,...,E ) h ig i1 ik · i1 ik i1<...

Now let g denote the Levi-Civita connection on the tangent bundle. We fix some 3-form T on M and some s R, and∇ we are studying the connection with totally anti-symmetric torsion as in (22). ∈ ∇ As in section 2 we fix some point p M, and we extend any tangent vectors X,Y,Z,W TpM to vector fields again denoted by X,Y,Z,W being synchronous∈ in p. ∈

20 We recall the Ricci decomposition for Riemg (compare [Bes87, Chapter 1.G]): Let the tracefree part of the Ricci curvature be denoted by b(X, Y ) = ricg(X, Y ) 1 Rg X, Y . Then, one has − n h i Riemg(X, Y )Z, W = 1 Rg ( X, W Y,Z Y, W X,Z ) h i n(n−1) h i h i − h i h i + 1 b(X, W ) Y,Z b(Y, W ) X,Z + X, W b(Y,Z) Y, W b(X,Z) n−2 h i− h i h i − h i + C(X, Y )Z, W , (59) h i where the (3, 1)-tensor C is the Weyl tensor of the Riemannian metric g. The Weyl tensor possesses all the symmetries of the Riemann tensor Riemg (e.g. the Bianchi identity holds), and in addition the contraction of C(X, Y )Z, W taken over any two slots is zero. Furthermore, it should be noted that this composition into scalar hcurvature, trace-freei Ricci curvature and Weyl curvature is orthogonal with respect to the usual scalar product in the space of (4, 0)-tensors (see [Bes87, Thm. 1.114]).

A.1 Proof of Proposition 3.1 Proof of Proposition 3.1. First, consider some Riemannian manifold (M,g) of arbitrary dimension n with Levi-Civita connection g. We ﬁx some 3-form T on M, and we are studying the connection deﬁned as in (22). ∇ ∇ The symmetric part of riem is

S 1 riem (X,Y,Z,W )= 2 (riem(X,Y,Z,W ) + riem(Z,W,X,Y )) g s = riem (X,Y,Z,W )+ 2 dT (X,Y,Z,W ) + s2 T (X,Z, )♯,T (Y, W, )♯ T (Y,Z, )♯,T (X, W, )♯ , (60) · · − · · where we have used (23) and (26). In the following we will abbreviate

C = C(E , E )E , E , riem(g/S/A) = riem(g/S/A)(E , E , E , E ), ijkl h i j k li ijkl i j k l (g/S/A) (g/S/A) ricij = ric (Ei, Ej ), bij = b(Ei, Ej ),

T = T (E , E , )♯, dT = dT (E , E , E , E ) ij i j · ijkl i j k l for all indices i,j,k,l. 2 We will calculate riemS 2 = n riemS by means of (60). Three cross terms appear, one of them k k i,j,k,l=1 ijkl vanishes in the case of arbitraryP dimensionn: We choose some orthonormal frame E1,...,En, which is deﬁned on some neighbourhood of p and is synchronous in p. Using the Bianchi identity we compute

n n 1 g g g g 0= 3 riemijkl + riemjkil + riemkijl dTijkl = riemijkl dTijkl. (61) i,j,k,lX=1 i,j,k,lX=1 

From now on we restrict to the case of dimension n = 4. We note that dT = 0 is only possible if i,j,k,l ijkl 6 ∈ 1,..., 4 are pairwise distinct. In that case one has T ,T = 4 T (E , E , E ) T (E , E , E )=0 as { } h ij kli m=1 i j m k l m any m 1,..., 4 is equal to one of i,j,k,l. Hence we get P ∈{ } 4 0= dT T ,T T ,T . (62) ijkl · h ik jli − h jk ili i,j,k,lX=1

21 In order to identify the third cross term we impose the Ricci decomposition (59) which now reads as

riemg = 1 Rg (δ δ δ δ )+ 1 b δ b δ + δ b δ b + C ijkl 12 ij jk − jl ik 2 il jk − jl ik il jk − jl ik ijkl and we get

riemg ( T ,T T ,T ) = 1 Rg ( T ,T + T ,T ) ijkl h ik jli − h jk ili 12 h ij jii h ji ij i i,j,k,lX Xi,j

