Dirac-Witten Operators and the Kastler-Kalau-Walze type theorem for manifolds with boundary

In this paper, we obtain two Lichnerowicz type formulas for the Dirac-Witten operators. And we give the proof of Kastler-Kalau-Walze type theorems for the Dirac-Witten operators on 4-dimensional and 6- dimensional compact manifolds with (resp.without) boundary


Introduction
Until now, many geometers have studied noncommutative residues. In [6,21], authors found noncommutative residues are of great importance to the study of noncommutative geometry. In [2], Connes used the noncommutative residue to derive a conformal 4-dimensional Polyakov action analogy. Connes showed us that the noncommutative residue on a compact manifold M coincided with the Dixmier's trace on pseudodifferential operators of order −dimM in [3]. And Connes claimed the noncommutative residue of the square of the inverse of the Dirac operator was proportioned to the Einstein-Hilbert action. Kastler [10] gave a brute-force proof of this theorem. Kalau and Walze proved this theorem in the normal coordinates system simultaneously in [9] . Ackermann proved that the Wodzicki residue of the square of the inverse of the Dirac operator Wres(D −2 ) in turn is essentially the second coefficient of the heat kernel expansion of D 2 in [1].
On the other hand, Wang generalized the Connes' results to the case of manifolds with boundary in [16,17], and proved the Kastler-Kalau-Walze type theorem for the Dirac operator and the signature operator on lower-dimensional manifolds with boundary [18]. In [18,19], Wang computed Wres[π + D −1 • π + D −1 ] and Wres[π + D −2 • π + D −2 ], where the two operators are symmetric, in these cases the boundary term vanished. But for Wres[π + D −1 •π + D −3 ], Wang got a nonvanishing boundary term [15], and give a theoretical explanation for gravitational action on boundary. In others words, Wang provides a kind of method to study the Kastler-Kalau-Walze type theorem for manifolds with boundary. In [11], López and his collaborators introduced an elliptic differential operator which is called the Novikov operator. In [20], Wei and Wang proved Kastler-Kalau-Walze type theorem for modified Novikov operators on compact manifolds. In [24], in order to prove the nonsymmetric positive mass theorem, Zhang introduced the Dirac-Witten operator. The motivation of this paper is to prove the Kastler-Kalau-Walze type theorem for the Dirac-Witten operators.
The paper is organized in the following way. In Section 2, by using the definition of Dirac-Witten operators, we compute the Lichnerowicz formulas for the Dirac-Witten operators. In Section 3 and in Section 4, we prove the Kastler-Kalau-Walze type theorem for 4-dimensional and 6-dimensional manifolds with boundary for the Dirac-Witten operators respectively.

The Dirac-Witten Operators and their Lichnerowicz formulas
Firstly we introduce some notations about the Dirac-Witten Operators. Let M be a n-dimensional (n ≥ 3) oriented compact spin Riemannian manifold with a Riemannian metric g M . And let ∇ L be the Levi-Civita connection about g M . In the local coordinates {x i ; 1 ≤ i ≤ n} and the fixed orthonormal frame {e 1 , · · · , e n }, the connection matrix (ω s,t ) is defined by ∇ L (e 1 , · · · , e n ) = (e 1 , · · · , e n )(ω s,t ). (2.1) Let c(e j ) be the Clifford action. Suppose that ∂ i is a natural local frame on T M and (g ij ) 1≤i,j≤n is the inverse matrix associated to the metric matrix (g ij ) 1≤i,j≤n on M . By [18], we have the Dirac operator where f 1 , f 2 is a complex number and p uv = p(e u , e v ), p is a (0, 2)-tensor. Then when f 1 = Theorem 2.1. The following equalities hold: where s is the scalar curvature.
Let M be a smooth compact oriented spin Riemannian n-dimensional manifolds without boundary and N be a vector bundle on M . If P is a differential operator of Laplace type, then it has locally the form where ∂ i is a natural local frame on T M and (g ij ) 1≤i,j≤n is the inverse matrix associated to the metric matrix (g ij ) 1≤i,j≤n on M , and A i and B are smooth sections of End(N ) on M (endomorphism). If a Laplace type operator P satisfies (2.7), then there is a unique connection ∇ on N and a unique endomorphism E such that where ∇ L is the Levi-Civita connection on M . Moreover (with local frames of T * M and N ), ∇ ∂i = ∂ i + ω i and E are related to g ij , A i and B through Then the Dirac-Witten operators D and D * can be written as (2.10) By [10], we have By (2.10), we have then we obtain (2.14) Similarly, we have (2.15) By (2.6), (2.7), (2.8) and (2.14), we have (2.16) Since E is globally defined on M , taking normal coordinates at (2.18) Similarly, we have where Φ 2 (∆) denotes the integral over the diagonal part of the second coefficient of the heat kernel expansion of ∆. Now let ∆ = D * D and D * D = ∆ − E, then we have where Wres denote the noncommutative residue. By computations, we have tr c(e j )∇ * T * M ej Then by (2.23), we get (2.24) If M is a n-dimensional compact oriented spin manifolds without boundary, and n is even, then we get the following equalities : (2.25) where s is the scalar curvature.

