# Spectral Torsion

Ludwik Dąbrowski(1) (1) SISSA (Scuola Internazionale Superiore di Studi
Avanzati),
Via Bonomea 265, 34136 Trieste, Italy<EMAIL_ADDRESS>,  Andrzej Sitarz(2)
and  Paweł Zalecki(2) (2) Institute of Theoretical Physics, Jagiellonian
University,
prof. Stanisława Łojasiewicza 11, 30-348 Kraków, Poland.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We introduce a trilinear functional of differential one-forms for a finitely
summable regular spectral triple with a noncommutative residue. We demonstrate
that for a canonical spectral triple over a closed spin manifold it recovers
the torsion of the linear connection. We examine several spectral triples,
including Hodge-de Rham, Einstein-Yang-Mills, almost-commutative two-sheeted
space, conformally rescaled noncommutative tori, and quantum $SU(2)$ group,
showing that the third one has a nonvanishing torsion if nontrivially coupled.

###### Key words and phrases:

spectral geometry, noncommutative geometry, torsion, noncommutative residue.

###### 2010 Mathematics Subject Classification:

58B34, 46L87, 58J42, 83C65, 58J50

This work is supported by the Polish National Science Centre grant
2020/37/B/ST1/01540

##  1\. Introduction

The existence and uniqueness of a metric-compatible linear connection with
vanishing torsion is one of the fundamental theorems of Riemannian geometry.
Torsion appears quite naturally as a vector-valued two-form in this approach,
and the assumption that it identically vanishes leads to the Levi-Civita
connection, which solely depends on the metric. In the background of general
relativity lies the torsion-free condition of Riemannian geometry, which very
accurately describes the gravitational interaction of bodies. Torsion, on the
other hand, has a physical interpretation as the quantity that measures the
twisting of reference frames along geodesics and has been considered an
independent field in physics in Einstein-Cartan theory [4]. Torsion, unlike
gravitational fields, must vanish in a vacuum and does not propagate; however,
its existence causes nonlinear interactions of matter with spin and has the
potential to change the standard singularity theorems of General Relativity
(see [16, 21] for a review of physical theories).

The emergence of noncommutative geometry [5, 6], which generalises standard
notions of differential geometry to an algebraic (or operator-algebraic)
setup, has raised new questions about the concepts of linear connections,
metric, and torsion. A transparent link between different approaches, ranging
from purely algebraic noncommutative differential geometry to operator
algebraic spectral triple formalism, and a unifying view of metric, linear
connection, and torsion has yet to be established. Although the algebraic
concepts of linear connections, torsion, and metric compatibility have been
achieved, the existence of the Levi-Civita connection can only be proven in
special cases (see [1, 2, 3] and the references therein).

On the other hand, the spectral triple approach, with the Dirac operator being
the fundamental object of geometry, has not yet been able to determine whether
the constructed Dirac operators correspond to the Levi-Civita connection or
whether they contain a nonvanishing torsion. So far, for that purpose, one
could only try to minimise the spectral functional corresponding to the
integrated scalar curvature while keeping the metric defined by the Dirac
operator unchanged. Even in the classical case of manifolds, this can be a
difficult task. (see computations of spectral action for a nonvanishing
torsion [15, 20, 18]), which becomes even harder in a genuinely noncommutative
situation [22].

The aim of this paper is to propose a plain, purely spectral method that
allows to determine the torsion as the density of the torsion functional and
impose (if possible) the torsion-free condition for regular finitely summable
spectral triples.

##  2\. Dirac operator with torsion

Let us assume that $M$ is a closed spin manifold of dimension $n$, with the
metric tensor $g_{ij}$. In terms of a (local) basis $e_{i}$ of orthonormal
vector fields on $M$ the covariant derivative can be written as

(2.1) $\nabla_{e_{i}}e_{j}=\omega_{ijk}e_{k}.$

The requirement that $\nabla$ is metric, that is for any vector fields
$X,Y,Z$,

${\mathcal{L}_{X}}g(Y,Z)=g(\nabla_{X}Y,Z)+g(Y,\nabla_{X}Z),$

allows to determine the connection explicitly through the formula,

$\displaystyle 2g(\nabla_{X}Y,Z)=$
$\displaystyle\;g([X,Y],Z)+g([Z,X],Y)+g([Z,Y],X)$
$\displaystyle+g(T(X,Y),Z)+g(T(Z,X),Y)+g(T(Z,Y),X),$

where $T$ is the torsion tensor,

$T(X,Y)=\nabla_{X}Y-\nabla_{Y}X-[X,Y].$

Therefore, the spin connection coefficients $\omega_{ijk}$ can be split into
the Levi-Civita connection, $\omega^{LC}_{ijk}$ and the contorsion part,
$\tau_{ijk}$. The first can be expressed explicitly in the basis $e_{i}$
through the structure constants $c_{ijk}$ of the commutators of orthonormal
vector fields,

(2.2)
$\omega^{LC}_{ijk}:=\frac{1}{2}(c_{ijk}+c_{kij}+c_{kji}),\qquad[e_{i},e_{j}]=c_{ijk}e_{k},$

and the remaining part of the connection, the contorsion tensor, is related to
the torsion of the connection,

$\tau_{ijk}={\mathchoice{{\tfrac{1}{2}}}{{\tfrac{1}{2}}}{{\scriptstyle\frac{1}{2}}}{{\scriptstyle\frac{1}{2}}}}(T_{ijk}+T_{kij}+T_{kji}),$

where $T(e_{i},e_{j})=T_{ijk}e_{k}$.

