Abstract
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.
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1 Introduction
Until now, many geometers have studied noncommutative residues. In [5, 16], 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-\({{\mathrm{dim}}}M\) 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 [7] gave a brute-force proof of this theorem. Kalau and Walze proved this theorem in the normal coordinates system simultaneously in [6]. Ackermann proved that the Wodzicki residue of the square of the inverse of the Dirac operator \({\mathrm{Wres}}(D^{-2})\) 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 [10, 11], and proved the Kastler–Kalau–Walze type theorem for the Dirac operator and the signature operator on lower-dimensional manifolds with boundary [12]. In [12, 13], Wang computed \(\widetilde{{\mathrm{Wres}}}[\pi ^+D^{-1}\circ \pi ^+D^{-1}]\) and \(\widetilde{{\mathrm{Wres}}}[\pi ^+D^{-2}\circ \pi ^+D^{-2}]\), where two operators are symmetric. And in these cases, the boundary term vanished. But for \(\widetilde{{\mathrm{Wres}}}[\pi ^+D^{-1}\circ \pi ^+D^{-3}]\), Wang got a nonvanishing boundary term [14], and give a theoretical explanation for gravitational action on boundary. And then, Wang provides a kind of method to study the Kastler–Kalau–Walze type theorem for manifolds with boundary. In [8], López and his collaborators introduced an elliptic differential operator, which is called the Novikov operator. In [15], Wei and Wang proved Kastler–Kalau–Walze type theorem for modified Novikov operators on compact manifolds. In [17], 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 Sect. 2, by using the definition of the Dirac–Witten operators, we compute the Lichnerowicz formulas for the Dirac–Witten operators. In Sects. 3 and 4, we prove the Kastler–Kalau–Walze type theorem for 4-dimensional and 6-dimensional manifolds with boundary for the Dirac–Witten operators respectively.
2 The Dirac–Witten Operators and Their Lichnerowicz Formulas
Firstly we introduce some notations about the Dirac–Witten operators. Let M be a n-dimensional (\(n\ge 3\)) oriented compact spin Riemannian manifold with a Riemannian metric \(g^{M}\). And let \(\nabla ^L\) be the Levi-Civita connection about \(g^M\). In the local coordinates \(\{x_i; 1\le i\le n\}\) and the fixed orthonormal frame \(\{e_1,\cdots ,e_n\}\), the connection matrix \((\omega _{s,t})\) is defined by
Let \(c(e_j)\) be the Clifford action. Suppose that \(\partial _{i}\) is a natural local frame on TM and \((g^{ij})_{1\le i,j\le n}\) is the inverse matrix associated to the metric matrix \((g_{ij})_{1\le i,j\le n}\) on M. By [12], we have the Dirac operator
Then the Dirac–Witten operators \({\widetilde{D}}\) and \({{\widetilde{D}}}^*\) are defined by
where \(f_1,f_2\) are complex numbers, p is a (0, 2)-tensor and \(p_{uv}=p(e_u,e_v)\). Then when \(f_1=\frac{\sqrt{-1}}{2}\), \(f_2=-\frac{\sqrt{-1}}{2}\sum _{i}p_{ii}\), \({\widetilde{D}}\) is the Dirac–Witten operator defined by [17].
Then, we get the following Lichnerowicz formulas,
Theorem 2.1
The following equalities hold:
where s is the scalar curvature.
Proof
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 \(\partial _{i}\) is a natural local frame on TM and \((g^{ij})_{1\le i,\,j\le n}\) is the inverse matrix associated to the metric matrix \((g_{ij})_{1\le i,\,j\le n}\) on M, and \(A^{i}\) and B are smooth sections of \(\text {End}(N)\) on M (endomorphism). If a Laplace type operator P satisfies (2.5), then there is a unique connection \(\nabla \) on N and a unique endomorphism E such that
where \(\nabla ^{L}\) is the Levi-Civita connection on M. Moreover (with local frames of \(T^{*}M\) and N), \(\nabla _{\partial _{i}}=\partial _{i}+\omega _{i} \) and E is related to \(g^{ij}\), \(A^{i}\) and B through
where \(\Gamma _{ kl}^{j}\) are the Christoffel coefficients of \(\nabla ^{L}\).
