1. INTRODUCTION

The noncommutative residue which is found in [5,19] plays a prominent role in noncommutative geometry. For this reason, it has been studied extensively by geometers. Connes derived a conformal 4-dimensional Polyakov action analogy by the noncommutative residue in [2]. Connes proved that the noncommutative residue on a compact manifold M coincided with the Dixmier’s trace on pseudodifferential operators of order −dimM in [3].

On the other hand, Connes had also observed 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 type theorem now. Kastler gave a brute-force proof of this theorem in [8]. Kalau and Walze [7] proved this theorem in the normal coordinates system simultaneously. And then, for the Dirac operator D, Ackermann proved that the Wodzicki residue Wres(D⁻²) in turn was essentially the second coefficient of the heat kernel expansion of D² in [1].

On the other hand, Wang generalized the Connes’ results to the case of manifolds with boundary in [13,14], and proved the Kastler-Kalau-Walze type theorems for the Dirac operator and the signature operator on lower-dimensional manifolds with boundary [15]. In [15,16], Wang computed Wres˜[π+D-1∘π+D-1] and Wres˜[π+D-2∘π+D-2] about symmetric operators and under the circumstances the boundary term vanished. Moreover, Wang got a nonvanishing boundary term for Wres˜[π+D-1∘π+D-3] in [17] and gave a theoretical explanation for the gravitational action on boundary. In others words, Wang provided a kind of method to study the Kastler-Kalau-Walze type theorem for manifolds with boundary.

In [6], Iochum and Levy computed heat kernel coefficients for Dirac operators with one-form perturbations and proved that there were no tadpoles for compact spin manifolds without boundary. Recently, we studied the Lichnerowicz-type formulas for modified Novikov operators. We proved Kastler-Kalau-Walze-type theorems for modified Novikov operators on compact manifolds with (respectively without) boundary in [18]. In [10], Barbara Opozda introduced statistical de Rham Hodge operators on manifolds with statistical structure. The aim of this paper is to prove the Kastler-Kalau-Walze type theorem for statistical de Rham Hodge operators on manifolds without (with) boundary, and also give the Lichnerowicz formulas about statistical de Rham Hodge operators.

The paper is organized as follows: in Section 2, we give the definition of statistical de Rham Hodge operators and their Lichnerowicz formulas. We also give the basic facts and formulas about the noncommutative residue for manifolds with boundary. In Section 3 and in Section 4, we give some expressions and symbols of operators associate with statistical de Rham Hodge operators. Moveover, we also prove the Kastler- Kalau-Walze type theorems for statistical de Rham Hodge operators on 4-dimensional and 6-dimensional manifolds with boundary.

2. STATISTICAL de RHAM HODGE OPERATORS AND THEIR LICHNEROWICZ FORMULAS

In this section, we prepare some basic notions about statistical de Rham Hodge operators. For details of the geometry of statistical manifolds, see [10].

Let M be a n-dimensional (n ≥ 3) Riemannian manifold with a positive definite Riemannian metric g, and ∇^ be a connection. We assume that M is oriented. Let ∇^L be the Levi-Civita connection for g and Vol_M be the volume form determined by g.

For all X, Y, Z ∈ T_xM, x ∈ M, if ∇^ is satisfying the following Codazzi condition:

we call a structure (g, ∇^) is a statistical structure, and a connection ∇^ is a statistical connection for g. Moreover, we define a statistical manifold with statistical structure by (M, g, ∇^).

We suppose that (M, g, ∇^) is statistical manifold. Let K be tensor field, where K: Γ(TM) × Γ(TM) → Γ(TM) and Γ(TM) denotes the algebra of smooth vector fields on M. Let K be the difference tensor between ∇^ and ∇^L, that is

Let K(X, Y) stand for K_XY. The de Rham derivative d is a differential operator on C^∞(M; ∧^*T^*M). Then we have the de Rham coderivative δ = d^* and the symmetric operator D = d + δ. The standard Hodge Laplacian is defined by

For a statistical manifold (M, g, ∇^), we will investigate a (Lichnerowicz) Laplacian relative to the connection ∇^. If f is a function, then we set

We now extend the definition (2.4) and set E = trace_g K(·,·). For any differential form ν, we set

where l(E) is the contraction operator.

By the above definition and Lemma 6.4 in [10], we have

For v ∈ Γ(TM), we define the generalized statistical de Rham Hodge operators by

where λ_i is a real number and v^* = g(v, ·).

In the local coordinates {x_i; 1 ≤ i ≤ n} and the fixed orthonormal frame {e₁, ···, e_n}, the connection matrix (ω_s,t) is defined by

Let ε(e_j*), l(e_j) be the exterior and interior multiplications respectively and c(e_j) be the Clifford action. Write

Moreover, we assume that ∂_i is a natural local frame on TM and (g^ij)_1≤i,j≤n is the inverse matrix associated with the metric matrix (g_ij)_1≤i,j≤n on M. The statistical de Rham Hodge operators D_i and Di*(i=1,2) are defined by

Let g^ij = g(dx_i, dx_j), ξ = ∑_j ξ_jdx_j and ∇∂iL∂j=∑kΓijk∂k, we denote that

Then the statistical de Rham Hodge operators D_i and Di* can be written as

On the other hand, we recall some basic facts and formulas about Boutet de Monvel’s calculus and the definition of the noncommutative residue for manifolds with boundary (for details see Section 2 in [15]).

