Wodzicki residue and minimal operators on a noncommutative 4-dimensional torus

Abstract

We compute the Wodzicki residue of the inverse of a conformally rescaled Laplace operator over a 4-dimensional noncommutative torus. We show that the straightforward generalisation of the Laplace-Beltrami operator to the noncommutative case is not the minimal operator.

1 Introduction

Noncommutative geometry as proposed in [2] aims to use geometric methods to study noncommutative algebras in a similar way that differential geometry is used to study spaces. One of the appealing potential applications is its use in physics to describe the structure of space-time and fundamental interactions at high energies. Although the construction of basic data in noncommutative geometry is equivalent in the classical case to the standard data of Riemannian geometry [1] in the genuine noncommutative examples this aspect has not been sufficiently explored until recently.

In a series of papers first [3, 4] and [7–12] a conformally rescaled metric has been proposed and studied for the noncommutative two and four-tori. This led to the expressions of Gauss-Bonnet theorem and formulae for the noncommutative counterpart of scalar curvature. Independently, another class of Dirac operators and metrics with the geometric interpretation as arising from the $U(1)$ connections on noncommutative circle bundles has been proposed by the author and Dabrowski in [5]. Following this lead, a more general type of metric on the two torus has been suggested and studied perturbatively for the two-torus [6].

However, in all the mentioned approaches one major assumption was made. The second term of the heat-kernel asymptotics of the rescaled Laplace operator (or square of the Dirac operator) was identified as a linear functional of the scalar curvature. In Riemannian geometry this is certainly true, provided that the Laplace operator (or Dirac operator) comes from the Levi-Civita connection. In the presence of the nontrivial torsion the term is modified and includes the integral of the square of the torsion, as has been observed already in the early computations of Seeley-Gilkey-de Witt coefficients. As in the noncommutative geometry there is no implicit notion of torsion, one may wonder whether the rescaled Laplace operators are those, which minimise the second term of the heat-kernel expansion for a fixed metric. Additionally, in the classical case the computation of second heat kernel coefficient is closely related to the computations of the Wodzicki residue of a certain power of the Dirac operator.

As it has been shown [12] and more generally in [15] Wodzicki residue exists also in the case of the pseudodifferential calculus over noncommutative tori.

In this note we shall address the question of the minimal operators (using the Wodzicki residue to check minimality) and compute the Wodzicki residue for a class of operators on the noncommutative torus.

2 Noncommutative tori and their pseudodifferential calculus

We use the usual presentation of the algebra of $d$-dimensional noncommutative torus as generated by $d$ unitary elements $U_i$, $i=1,\ldots ,d$, with the relations

$$\begin{aligned} U_j U_k = e^{2\pi i\theta _{jk}} U_k U_j, \end{aligned}$$

where $0< \theta _{jk} < 1$ is real. The smooth algebra $\mathcal {A}(\mathbb {T}^d_\theta )$ is then taken as an algebra of elements

$$\begin{aligned} a = \sum _{\beta \in \mathbb {Z}^d} a_{\beta } U^\beta , \end{aligned}$$

where $a_{\beta }$ is a rapidly decreasing sequence and

$$\begin{aligned} U^\beta = U_1^{\beta _1} \cdots U_d^{\beta _d}. \end{aligned}$$

The natural action of $U(1)^d$ by automorphisms, gives, in its infinitesimal form, two linearly independent derivations on the algebra: given on the generators as:

$$\begin{aligned} \delta _k (U_j) = \delta _{jk} U_j, \;\;\;\; \forall j,k=1,\ldots ,d. \end{aligned}$$

(2.1)

where $\delta _{jk}$ denotes the Kronecker delta.

The canonical trace on $\mathcal {A}(\mathbb {T}^d_\theta )$ is

$$\begin{aligned} \mathfrak {t}(a) = \alpha _{{\mathbf 0}}, \end{aligned}$$

where ${\mathbf 0} = \{0,0,\ldots ,0\} \in \mathbb {Z}^d$. The trace is invariant with respect to the action of $U(1)^d$, hence

$$\begin{aligned} \mathfrak {t}(\delta _j(a)) = 0, \quad \forall j =1,\ldots ,d, \forall a \in \mathcal {A}(\mathbb {T}^d_\theta ). \end{aligned}$$

By $\mathcal {H}$ we denote the Hilbert space of the GNS construction with respect to the trace $\mathfrak {t}$ on the $C^*$ completion of $\mathcal {A}(\mathbb {T}^d_\theta )$ and $\pi$ the associated faithful representation. The elements of the smooth algebra $\mathcal {A}(\mathbb {T}^d_\theta )$ act on $\mathcal {H}$ as bounded operators by left multiplication, whereas the derivations $\delta _i$ extend to densely defined selfadjoint operators on $\mathcal {H}$ with the smooth elements of the Hilbert space, $\mathcal {A}(\mathbb {T}^d_\theta )$, in their common domain.