Let \(g^{ij}=g(dx_{i},dx_{j})\), \(\xi =\sum _{k}\xi _{j}dx_{j}\) and \(\nabla ^L_{\partial _{i}}\partial _{j}=\sum _{k}\Gamma _{ij}^{k}\partial _{k}\), we denote that
Then the Dirac–Witten operators \({\widetilde{D}}\) and \({{\widetilde{D}}}^*\) can be written as
By [7], we have
By (2.10), we have
then we obtain
Similarly, we have
By (2.6), (2.7), (2.8) and (2.14), we have
Since E is globally defined on M, taking normal coordinates at \(x_0\), we have \(\sigma ^{i}(x_0)=0\), \(\partial ^{j}[c(\partial _{j})](x_0)=0\), \(\Gamma ^k(x_0)=0\), \(g^{ij}(x_0)=\delta ^j_i\), then
Similarly, we have
by (2.5), we get Theorem 2.1. \(\square \)
From [1], we know that the noncommutative residue of an appropriate power of a generalized laplacian \({\overline{\Delta }}\) is expressed as
where \(\Phi _{2}({\overline{\Delta }})\) denotes the integral over the diagonal part of the second coefficient of the heat kernel expansion of \({\overline{\Delta }}\). Now let \({\overline{\Delta }}={{\widetilde{D}}}^*{{\widetilde{D}}}\) and \({{\widetilde{D}}}^*{{\widetilde{D}}}=\Delta -E\), then we have
where \({\mathrm{Wres}}\) denotes the noncommutative residue. Then,
By
Similarly, we have
By (2.21), (2.22) and (2.27), we can get the following theorem,
Theorem 2.2
If M is a n-dimensional compact oriented spin manifolds without boundary, and n is even, then we get the following equalities:
where s is the scalar curvature.
3 A Kastler–Kalau–Walze Type Theorem for 4-Dimensional Manifolds with Boundary
We firstly recall 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 Sect. 2 in [12]).
Let \(U\subset M\) be a collar neighborhood of \(\partial M\) which is diffeomorphic with \(\partial M\times [0,1)\). By the definition of \(h(x_n)\in C^{\infty }([0,1))\) and \(h(x_n)>0\), there exists \({\widehat{h}}\in C^{\infty }((-\varepsilon ,1))\) such that \({\widehat{h}}|_{[0,1)}=h\) and \({\widehat{h}}>0\) for some sufficiently small \(\varepsilon >0\). Then there exists a metric \(g'\) on \({\widetilde{M}}=M\bigcup _{\partial M}\partial M\times (-\varepsilon ,0]\) which has the form on \(U\bigcup _{\partial M}\partial M\times (-\varepsilon ,0 ]\)
such that \(g'|_{M}=g\). We fix a metric \(g'\) on the \({\widetilde{M}}\) such that \(g'|_{M}=g\).
Let Fourier transformation \(F'\) be
and let
where \(\Phi (\mathbf{R})\) denotes the Schwartz space and \(\Phi (\widetilde{\mathbf{R}^+}) =r^+\Phi (\mathbf{R})\), \(\Phi (\widetilde{\mathbf{R}^-}) =r^-\Phi (\mathbf{R})\).