The lift of the covariant derivative to Dirac spinor fields is,

(2.3)
$\nabla^{(s)}_{e_{i}}=e_{i}-\frac{1}{4}\omega_{ijk}\gamma^{j}\gamma^{k},$

with $\gamma^{j}$ are the usual generators of Clifford algebra. The Dirac
operator on a spin manifold is, in a local basis of orthonormal frames, a
first-order differential operator,

(2.4) $D^{(s)}=i\gamma^{j}\nabla^{(s)}_{e_{j}},$

and is unique for a given metric and contorsion tensor. The canonical Dirac
operator, $D$, is the one for vanishing contorsion and we call it torsion-free
Dirac operator. The torsion tensor can be expressed using the contorsion,

$T_{ijk}=\tau_{ijk}-\tau_{jik}.$

Note the antisymmetry $\tau_{ijk}=-\tau_{ikj}$ while $T_{ijk}=-T_{kij}$. The
torsion tensor can be split into three parts, the vector part, totally
antisymmetric part and the Cartan part, however the Cartan part does not
contribute at all to the Dirac operator and the vector part has to vanish if
the resulting Dirac operator has to be self-adjoint [13]. Therefore, out of
the full torsion only the antisymmetric part appears in the selfadjoint Dirac
operator and hence we shall assume that torsion (and contorsion) are
antisymmetric tensors. Therefore, the Dirac operator with a fully antisymetric
torsion $T$ is,

(2.5)
$D_{T}=i\gamma^{j}\bigl{(}\nabla^{(s,LC)}_{e_{j}}-\frac{1}{8}T_{jkl}\gamma^{k}\gamma^{l}\bigr{)}=D-\frac{i}{8}T_{jkl}\gamma^{j}\gamma^{k}\gamma^{l},$

where $D$ is the standard Dirac operator, which originates from the Levi-
Civita connection. Note that since $[D_{T},f]=[D,f]$ for any function $f\in
C^{\infty}(M)$, the algebra of differential forms remains the same. We will
denote $\hat{u}$ the Clifford multiplication by the one-form $u$. Let
$\mathcal{W}$ denote the Wodzicki residue [25, 14]. We introduce the following
definition of the torsion functional of which the name is explained by the
subsequent torsion recovery theorem.

######  Definition 2.1.

For $D_{T}$ given by 2.5 the trilinear functional of differential one-forms
$u,v,w$

(2.6)
${\mathcal{T}}(u,v,w)=\mathcal{W}\bigl{(}\hat{u}\hat{v}\hat{w}D_{T}|D_{T}|^{-n}\bigr{)},$

is called torsion functional.

As our first main result we state,

######  Theorem 2.2.

The torsion functional for the Dirac operator $D_{T}$ on a Riemannian closed
spin manifold of dimension $n$ is,

(2.7)
${\mathcal{T}}(u,v,w)=-2^{m}iV(S^{n-1})\int\limits_{M}vol_{g}\,u_{a}v_{b}w_{c}T_{abc}.$

where $m=\frac{n}{2}$ for even $n$ and $m=\frac{n+1}{2}$ for odd $n>1$.

An immediate consequence is,

######  Corollary 2.3.

The Dirac operator $D_{T}$ is torsion free iff the torsion functional
vanishes:

$T\equiv 0\;\;\Leftrightarrow\;\;{\mathcal{T}}(\hat{u},\hat{v},\hat{w})=0.$

To prove the theorem we employ the calculus of pseudodifferential operators
and expansion of the symbols in normal coordinates.

###  2.1. Normal coordinates and symbols

Let us recall that in the normal coordinates [19] the metric and the Levi-
Civita connection have a Taylor expansion at a given point (${\bf x}=0$) on
the manifold,

(2.8) $\displaystyle g_{ab}=\delta_{ab}-\frac{1}{3}R_{acbd}x^{c}x^{d}+o({\bf
x^{2}}),$ $\displaystyle
g^{ab}=\delta_{ab}+\frac{1}{3}R_{acbd}x^{c}x^{d}+o({\bf x^{2}}),$
$\displaystyle\sqrt{\hbox{det}(g)}=1-\frac{1}{6}\mathrm{Ric}_{ab}x^{a}x^{b}+o({\bf
x^{2}}),$
$\displaystyle\Gamma^{a}_{bc}(x)=-\frac{1}{3}(R_{abcd}+R_{acbd})x^{d}+o({\bf
x^{2}}),$

where $R_{acbd}$ and $\mathrm{Ric}_{ab}$ are the components of the Riemann and
Ricci tensors, respectively, at the point with ${\bf x}=0$ and we use the
notation $o({\bf x^{k}})$ to denote that we expand a function up to the
polynomial of order $k$ in the normal coordinates.

As a next step, we compute the symbols of $D_{T}$ and its inverses as
pseudodifferential operators (see Appendix). We start with the symbol of
$D_{T}$ expanded in normal coordinates up to $o({\bf 1})$:

(2.9)
$\sigma(D_{T})=i\gamma^{j}\biggl{(}i\xi_{j}{-\frac{1}{8}T_{jps}\gamma^{p}\gamma^{s}}\biggr{)}+o({\bf
1}).$

A Clifford multiplication by a differential form $v$ can be expanded in local
normal coordinates around a point on a manifold, as:

$\hat{v}=v_{a}\gamma^{a}+o({\bf 1}).$

We can now proceed with the proof of the theorem.

Proof of Theorem 2.2: First of all, observe that since each one-form is a
zero-differential operator then effectively we need to compute the symbol of
order $-n$ of $D_{T}|D_{T}|^{-n}$. We start with the even dimension case
$n=2m$, postponing the case of odd dimensions till later. To compute the two
leading symbols of $D_{T}^{-2k}$, denoted by
$\mathfrak{c}_{2m}+\mathfrak{c}_{2m+1}$ we need the two leading symbols of
$\sigma(D_{T}^{2})={\mathfrak{a}}_{2}+{\mathfrak{a}}_{1}+{\mathfrak{a}}_{0}$.