+ 1 b T ,T + b T ,T 2 il h ij jli jl h ji ili Xi,j,l i,j,,lX

+ b T ,T + b T ,T jk h ik jii ik h jk ij i i,j,kX i,j,kX 

+ C ( T ,T T ,T ) ijkl h ik jli − h jk ili i,j,k,lX

= 1 Rg T 2 + 2 B(T )+C(T ) − 6 k k where we have set

B(T ) = b T ,T = ricg T ,T 1 Rg T ,T ik h ij jki ik h ij jki− 4 h ij jii i,j,kX i,j,kX Xi,j

= ricg T ,T + 1 Rg T 2 (63) ik h ij jki 4 k k i,j,kX

C(T ) = C ( T ,T T ,T ) (64) ijkl h ik jli − h jk ili i,j,k,lX It follows that

riemS 2 = riemg 2 + 1 s2 dT 2 1 s2 Rg T 2 +4s2 B(T )+ s2 C(T ) k k k k 4 k k − 3 k k 4 +s4 ( T ,T T ,T )2 h ik jli − h jk ili i,j,k,lX=1

= riemg 2 + 1 s2 dT 2 1 s2 Rg T 2 +4s2 B(T )+ 1 s4 T 4 k k 4 k k − 3 k k 3 k k where for the last step we use the following technical Lemmas A.1 and A.2. 

Lemma A.1 Let M be a 4-dimensional Riemannian manifold and p M. For any T 3 T ∗M we consider ∈ ∈ p C(T ) as deﬁned (64). Then one has C(T )=0. V Proof. Let E , E , E , E be an orthogonal basis of T M. For T,S 3 T ∗M we deﬁne 1 2 3 4 p ∈ p V 4 c(T,S)=2 C T ,S . · ijkl h ik jli i,j,k,lX=1

The Weyl tensor satisﬁes C = C , and hence c is a symmetric bilinear form. For T 3 T ∗M we have ijkl jilk ∈ p V c(T,T )= C T ,T C T ,T = C ( T ,T T ,T )=C(T ). ijkl h ik jli− jikl h ik jli jikl h ik jli − h jk ili i,j,k,lX i,j,k,lX i,j,k,lX

22 Hence, the Lemma will be proved if we show that c 0. We will check this on the basis of 3 T ∗M consisting of ≡ p E ♭ E ♭ E ♭, E ♭ E ♭ E ♭, E ♭ E ♭ E ♭, E ♭ E ♭ E ♭. V 1 ∧ 2 ∧ 3 2 ∧ 3 ∧ 4 1 ∧ 3 ∧ 4 1 ∧ 2 ∧ 4 For t ,t R we consider T ,T 3 T ∗M deﬁned by 1 2 ∈ 1 2 ∈ p V T := t E ♭ E ♭ E ♭,T := t E ♭ E ♭ E ♭. 1 1 · 1 ∧ 2 ∧ 3 2 2 · 2 ∧ 3 ∧ 4 We will verify that c(T1,T1)=0 and c(T1,T2)=0, this will sufﬁce because the computations are the same if one inserts any elements of the above basis into c. For a 1, 2 Table 1 gives (T ) := T (E , E , )♯ explicitly. ∈{ } a ij a i j · ❅ j ❅ j 1 2 34 12 3 4 i❅❅ i❅❅ 1 0 t E t E 0 1 00 0 0 (T ) : 1 3 1 2 (T ) : 1 ij 2 t E − 0 t E 0 2 ij 2 0 0 t E t E 1 3 − 1 1 − 2 4 2 3 3 t1E2 t1E1 0 0 3 0 t2E4 0 t2E2 4 − 0 0 00 4 0 t E t E − 0 − 2 3 2 2 Table 1: (T ) := T (E , E , )♯ for a 1, 2 . a ij a i j · ∈{ } Using Table 1 we compute

4 3 3 c(T ,T )= C T ,T = C t2 (δ δ δ δ )= t2 C (65) 1 1 ijkl h ik jli ijkl 1 ij kl − il jk − 1 ijji i,j,k,lX=1 i,j,k,lX=1 i,jX=1