We define \(H^+=F'(\Phi (\widetilde{\mathbf{R}^+}));~ H^-_0=F'(\Phi (\widetilde{\mathbf{R}^-}))\) which satisfies \(H^+\bot H^-_0\). We have the following property: \(h\in H^+~(H^-_0)\) if and only if \(h\in C^\infty (\mathbf{R})\) which has an analytic extension to the lower (upper) complex half-plane \(\{{\mathrm{Im}}\xi <0\}~(\{{\mathrm{Im}}\xi >0\})\) such that for all nonnegative integer l,
as \(|\xi |\rightarrow +\infty ,{\mathrm{Im}}\xi \le 0~({\mathrm{Im}}\xi \ge 0)\). Let \(H'\) be the space of all polynomials and \(H^-=H^-_0\bigoplus H';~H=H^+\bigoplus H^-.\) Denote by \(\pi ^+~(\pi ^-)\) respectively the projection on \(H^+~(H^-)\). For calculations, we take \(H={{\widetilde{H}}}=\{\)rational functions having no poles on the real axis\(\}\) (\({\tilde{H}}\) is a dense set in the topology of H). Then on \({\tilde{H}}\),
where \(\Gamma ^+\) is a Jordan close curve included \({\mathrm{Im}}(\xi )>0\) surrounding all the singularities of h in the upper half-plane and \(\xi _0\in \mathbf{R}\). Similarly, define \(\pi '\) on \({\tilde{H}}\),
So, \(\pi '(H^-)=0\). For \(h\in H\bigcap L^1(\mathbf{R})\), \(\pi 'h=\frac{1}{2\pi }\int_{\mathbf{R}}h(v)dv\) and for \(h\in H^+\bigcap L^1(\mathbf{R})\), \(\pi 'h=0\).
Let M be a n-dimensional compact oriented spin manifold with boundary \(\partial M\). Denote by \({\mathcal {B}}\) Boutet de Monvel’s algebra, we recall the main theorem in [4, 12].
Theorem 3.1
[4] (Fedosov–Golse–Leichtnam–Schrohe) Let X and \(\partial X\) be connected, \({\mathrm{dim}}X=n\ge 3\), \(A=\left( \begin{array}{lcr}\pi ^+P+G &{} K \\ T &{} S \end{array}\right) \) \(\in {\mathcal {B}}\), and denote by p, b and s the local symbols of P, G and S respectively. Define:
where \({{{\widetilde{{\mathrm{Wres}}}}}}\) denotes the noncommutative residue of an operator in the Boutet de Monvel’s algebra.
Then a) \({{{\widetilde{{\mathrm{Wres}}}}}}([A,B])=0 \), for any \(A,B\in {\mathcal {B}}\); b) It is a unique continuous trace on \({\mathcal {B}}/{\mathcal {B}}^{-\infty }\).
Definition 3.2
[12] Lower dimensional volumes of spin manifolds with boundary are defined by
By [12], we get
and
where the sum is taken over \(r+l-k-|\alpha |-j-1=-n,~~r\le -p_1,l\le -p_2\) and \(\widetilde{{\mathrm{Wres}}}\) denotes the noncommutative residue for manifolds with boundary.
Since \([\sigma _{-n}(D^{-p_1-p_2})]|_M\) has the same expression as \(\sigma _{-n}(D^{-p_1-p_2})\) in the case of manifolds without boundary, so locally we can compute the first term by [6, 7, 9, 12].
For any fixed point \(x_0\in \partial M\), we choose the normal coordinates U of \(x_0\) in \(\partial M\) (not in M) and compute \(\Phi (x_0)\) in the coordinates \({\widetilde{U}}=U\times [0,1)\subset M\) and the metric \(\frac{1}{h(x_n)}g^{\partial M}+dx_n^2.\) The dual metric of \(g^M\) on \({\widetilde{U}}\) is \({h(x_n)}g^{\partial M}+dx_n^2.\) Write \(g^M_{ij}=g^M(\frac{\partial }{\partial x_i},\frac{\partial }{\partial x_j});~ g_M^{ij}=g^M(dx_i,dx_j)\), then
and
From [12], we can get three lemmas.
Lemma 3.3
[12] With the metric \(g^{M}\) on M near the boundary
where \(\xi =\xi '+\xi _{n}dx_{n}\).