2.1 Pseudodifferential operators on $\mathbb {T}^d_\theta$

The symbol calculus defined in [4] and developed further in [3] (see also [15]) is easily generalised to the $d$-dimensional case and to the operators defined above. We shall briefly review the basic definitions and methods used further in the note. Let us recall that a differential operator of order at most $n$ is of the form

$$\begin{aligned} P = \sum _{0 \le k \le n} \sum _{|\beta _k|=k} a_{\beta _k} \delta ^{\beta _k}, \end{aligned}$$

where $a_{\beta _k}$ are assumed to be in the algebra $\mathcal {A}(\mathbb {T}^d_\theta )$, $\beta _k \in \mathbb {Z}^d$ and:

$$\begin{aligned} |\beta _k| = \beta _1+ \cdots + \beta _d, \;\;\;\;\; \delta ^\beta = \delta _1^{\beta _1} \cdots \delta _d^{\beta _d}. \end{aligned}$$

Its symbol is:

$$\begin{aligned} \rho ({P}) = \sum _{0 \le k \le n} \sum _{|\beta _k|=k} a_{\beta _k} \xi ^{\beta _k}, \end{aligned}$$

where

$$\begin{aligned} \xi ^{\beta } = \xi _1^{\beta _1} \cdots \xi _d^{\beta _d}. \end{aligned}$$

On the other hand, let $\rho$ be a symbol of order $n$, which is assumed to be a $C^\infty$ function from $\mathbb {R}^d$ to $\mathcal {A}(\mathbb {T}^d_\theta )$, which is homogeneous of order $n$, satisfying certain bounds (see [4] for details). With every such symbol $\rho$ there is associated an operator $P_\rho$ on a dense subset of $\mathcal {H}$ spanned by elements $a \in \mathcal {A}(\mathbb {T}^d_\theta )$:

$$\begin{aligned} P_\rho (a) = \frac{1}{(2\pi )^d} \int \limits _{\mathbb {R}^d \times \mathbb {R}^d} e^{-i \sigma \cdot \xi } \rho (\xi ) \alpha _\sigma (a)\, d\sigma d \xi , \end{aligned}$$

where

$$\begin{aligned} \alpha _\sigma (U^\alpha ) = e^{i \sigma \cdot \alpha } U^\alpha , \;\;\; \sigma \in \mathbb {R}^d, \alpha \in \mathbb {Z}^d. \end{aligned}$$

For two operators $P,Q$ with symbols:

$$\begin{aligned} \rho (P) = \sum p_\alpha \xi ^\alpha , \;\;\; \rho (Q) = \sum q_\beta \xi ^\beta , \end{aligned}$$

we use the formula, which follows directly from the same computations as in the case of classical calculus of pseudodifferential operators:

$$\begin{aligned} \rho (P Q) = \sum _{\gamma \in \mathbb {N}^{d}} \frac{1}{\gamma !} \partial _\xi ^\gamma (\rho (P))\delta ^\gamma (\rho (Q)), \end{aligned}$$

(2.2)

where $\gamma ! = \gamma _1 ! \cdots \gamma _d !$.

2.2 Wodzicki residue

In this part we shall provide an elementary proof that there exists a trace on the above defined algebra of symbols on the $d$-dimensional noncommutative torus:

Proposition 2.1

Let $\rho = \sum _{j \le k} \rho _j(\xi )$ be a symbol over the noncommutative torus $\mathcal {A}(\mathbb {T}^d_\theta )$. Then the functional:

$$\begin{aligned} \rho \mapsto \int \limits _{S^{d-1}} \mathfrak {t} \left( \rho _{-d} (\xi ) \right) d\xi , \end{aligned}$$

is a trace over the algebra of symbols.