The Weyl tensor is trace-free in any pair of indices. Hence for any i 1, 2, 3 we get C = 3 C . ∈ { } − i44i j=1 ijji Inserting this into (65) and using C4444 =0 we obtain P 4 2 c(T1,T1)= t1 Ci44i =0 Xi=1 as the Weyl tensor is trace-free in any pair of indices. With table 1 we obtain 4 c(T ,T ) = 2 C (T ) , (T ) 1 2 ijkl h 1 ik 2 jli i,j,k,lX=1 = 2 t t (C + C + C + C ) − 1 2 1224 2412 1334 3413 = 4 t t (C + C ) − 1 2 1224 1334 4 = 4 t t C − 1 2 1ii4 Xi=1 = 0 where we used the symmetry Cijkl = Cklij of the Weyl tensor in the second step and C1114 = C1444 = 0 in the third step, the last step is achieved as the Weyl tensor is trace-free.

Lemma A.2 Let M be a Riemannian manifold of dimension 4, let p M and let E1, E2, E3, E4 be an orthonormal basis of T M. Then for any T 3 T ∗M one has ∈ p ∈ p V 4 1 ( T ,T T ,T )2 = T 4 . h ik jli − h jk ili 3 k k i,j,k,lX=1

23 Proof. Choosing the parameters properly we have T = T1 + T2 + T3 + T4 with T1 and T2 as in Lemma A.1 and

T := t E ♭ E ♭ E ♭,T := t E ♭ E ♭ E ♭. 3 3 · 1 ∧ 3 ∧ 4 4 4 · 1 ∧ 2 ∧ 4

Deﬁning (Ta)ij as in Lemma A.1 we summarise (T3)ij and (T4)ij in Table 2.

❅ j ❅ j 123 4 1 234 i❅❅ i❅❅ 1 0 0 t E t E 1 0 t E 0 t E (T ) : 3 4 3 3 (T ) : 4 4 4 2 3 ij 2 000− 0 4 ij 2 t E − 0 0 t E 4 4 − 4 1 3 t3E4 0 0 t3E1 3 0 000 4 t E 0 t E − 0 4 t E t E 0 0 − 3 3 3 1 − 4 2 4 1 Table 2: (T ) := T (E , E , )♯ for a 3, 4 . a ij a i j · ∈{ } 2 In order to calculate T 4 = T 2 we note that k k k k 4 0 if a = b 6 (Ta)ij , (Tb)ij =  h i  2 i,jX=1 6 ta if a = b and therefore 

4 4 4 T 2 = T ,T = (T ) , (T ) =6 t2. (66) k k h ij ij i h a ij a ij i a i,jX=1 i,j,aX=1 Xa=1

4 Next, we also want to express ( T ,T T ,T )2 in terms of t ,...,t . h ik jli − h jk ili 1 4 i,j,k,lP=1 2 We note that each summand ( Tik,Tjl Tjk,Til ) remains the same after interchanging i and j and is zero if i = j. Noticing the same for theh indicesi −k and h l, wei restrict ourselves to the case i < j, k

( T ,T T ,T )2 = ( T ,T )2 = ( (T ) , (T ) + (T ) , (T ) )2 = (t2 + t2)2, h 11 22i − h 21 12i h 12 12i h 1 12 1 12i h 4 12 4 12i 1 4 where we have used that 0 if a = b 6 (Ta)ij , (Tb)ij =  h i  2 ta if a = b. For (i, j)=(1, 2) and (k,l)=(1, 3) we ﬁnd 

( T ,T T ,T )2 = ( T ,T )2 = ( (T ) , (T ) )2 = t2t2, h 11 23i − h 21 13i h 12 13i h 4 12 3 13i 3 4 and for (i, j)=(1, 2) and (k,l)=(3, 4) we have

( T ,T T ,T )2 =0. h 13 24i − h 23 14i The same considerations apply to the other summands, and we give the results for the cases i < j, k