Lemma 3.4
[12] With the metric \(g^{M}\) on M near the boundary
where \((\omega _{s,t})\) denotes the connection matrix of Levi-Civita connection \(\nabla ^L\).
Lemma 3.5
[12]
By (3.6) and (3.7), we firstly compute
where
the sum is taken over \(r+l-k-j-|\alpha |=-3,~~r\le -1,l\le -1\) and \(\widetilde{{\mathrm{Wres}}}\) denotes the noncommutative residue for manifolds with boundary.
By Theorem 2.2, we can compute the interior of \(\widetilde{{\mathrm{Wres}}}[\pi ^+{{\widetilde{D}}}^{-1}\circ \pi ^+({{\widetilde{D}}}^*)^{-1}]\), so
Now we need to compute \(\int _{\partial M} \Phi \). Since, some operators have the following symbols.
Lemma 3.6
The following identities hold:
where \(\xi =\Sigma _{i=1}^n\xi _id_{x_i}\) denotes the cotangent vector.
Write
By the composition formula of pseudodifferential operators, we have
so
Lemma 3.7
The following identities hold:
When \(n=4\), then \({\mathrm{tr}}_{\wedge ^*T^*M}[{{ \texttt {id}}}]={\mathrm{dim}}(\wedge ^*(4))=4\), where \({\mathrm{tr}}\) as shorthand of \({\mathrm{trace}}\), the sum is taken over \( r+l-k-j-|\alpha |=-3,~~r\le -1,l\le -1,\) then we have the following five cases:
Case (a) (I) \(r=-1,~l=-1,~k=j=0,~|\alpha |=1\)
By (3.18), we get
By Lemma 3.3, for \(i<n\), then
so \(\Phi _1=0\).
Case (a) (II) \(r=-1,~l=-1,~k=|\alpha |=0,~j=1\)
By (3.18), we get
By Lemma 3.7, we have
Similarly we have,
By (3.29), then
By the relation of the Clifford action and \({\mathrm{tr}}{AB}={\mathrm{tr }}{BA}\), we have the equalities:
By (3.31), we have
Similarly, we have
Then
where \({{\Omega _{3}}}\) is the canonical volume of \(S^{3}\).
Case (a) (III) \(r=-1,~l=-1,~j=|\alpha |=0,~k=1\)
By (3.18), we get
By Lemma 3.7, we have
Similar to case a) II), we have
and
So we have
Case (b) \(r=-2,~l=-1,~k=j=|\alpha |=0\)
By (3.18), we get
By Lemma 3.7 we have
where
We denote
Then
And
Since
Then, we have
where
and
where \(Q=c_0c(dx_n)\) and \(c_0=-\frac{3}{4}h'(0)\).
Then, we have
Then, we have
Case (c) \(r=-1,~l=-2,~k=j=|\alpha |=0\)
By (3.18), we get
By (3.5) and (3.6), Lemma 3.7, we have
Since
where
then
by \(c(\xi )=c(\xi ')+\xi _nc(dx_n)\), \(|\xi |^2=1+\xi _n^2\) and direct derivation, we get
We denote
then
then
By \(\int _{|\xi '|=1}\xi _{1}\cdot \cdot \cdot \xi _{2d+1}\sigma (\xi ')=0\), (3.59) and (3.62), we have
Then,
So \(\Phi =\sum _{i=1}^5\Phi _i=0\).
By (3.17), (3.19) and (3.68), we can get
Theorem 3.8
Let M be a 4-dimensional compact oriented spin manifolds with the boundary \(\partial M\) and the metric \(g^M\) as above, \({{\widetilde{D}}}\) and \({{\widetilde{D}}}^*\) be the Dirac–Witten operators on \({\widetilde{M}}\) , then
where s is the scalar curvature.
4 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 [14], we know that
where
and the sum is taken over \(r+\ell -k-j-|\alpha |-1=-6, \ r\le -1, \ell \le -3\).