(2.10) $\displaystyle{\mathfrak{a}}_{2}=$ $\displaystyle||\xi||^{2}+o({\bf
x}),$ $\displaystyle{\mathfrak{a}}_{1}=$
$\displaystyle{-\frac{1}{8}iT_{abc}\bigl{(}\gamma^{j}\gamma^{a}\gamma^{b}\gamma^{c}+\gamma^{a}\gamma^{b}\gamma^{c}\gamma^{j}\bigr{)}\xi_{j}}+o({\bf
1}).$

Then, using results from [11] (Lemma A.1) we have:

(2.11) $\displaystyle\mathfrak{c}_{2m}=||\xi||^{-2m}+o(\bf x),$
$\displaystyle\mathfrak{c}_{2m+1}={\frac{1}{8}im||\xi||^{-2m-2}T_{abc}\bigl{(}\gamma^{j}\gamma^{a}\gamma^{b}\gamma^{c}+\gamma^{a}\gamma^{b}\gamma^{c}\gamma^{j}\bigr{)}\xi_{j}}+o({\bf
1}).$

We are now ready to compute the symbol of order $-n$ of
$\hat{u}\hat{v}\hat{w}D_{T}|D_{T}|^{-n}$:

(2.12) $\displaystyle\sigma_{-n}(\hat{u}$
$\displaystyle\hat{v}\hat{w}D_{T}|D_{T}|^{-n})=-\frac{1}{8}u_{a}\gamma^{a}v_{b}\gamma^{b}v_{c}\gamma^{c}\biggl{(}iT_{jps}\gamma^{j}\gamma^{p}\gamma^{s}||\xi||^{-2m}$
$\displaystyle+(-\gamma^{k})\biggl{(}-im||\xi||^{-2m-2}T_{psr}\bigl{(}\gamma^{j}\gamma^{p}\gamma^{s}\gamma^{r}+\gamma^{p}\gamma^{s}\gamma^{r}\gamma^{j}\bigr{)}\xi_{j}\xi_{k}\biggr{)}+o({\bf
1}).$

Then, we proceed with the calculation of the Wodzicki residue.

(2.13)
$\displaystyle\mathcal{W}\bigl{(}\hat{u}\hat{v}\hat{w}D_{T}|D_{T}|^{-n}\bigr{)}$
$\displaystyle=\int_{M}\hbox{Tr}_{Cl}\,\int\limits_{||\xi||=1}\sigma_{-n}(\hat{u}\hat{v}\hat{w}D_{T}|D_{T}|^{-n})$
$\displaystyle=-2^{m}iV(S^{n-1})\int\limits_{M}vol_{g}\,u_{a}v_{b}w_{c}T_{abc}.$

This ends the proof for even $n$. To see that the proof holds also for odd $n$
we apply the usual technique to compute the symbol of
$|D_{T}|=\sqrt{D_{T}^{2}}$ following [23, Chapter XIII §1]. It follows that
the symbol $|D_{T}|$ is, as a sum of homogenous symbols, of which we need only
the first two (up to relevant order in normal coordinates):

$\sigma(|D_{T}|)=\biggl{(}|\xi|+o(\mathbf{x})\biggl{)}+\biggl{(}-\frac{1}{8}iT_{abc}\bigl{(}\gamma^{j}\gamma^{a}\gamma^{b}\gamma^{c}+\gamma^{a}\gamma^{b}\gamma^{c}\gamma^{j}\bigr{)}\xi_{j}|\xi|^{-1}+o({\bf
1})\biggr{)}+\cdots.$

Then the rest of the proof is the same as in the case of even dimensions.
$\qquad\Box$

######  Remark 2.4.

Note that the lowest odd dimension in which there can be a nonvanishing
torsion is $n=3$. In the lowest possible even dimension $n=4$ the torsion
functional can also be written as a functional over a single one-form:

(2.14)
${\mathcal{T}}(u)=\mathcal{W}\bigl{(}\gamma\hat{u}D_{T}|D_{T}|^{-n}\bigr{)}.$

##  3\. Noncommutative torsion

The spectral definition of torsion, Def.2.1, can be readily extended to the
noncommutative case of spectral triples. Let $({\mathcal{A}},D,{\mathcal{H}})$
be a $n$-summable unital spectral triple, $\Omega^{1}_{D}({\mathcal{A}})$ be
the ${\mathcal{A}}$ bimodule of one forms generated by ${\mathcal{A}}$ and
$[D,{\mathcal{A}}]$, which by definition consists of bounded operators on
${\mathcal{H}}$. We assume that there exists a generalised algebra of
pseudodifferential operators, which contains ${\mathcal{A}}$, $D$,
$|D|^{\ell}$ for $\ell\in\mathbb{Z}$, and there exists a tracial state
$\mathcal{W}$ on it, called a noncommutative residue. Note that these
assumptions follow if the spectral triple is finitely summable and regular,
(c.f. [17] and [24]). Moreover, we assume that the noncommutative residue
identically vanishes on $T|D|^{-k}$ for any $k>n$ (c.f. [7]) and a zero-order
operator $T$ (i.e., an operator in the algebra generated by ${\mathcal{A}}$
and $\Omega^{1}_{D}({\mathcal{A}})$). We introduce the following definition of
torsion functional for spectral triples.

######  Definition 3.1.

Torsion functional for the spectral triple $({\mathcal{A}},D,{\mathcal{H}})$
is a trilinear functional of $u,v,w\in\Omega^{1}_{D}({\mathcal{A}})$ defined
by,

$\mathcal{T}(u,v,w)=\mathcal{W}(uvwD|D|^{-n}).$

We say that the spectral triple is torsion-free (or, for simplicity, the Dirac
operator $D$ is torsion-free) if $\mathcal{T}$ vanishes identically.

We recall a stronger condition of spectrally closed spectral triples defined
in [11, Definition 5.5] which, in particular, implies the torsion-free
condition stated above.

######  Definition 3.2.

We say that a spectral triple with a trace on the generalised algebra of
pseudodifferential operators is spectrally closed if for any zero-order
operator $P$ (from the algebra generated by ${\mathcal{A}},$ and
$\Omega^{1}_{D}({\mathcal{A}})$ ) the following holds:

$\mathcal{W}(PD|D|^{-n})=0.$

######  Lemma 3.3 ([11, Lemma 5.6]).

The classical spectral triple over a closed spin-c manifold $M$ of dimension
$n=2m$ is spectrally closed in the above sense and hence is torsion-free.

##  4\. Examples

The examples of spectral triples presented below are all significant cases of
various noncommutative geometries to which we apply the proposed spectral
definition of torsion and compute the torsion functional.

###  4.1. Hodge-Dirac.

Let us start with a purely classical example of a Hodge-Dirac spectral triple
over an oriented, closed Riemannian manifold, where the Hilbert space are
square-summable differential forms and the Dirac is $D=d+\delta$, where
$\delta=\ast d\ast$, with $\ast$ being the Hodge star operator. In [12] we
demonstrated that such spectral triple is spectrally closed and therefore
torsion-free.