Although the above proposition as well as the proof are not new, we provide it to make the paper self-contained.

Let us start with a simple lemma about homogeneous functions.

Lemma 2.2

Let $f$ be a smooth function on $\mathbb {R}^d {\setminus } \{0\}$, homogeneous of degree $\rho$. Then

$$\begin{aligned} \int \limits _{S^{d-1}} \partial _{\xi ^i} f(\xi ) d\xi =0, \end{aligned}$$

holds for every $1 \le i \le d$ if and only if $\rho =1-d$.

Proof

The $\Rightarrow$ part is trivial, as it is sufficient to take $f_j(\xi ) = \xi ^j (\xi ^2)^{\frac{\rho -1}{2}}$. Then:

$$\begin{aligned} \partial _i ( \xi ^j (\xi ^2)^{\frac{\rho }{2}-1} ) = \delta ^{ij} (\xi ^2)^{\frac{\rho -1}{2}} + 2 \frac{\rho -1}{2} \xi ^j \xi ^i (\xi ^2)^{\frac{\rho -1}{2}-1}, \end{aligned}$$

which, when restricted to the sphere $\xi ^2=1$ and integrated, gives:

$$\begin{aligned} V(d) \left( 1 + (\rho -1) \frac{1}{d} \right) , \end{aligned}$$

where $V(d)$ is the volume of $d\!-\!1$ dimensional sphere. This vanishes only if $\rho =1\!-\!d$.

Assume now that we have a homogeneous function on $\mathbb {R}^d$ of degree $\rho$, denoted $f$. Observe that using the freedom of the choice of coordinates we can safely assume that $i=1$. Using the spherical coordinates $r,\phi _1,\ldots ,\phi _{d-1}$:

$$\begin{aligned} \xi ^1 = r \sin \phi _1, \xi ^2 = r \cos \phi _1 \sin \phi _2, \ldots , \xi ^d = r \cos \phi _1 \cdots \cos \phi _{d-2} \cos \phi _{d-1}, \end{aligned}$$

we know the volume form on the sphere:

$$\begin{aligned} \omega = (\sin \phi _1)^{d-2} (\sin \phi _2)^{d-3} \cdots \sin \phi _{d-2} \, d\phi _1 \cdots d\phi _{d-1}, \end{aligned}$$

and we can express the partial derivative $\frac{\partial }{\partial x^1}$ as:

$$\begin{aligned} \frac{\partial }{\partial \xi ^1} = \cos \phi _1 \frac{\partial }{\partial r} -\frac{\sin \phi _1}{r} \frac{\partial }{\partial \phi _1}. \end{aligned}$$

Since the function $f$ is homogeneous in $r$ of order $\alpha$ we have:

$$\begin{aligned} \frac{\partial f}{\partial \xi ^1} = \cos \phi _1 \, \frac{\alpha }{r} f - \frac{\sin \phi _1}{r}\, \frac{\partial f}{\partial \phi _1}. \end{aligned}$$

Consider now the following function in the coordinates $\phi _1,\ldots \phi _{d-1}$:

$$\begin{aligned} \omega \left( \frac{\partial f}{\partial \xi ^1} \right) _{|_{r=1}}&= \left( \alpha (\sin \phi _1)^{d-2} (\cos \phi _1) f_{|_{r=1}} - (\sin \phi _1)^{d-1} \frac{\partial f_{|_{r=1}}}{\partial \phi _1}\right) \\&\times (\sin \phi _2)^{d-3} \cdots \sin \phi _{d-2} = \cdots \end{aligned}$$

If $\alpha = (1\!-\!d)$ it could be written as:

$$\begin{aligned} \cdots = \frac{\partial }{\partial \phi _1} \left( - (\sin \phi _1)^{d-1} f_{|_{r=1}} \right) (\sin \phi _2)^{d-3} \cdots (\sin \phi _{d-2}). \end{aligned}$$

