A Kastler-Kalau-Walze type theorem and the spectral action for perturbations of Dirac operators on manifolds with boundary

In this paper, we prove a Kastler-Kalau-Walze type theorem for perturbations of Dirac operators on compact manifolds with or without boundary. As a corollary, we give two kinds of operator-theoretic explanations of the gravitational action on boundary. We also compute the spectral action for Dirac operators with two-form perturbations on $4$-dimensional compact manifolds.


Introduction
The noncommutative residue found in [Gu] and [Wo] plays a prominent role in noncommutative geometry. In [Co1], Connes used the noncommutative residue to derive a conformal 4-dimensional Polyakov action analogy. In [Co2], Connes proved that the noncommutative residue on a compact manifold M coincided with the Dixmier's trace on pseudodifferential operators of order −dimM . Several years ago, Connes made a challenging observation that the noncommutative residue of the square of the inverse of the Dirac operator was proportional to the Einstein-Hilbert action, which is called the Kastler-Kalau-Walze theorem now. In [Ka], Kastler gave a brute-force proof of this theorem. In [KW], Kalau and Walze proved this theorem by the normal coordinates way simultaneously. In [Ac], Ackermann gave a note on a new proof of this theorem by the heat kernel expansion way. The Kastler-Kalau-Walze theorem had been generalized to some cases, for example, Dirac operators with torsion [AT], CR manifolds [Po], R n [BC1] (see also [BC2], [Ni]).
On the other hand, Fedosov et al. defined a noncommutative residue on Boutet de Monvel's algebra and proved that it was the unique continuous trace in [FGLS]. In [Sc], Schrohe gave the relation between the Dixmier trace and the noncommutative residue for manifolds with boundary. In [Wa1], [Wa2], we gave an operator theoretic explaination of the gravitational action for manifolds with boundary and proved a Kastler-Kalau-Walze type theorem for Dirac operators and signature operators on manifolds with boundary.

A Kastler-Kalau-Walze type theorem for perturbations of Dirac operators on manifolds without boundary
Let M be a smooth compact Riemannian n-dimensional manifold without boundary and V be a vector bundle on M . Recall that a differential operator P is of Laplace type if it has locally the form where ∂ i is a natural local frame on T M and g i,j = g(∂ i , ∂ j ) and (g ij ) 1≤i,j≤n is the inverse matrix associated to the metric matrix (g i,j ) 1≤i,j≤n on M , and A i and B are smooth sections of End(V ) on M (endomorphism). If P is a Laplace type operator of the form (2.1), then (see [Gi]) there is a unique connection ∇ on V and an unique endomorphism E such that where ∇ L denotes the Levi-civita connection on M . Moreover (with local frames of T * M and V ), ∇ ∂ i = ∂ i + ω i and E are related to g ij , A i and B through where Γ k ij are the Christoffel coefficients of ∇ L . Now we let M be a n-dimensional oriented spin manifold with Riemannian metric g. We recall that the Dirac operator D is locally given as follows in terms of orthonormal frames e i , 1 ≤ i ≤ n and natural frames ∂ i of T M : one has where c(e i ) denotes the Clifford action which satisfies the relation (2.7) By (6a) in [Ka], we have where s is the scalar curvature. Let Ψ be a smooth differential form on M and we also denote the associated Clifford action by Ψ. We will compute D 2 Ψ := (D + Ψ) 2 . We note that (D + Ψ) 2 = D 2 + DΨ + ΨD + Ψ 2 , (2.9) (2.10) By (2.8)-(2.10), we have For a smooth vector field X on M , let c(X) denote the Clifford action. So (2.14) Since E is globally defined on M , so we can perform computations of E in normal coordinates. Taking normal coordinates about x 0 , then We get the following Lichnerowicz formula: Proposition 2.1 Let Ψ be a smooth differential form on M and D Ψ := D + Ψ, then (2.16) where ∇ ∂ i is defined by (2.14) and setting X = ∂ i .
We see two special cases of Proposition 2.1. When Ψ = f where f is a smooth function on M , we have (2.17) (2.18) Let η = a i e i be a one-form where a i is a smooth real function and e i be a dual orthonormal frame by parallel transport along geodesic and X = a i e i be the dual vector field of η. When Ψ = √ −1c(η), by (2.14), we have ∇ Y = ∇ S Y + √ −1g(X, Y ) where Y is a smooth vector field on M . By e j (c(e i )) = 0 and de l (x 0 ) = 0 (see [BGV,Lemma 4.13]), we have Corollary 2.3 For a one-form η and the Clifford action c(η), we have (2.20) When Ψ is a two-form, we let Ψ = 2 k<l a kl e k ∧ e l = a kl e k ∧ e l , where a kl = −a lk , and c(Ψ) = a kl c(e k )c(e l ). So where ω st (e i ) denotes the connection coefficient. By (2.15), (2.26) By (2.22) and (2.24)-(2.26), we have and Corollary 2.4 let Ψ = a kl e k ∧e l and a kl = −a lk ,then For a general differential form Ψ, by (2.15) and Tr(AB) = Tr(BA), we have (2.28) By the Kastler-Kalau-Walze theorem (see [Ka], [KW]), we have where Wres denotes the noncommutative residue (see [Wo]). By (2.28) and (2.29), we have Theorem 2.5 For even n-dimensional compact spin manifolds without boundary and a general form Ψ, the following equality holds: (2.30) By Corollary 2.2, we have Corollary 2.6 For even n-dimensional compact spin manifolds without boundary and a smooth function f on M , the following equality holds: (2.31)