###  4.2. Einstein-Yang-Mills.

Consider a closed spin manifold $M$ of even dimension $n=2m$ and a spectral
triple

$\left(C^{\infty}(M)\otimes M_{N}(\mathbb{C}),\widetilde{D},L^{2}(S)\otimes
M_{N}(\mathbb{C})\right).$

Here, the algebra $M_{N}(\mathbb{C})$ acts by left multiplications on
$M_{N}(\mathbb{C})$ regarded as a Hilbert space with the scalar product
$\langle M,M^{\prime}\rangle=\hbox{Tr}(M^{*}M^{\prime})$ . Also,

$\widetilde{D}=D\otimes\hbox{id}_{N}+A+JAJ^{-1}$

is a fluctuation of $D\otimes\hbox{id}_{N}$, where $D$ is the standard Dirac
operator on $M$,

$A\\!=\\!A^{*}\\!\in\\!\,\Omega^{1}_{D}=\bigl{\\{}\alpha[D\otimes\hbox{id}_{N},\beta]\;|\;\alpha,\beta\in
C^{\infty}(M)\otimes M_{N}(\mathbb{C})\bigr{\\}},$

and $J=C{\otimes}*$, with $C$ being the charge conjugation on spinors in $S$.
Note that $A$ equals $i\gamma^{a}X_{a}$, where $X_{a}=-X_{a}^{*}\in
C^{\infty}(M,M_{N}(\mathbb{C}))$ acts on $M_{N}(\mathbb{C})$ by left
multiplication, while $JAJ^{-1}$ equals $i\gamma^{a}$ composed with the right
multiplication by $X_{a}^{*}$. Therefore, the fluctuation $A+JAJ^{-1}$ amounts
to the action of $i\gamma^{a}\mathrm{Ad}_{X_{a}}$. It follows that the center
of this algebra, $C^{\infty}(M,\mathbb{C}1_{N})$, acts trivially, so we just
can consider $X_{a}$ such that $\langle X_{a},1_{N}\rangle=0$, that is,
traceless matrix valued functions, $X_{a}\in C^{\infty}(M,su(N))$. Moreover,
we need only the expansion of $X_{a}$ and so of the operator
$i\gamma^{a}\mathrm{Ad}_{X_{a}}$ in normal coordinates up to $o({\bf 1})$.

It is known that this spectral triple is finitely summable and regular. To
compute the torsion functional for $\widetilde{D}$ we start (slightly abusing
the notation) with its symbol expanded as:

(4.1)
$\sigma(\widetilde{D})=\gamma^{a}\bigl{(}-\xi_{a}{+\,i\,\mathrm{Ad}_{X_{a}}\bigr{)}+o({\bf
1})}.$

It is not difficult to see that the bimodule $\Omega^{1}_{\widetilde{D}}$
spanned by $\alpha[\widetilde{D},\beta]$, $\alpha,\beta\in
C^{\infty}(M,M_{N}(\mathbb{C}))$, is also spanned just by the elements
$u=\gamma^{a}v_{a}$, where we can expand $v_{a}\in M_{N}(\mathbb{C})+o({\bf
1})$. We now compute the symbols of
$\sigma(uvw\widetilde{D})=\mathfrak{s}_{1}+\mathfrak{s}_{0}$ for
$u,v,w\in\Omega^{1}_{\widetilde{D}}$:

(4.2) $\displaystyle\mathfrak{s}_{1}$
$\displaystyle=-\gamma^{a}\gamma^{b}\gamma^{c}\gamma^{d}u_{a}v_{b}w_{c}\,\xi_{d},$
$\displaystyle\mathfrak{s}_{0}$
$\displaystyle=i\gamma^{a}\gamma^{b}\gamma^{c}\gamma^{d}u_{a}v_{b}w_{c}\,\mathrm{ad}_{X_{d}}.$

Next, to compute the two leading symbols of $\widetilde{D}^{-2m}$,
$\sigma(\widetilde{D}^{-2m})=\mathfrak{c}_{2m}+\mathfrak{c}_{2m+1}$, we need
the two leading symbols of $\widetilde{D}^{2}$,
$\sigma(\widetilde{D}^{2})={\mathfrak{a}}_{2}+{\mathfrak{a}}_{1}+\ldots$,

(4.3) $\displaystyle{\mathfrak{a}}_{2}=$ $\displaystyle||\xi||^{2}+o({\bf
x}),$ $\displaystyle{\mathfrak{a}}_{1}=$
$\displaystyle{-2\mathrm{Ad}_{X_{a}}\xi_{a}+o(\bf 1),}$

and the two leading symbols of $\widetilde{D}^{-2}$,
$\sigma(\widetilde{D}^{-2})={\mathfrak{b}}_{2}+{\mathfrak{b}}_{3}+\ldots$,

(4.4) $\displaystyle{\mathfrak{b}}_{2}=$ $\displaystyle||\xi||^{-2}+o({\bf
x}),$ $\displaystyle{\mathfrak{b}}_{3}=$
$\displaystyle+{2\mathrm{Ad}_{X_{a}}\xi_{a}||\xi||^{-4}+o(\bf 1).}$

Then, using [11, Lemma A.1] we have:

(4.5) $\displaystyle\mathfrak{c}_{2m}=||\xi||^{-2m}+o(\bf x),$
$\displaystyle\mathfrak{c}_{2m+1}={n\mathrm{Ad}_{X_{a}}\xi_{a}||\xi||^{-n-2}.}$

Therefore, since $\partial_{a}\mathfrak{c}_{2m}=0$,

$\sigma_{-n}\left(uvw\widetilde{D}\|\widetilde{D}\|^{-n}\right)=\mathfrak{s}_{0}\mathfrak{c}_{n}+\mathfrak{s}_{1}\mathfrak{c}_{n+1},$

where both terms on the right hand side contain linearly
$\mathrm{ad}_{X_{a}}$. It turns out however, that the trace of
$\mathrm{ad}_{X}$, $X\in M_{N}(\mathbb{C})$ , considered as a linear operator
on $M_{N}(\mathbb{C})$ vanishes. This could be checked explicitly, using the
basis $E_{\alpha\beta}$, $\alpha,\beta=1,\ldots,N$ of $M_{N}(\mathbb{C})$,
where the matrix $E_{\alpha\beta}$ has the $\alpha,\beta$ matrix entry $1$ and
otherwise $0$. We compute:

$Ad_{E_{\mu\nu}}E_{\alpha\beta}=E_{\mu\beta}\delta_{\alpha\nu}-E_{\alpha\nu}\delta_{\mu\beta}=K_{\mu\nu;\alpha\beta,\rho\tau}E_{\rho\tau},$

where $K$ as a matrix is,

$K_{\mu\nu;\alpha\beta,\rho\tau}=\delta_{\mu,\rho}\delta_{\beta,\tau}\delta_{\nu,\alpha}-\delta_{\alpha,\rho}\delta_{\nu,\tau}\delta_{\mu,\beta}.$

Thus,

$\hbox{Tr}\,\mathrm{ad}_{E_{\mu\nu}}=K_{\mu\nu;\alpha\beta,\alpha\beta}=N(\delta_{\mu,\alpha}\delta_{\nu,\alpha}-\delta_{\nu,\alpha}\delta_{\mu,\alpha})=0.$

Therefore, the spectral torsion functional ${\mathcal{T}}(u,v,w)$ for
$\widetilde{D}$ vanishes and so $\widetilde{D}$ is torsion free.

###  4.3. Almost commutative $\bf M\times\mathbb{Z}_{2}$

We assume that $M$ is a closed spin manifold and
$(C^{\infty}(M),D,{\mathcal{H}})$ is an even spectral triple of dimension $n$
with the standard Dirac operator $D$ and a grading $\gamma$. We know that it
is spectrally closed, so, in particular, $D$ is torsion-free. We consider the
usual double-sheet spectral triple,
$(C^{\infty}(M)\otimes\mathbb{C}^{2},{\mathcal{D}},{\mathcal{H}}\otimes\mathbb{C}^{2})$,
with the Dirac operator,

${\mathcal{D}}=\left(\begin{array}[]{cc}D&\gamma\Phi\\\
\gamma\Phi^{*}&D\end{array}\right),$

where $\Phi\in\mathbb{C}$. The bimodule of one forms, associated to
${\mathcal{D}}$ consists of the following operators,

(4.6) $\omega=\begin{pmatrix}w^{+}&\Phi\gamma f^{+}\\\ \Phi^{*}\gamma
f^{-}&w^{-}\end{pmatrix},$

where $w^{\pm}\in\Omega^{1}_{D}(M)$ and $f^{\pm}\in C^{\infty}(M)$. We shall
denote the diagonal part as $\omega^{d}$ and the off-diagonal part as
$\omega^{o}$.

In order to calculate
$\mathcal{W}(\omega_{1}\omega_{2}\omega_{3}\mathcal{D}\mathcal{D}^{-2m})$ for
one-forms $\omega_{1},\omega_{2},\omega_{3}$ let us observe that due to
linearity it suffices to do it separately for diagonal and off-diagonal parts.
Moreover, the algebra rules in this extended Clifford algebra are obvious and
we restrict ourselves to the four possible cases. Furthermore, since
$\mathcal{D}^{-2m}=D^{-2m}\mathbf{1}(1-2m|\Phi|^{2}D^{-2}+\dots)$ we note that
in the Wodzicki residue formula $\mathcal{D}^{-2m}$ can be replaced by
$D^{-2m}$.

By a straightforward calculation we get,

(4.7)
$\displaystyle\mathcal{W}\bigl{(}\omega_{1}^{d}\omega_{2}^{d}\omega_{3}^{d}\mathcal{D}\mathcal{D}^{-2m}\bigr{)}$
$\displaystyle=0,$
$\displaystyle\mathcal{W}\bigl{(}\omega_{1}^{d}\omega_{2}^{d}\omega_{3}^{o}\mathcal{D}\mathcal{D}^{-2m}\bigr{)}$
$\displaystyle=\mathcal{W}\bigl{(}|\Phi|^{2}(w_{1}^{+}w_{2}^{+}f_{3}^{+}+w_{1}^{-}w_{2}^{-}f_{3}^{-})D^{-2m}\bigr{)}$
$\displaystyle=|\Phi|^{2}\bigl{(}\mathcal{g}(w_{1}^{+},w_{2}^{+})f_{3}^{+}+\mathcal{g}(w_{1}^{-},w_{2}^{-})f_{3}^{-}\bigr{)},$
$\displaystyle\mathcal{W}\bigl{(}\omega_{1}^{d}\omega_{2}^{o}\omega_{3}^{o}\mathcal{D}\mathcal{D}^{-2m}\bigr{)}$
$\displaystyle=\mathcal{W}\bigl{(}|\Phi|^{2}(w_{1}^{+}f_{2}^{+}f_{3}^{-}+w_{1}^{-}f_{2}^{-}f_{3}^{+})DD^{-2m}\bigr{)}=0,$
$\displaystyle\mathcal{W}\bigl{(}\omega_{1}^{o}\omega_{2}^{o}\omega_{3}^{o}\mathcal{D}\mathcal{D}^{-2m}\bigr{)}$
$\displaystyle=\mathcal{W}\bigl{(}|\Phi|^{4}(f_{1}^{+}f_{2}^{-}f_{3}^{+}+f_{1}^{-}f_{2}^{+}f_{3}^{-})D^{-2m}\bigr{)}$
$\displaystyle=|\Phi|^{4}\mathcal{V}(f_{1}^{+}f_{2}^{-}f_{3}^{+}+f_{1}^{-}f_{2}^{+}f_{3}^{-}).$

where $\mathcal{g}$ is the metric functional on $M$ and the functional
$\mathcal{V}(f)$ is proportional to $\int_{M}f$. Note that the first equation
is a consequence of $D$ being torsion-free, whereas the third equation holds
if $\mathcal{W}(\omega DD^{-2m})=0$, which is the consequence of the spectral
triple on $M$ being spectrally closed.

As a consequence, we see that the product of the standard spectral triple over
$M$ with a finite discrete one does not satisfy the torsion-free condition
unless $\Phi=0$. Note that the generalisation of the model with $\Phi$ being a
complex-valued function on $M$ does not change the conclusion.