Since the integral of a function $f$ over the sphere $S^{d-1}$ in the spherical coordinates is:

$$\begin{aligned} \int \limits _{S^{d-1}} F&= \int \limits _0^{2\pi } d\phi _1 \int \limits _0^\pi d\phi _2 \!\cdots \! \int \limits _0^\pi d\phi _{d-1}(\sin \phi _1)^{d\!-\!2} (\sin \phi _2)^{d-3} \!\cdots \! \sin \phi _{d-2} \, f(\phi _1,\ldots ,\phi _{d-1}), \end{aligned}$$

we have that:

$$\begin{aligned}&\int \limits _{S^{d-1}} \left( \frac{\partial f}{\partial \xi ^1} \right) _{|_{r=1}} \\&\quad \!=\! \int \limits _0^\pi \!\! d\phi _{d-1} \int \limits _0^\pi \!\! d\phi _{d-2} \sin \phi _{d-2} \cdots \int \limits _0^\pi \!\! d\phi _2 (\sin \phi _2)^{d-3} \int \limits _0^{2\pi } \!\! d\phi _1 \! \! \left( \frac{\partial }{\partial \phi _1} \left( \!-\! (\sin \phi _1)^{d-1} f_{|_{r=1}} \right) \right) \!=\! 0. \end{aligned}$$

$\square$

Proof of Proposition 2.1

Let $\rho$ and $\sigma$ be two symbols, that is $C^\infty$ maps from $\mathbb {R}^d$ to $\mathcal {A}(\mathbb {T}^d_\theta )$, which are decomposed into the sum of homogeneous symbols:

$$\begin{aligned} \sigma = \sum _{j \le S} \sigma _j, \;\;\;\; \rho = \sum _{j \le R} \rho _j. \end{aligned}$$

We shall prove that the the Wodzicki residue is a trace, that is:

$$\begin{aligned} \hbox {Wres}(\rho \sigma ) = \hbox {Wres}(\sigma \rho ). \end{aligned}$$

Using the formula for the product (2.2) we have:

$$\begin{aligned} (\rho \sigma )_{-d} = \sum _{\mathop {|\gamma |+j+k=-d}\limits ^{\gamma ,j,k}} \frac{1}{\gamma !} \partial _\xi ^\gamma (\rho _j) \delta ^\gamma (\sigma _k), \end{aligned}$$

and since derivation in $\xi$ decreases the degree of homogeneity by 1, the sum is necessarily finite.

First we shall prove the trace property for $|\gamma |=0$ and $|\gamma |=1$. If $|\gamma |=0$ we have:

$$\begin{aligned} \mathfrak {t} \left( \rho _j \sigma _k \right) = \mathfrak {t} \left( \sigma _k \rho _j \right) , \end{aligned}$$

since $\mathfrak {t}$ is a trace on the algebra $\mathcal {A}(\mathbb {T}^d_\theta )$.

Next, if $|\gamma |=1$, we have:

$$\begin{aligned} \int \limits _{S^{d-1}} d\xi \, \mathfrak {t}\left( \partial _{\xi ^i}(\rho _j) \delta _i(\sigma _k) \right)&= \int \limits _{S^{d-1}} d\xi \, \mathfrak {t}\left( - \delta _i(\partial _{\xi ^i}(\rho _j)) \sigma _k \right) \\&= \int \limits _{S^{d-1}} d\xi \, \mathfrak {t}\left( - \partial _{\xi ^i}(\delta _i(\rho _j)) \sigma _k \right) \\&= \int \limits _{S^{d-1}} d\xi \, \mathfrak {t}\left( \delta _i(\rho _j) \partial _{\xi ^i}(\sigma _k) \right) \\&= \int \limits _{S^{d-1}} d\xi \, \mathfrak {t}\left( \partial _{\xi ^i}(\sigma _k) \delta _i(\rho _j) \right) , \end{aligned}$$

where we have used that $\mathfrak {t}$ is invariant with respect to $U(1)$ symmetry generated by $\delta _i$, the fact that $\delta _i$ and $\partial _{\xi ^i}$ commute, then Lemma 2.2 (which we can use because the product $\sigma _j \rho _j$ is homogeneous of degree $1\!-\!d$), and finally the trace property of $\mathfrak {t}$.

For any $|\gamma |>1$ we repeat the above argument sufficient number of times. $\square$

3 Laplace-type operator of a conformally rescaled metric

In this section we shall fix our attention on a family of Laplace-type operators, which originate from a conformally rescaled fixed metric on manifold. We begin with the classical situation. Let us take a closed Riemannian manifold $M$ of dimension $d$ with a fixed metric tensor $g$. If $h$ is a positive smooth function on $M$ then we take the conformally rescaled metric to be given by:

$$\begin{aligned} g_{ab} \rightarrow h^2 g_{ab} = \tilde{g}_{ab}, \end{aligned}$$

where $g_{ab}$ is the original metric tensor.