By Corollary 2.3, we have
Corollary 2.7 For even n-dimensional compact spin manifolds without boundary and a one-form Ψ, the following equality holds: (2.32) By Corollary 2.4 and (2.29), we have Corollary 2.8 For even n-dimensional compact spin manifolds without boundary and a two-form Ψ, the following equality holds: (2.33)

A Kastler-Kalau-Walze type theorem for perturbations of Dirac operators on manifolds with boundary
We now let M be a compact 4-dimensional spin manifold with boundary ∂M and U ⊂ M be the collar neighborhood of ∂M which is diffeomorphic to ∂M × [0, 1). and we shall compute the noncommutative residue for manifolds with boundary of (π + D −1 Ψ ) 2 . That is, we shall compute Wres[(π + D −1 Ψ ) 2 ] (for the related definitions, see [Wa1]) and we take the metric as in [Wa1]. Let (x ′ , x n ) ∈ U where x ′ ∈ ∂M and x n denotes the normal direction coordinate. By (2.2.4) in [Wa1], we have where the sum is taken over r (2.36) So we only need to compute ∂M Φ. In analogy with Lemma 2.1 of [Wa1], we can prove the following useful result.
Lemma 2.9 The symbolic calculus of pseudodifferential operators yields Similar to the computations in section 2.2.2 in [Wa1], we can split Φ into the sum of five terms. Since q −1 (D −1 Ψ ) = q −1 (D −1 ), then terms (a)(I), (II),(III) in our case are the same as the terms (a)(I), (II),(III) in [Wa1], so Then we only need to compute the term (b) and the term (c). By Lemma 2.9, (2.43) (2.44) By (2.43) and (2.44) and (2.46) Then the sum of terms (b) and (c) is zero and Φ is zero. Then we get Theorem 2.10 Let M be a 4-dimensional compact spin manifold with boundary ∂M and the metric g M (see (1.3) in [Wa1]). Let Ψ be a general differential form on M . Then In [Wa2], we proved a Kastler-Kalau-Walze theorem associated to Dirac operators for 6-dimensional spin manifolds with boundary. In fact, our computations hold for general Laplacians. This implies Proposition 2.11 ( [Wa2]) Let M be a 6-dimensional compact Riemannian manifold with boundary ∂M and the metric g M (see (1.3) in [Wa1]). Let ∆ be a general Laplacian acting on sections of the vector bundle V . Then (2.51) Since D 2 Ψ is a general Laplacian, then we get Corollary 2.12 Let M be a 6-dimensional compact spin manifold with boundary ∂M and the metric g M . Let Ψ be a general differential form on M . Then (2.52) In the above two cases, the boundary terms vanish. In the following, we will give a boundary term nonvanishing case and compute Wres((D Ψ D) −1 ). We have and (2.55) Similar to the proof of (2.15), we have (2.58) When Ψ is a one-form, we can get the following corollary: Corollary 2.14 Let M be a 4-dimensional compact spin manifold without boundary and let Ψ be a one-form on M . Then (2.59) Now we compute Wres[π + D −1 Ψ π + D −1 ]. We have that terms (a) and (b) are the same as in Theorem 2.10, and since term (c) = − 9 8 πh ′ (0)Ω 3 dx ′ , we get Remark. When Ψ is not a one-form, then the boundary term vanishes. When Ψ = Kdx n near the boundary where K is the extrinsic curvature, then the boundary term is proportional to the gravitational action on the boundary. In fact, the reason of the boundary term being not zero is that π + D Ψ and π + D are not symmetric. In the following, we will compute the more general case, i.e. Wres[f D −1 gD −1 ] for nonzero smooth functions f, g and prove a Kastler-Kalau-Walze type theorem for conformal Dirac operators. When f = g = e −2h , we get the expression of Wres[(e h De h ) −2 ]. We have

