Spectral Torsion

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 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.


I
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.

D
Let us assume that is a closed spin manifold of dimension , with the metric tensor .In terms of a (local) basis of orthonormal vector fields on the covariant derivative can be written as The requirement that ∇ is metric, that is for any vector fields , , , allows to determine the connection explicitly through the formula, where is the torsion tensor, Therefore, the spin connection coefficients can be split into the Levi-Civita connection, and the contorsion part, .The first can be expressed explicitly in the basis through the structure constants of the commutators of orthonormal vector fields, and the remaining part of the connection, the contorsion tensor, is related to the torsion of the connection, where ( , ) = .The lift of the covariant derivative to Dirac spinor fields is, with 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) and is unique for a given metric and contorsion tensor.The canonical Dirac operator, , is the one for vanishing contorsion and we call it torsion-free Dirac operator.The torsion tensor can be expressed using the contorsion, = − .
Note the antisymmetry = − while = − .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 is, where is the standard Dirac operator, which originates from the Levi-Civita connection.Note that since [ , ] = [ , ] for any function ∈ ∞ ( ), the algebra of differential forms remains the same.We will denote ˆ the Clifford multiplication by the one-form .Let 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 given by 2.5 the trilinear functional of differential one-forms , , As our first main result we state,

Theorem 2.2. The torsion functional for the Dirac operator on a Riemannian closed spin manifold of dimension is,
∫ .

Corollary 2.3. The Dirac operator is torsion free iff the torsion functional vanishes:
To prove the theorem we employ the calculus of pseudodifferential operators and expansion of the symbols in normal coordinates.

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 (x = 0) on the manifold, (2.8) where and Ric are the components of the Riemann and Ricci tensors, respectively, at the point with x = 0 and we use the notation (x k ) to denote that we expand a function up to the polynomial of order in the normal coordinates.
As a next step, we compute the symbols of and its inverses as pseudodifferential operators (see Appendix).We start with the symbol of expanded in normal coordinates up to (1): (2.9) A Clifford multiplication by a differential form can be expanded in local normal coordinates around a point on a manifold, as: .
We can now proceed with the proof of the theorem.
Proof of Theorem .First of all, observe that since each one-form is a zero-differential operator then effectively we need to compute the symbol of order − of | | − .We start with the even dimension case = 2 , postponing the case of odd dimensions till later.To compute the two leading symbols of −2 , denoted by 2 + 2 +1 we need the two leading symbols of .
Then, we proceed with the calculation of the Wodzicki residue. (2.13) ∫ .
This ends the proof for even .
Then the rest of the proof is the same as in the case of even dimensions.
Remark 2.4.Note that the lowest odd dimension in which there can be a nonvanishing torsion is = 3.In the lowest possible even dimension = 4 the torsion functional can also be written as a functional over a single one-form:

N
The spectral definition of torsion, Def.2.1, can be readily extended to the noncommutative case of spectral triples.Let (A, , H) be a -summable unital spectral triple, Ω 1 (A) be the A bimodule of one forms generated by A and [ , A], which by definition consists of bounded operators on H.We assume that there exists a generalised algebra of pseudodifferential operators, which contains A, , | | ℓ for ℓ ∈ Z, and there exists a tracial state 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 | | − for any > (c.f.[7]) and a zero-order operator (i.e., an operator in the algebra generated by A and Ω 1 (A)).We introduce the following definition of torsion functional for spectral triples.We say that the spectral triple is torsion-free (or, for simplicity, the Dirac operator is torsionfree) if 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 (from the algebra generated by A, and Ω 1 (A) ) the following holds:

E
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 = + , where = * * , with * 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 of even dimension = 2 and a spectral triple Here, the algebra (C) acts by left multiplications on (C) regarded as a Hilbert space with the scalar product , ′ = Tr( * ′ ) .Also, is a fluctuation of ⊗ id , where is the standard Dirac operator on , and = ⊗ * , with being the charge conjugation on spinors in .Note that equals , where = − * ∈ ∞ ( , (C)) acts on (C) by left multiplication, while −1 equals composed with the right multiplication by * .Therefore, the fluctuation + −1 amounts to the action of Ad .It follows that the center of this algebra, ∞ ( , C1 ), acts trivially, so we just can consider such that , 1 = 0, that is, traceless matrix valued functions, ∈ ∞ ( , ( )).Moreover, we need only the expansion of and so of the operator Ad in normal coordinates up to (1).It is known that this spectral triple is finitely summable and regular.To compute the torsion functional for we start (slightly abusing the notation) with its symbol expanded as: It is not difficult to see that the bimodule Ω 1 spanned by [ , ], , ∈ ∞ ( , (C)), is also spanned just by the elements = , where we can expand ∈ (C) + (1).We now compute the symbols of ( ) = 1 + 0 for , , ∈ Ω 1 : Next, to compute the two leading symbols of −2 , ( −2 ) = 2 + 2 +1 , we need the two leading symbols of 2 , ( 2 ) = 2 + 1 + . .., and the two leading symbols of −2 , ( −2 ) = 2 + 3 + . .., Then, using [11,Lemma A.1] we have: Therefore, since 2 = 0, where both terms on the right hand side contain linearly ad .It turns out however, that the trace of ad , ∈ (C) , considered as a linear operator on (C) vanishes.This could be checked explicitly, using the basis , , = 1, . . ., of (C), where the matrix has the , matrix entry 1 and otherwise 0. We compute: where as a matrix is, ; , = , , , − , , , . Thus, Therefore, the spectral torsion functional T( , , ) for vanishes and so is torsion free.