Lemma 3.1

Let $\Delta$ be the usual Laplace operator on $M$ with the metric given by the metric tensor $g_{ab}$ and $\mathcal {H}$ be the Hilbert space of $L^2(M,g)$ (where the measure is taken with respect to the metric $g_{ab}$). Let $\tilde{\Delta }$ be the Laplace operator on $M$ with the conformally rescaled metric acting on the Hilbert space $\tilde{\mathcal {H}} = L^2(M, \tilde{g})$.

Then $\tilde{\Delta }$ is unitarily equivalent to $\Delta _h = h^{\frac{d}{2}} \tilde{\Delta } h^{-\frac{d}{2}}$ acting on $\mathcal {H}$. Moreover, the operator $\Delta _h$ written in local coordinates is:

$$\begin{aligned} \Delta _h = h^{-2} \Delta - 2 h^{-3} g^{ab} (\partial _a h) \partial _b + h^{\frac{d}{2}-2} (\Delta h^{-\frac{d}{2}}). \end{aligned}$$

Proof

Let $U: \Psi \rightarrow h^{-\frac{d}{2}} \Psi$ be a map between Hilbert spaces $U: \mathcal {H}\rightarrow \mathcal {H}_h$. Since:

$$\begin{aligned} || \Psi ||_\mathcal {H}= \int \limits _M \sqrt{g} |\Psi |^2 = \int \limits _M \sqrt{g h^{2d}} | h^{-\frac{d}{2}} \psi |^2= ||U \Psi ||_{\tilde{H}}. \end{aligned}$$

the map $U$ is unitary. Then $\Delta _h = U^{-1} \tilde{\Delta } U$ is an operator on $\mathcal {H}$ unitary equivalent to $\tilde{\Delta }$. The formula in local coordinates follows by explicit computations. $\square$

Next we shall restrict ourselves now to the case when $M$ is a $d$-dimensional torus, $\mathbb {T}^d$, and the metric we begin with is a constant, flat metric.

3.1 The case of $d$-dimensional torus

Consider a flat $d$-dimensional torus, $\mathbb {T}^d = (S^1)^d$, with a constant diagonal metric $g_{ab} = \delta _{ab}$. We take the usual system of coordinates on the torus (each circle parametrised by an angle) and from now on we assume that $\delta _a$ are the associated derivations. Take as $\mathcal {H}_0$ the Hilbert space of square summable functions with respect to the flat metric measure. An immediate consequence of Lemma 3.1 is:

Lemma 3.2

Let $h$ be a positive smooth function on the torus $\mathbb {T}^n$. The following operator on $\mathcal {H}_0$:

$$\begin{aligned} \Delta _h = \sum _{a=1}^n h^{-\frac{d}{2}} \partial _a( h^{d-2} \partial _a ) h^{-\frac{d}{2}} , \end{aligned}$$

is unitarily equivalent to the Laplace operator of the conformally rescaled metric $h^2 \delta _{ab}$.

Proof

This Laplace operator on the torus with the metric $g_{ab}= h^2 \delta _{ab}$ is:

$$\begin{aligned} \tilde{\Delta } = \sum _{a=1}^n h^{-d} \delta _a \left( h^{d-2} \delta _a \right) . \end{aligned}$$

Applying Lemma 3.1 we get the above explicit formula for $\Delta _h$. $\square$

This formula has been generalised to the noncommutative case in dimension 4 [10] in order to compute the curvature of the conformally rescaled Laplace operator on the noncommutative four-torus. However, even though the noncommutative generalisation of the above prescription for the conformally rescaled Laplace operator makes sense, it does not exclude the possibility that the quantity computed is not exactly the scalar curvature. The reason for this is the existence of torsion and the possibility that the above operator might not be torsion-free in the noncommutative generalisation.

