Sub-signature Operators and the Kastler-Kalau-Walze Type Theorem for Five Dimensional Manifolds with Boundary

In this paper, we prove the Kastler-Kalau-Walze type theorems for the sub-signature operators on 5-dimensional manifolds with boundary.


Introduction
The noncommutative residue found in [1,2] plays a prominent role in noncommutative geometry.For arbitrary closed compact n−dimensional manifolds, the non- commutative residue was introduced by Wodzicki in [1] using the theory of zeta functions of elliptic pseudodifferential operators.In [3], Connes used the noncommutative residue to derive a conformal 4−dimensional Polyakov action analogue.Furthermore, 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 in [4].In [5], Kastler gave a brute-force proof of this theorem.In [6], Kalau and Walze proved this theorem in the normal coordinates system simultaneously, which is called the Kastler-Kalau-Walze theorem now.
An important application of Riemannian geometry is to allow us to define the volume element of a Riemannian manifold (M n , g) .The noncommutative resi- due of Wodzicki [1] and Guillemin [2] is a trace on the algebra of (integer order) ΨDOs on M.An important feature is that it allows us to extend to all ΨDOs by the Dixmier trace, which plays the role of the integral in the framework of noncommutative geometry.Fedosov etc. defined a noncommutative residue on Boutet de Monvel's algebra and proved that it was a unique continuous trace in [7].In [8], 1 3 Journal of Nonlinear Mathematical Physics (2023) 30:1032-1068 Schrohe gave the relation between the Dixmier trace and the noncommutative residue for manifolds with boundary.For a spin manifold M with boundary M , by the composition formula in Boutet de Monvel's algebra and the definition of Wres [16], Wres[( + D −1 ) 2 ] should be the sum of two terms from interior and boundary of M, where + D −1 is an element in Boutet de Mouvel's algebra [17].

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) and proved a Kastler-Kalau-Walze type theorem for the Dirac operator on 5-dimensional (resp.7-dimensional) manifolds with boundary.In [12,13], Zhang introduced the sub-signature operators and proved a local index formula for these operators.Wu and Wei and Wang proved the Kastler-Kalau-Walze type theorem for the sub-signature operators on 4-dimensional and 6-dimensional compact manifolds in [15].In the present paper, we shall restrict our attention to the cases Wres[ + (D t −1 )• + (D t −1 )] and Wres[ + (D t −1 )• + ((D t * ) −1 )] for 5-dimensional manifolds with boundary, where the sub-signature operators D t , D t * are defined in (2.8), (2.9).The paper is organized in the following way.In Sect.2, we recall the definition of the sub-signature operators and define lower dimensional volumes of compact Riemannian manifolds with boundary.In Sect.3, we compute Wres[ + (D t −1 )• + (D t −1 )] , Wres[ + (D t −1 )• + ((D t * ) −1 )] and get the Kastler-Kalau- Walze type theorem for the sub-signature operators on 5-dimensional spin manifolds with boundary.

Sub-signature Operators and Lower-Dimensional Volumes of Spin Manifolds with Boundary
Firstly, we introduce some notations about the sub-signature operators.Let M be an n-dimensional ( n ≥ 3 ) oriented compact Riemannian manifold with a Riemannian metric g TM .And let F be a subbundle of TM, F ⊥ be the subbundle of TM orthogonal to F. Then we have the following orthogonal decomposition: where g F and g F ⊥ are the induced metric on F and F ⊥ .Let ∇ L denote the Levi-civita connection about g TM .In the local coordinates x i ; 1 ≤ i ≤ n and the fixed orthonormal frame {e 1 , ⋯ , e n } , the connection matrix ( s,t ) is defined by Let (e * j ), (e * j ) be the exterior and interior multiplications respectively, where e * j = g TM (e j , ⋅) .Write (1) (3) ĉ(e j ) = (e * j ) + (e * j ); c(e j ) = (e * j ) − (e ĉ(e i )ĉ(e j ) + ĉ(e j )ĉ(e i ) = 2g TM (e i , e j ); c(e i )c(e j ) + c(e j )c(e i ) = −2g TM (e i , e j ); c(e i )ĉ(e j ) + ĉ(e j )c(e i ) = 0. ( Journal of Nonlinear Mathematical Physics (2023) 30:1032-1068 where DF is the sub-signature operator defined in Proposition 2.2 in [14], so we call that D t is the sub-signature operator.
Theorem 2.1 [7] (Fedosov-Golse-Leichtnam-Schrohe) Let M and M be connected, Definition 2.2 [17] Lower dimensional volumes of spin manifolds with boundary are defined by By [17], we get and where the sum is taken over r

A Kastler-Kalau-Walze Type Theorem for 5-Dimensional Spin Manifolds with Boundary
In this section, we compute the lower dimensional volume for D t and D t * on 5-dimensional compact spin manifolds with boundary and get a Kastler-Kalau-Walze type theorem in this case.Proposition 3.1 [18] The following identity holds: Firstly, we compute Vol (1,1)   5 for D t on 5-dimensional spin manifolds with boundary, then we have Write By the composition formula of pseudodifferential operators, we have so (15) Since Ψ is a global form on M , so for any fixed point x 0 ∈ M , we choose the nor- mal coordinates U of x 0 in M (not in M) and compute Ψ(x 0 ) in the coordinates and Let { ẽ1 , ⋅ ⋅ ⋅, ẽn } be an orthonormal frame field in U about g M which is parallel along geodesics and ẽi = In the following, since the global form Ψ is independent of the choice of the local frame, so we can compute tr S(TM) in the frame {[ , be the Clifford action.By [19], then then we have c(e i ) = 0 in the above frame.
From [17], we can get three lemmas.Lemma 3.2 [17] With the metric g TM on M near the boundary where ( s,t ) denotes the connection matrix of Levi-Civita connection ∇ L .Lemma 3.3 [17] When i < n, then Lemma 3.4 [17] With the metric g TM on M near the boundary where = � + n dx n .

Then,
where Ω 3 is the canonical volume of S 3 .

So
Similarly, we obtain And (84) 1 3 Journal of Nonlinear Mathematical Physics (2023) 30:1032-1068 Combining these assertions, we see By ( 14), we get By (2.2.29) in [17], we have In the orthonormal frame filed, we have (91) (92) (94) By the relation of the Clifford action and tr(AB) = tr(BA) , then we have the equalities: we obtain Therefore, 1 3 Journal of Nonlinear Mathematical Physics (2023) 30:1032-1068 By the Leibniz rule, trace property and " ++ " and " −− " vanishing after the integra- tion over n in [7], then By Lemma 3.4, we have Now Ψ is the sum of the Ψ (1,2,⋅⋅⋅,15) , then we obtain Hence, Theorem 3.8 Let M be a 5-dimensional compact oriented manifold with the boundary M , and sub-signature operator (100 Next we recall the Einstein-Hilbert action for manifolds with boundary (see [17] or [18]), where and K ij is the second fundamental form or extrinsic curvature.Take the metric in Sect.2, then by Lemma A.2 in [17], then Therefore, we have Theorem 3.9 Let M be a 5-dimensional compact oriented manifold with the boundary M , and sub-signature operator The following identity holds where s M , s M are respectively scalar curvatures on M and M.

So
Similarly,

And
Combining these assertions, we see (154) and denote by p, b and s the local sym- bols of P, G and S respectively.Define: where Wres denotes the noncommutative residue of an operator in the Boutet de Monvel's algebra.Then a) Wres([ Ã, B]) = 0 , for any Ã, B ∈ B; b) It is a unique continuous trace on B∕B −∞ .