Now we drop the condition i < j, k

4 ( T ,T T ,T )2 = 4 (t2 + t2)2 + 16 t2 t2 h ik jli − h jk ili a b a b i,j,k,lX=1 Xa=6 b Xa=6 b

24 P PP (k,l) PP (1, 2) (1, 3) (1, 4) (2, 3) (2, 4) (3, 4) (i, j) PPP

2 2 2 2 2 2 2 2 2 2 2 (1, 2) (t1 + t4) t3t4 t1t3 t2t4 t1t2 0

2 2 2 2 2 2 2 2 2 2 2 (1, 3) t3t4 (t1 + t3) t1t4 t2t3 0 t1t2

2 2 2 2 2 2 2 2 2 2 2 (1, 4) t1t3 t1t4 (t3 + t4) 0 t2t3 t2t4

2 2 2 2 2 2 2 2 2 2 2 (2, 3) t2t4 t2t3 0 (t1 + t2) t1t4 t1t3

2 2 2 2 2 2 2 2 2 2 2 (2, 4) t1t2 0 t2t3 t1t4 (t2 + t4) t3t4

2 2 2 2 2 2 2 2 2 2 2 (3, 4) 0 t1t2 t2t4 t1t3 t3t4 (t2 + t3)

Table 3: The summands ( T ,T T ,T )2 for i < j, k

4 4 2 2 2 2 = 12 ta +8 tatb + 16 tatb Xa=1 Xa=6 b Xa=6 b

4 4 2 2 = 12 ta +2 tatb Xa=1 Xa=6 b

4 2 2 = 12 ta , Xa=1 from which in combination with (66) the claim follows. 

Lemma A.3 The square of the anti-symmetric part of the Riemann tensor of is pointwise given by ∇ n n A 2 2 g 2 2 g g riem = s T (Ej , Ek, El) + s T (Ej , Ek, El) T (Ei, Ek, El) ∇Ei ∇Ei ∇Ej i,j,k,lX=1 i,j,k,lX=1

Proof. The anti-symmetric part of riem is then given by

riemA(X,Y,Z,W ) = 1 (riem(X,Y,Z,W ) riem(Z,W,X,Y )) 2 − s g g = 2 ( X T ) (Y,Z,W ) ( Y T ) (X,Z,W ) ∇ − ∇ g g ( Z T ) (X,Y,W ) + ( W T ) (X,Y,Z) . (67) − ∇ ∇ The lemma follows by direct calculation. This shows in particular that riemA does only dependon the torsion T and not on the Riemannian curvature Riemg.

A.2 The square of the norm of the Ricci curvature Now we decompose the Ricci curvature into its symmetric and its anti-symmetric components

ric(X, Y ) = ricS(X, Y ) + ricA(X, Y ).

25 With (24) we have

n ricS(X, Y ) = 1 (ric(X, Y ) + ric(Y,X)) = ricg(X, Y ) s2 T (X, E , )♯,T (Y, E , )♯ (68) 2 − i · i · Xi=1 and

n A 1 g ric (X, Y ) = (ric(X, Y ) ric(Y,X)) = s T (X,Y,Ei)= s δT (X, Y ). (69) 2 − ∇Ei − Xi=1 To evaluate ric 2 of the Ricci curvature (24) we proceed as in the evaluation of Riem 2 and get k k k k n n n n 2 2 ric 2 = (ric(E , E ))2 = ric2 = ricS + ricA k k i j ij i,j ij i,jX=1 i,jX=1 i,jX=1 i,jX=1 = ricS 2 + ricA 2. (70) k k k k We ﬁnd

Theorem A.4 Let M be a 4-dimensional manifold with Riemannian metric g and connection as given in (22). Then the norm of the Ricci tensor of is given by ∇ ∇ ric 2 = ricg 2 + 1 s4 T 4 1 s2 Rg T 2 +2s2 B(T )+ s2 δT 2 k k k k 3 k k − 2 k k k k with B(T ) as deﬁned in (63). Proof. In order to compute the squared norm ric 2 of the Ricci curvature deﬁned in (24) we proceed as in the evaluation of Riem 2. With (68) and (69) wek get k k k n n n n 2 2 ric 2 = (ric(E , E ))2 = ric2 = ricS + ricA . k k i j ij ij ij i,jX=1 i,jX=1 i,jX=1 i,jX=1