###  4.4. Conformally rescaled noncommutative tori

Consider the spectral triple on the noncommutative $n$-torus,
$({\mathcal{A}},{\mathcal{H}},D_{k})$, where
$\mathcal{A}=C^{\infty}(\mathbb{T}^{n}_{\theta})$ is the $\delta_{j}$-smooth
subalgebra for the standard derivations $\delta_{j}$, $j=1,\dots,n$. Next,
${\mathcal{H}}=L^{2}(\mathbb{T}^{n}_{\theta},\tau)$, where $\tau$ is the
standard trace which anihilates $\delta_{j}$. Also, $D_{k}=kDk$ is the
conformal rescaling of the standard (flat) Dirac operator
$D=\sum_{j}\gamma^{j}\delta_{j}$ with $\gamma^{j}$ being the usual gamma
matrices and the conformal factor $k>0$ is from the copy ${\mathcal{A}}^{o}$
of ${\mathcal{A}}$ in the commutant of $\mathcal{A}$. The bimodule of one-
forms, generated by the commutators $[D_{k},{\mathcal{a}}]$,
${\mathcal{a}}\in{\mathcal{A}}$, is a free left module generated by
$k^{2}\gamma^{j}$. Using the extension of the calculus of pseudodifferential
operators over noncommutative tori, following [8], to
$\hat{{\mathcal{A}}}$-valued symbols (where $\hat{{\mathcal{A}}}$ is the
algebra generated by ${\mathcal{A}}$ and ${\mathcal{A}^{o}}$) and of the
analogue of the Wodzicki residue, we defined and compute various spectral
functionals, as the metric and Einstein functionals [11, Section 5.2.1] in
dimension 2 and 4. In particular, we demonstrated there that strictly
irrational noncommutative 2- and 4-dimensional tori with a conformally
rescaled Dirac operator are spectrally closed. We extend here this result to
any dimension:

######  Theorem 4.1.

The spectral triple
$\bigl{(}C^{\infty}(\mathbb{T}^{n}_{\theta}),{\mathcal{H}},D_{k}\bigr{)}$ for
strictly irrational noncommutative torus of dimension $n$ is spectrally
closed, that is for any $T$, operator of order $0$ (that is generated by
$a,[D_{k},b]$, $a,b\in\mathcal{A}$)

$\mathcal{W}\bigl{(}TD_{k}|D_{k}|^{-n}\bigr{)}=0.$

Proof.  The proof is based on a simple observation that the symbol of order
$-n$ of $TD_{k}|D_{k}|^{-n}$ is a sum of terms that are of the form
$T_{\alpha,j,\beta}(\xi)k^{\alpha}\delta_{j}(k)k^{\beta}$, for $j=1,\ldots,n$,
some integers $\alpha,\beta$ and $C^{\infty}(\mathbb{T}^{n}_{\theta})$-valued
functions $T_{\alpha,j,\beta}(\xi)$, which are homogeoneous in $\xi$ of degree
$-n$. Since for strictly irrational tori the trace over the full algebra
$\mathcal{A}\mathcal{A}^{o}$ factorizes, then it is sufficient to show that
$\tau(k^{\alpha}\delta_{j}(k)k^{\beta})=0$ for any $j,\alpha,\beta$. For that
we write $k=e^{h}$ and observe that

$\tau(k^{\alpha}\delta_{j}(k)k^{\beta})=\tau\bigl{(}e^{(\alpha+\beta)h}\delta_{j}(e^{h})\bigr{)}$

vanishes since for $\alpha+\beta\not=-1$ it equals

$\frac{1}{\alpha+\beta+1}\tau\bigl{(}\delta_{j}(e^{(\alpha+\beta+1)h})\bigr{)}$

while for $\alpha+\beta=-1$ it equals $\tau\bigl{(}\delta_{j}(h))\bigr{)}$.
This ends the proof. $\qquad\Box$ As an immediate consequence of spectral
closedness, we have:

######  Corollary 4.2.

The Dirac operator $D_{k}$ over the $n$-dimensional strictly irrational
noncommutative torus $\mathbb{T}^{n}_{\theta}$ is torsion free.

###  4.5. Quantum $SU(2)$.

We briefly recall the construction of the spectral triple
$\bigl{(}\mathcal{A}(SU_{q}(2)),\mathcal{H},D\bigr{)}$ of [9]. Let
$\mathcal{A}=\mathcal{A}(SU_{q}(2))$ be the $*$-algebra generated by $a$ and
$b$, subject to the commutation rules:

(4.8) $\displaystyle ba=qab,\qquad b^{*}a=qab^{*},\qquad bb^{*}=b^{*}b,$
$\displaystyle a^{*}a+q^{2}b^{*}b=1,\qquad aa^{*}+bb^{*}=1,$

where $0<q<1$. The Hilbert space $\mathcal{H}$ has an orthonormal basis $|j\mu
n{\mathord{\uparrow}}\rangle$ for
$j=0,{\mathchoice{{\tfrac{1}{2}}}{{\tfrac{1}{2}}}{{\scriptstyle\frac{1}{2}}}{{\scriptstyle\frac{1}{2}}}},1,\dots$,
$\mu=-j,\dots,j$ and $n=-j^{+},\dots,j^{+}$; together with $|j\mu
n{\mathord{\downarrow}}\rangle$ for
$j={\mathchoice{{\tfrac{1}{2}}}{{\tfrac{1}{2}}}{{\scriptstyle\frac{1}{2}}}{{\scriptstyle\frac{1}{2}}}},1,\dots$,
$\mu=-j,\ldots,j$ and $n=-j^{-},\dots,j^{-}$, where
$j^{+}=j+{\mathchoice{{\tfrac{1}{2}}}{{\tfrac{1}{2}}}{{\scriptstyle\frac{1}{2}}}{{\scriptstyle\frac{1}{2}}}}$
and
$j^{-}=j-{\mathchoice{{\tfrac{1}{2}}}{{\tfrac{1}{2}}}{{\scriptstyle\frac{1}{2}}}{{\scriptstyle\frac{1}{2}}}}$.
For the explicit formulas defining the representation of $\mathcal{A}$ on
$\mathcal{H}$ we refer to [9] or [10]. The equivariant Dirac operator is
chosen to be isospectral to the classical Dirac over $SU(2)$;

(4.9) $D|j\mu
n{\mathord{\uparrow}}\rangle=(2j+{\mathchoice{{\tfrac{3}{2}}}{{\tfrac{3}{2}}}{{\scriptstyle\frac{3}{2}}}{{\scriptstyle\frac{3}{2}}}})|j\mu
n{\mathord{\uparrow}}\rangle,\qquad D|j\mu
n{\mathord{\downarrow}}\rangle=-(2j+{\mathchoice{{\tfrac{1}{2}}}{{\tfrac{1}{2}}}{{\scriptstyle\frac{1}{2}}}{{\scriptstyle\frac{1}{2}}}})|j\mu
n{\mathord{\downarrow}}\rangle.$

In [10] it was shown that the triple is regular and its dimension spectrum of
the spectral triple was determined, together with a an explicit formula that
computes respective noncommutative residue. In particular, the following
theorem was proven.