3.2 Laplace-type operators on noncommutative tori

We shall investigate the family of operators, which are noncommutative generalisations of the above Laplace operator and which differ from them only by terms of lower order. This guarantees that their principal symbol is unchanged and hence using the natural (albeit naive) notion of noncommutative metric we could say both operators determine the same metric. We begin with the following definition:

Definition 3.3

Let us take $h \in \mathcal {A}(\mathbb {T}^d_\theta )$ to be a positive element with a bounded inverse and take the following densely defined operator on $\mathcal {H}$:

$$\begin{aligned} \Delta _h = \sum _{a=1}^n h^{-\frac{d}{2}} \delta _a ( h^{d-2} \delta _a ) h^{-\frac{d}{2}} , \end{aligned}$$

(3.1)

to be the Laplace operator on $d$-dimensional noncommutative torus.

In both cases of $d=2$ and $d=4$ this has been studied as the Laplace operator of the conformally rescaled noncommutative torus.

The family which we intend to investigate now is,

Definition 3.4

A generalised family of Laplace operator for the conformally rescaled metric over the torus has a form:

$$\begin{aligned} \Delta = \Delta _h + \sum _{a=1}^n \left( T^a \delta _a + \frac{1}{2} \delta _a(T^a) \right) + X , \end{aligned}$$

(3.2)

where $T^a$ and $X$ are some selfadjoint elements of $\mathcal {A}(\mathbb {T}^d_\theta )$. This form could be rewritten as:

$$\begin{aligned} \Delta = h^{-2} \Big ( \sum _a \delta _a^2 \Big ) + \sum _a Y^a \delta _a + \Phi , \end{aligned}$$

(3.3)

where

$$\begin{aligned} Y^a = h^{\frac{d}{2}-2} \delta _a (h^{-\frac{d}{2}})+ h^{-\frac{d}{2}} \delta _a (h^{\frac{d}{2}-2}) + T^a, \end{aligned}$$

and

$$\begin{aligned} \Phi = h^{\frac{d}{2}-2} \left( \sum _a \delta _a^2 (h^{-\frac{d}{2}}) \right) +h^{-\frac{d}{2}} \left( \sum _a \delta _a(h^{d-2}) \delta _a(h^{-\frac{d}{2}}) \right) + \frac{1}{2}\sum _a \delta _a T^a + X. \end{aligned}$$

The above Laplace-type operator (3.3) has the following symbol:

$$\begin{aligned} \rho (\Delta ) = h^{-2} \xi ^2 + \sum _a Y^a \xi _a + \Phi , \end{aligned}$$

(3.4)

From now on, we shall work only with the symbol $\rho (\Delta )$ (3.4).

4 Wodzicki residue in dimension 4

Our aim will be to compute the Wodzicki residue of $\Delta ^{-\frac{d}{2}+k}$ in the case of $d=4$ for $k=0,1$. As the only difficulty in considering the general case is purely computational, we postpone it for future work, concentrating instead on the example case of $d=4$ (which is most relevant for physics). The significance of these computations lies in the classical relation between Wodzicki residue of the inverse of the Laplace operator (in the sense of pseudodifferential calculus) and the scalar of curvature in the classical case [11]. We shall see, whether this extends to the noncommutative case. For simplicity we keep $X=0$, focusing on the parameters $h$ and $T_a$.

4.1 The symbol of $\Delta ^{-2}$ and $\Delta ^{-1}$

We fix here $d=4$. To simplify the notation above and in the remaining part of the note we use the convenient Einstein notation (implicit summation over repeated indices).

Proposition 4.1

The Wodzicki residue of $\Delta ^{-2}$ depends only on $h$:

$$\begin{aligned} \hbox {Wres}(\Delta ^{-2}) = 2\pi ^2 \, \mathfrak {t}(h^{4}). \end{aligned}$$

whereas for $\Delta ^{-1}$:

$$\begin{aligned} \hbox {Wres}(\Delta ^{-1}) \!=\! \frac{\pi ^2}{2} \left( \mathfrak {t}( h^2 T_a h^2 T_a h^2 ) \!+\! \mathfrak {t}( h^2 [T_a, \delta _a(h^2)]) \!-\! \mathfrak {t}( \delta _a(h^2) h^{-2} \delta _a(h^2)) \right) .\qquad \end{aligned}$$