2 We restrict ourselves again to dimension n =4 and calculate ricS 2 = n ricS with (68). We ﬁnd k k i,j=1 ij P 4 4 4 4 2 2 ricS = ricg 2s2 ricg T ,T + s4 T ,T T ,T ij ij − ij h ik jki h ik jki h il jli i,jX=1 i,jX=1 i,j,kX=1 i,j,k,lX=1

= ricg 2 1 s2 Rg T 2 +2s2 B(T )+ 1 s4 T 4 k k − 2 k k 3 k k where we used Lemmas A.5 and A.6 in the last step. We conclude the proof noting that

n n 2 ricA = s2 (δT (E , E ))2 = s2 δT 2. ij i j k k i,jX=1 i,jX=1 

Lemma A.5 Let M be an n-dimensional Riemannian manifold. For any 3-form T one has

n 1 ricg T ,T = B(T )+ Rg T 2. ij h ik jki − n k k i,j,kX=1

26 Proof. We use the decomposition of the Ricci curvature into its trace-free part and its trace (59) and ﬁnd

n n 1 n ricg T ,T = b(E , E ) T ,T + Rgδ T ,T ij h ik jki i j h ik jki n ij h ik jki i,j,kX=1 i,j,kX=1 i,j,kX=1

n 1 n = b(E , E ) T ,T + Rg T ,T − i j h ik kj i n h ik iki i,j,kX=1 i,kX=1 1 = B(T )+ Rg T 2 − n k k with B(T ) as deﬁned in Theorem 3.1. 

Lemma A.6 Let M be an 4-dimensional Riemannian manifold. For any 3-form T one has

4 T ,T T ,T = 1 T 4. h ik jki h il jli 3 k k i,j,k,lX=1

Proof. We compute:

4 4 T ,T T ,T = T (E , E , E )T (E , E , E )T (E , E , E )T (E , E , E ) h ik jki h il jli i k n j k n i l m j l m i,j,k,lX=1 i,j,k,l,n,mX =1

4 = T (En, Ek, Ei)T (Em, El, Ei)T (En, Ek, Ej )T (Em, El, Ej ) i,j,k,l,n,mX =1

4 = T ,T 2 . (71) h kn lmi k,l,n,mX=1

We decompose T = T1 + T2 + T3 + T4 according to the deﬁnitions of the Ta in Lemmas A.1 and A.2. Following precisely the arguments of Lemma A.2 we obtain for (71) from the Tables 1 and 2 the summands given in Table 3. It follows that

4 4 2 ( T ,T )2 = 12 t2 . h kn lmi a k,l,n,mX=1 Xa=1 Combining this with (66) (71) ﬁnishes the proof. 

A.3 Proofs of Theorem 3.4 and Lemma 3.6

Proof of Theorem 3.4. Choose an orthonormal basis E1, .., E4 of TpM at a point p and express the curvature 2-form in this basis

4 Riem = Riem (E , E )(E )♭ (E )♭ ij ij n m n ∧ m n,mX=1

4 = Riem(E , E )E , E (E )♭ (E )♭ h i j n mi n ∧ m n,mX=1

27 4 = riem (E )♭ (E )♭ ijnm n ∧ m n,mX=1 where riem is the (4, 0)-tensor deﬁned in (29). Now the 4-form K reads

4 1 ♭ ♭ ♭ ♭ K = 2 ǫ riem riem (E ) (E ) (E ) (E ) 32π ijkl ijnm klsp n ∧ m ∧ s ∧ p i,..,pX=1

4 1 = 32π2 ǫijklǫnmsp riemijnm riemklsp dvol. i,..,pX=1

With the standard equality δin δim δis δip δjn δjm δjs δjp  ǫ ǫ = det ijkl nmsp δ δ δ δ  kn km ks kp δln δlm δls δlp    and ricij = ric(Ei, Ej ) one ﬁnds