A Kastler-Kalau-Walze type theorem for conformal perturbations of Dirac operators
where wres denotes the residue density and we note that the Kastler-Kalau-Walze theorem holds at the residue density level. Some computations show that Since E is globally defined, we can compute it in the normal coordinates. Then we have (3.6) and Similarly, Remark. In Theorem 3.1, when f = g = e −2h , we get a Kastler-Kalau-Walze theorem for conformal Dirac operators. In fact, Theorem 3.1 holds true for any choice of the smooth functions f and g, since we can prove (3.11) by means of the symbolic calculus of pseudodifferential operators without using (3.2), and it is not essential that f and g are nowhere vanishing.
Now we consider manifolds with boundary and we will compute Wres[π + (f D −1 )π + (gD −1 )]. As in [Wa1], we have five terms. (3.14) As in [Wa1], we have (3.15) and (3.16) So the sum of terms (b) and (c) is zero. Then we obtain By the definition of the noncommutative residue for manifolds with boundary, we have that the interior term of Wres[π + (f D −1 )π + (gD −1 )] equals Wres[f D −1 gD −1 ]. Then by Theorem 3.1 and (3.17), we get Theorem 3.2 Let M be a 4-dimensional compact spin manifold with boundary. Then (3.18) When f = 1, g = x n K near the boundary, we have that the boundary term is proportional to the gravitational action on the boundary.

The spectral action for perturbations of Dirac operators
In [IL], Iochum and Levy computed heat kernel coefficients for Dirac operators with one form perturbations and proved that there are no tadpoles for compact spin manifolds without boundary. In [SZ], they investigated the spectral action for scalar perturbations of Dirac operators. In [HPS], Hanisch, Pfäffle and Stephan derived a formula for the gravitational part of the spectral action for Dirac operators on 4-dimensional spin manifolds with totally anti-symmetric torsion. In fact Dirac operators with totally anti-symmetric torsion are three form perturbations of Dirac operators. In this section, we will give some details on the spectral action for Dirac operators with scalar perturbations. We also compute the spectral action for Dirac operators with two-form perturbations on 4-dimensional compact spin manifolds without boundary.
For the perturbed self-adjoint Dirac operator D Ψ , we will calculate the bosonic part of the spectral action. It is defined to be the number of eigenvalues of D Ψ in the interval [−∧, ∧] with ∧ ∈ R + . As in [CC1], it is expressed as Here Tr denotes the operator trace in the L 2 completion of Γ(M, S(T M )), and F : R + → R + is a cut-off function with support in the interval [0, 1] which is constant near the origin. Let dim M = n. By Lemma 1.7.4 in [Gi], we have the heat trace asymptotics for t → 0, One uses the Seeley-deWitt coefficients a 2m (D 2 Ψ ) and t = ∧ −2 to obtain an asymptotics for the spectral action when dim M = 4 [CC1] with the first three moments of the cut-off function which are given by F 4 = ∞ 0 sF (s)ds, F 2 = ∞ 0 F (s)ds and F 0 = F (0). Let (4.4) We use [Gi,Thm 4.1.6] to obtain the first three coefficients of the heat trace asymptotics: (4.7) When Ψ = f , by (2.17) and (4.6), By (2.23), we obtain n i,j,s,t,s 1 ,t 1 =1 By (4.12)-(4.17), we obtain By (4.7) (4.9) and (4.18), we get Proposition 4.1 ( [SZ])The following equality holds In the following, we assume that dim M = 4 and d = 4. We let Ψ be a two-form, namely Ψ = k,l a kl e k ∧ e l where a kl = −a lk . We may consider √ −1Ψ for selfadjoint perturbed Dirac operators. By Corollary 2.4, we obtain (4.23) Then (4.24) Now we can compute Tr(E 2 ).