Proof

Let us compute the relevant symbols of both operators. Observe, that for a differential operator of degree $2$ its symbol (split into part of homogeneous degrees) reads:

$$\begin{aligned} \rho (\Delta ) = a_2 + a_1 + a_0. \end{aligned}$$

Here we have:

$$\begin{aligned} a_2&= h^{-2} \xi ^2 , \nonumber \\ a_1&= \left( \delta _a (h^{-2}) + T_a \right) \xi ^a, \nonumber \\ a_0&= \delta _a \delta _a (h^{-2}) + h^{-2} \left( \delta _a(h^{2}) \delta _a(h^{-2}) \right) + \frac{1}{2} \delta _a T^a. \end{aligned}$$

(4.1)

To see the first result, it is sufficient to observe that the symbol $\rho (\Delta ^{-2})$ starts with a homogeneous symbol of order $-4$, which is exactly $h^4 |\xi |^{-4}$. Hence, computing the Wodzicki residue as in the proposition (2.1) gives the above result.

To compute the Wodzicki residue of $\Delta ^{-1}$ we need to calculate further terms of its symbol using the parametrix in the pseudodifferential calculus. For a pseudodifferential operator of order $-2$ we have:

$$\begin{aligned} b = b_0 + b_1 + b_2 + \cdots , \end{aligned}$$

where each $b_k$ is homogeneous of degree $-2-k$ and could be iteratively computed from the following sequence of identities, which arise from comparing homogeneous terms of the product $\Delta ^{-1}$ and $\Delta$ using the following formulae, which follow directly from the product rules of pseudodifferential operators:

$$\begin{aligned}&b_0 a_2 = 1, \nonumber \\&b_1 a_2 + b_0 a_1 + \partial _k(b_0) \delta _k(a_2) = 0, \nonumber \\&b_2 a_2 + b_1 a_1 + b_0 a_0 + \partial _k(b_0) \delta _k(a_1)+ \partial _k(b_1) \delta _k(a_2)\nonumber \\&\quad + {\frac{1}{2}}\partial _k \partial _j (b_0) \delta _k \delta _j (a_2) = 0, \end{aligned}$$

(4.2)

where $\delta _k$ ($k=1,\ldots , 4$) are the standard derivations on the noncommutative torus and $\partial _j$ ($j=1,\ldots , 4$) are partial derivatives with respect to $\xi _j$.

The relations could be solved explicitly (compare [4]), giving:

$$\begin{aligned} b_0&= ( a_2 )^{-1}, \nonumber \\ b_1&= -\left( b_0 a_1 b_0 + \partial _k(b_0) \delta _k(a_2) b_0 \right) , \nonumber \\ b_2&= \!-\! \Bigg ( b_0 a_0 b_0 \!+\! b_1 a_1 b_0 \!+\! \partial _j(b_0) \delta _j(a_1) b_0 \!+\! \partial _j(b_1)\delta _j(a_2)b_0\nonumber \\&\qquad + {\frac{1}{2}}\partial _{jk}(b_0)\delta _j \delta _k(a_2) b_0 \Bigg ), \end{aligned}$$

(4.3)

For the pseudodifferential operator (4.1) we obtain first:

$$\begin{aligned} b_1(T,h) =&\left( - h^2 (\delta _a(h^{-2}) +T_a) h^2 + 2 h^2 \delta _a(h^{-2}) h^2 \right) |\xi |^{-4} \xi ^a \\ =&\left( h^2 (\delta _a(h^{-2}) - T_a) h^2 \right) |\xi |^{-4} \xi ^a, \end{aligned}$$

and then:

$$\begin{aligned} b_2(T,h)(\xi ) =&|\xi |^{-6} \xi ^a \xi ^b \left( h^2 T_a h^2 T_b h^2 + h^2 T_a \delta _b(h^2) + 2 h^2 \delta _a(T^1) h^2 \right. \\&\left. + 3 \delta _a(h^2) T^a h^2- \delta _a(h^2) h^{-2} \delta _b(h^2)+ 2 \delta _a \delta _b (h^2) \right) \\&+ |\xi |^{-4} \left( - \frac{1}{2} h^2 \delta _a(T_a) h^2 - \delta _a(h^2) T^a h^2 \right) \end{aligned}$$

Finally, taking trace, using the Leibniz rule, the fact that the trace is closed and integrating over $3$-dimensional sphere we obtain the result (4.1). $\square$

Before we pass to the interpretation of the above result, let us consider the classical limit $\theta =0$.