By Theorem 2.2, we compute the interior term of (4.1), then
Next, we compute \(\int _{\partial M} \Psi \). By (2.12), we get
Then,
Then, by (4.5), we obtain
Lemma 4.1
The following identities hold:
where \(\xi =\Sigma _{i=1}^n\xi _id_{x_i}\) denotes the cotangent vector.
Write
By the composition formula of pseudodifferential operators, we have
by (4.8), we have
By Lemma 4.1, we have some symbols of operators.
Lemma 4.2
The following identities hold:
When \(n=6\), then \({\mathrm{tr}}_{\wedge ^*T^*M}[\texttt {id}]=8\), where \({\mathrm{tr}}\) as shorthand of \({\mathrm{trace}}\). Since the sum is taken over \(r+\ell -k-j-|\alpha |-1=-6, \ r\le -1, \ell \le -3\), then we have the sum of the following five cases:
Case (a) (I) \(r=-1, l=-3, j=k=0, |\alpha |=1\).
By (4.2), we get
By Lemma 4.2, for \(i<n\), we have
so \(\Psi _1=0\).
Case (a) (II) \(r=-1, l=-3, |\alpha |=k=0, j=1\).
By (4.2), we have
By direct derivation, we have
Since \(n=6\), \({\mathrm{tr}}[-\texttt {id}]=-8\). By the relation of the Clifford action and \({\mathrm{tr}}AB={\mathrm{tr}}BA\), then
By (3.29), (4.13) and (4.15), we get
Then we obtain
where \({{\Omega _{4}}}\) is the canonical volume of \(S^{4}\).
Case (a) (III) \(r=-1,l=-3,|\alpha |=j=0,k=1\).
By (4.2), we have
By direct derivation, we have
Combining (3.36) and (4.19), we have
Then
Case (b) \(r=-1,l=-4,|\alpha |=j=k=0\).
By (4.2), we have
In the normal coordinate, \(g^{ij}(x_{0})=\delta ^{j}_{i}\) and \(\partial _{x_{j}}(g^{\alpha \beta })(x_{0})=0\), if \(j<n\); \(\partial _{x_{j}}(g^{\alpha \beta })(x_{0})=h'(0)\delta ^{\alpha }_{\beta }\), if \(j=n\). So by [12], when \(k<n\), we have \(\Gamma ^{n}(x_{0})=\frac{5}{2}h'(0)\), \(\Gamma ^{k}(x_{0})=0\), \(\delta ^{n}(x_{0})=0\) and \(\delta ^{k}=\frac{1}{4}h'(0)c(e_{k})c(e_{n})\). Then, we obtain
Then by (4.24), we get
Case (c) \(r=-2,l=-3,|\alpha |=j=k=0\).
By (4.2), we have
By Lemma 4.1 and Lemma 4.2, we have
where
On the other hand,
By (4.27), (3.5) and (3.6), we have
We denote
Then, we obtain
Furthermore,
By \(c(\xi )=c(\xi ')+\xi _nc(dx_n)\), we have
where
Similarly, we have
By \(\int _{|\xi '|=1}\xi _{1}\ldots \xi _{2d+1}\sigma (\xi ')=0,\) we have
Now \(\Psi \) is the sum of the cases (a), (b) and (c), then
By (4.1), (4.3) and (4.39), we can get
Theorem 4.3
Let M be a 6-dimensional compact oriented spin manifold with the boundary \(\partial M\) and the metric \(g^M\) as above, \({{\widetilde{D}}}\) and \({{\widetilde{D}}}^*\) be the Dirac–Witten operators on \({\widetilde{M}}\) , then
where s is the scalar curvature.
Next, we prove the Kastler–Kalau–Walze type theorem for 6-dimensional manifold with boundary associated to \({{\widetilde{D}}}^{3}\). From [14], we know that
where \(\widetilde{{\mathrm{Wres}}}\) denote noncommutative residue on minifolds with boundary,
and the sum is taken over \(r+\ell -k-j-|\alpha |-1=-6, \ r\le -1, \ell \le -3\).