4 4 K 1 2 = 2 R 4 ricij ricji + riemijkl riemklij  dvol. 8π − j,kX=1 i,j,k,lX=1   Decomposing the Riemann and the Ricci curvature into their symmetric and anti-symmtric components we ﬁnd

4 4 4 4 4 ric ric = (ricS )2 (ricA )2 ricS ricA + ricA ricS ij ji ij − ij − ij ij ij ij j,kX=1 j,kX=1 j,kX=1 j,kX=1 j,kX=1

= ricS 2 ricA 2 k k −k k and by the same argument

4 riem riem = riemS 2 riemA 2 ijkl klij k k −k k i,j,k,lX=1

Thus we have 1 2 S 2 A 2 S 2 A 2 K = 2 R 4 ric +4 ric + riem riem dvol. 8π − k k k k k k −k k 

Corollary A.7 Let M be a closed 4-dimensional manifold with Riemannian metric g, an arbitrary orthogonal connection and g the Levi-Civita connection. Then ∇ ∇ (Rg)2 4 ricg 2 + riemg 2 dvol = R2 4 ricS 2 +4 ricA 2 Z − k k k k Z − k k k k M M + riemS 2 riemA 2 dvol k k −k k

Proof. The Euler characteristic is a topological invariant and does not depend on the choice of the connection in the representation of the Euler class. 

28 Proof of Lemma 3.6. Let riem, ric and R be the Riemann curvature, the Ricci curvature and the scalar curvature of . With the decomposition of the norm squared of the symmetric part of the Riemann curvature, theorem 3.1, the∇ the Ricci curvature, theorem A.4, and the scalar curvature,equation (25), we have from corollary A.7

0 = R2 (Rg)2 4 ( ricS 2 ricA 2 ricg 2)+ riemS 2 riemA 2 riemg 2 dvol Z − − k k −k k − k k k k −k k −k k M

2 g 2 4 4 4 4 4 2 g 2 2 2 2 = 2s R T + s T 3 s T +2s R T 8s B(T )+4s δT dvol ZM − k k k k − k k k k − k k 

2 + 1 s4 T 4 + 1 s2 dT 2 1 s2 Rg T 2 +4s2 B(T ) riemA dvol Z 3 k k 4 k k − 3 k k − M 

1 2 = s2 4 δT 2 1 Rg T 2 + 1 dT 2 4 B(T ) riemA dvol. Z k k − 3 k k 4 k k − − s2 M

For any s =0 follows the assertion of the lemma. 6 Remark A.8 At ﬁrst glance the cancelation of the term T 4 in the proof of Lemma 3.6 might surprise. But it has to vanish because it is the only term scaling in the 4kth powerk in the pre-factor s while all other terms scale quadradically in s.

References

[Ag06] I.Agricola, The Srn´ı lectures on non-integrable geometries with torsion. Arch. Math. (Brno) 42 (2006), 5–84. With an appendix by M. Kassuba. [AF04] I. Agricola & T. Friedrich, On the holonomy of connections with skew-symmetric torsion. Math. Ann. 328 (2004), no. 4, 711–748. [Ber70] M.Berger, Quelques formules de variation pour une structure riemannienne. Ann. Ec.´ Norm. Sup. (4) 3 (1970), 285–294. [Bes87] A. L. Besse, Einstein Manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer- Verlag, Berlin 1987. [Bl81] D. Bleecker, Gauge theory and variational principles. Global Analysis Pure and Applied Series, Addison-Wesley, Reading 1981. Unabriged republication, Dover, Mineola 2005.

[Ca23] E.´ Cartan, Sur les variet´ es´ a` connexionafﬁne et la theorie´ de la relativit´ eg´ en´ eralis´ ee´ (premiere` partie) Ann. Ec.´ Norm. Sup. 40 (1923), 325–412.

[Ca24] E.´ Cartan, Sur les variet´ es´ a` connexion afﬁne et la theorie´ de la relativit´ eg´ en´ eralis´ ee´ (premiere` partie, suite). Ann. Ec.´ Norm. Sup. 41 (1924), 1–25.