4.2 The commutative case

We assume here that $\theta =0$, so $h$ and $T_a$ are smooth functions on a torus, which commute with each other (and with their derivations).

Lemma 4.2

For the commutative torus the Wodzicki residue of $\Delta ^{-1}$ is:

$$\begin{aligned} \hbox {Wres}(\Delta ^{-1}) = 2\pi ^2 \int \limits _{\mathbb {T}^4} \left( h^{6} (T_a T_a) - \delta _a(h)\delta _a(h) \right) dV, \end{aligned}$$

and for a fixed $h$ the term reaches an absolute extremum if and only if $T_a=0$, which has the interpretation of torsion-free Laplace operator.

Observe that the classical results of Kastler and Kalau-Walze [13, 14] give (for Laplace-Beltrami operator):

$$\begin{aligned} \hbox {Wres}(\Delta ^{-1}) = 2 \pi ^2 \int \limits _M \sqrt{g} \, \left( \frac{1}{6} R \right) , \end{aligned}$$

where $R$ is the scalar curvature.

In the conformally rescaled metric the volume form and the curvature (in dimension $d=4$) are:

$$\begin{aligned} \sqrt{g} = h^4, \;\;\;\; R = 6 h^{-3} \delta _{aa}(h), \end{aligned}$$

so we obtain the same result.

4.3 Nonminimal operators and curvature

In the classical (commutative) situation the additional first-order term in the Laplace-type operator $\Delta$ contributes to the Wodzicki residue of $\Delta ^{-1}$ with a term proportional to $T_a T_a$. As already noted, for minimal operators (like Laplace-Beltrami) such term vanishes and, equivalently, one can say that minimal operators are such Laplace-type operators, which, at the fixed metric minimise the Wodzicki residue.

In the noncommutative case we have two terms, one which is quadratic in $T_a$ and the second one, which is linear in $T_a$ and involves a commutator with $\delta _a(h^2)$. Therefore one can clearly state the following corollary,

Lemma 4.3

A naive generalisation of Laplace-Beltrami operator for the conformally rescaled metric to the noncommutative case (in dimension $d=4$) as proposed in (3.1) is not a minimal operator in the above sense (does not minimise the Wodzicki residue of $\Delta ^{-1}$ with $h$ fixed).

Proof

Assume that $T_a=0$ minimises for a fixed $h$ the functional (4.1). If we consider a small perturbation of $T_a$, $T_a = \varepsilon t_a$, for some fixed $t_a$ we obtain a quadratic function of $\varepsilon$, which has a nonvanishing linear term provided that the commutator $[T_a, \delta _a(h^2)]$ does not vanish. Such function does not have minimum at $\varepsilon =0$, hence $T_a=0$ cannot be a minimum of the functional. $\square$

A significant consequence of this fact is that no simple identification of the scalar of curvature is possible: indeed, if have no means of identifying minimal (or unperturbed, or torsion-free) Laplace-type operators we cannot possibly recover the geometric invariants associated with the metric alone.

5 Conclusions and open problems

The computations aimed to show that the Wodzicki residue of the noncommutative generalisation of the Laplace-type operator on the 4-dimensional noncommutative torus with a conformally recalled metric is a nontrivial functional on the parameters $h$ and $T_a$. There are several interesting problems, which arise that are linked to our result.

First of all, computations of the Wodzicki residue are closely linked to heat-kernel coefficients. As the principal symbol fails to be scalar, this is not happening for the noncommutative tori. An interesting point is then to link the Wodzicki residue in this case to the respective heat-kernel coefficients, which then, in turn, appear in the spectral action.

To study the geometric notions like curvature and geometric constructions like scalar curvature (at least as a noncommutative analogue of the classical one) it is important to identify the class of minimal Laplace-type operators. Otherwise, the functional would not only depend on the metric but also on some additional data. Our result shows that the naive generalisation of minimal operator fails to be minimal in the noncommutative case (at least in the proposed sense).

A natural question, which arises in this context, is about a proper definition of Dirac and Laplace operators on noncommutative tori. Even though the flat situation appears to be quite well understood, even a slight deviation, like the conformal rescaling, discussed in this note, changes completely the picture. Unlike classical case we still cannot identify the components of such operators, which are of intrinsic geometric origin and distinguish them from the additional degrees of freedom (like torsion).