A Kastler-Kalau-Walze type theorem for 4-dimensional manifolds with boundary
We firstly recall that some basic facts and formulas about Boutet de Monvel's calculus and the definition of the noncommutative residue for manifolds with boundary which will be used in the following. For more details, (see in Section 2 in [18]). Let where Φ(R) denotes the Schwartz space and Φ( We have the following property: h ∈ H + (H − 0 ) if and only if h ∈ C ∞ (R) which has an analytic extension to the lower (upper) complex half-plane {Imξ < 0} ({Imξ > 0}) such that for all nonnegative integer l, where Γ + is a Jordan close curve included Im(ξ) > 0 surrounding all the singularities of h in the upper half-plane and ξ 0 ∈ R. Similarly, define π ′ onH, Let M be a n-dimensional compact oriented spin manifold with boundary ∂M . Denote by B Boutet de Monvel's algebra, we recall the main theorem in [4,18].
, and denote by p, b and s the local symbols of P, G and S respectively. Define: By [18], we get where the sum is taken over in the case of manifolds without boundary, so locally we can compute the first term by [10], [9], [18], [12].
For any fixed point x 0 ∈ ∂M , we choose the normal coordinates U of x 0 in ∂M (not in M ) and compute Φ(x 0 ) in the coordinates U = U × [0, 1) ⊂ M and the metric 1 and From [18], we can get three lemmas.
other cases, where (ω s,t ) denotes the connection matrix of Levi-Civita connection ∇ L . (3.16) By (3.6) and (3.7), we firstly compute 18) and the sum is taken over

By Theorem 2.2, we can compute the interior of Wres
Now we need to compute ∂M Φ. Since, some operators have the following symbols.
Lemma 3.6. The following identities hold: By the composition formula of pseudodifferential operators, we have Lemma 3.7. The following identities hold: case a) II) r = −1, l = −1, k = |α| = 0, j = 1 By (3.18), we get By Lemma 3.7, we have Similarly we have, By (3.29), then (3.32) By the relation of the Clifford action and trAB = trBA, we have the equalities: (3.35) where Ω 3 is the canonical volume of S 3 .

A Kastler-Kalau-Walze type theorem for 6-dimensional manifolds with boundary
Firstly, we prove the Kastler-Kalau-Walze type theorems for 6-dimensional manifolds with boundary. From [15], we know that and the sum is taken over By Theorem 2.2, we compute the interior term of (4.1), then M |ξ|=1 Next, we compute ∂M Ψ. By computations, we get (4.4) Then, we obtain Lemma 4.1. The following identities hold:  By the composition formula of pseudodifferential operators, we have by (4.7), we have (4.8) By Lemma 4.1, we have some symbols of operators.