Let U ⊂ M be a collar neighborhood of ∂M which is diffeomorphic with ∂M × [0, 1). f ∈ C^∞([0, 1)) means that there is f˜∈C∞((-ɛ,1)) such that f˜|[0,1)=f for a small positive number ε. Let h(x_n) ∈ C^∞([0, 1)) and h(x_n) > 0. By the definition of h(x_n) ∈ C^∞([0, 1)) and h(x_n) > 0, there exists h˜∈C∞((-ɛ,1)) such that h˜|[0,1)=h and h˜>0 for some sufficiently small ɛ > 0. Then there exists a metric g^ on M^=M∪∂M∂M×(-ɛ,0] which has the form on U ∪_∂M ∂M × (−ε, 0]

such that g^|M=g. We fix a metric g^ on the M^ such that g^|M=g.

Let

denote the Fourier transformation and φ(R+¯)=r+φ(R)(similarly define φ(R-¯)), where ϕ(R) denotes the Schwartz space and

We define H+=F(φ(R+¯)); H0-=F(φ(R-¯)) which are orthogonal to each other. We have the following property: h∈H+ (H0-) 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,

as |ξ|→ +∞, Imξ ≤ 0 (Imξ ≥ 0).

Let H′ be the space of all polynomials and H-=H0-⊕H′; H=H+⊕H-. Denote by π⁺(π⁻) respectively the projection on H⁺(H⁻). For calculations, we take H=H˜={rational functions having no poles on the real axis}( H˜ is a dense set in the topology of H). Then on H˜,

where Γ⁺ is a Jordan close curve included Im(ξ) > 0 surrounding all the singularities of h in the upper half-plane and ξ₀ ∈ R. Similarly, define π′ on H˜,

So, π′(H⁻) = 0. For h ∈ H ∩ L¹(R), π′h=12π∫Rh(v)dv and for h ∈ H⁺ ∩ L¹(R), π′h = 0.

Next we give the basic notions of Laplace type operators. Let M be a smooth compact oriented Riemannian n-dimensional manifolds without boundary and V′ be a vector bundle on M. Any differential operator P of Laplace type has locally the form

where Aⁱ and B are smooth sections of the endomorphism End(V′) on M. If P is a Laplace type operator with the form (2.18), then there is a unique connection ∇ on V′ and a unique endomorphism E′ such that where ∇^L is the Levi-Civita connection on M. Moreover (with local frames of T^∗M and V′), Δ_∂i = Δ_∂i + ω_i and E′ are related to g^ij, Aⁱ and B through where Γklj is the Christoffel coefficient of ∇^L.

By the above definitions, we establish the main theorem in this section. One has the following Lichneriowicz formulas.

The noncommutative residue of a generalized laplacian Δ is expressed by [1], as

where φ(Δ) denotes the integral over the diagonal part of the second coefficient of the heat kernel expansion of Δ. Since D₁D₂, D2*D1* and D2*D1 are generalized laplacian opeartors, we have where Wres is the noncommutative residue.

Similarly, we have

By Theorem 2.1 and its proof, we have

3. A KASTLER-KALAU-WALZE TYPE THEOREM FOR 4-DIMENSIONAL MANIFOLDS WITH BOUNDARY

In this section, we prove the Kastler-Kalau-Walze type theorem for 4-dimensional oriented compact manifold with boundary about statistical de Rham Hodge Operators.

Denote by M a n-dimensional manifold with boundary ∂M. We assume that M is compact and oriented. Let ℬ be Boutet de Monvel’s algebra, we now recall the main theorem in [4,15].

Next, we prove the Kastler-Kalau-Walze type theorem for 4-dimensional manifolds with boundary associated with D2*D1. By (3.3) and (3.4), we will compute

where and the sum is taken over r + l − k − j − |α| = −3, r ≤ −1, l ≤ −1.

Locally we can use Theorem 2.2 (2.44) to compute the interior of Wres˜[π+D1-1∘π+(D2*)-1], we have

So we only need to compute ∫_∂M Φ₃. From the remark above, now we can compute Φ₃ (see formula (3.89) for the definition of Φ₃). We use tr as shorthand of trace. Since n = 4, then tr_∧∗T∗M [id] = 16, since the sum is taken over r + l − k− j− |α| = −3, r ≤ −1, l ≤ − 1, then we have the following five cases:

case a) I) r = −1, l = −1, k = j = 0, |α|= 1.

By (3.89), we get

case a) II) r = −1, l = −1, k = |α|= 0, j = 1.

By (3.89), we get

case a) III) r = −1, l = −1, j = |α|= 0, k = 1.

By (3.89), we get

By Lemma 3.7, we have σ-1(Di-1)=σ-1((Di*)-1) (i=1,2). By (3.19)-(3.34), so case a) vanishes.

case b) r = −2, l = −1, k = j = |α|= 0.

By (3.89), we get

By Lemma 3.7, we have σ-1(Di-1)=σ-1((Di*)-1) (i=1,2). By (3.35)-(3.54), we have

case c) r = −1, l = −2, k = j = |α|= 0.

By (3.70), we get

By Lemma 3.7, we have σ-1(Di-1)=σ-1((Di*)-1) (i=1,2). By (3.83)-(3.87), we have

Since Φ₃ is the sum of the cases a), b) and c), so Φ₃ = 2(λ₁ + λ₂)π 〈dx_n, v〉Ω₃dx′.