######  Theorem 4.3.

If $T\in\Psi^{0}(\mathcal{A})$ is in the algebra of pseudodifferential
operators of order $0$ (as defined in [10]) then for
$P^{\mathord{\uparrow}},P^{\mathord{\downarrow}}$, projections on the
respective up and down part of the Hilbert space of spinors, the
noncommutative integral of
$P^{({\mathord{\uparrow}},{\mathord{\downarrow}})}T|D|^{-2}$ could be
explicitly computed as:

(4.10) $\displaystyle\mathop{\mathchoice{\hbox{\hbox to0.0pt{\hbox
to5.83331pt{\kern-1.00006pt\hbox{$\displaystyle\int$}\hfil}\hss}\hbox{$-$}\kern
1.00006pt}}{\hbox{\hbox to0.0pt{\hbox
to5.83331pt{\kern-0.50003pt\hbox{$\textstyle\int$}\hfil}\hss}\hbox{$\vbox{\hrule
width=4.0pt}$}\kern 1.00006pt}}{\hbox{\hbox to0.0pt{\hbox
to5.83331pt{\kern-0.50003pt\hbox{$\textstyle\int$}\hfil}\hss}\hbox{$\vbox{\hrule
width=4.0pt}$}\kern 1.00006pt}}{\hbox{\hbox to0.0pt{\hbox
to5.83331pt{\kern-0.50003pt\hbox{$\textstyle\int$}\hfil}\hss}\hbox{$\vbox{\hrule
width=4.0pt}$}\kern 1.00006pt}}}\nolimits P^{\mathord{\uparrow}}T|D|^{-2}$
$\displaystyle=\bigl{(}\tau_{1}\otimes\tau_{0}^{\mathord{\uparrow}}+\tau_{0}^{\mathord{\downarrow}}\otimes\tau_{1}\bigr{)}\bigl{(}\beta(T)\bigr{)},$
$\displaystyle\mathop{\mathchoice{\hbox{\hbox to0.0pt{\hbox
to5.83331pt{\kern-1.00006pt\hbox{$\displaystyle\int$}\hfil}\hss}\hbox{$-$}\kern
1.00006pt}}{\hbox{\hbox to0.0pt{\hbox
to5.83331pt{\kern-0.50003pt\hbox{$\textstyle\int$}\hfil}\hss}\hbox{$\vbox{\hrule
width=4.0pt}$}\kern 1.00006pt}}{\hbox{\hbox to0.0pt{\hbox
to5.83331pt{\kern-0.50003pt\hbox{$\textstyle\int$}\hfil}\hss}\hbox{$\vbox{\hrule
width=4.0pt}$}\kern 1.00006pt}}{\hbox{\hbox to0.0pt{\hbox
to5.83331pt{\kern-0.50003pt\hbox{$\textstyle\int$}\hfil}\hss}\hbox{$\vbox{\hrule
width=4.0pt}$}\kern 1.00006pt}}}\nolimits P^{\mathord{\downarrow}}T|D|^{-2}$
$\displaystyle=\bigl{(}\tau_{1}\otimes\tau_{0}^{\mathord{\downarrow}}+\tau_{0}^{\mathord{\uparrow}}\otimes\tau_{1}\bigr{)}\bigl{(}\beta^{0}(T)\bigr{)},$

where the noncommutative ingeral is, as usually,

$\mathop{\mathchoice{\hbox{\hbox to0.0pt{\hbox
to5.83331pt{\kern-1.00006pt\hbox{$\displaystyle\int$}\hfil}\hss}\hbox{$-$}\kern
1.00006pt}}{\hbox{\hbox to0.0pt{\hbox
to5.83331pt{\kern-0.50003pt\hbox{$\textstyle\int$}\hfil}\hss}\hbox{$\vbox{\hrule
width=4.0pt}$}\kern 1.00006pt}}{\hbox{\hbox to0.0pt{\hbox
to5.83331pt{\kern-0.50003pt\hbox{$\textstyle\int$}\hfil}\hss}\hbox{$\vbox{\hrule
width=4.0pt}$}\kern 1.00006pt}}{\hbox{\hbox to0.0pt{\hbox
to5.83331pt{\kern-0.50003pt\hbox{$\textstyle\int$}\hfil}\hss}\hbox{$\vbox{\hrule
width=4.0pt}$}\kern 1.00006pt}}}\nolimits
T:=\operatorname{Res}_{z=0}\mathrm{Tr}(T|D|^{-z}).$

The map $\beta^{0}$ is a projection on the two first legs of the zero-degree
part of a map $\rho$, which is a homomorphism

(4.11)
$\rho:\mathcal{B}\to\mathcal{A}(D^{2}_{q})\otimes\mathcal{A}(D^{2}_{q})\otimes\mathcal{A}(\mathbb{S}^{1}),$

where $\mathcal{B}$ is a subalgebra of $\Psi^{0}(\mathcal{A})$ generated by
diagonal parts of the generators $a,b$ and their commutators with $D$ and
$|D|$, $\mathcal{A}(D^{2}_{q})$ are quantum discs. Using the symbol map
$\sigma:\mathcal{A}(D^{2}_{q})\to\mathcal{A}(\mathbb{S}^{1})$ and the standard
irreducible representation of quantum discs $\pi$ the functionals
$\tau_{1},\tau_{0}^{\mathord{\uparrow}},\tau_{0}^{\mathord{\downarrow}}$ are
defined for $x\in\mathcal{A}(D^{2}_{q})$, as