By Theorem 2.2, we compute the interior term of (4.42), then
So we only need to compute \(\int _{\partial M} {\overline{\Psi }}\). Let us now turn to compute the specification of \({{\widetilde{D}}}^3\).
Then, we obtain
Lemma 4.4
The following identities hold:
Write
By the composition formula of pseudodifferential operators, we have
by (4.47), we have
By (4.44)–(4.48), we have some symbols of operators.
Lemma 4.5
The following identities hold:
When \(n=6\), then \({\mathrm{tr}}_{\wedge ^*T^*M}[\texttt {id}]=8\), where \({\mathrm{tr}}\) as shorthand of \({\mathrm{trace}}\). Since the sum is taken over \(r+\ell -k-j-|\alpha |-1=-6, \ r\le -1, \ell \le -3\), then we have the following five cases:
Case (a) (I) \(r=-1, l=-3, j=k=0, |\alpha |=1\).
By (4.42), we get
Case (a) (II) \(r=-1, l=-3, |\alpha |=k=0, j=1\).
By (4.42), we have
Case (a) (III) \(r=-1,l=-3,|\alpha |=j=0,k=1\).
By (4.42), we have
By Lemma 4.2 and Lemma 4.5, we have \(\sigma _{-3}(({{\widetilde{D}}}^*{{\widetilde{D}}}{{\widetilde{D}}}^*)^{-1})=\sigma _{-3}({{\widetilde{D}}}^{-3})\), by (4.11)- (4.21), we obtain
where \({{\Omega _{4}}}\) is the canonical volume of \(S^{4}.\)
Case (b) \(r=-1,l=-4,|\alpha |=j=k=0\).
By (4.42), we have
In the normal coordinate, \(g^{ij}(x_{0})=\delta ^{j}_{i}\) and \(\partial _{x_{j}}(g^{\alpha \beta })(x_{0})=0\), if \(j<n\); \(\partial _{x_{j}}(g^{\alpha \beta })(x_{0})=h'(0)\delta ^{\alpha }_{\beta }\), if \(j=n\). So by [12], when \(k<n\), we have \(\Gamma ^{n}(x_{0})=\frac{5}{2}h'(0)\), \(\Gamma ^{k}(x_{0})=0\), \(\delta ^{n}(x_{0})=0\) and \(\delta ^{k}=\frac{1}{4}h'(0)c(e_{k})c(e_{n})\). Then, we obtain
By (4.55), we have
Case (c) \(r=-2,l=-3,|\alpha |=j=k=0\).
By (4.42), we have
By Lemma 4.2 and Lemma 4.5, we have \(\sigma _{-3}(({{\widetilde{D}}}^*{{\widetilde{D}}}{{\widetilde{D}}}^*)^{-1})=\sigma _{-3}({{\widetilde{D}}}^{-3})\), by (4.26)- (4.38), we obtain
Now \({\overline{\Psi }}\) is the sum of the cases (a), (b) and (c), then
By (4.41), (4.43) and (4.58), we can get
Theorem 4.6
Let M be a 6-dimensional compact oriented spin manifold with the boundary \(\partial M\) and the metric \(g^M\) as above, \({{\widetilde{D}}}\) be the Dirac-Witten operator on \({\widetilde{M}}\) , then
where s is the scalar curvature.
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The author was supported in part by NSFC No.11771070. The author thanks the referee for his (or her) careful reading and helpful comments.
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This research was funded by National Natural Science Foundation of China: No.11771070.
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Wu, T., Wang, J. & Wang, Y. Dirac–Witten Operators and the Kastler–Kalau–Walze Type Theorem for Manifolds with Boundary. J Nonlinear Math Phys 29, 1–40 (2022). https://doi.org/10.1007/s44198-021-00009-6
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DOI: https://doi.org/10.1007/s44198-021-00009-6