[Ca25] E.´ Cartan, Sur les variet´ es´ a` connexion afﬁne et la theorie´ de la relativit´ e´ gen´ eralis´ ee´ (deuxieme` partie). Ann. Ec.´ Norm. Sup. 42 (1925), 17–88. [CC97] A. Chamseddine & A. Connes, The spectral action principle. Comm. Math. Phys. 186 (1997), no. 3, 731–750. [CC10] A. Chamseddine & A. Connes, Noncommutative geometry as a framework for uniﬁcation of all fun- damental interactions including gravity. Part I. Fortschritte der Physik 58 (2010), 553–600. [Co94] A.Connes, Noncommutative geometry. Academic Press, San Diego 1994.

29 [Co95] A.Connes, Brisure de symetrie´ spontaneeet´ geometrie´ du point de vue spectral. Séminaire Bourbaki, Vol. 1995/96. Astérisque 241 (1997), Exp. No. 816, 5, 313-349 [Co96] A. Connes, Gravity coupled with matter and the foundation of noncommutative geometry. Comm. Math. Phys. 183 (1996), no. 1, 155–176. [Co08] A.Connes, On the spectral characterization of manifolds. arXiv: 0810.2088. [CM08] A. Connes, M. Marcolli, Noncommutative geometry, quantum fields and motives. AMS, Provi- dence; Hindustan Book Agency, New Delhi 2008. [Fr00] T.Friedrich, Dirac Operators in Riemannian Geometry. AMS, Providence 2000.

[FS79] T. Friedrich & S. Sulanke, Ein Kriterium fur¨ die formale Selbstadjungiertheit des Dirac-Operators. Colloq. Math. 40 (1978/79), no. 2, 239–247. [Gi95] P.B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem. Sec- ond edition, CRC Press, Boca Raton 1995. [GS87] M. Göckeler & T. Schücker, Differential Geometry, Gauge Theories, and Gravity. Cambridge Monographs on Mathematical Physics, Cambridge University Press 1987. [Go80] W. H. Goldthorpe, Spectral Geometry and SO(4) Gravity in a Riemann-Cartan Spacetime. Nucl. Phys. B 170 (1980),307–328. [GVF01] J. M. Gracia-Bond´ıa, J. C. Várilly, H. Figueroa, Elements of Noncommutative Geometry. Birkhäuser, Boston 2001. [Gr86] G.Grensing, Induced Gravity For Nonzero Torsion. Phys. Lett. 169B (1986), 333–336. [HPS10] F. Hanisch, F. Pfäffle & C. A. Stephan, The Spectral Action for Dirac Operators with skew-symmetric Torsion. Comm. Math. Phys. 300 (2010), no. 3, 877–888. [HHKN76] F. W. Hehl, P. von der Heyde, G. D. Kerlick & J. N. Nester, General Relativity with Spin and Torsion: Foundations and Prospects. Rev. Mod. Phys. 48 (1976), 393–416. [ILV10] B. Iochum,C. Levy & D. Vassilevich, Spectral action for torsion with and without boundaries. arXiv: 1008.3630. [KN69] S. Kobayashi & K. Nomizu, Foundations of Differential Geometry, Interscience Publishers, New York 1969. [LM89] H. B. Lawson& M.-L. Michelsohn, Spin Geometry. Princeton University Press, Princeton 1989. [Ob83] Yu.N.Obukhov, Spectral Geometry Of The Riemann-Cartan Space-Time. Nucl. Phys. B 212 (1983), 237–254. [OVEH97] Yu. N. Obukhov, E. J. Vlachynsky,W. Esser & F. W. Hehl, Effective Einstein theory from metric-affine gravity models via irreducible decompositions. Phys. Rev. D 56 (1997), 7769-7778. [Sh02] I.L.Shapiro, Physical Aspects of the Space-Time Torsion. Phys. Rept. 357 (2002), 113–213. [SZ10] A.Sitarz&A.Zajac, Spectral action for scalar perturbations of Dirac operators. arXiv: 1012.3860. [TV83] F. Tricerri & L. Vanhecke, Homogeneous structures on Riemannian manifolds. London Math. Soc. Lecture Notes Series, vol. 83, Cambridge University Press, Cambridge 1983.