(4.12) $\displaystyle\tau_{1}(x)$
$\displaystyle:=\frac{1}{2\pi}\int_{S^{1}}\sigma(x),$
$\displaystyle\tau_{0}^{\mathord{\uparrow}}(x)$
$\displaystyle:=\lim_{N\to\infty}\hbox{Tr}_{N}\pi(x)-(N+{\mathchoice{{\tfrac{3}{2}}}{{\tfrac{3}{2}}}{{\scriptstyle\frac{3}{2}}}{{\scriptstyle\frac{3}{2}}}})\tau_{1}(x),$
$\displaystyle\tau_{0}^{\mathord{\downarrow}}(x)$
$\displaystyle:=\lim_{N\to\infty}\hbox{Tr}_{N}\pi(x)-(N+{\mathchoice{{\tfrac{1}{2}}}{{\tfrac{1}{2}}}{{\scriptstyle\frac{1}{2}}}{{\scriptstyle\frac{1}{2}}}})\tau_{1}(x).$

Using the above theorem we can state as a simple corollary,

######  Corollary 4.4.

The standard equivariant spectral triple over $\mathcal{A}(SU_{q}(2))$ is
spectrally closed and, in particular, the Dirac operator (4.9) is torsion-
free.

Proof.  We compute,

$\mathop{\mathchoice{\hbox{\hbox to0.0pt{\hbox
to5.83331pt{\kern-1.00006pt\hbox{$\displaystyle\int$}\hfil}\hss}\hbox{$-$}\kern
1.00006pt}}{\hbox{\hbox to0.0pt{\hbox
to5.83331pt{\kern-0.50003pt\hbox{$\textstyle\int$}\hfil}\hss}\hbox{$\vbox{\hrule
width=4.0pt}$}\kern 1.00006pt}}{\hbox{\hbox to0.0pt{\hbox
to5.83331pt{\kern-0.50003pt\hbox{$\textstyle\int$}\hfil}\hss}\hbox{$\vbox{\hrule
width=4.0pt}$}\kern 1.00006pt}}{\hbox{\hbox to0.0pt{\hbox
to5.83331pt{\kern-0.50003pt\hbox{$\textstyle\int$}\hfil}\hss}\hbox{$\vbox{\hrule
width=4.0pt}$}\kern 1.00006pt}}}\nolimits
TD|D|^{-3}=\mathop{\mathchoice{\hbox{\hbox to0.0pt{\hbox
to5.83331pt{\kern-1.00006pt\hbox{$\displaystyle\int$}\hfil}\hss}\hbox{$-$}\kern
1.00006pt}}{\hbox{\hbox to0.0pt{\hbox
to5.83331pt{\kern-0.50003pt\hbox{$\textstyle\int$}\hfil}\hss}\hbox{$\vbox{\hrule
width=4.0pt}$}\kern 1.00006pt}}{\hbox{\hbox to0.0pt{\hbox
to5.83331pt{\kern-0.50003pt\hbox{$\textstyle\int$}\hfil}\hss}\hbox{$\vbox{\hrule
width=4.0pt}$}\kern 1.00006pt}}{\hbox{\hbox to0.0pt{\hbox
to5.83331pt{\kern-0.50003pt\hbox{$\textstyle\int$}\hfil}\hss}\hbox{$\vbox{\hrule
width=4.0pt}$}\kern 1.00006pt}}}\nolimits
T(P^{\mathord{\uparrow}}-P^{\mathord{\downarrow}})|D|^{-3}=\bigl{(}\tau_{1}\otimes(\tau_{0}^{\mathord{\uparrow}}-\tau_{0}^{\mathord{\downarrow}})+(\tau_{0}^{\mathord{\downarrow}}-\tau_{0}^{\mathord{\uparrow}})\otimes\tau_{1}\bigr{)}\bigl{(}\beta(T)\bigr{)},$

where we used (4.10). Then, from the definition of traces we see that:

$\tau_{0}^{\mathord{\uparrow}}(x)-\tau_{0}^{\mathord{\downarrow}}(x)=-\tau_{1}(x),$

and therefore

$\mathop{\mathchoice{\hbox{\hbox to0.0pt{\hbox
to5.83331pt{\kern-1.00006pt\hbox{$\displaystyle\int$}\hfil}\hss}\hbox{$-$}\kern
1.00006pt}}{\hbox{\hbox to0.0pt{\hbox
to5.83331pt{\kern-0.50003pt\hbox{$\textstyle\int$}\hfil}\hss}\hbox{$\vbox{\hrule
width=4.0pt}$}\kern 1.00006pt}}{\hbox{\hbox to0.0pt{\hbox
to5.83331pt{\kern-0.50003pt\hbox{$\textstyle\int$}\hfil}\hss}\hbox{$\vbox{\hrule
width=4.0pt}$}\kern 1.00006pt}}{\hbox{\hbox to0.0pt{\hbox
to5.83331pt{\kern-0.50003pt\hbox{$\textstyle\int$}\hfil}\hss}\hbox{$\vbox{\hrule
width=4.0pt}$}\kern 1.00006pt}}}\nolimits
TD|D|^{-3}=\bigl{(}\tau_{1}\otimes(-\tau_{1})+(\tau_{1})\otimes\tau_{1}\bigr{)}\bigl{(}\beta(T)\bigr{)}=0.\qquad\qquad\Box$

##  5\. Final remarks

We introduced a tangible notion of torsion for finite summable, regular
spectral triples with a generalised noncommutative trace via a spectral
trilinear functional of Dirac one-forms. It allows a direct verification
whether the spectral triple (or the Dirac operator) is torsion-free. Four of
the discussed examples are indeed torsion-free except the simplest case of an
almost-commutative geometry. We conjecture that the spectral triples over
almost-commutative geometries with non-simple internal algebra, nontrivially
coupled by $D$, have a non-vanishing torsion. We hope that in many other
cases, the spectral torsion functional will enable a more advanced study of
the families of Dirac operators.

Acknowledgements: LD acknowledges that this work is part of the project Graph
Algebras partially supported by EU grant HORIZON-MSCA-SE-2021 Project
101086394.

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