Almost commutative M×Z 2 .
We assume that is a closed spin manifold and ( ∞ ( ), , H) is an even spectral triple of dimension with the standard Dirac operator and a grading .We know that it is spectrally closed, so, in particular, is torsion-free.We consider the usual double-sheet spectral triple, ( ∞ ( ) ⊗ C 2 , D, H ⊗ C 2 ), with the Dirac operator, where Φ ∈ C. The bimodule of one forms, associated to D consists of the following operators, (4.6) where ± ∈ Ω 1 ( ) and ± ∈ ∞ ( ).We shall denote the diagonal part as and the off-diagonal part as .
In order to calculate W( 1 2 3 DD −2 ) for one-forms 1 , 2 , 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 D −2 = −2 1(1 − 2 |Φ| 2 −2 + . . . ) we note that in the Wodzicki residue formula D −2 can be replaced by −2 .
By a straightforward calculation we get, (4.7) where g is the metric functional on and the functional V( ) is proportional to ∫ .Note that the first equation is a consequence of being torsion-free, whereas the third equation holds if W( −2 ) = 0, which is the consequence of the spectral triple on being spectrally closed.
As a consequence, we see that the product of the standard spectral triple over with a finite discrete one does not satisfy the torsion-free condition unless Φ = 0. Note that the generalisation of the model with Φ being a complex-valued function on does not change the conclusion.4.4.Conformally rescaled noncommutative tori.Consider the spectral triple on the noncommutative -torus, (A, H, ), where A = ∞ (T ) is the -smooth subalgebra for the standard derivations , = 1, . . ., .Next, H = 2 (T , ), where is the standard trace which anihilates .Also, = is the conformal rescaling of the standard (flat) Dirac operator = with being the usual gamma matrices and the conformal factor > 0 is from the copy A of Ain the commutant of A. The bimodule of one-forms, generated by the commutators [ , a], a ∈ A, is a free left module generated by 2 .Using the extension of the calculus of pseudodifferential operators over noncommutative tori, following [8], to Â-valued symbols (where Â is the algebra generated by A and A ) 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 ∞ (T ), H, for strictly irrational noncommutative torus of dimension is spectrally closed, that is for any , operator of order 0 (that is generated by Proof.The proof is based on a simple observation that the symbol of order − of | | − is a sum of terms that are of the form , , ( ) ( ) , for = 1, . . ., , some integers , and ∞ (T )-valued functions , , ( ), which are homogeoneous in of degree − .Since for strictly irrational tori the trace over the full algebra AA factorizes, then it is sufficient to show that ( ( ) ) = 0 for any , , .For that we write = ℎ and observe that vanishes since for + ≠ −1 it equals 1 + + 1 ( ( + +1)ℎ ) while for + = −1 it equals (ℎ)) .This ends the proof.As an immediate consequence of spectral closedness, we have: Corollary 4.2.The Dirac operator over the -dimensional strictly irrational noncommutative torus T is torsion free.

Quantum
(2).We briefly recall the construction of the spectral triple A( ), H, of [9].Let A = A( (2)) be the * -algebra generated by and , subject to the commutation rules: For the explicit formulas defining the representation of A on H we refer to [9] or [10].The equivariant Dirac operator is chosen to be isospectral to the classical Dirac over (4.9) 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 ∈ Ψ 0 (A) is in the algebra of pseudodifferential operators of order 0 (as defined in [10]) then for ↑ , ↓ , projections on the respective up and down part of the Hilbert space of spinors, the noncommutative integral of (↑,↓) | | −2 could be explicitly computed as: where the noncommutative ingeral is, as usually, The map 0 is a projection on the two first legs of the zero-degree part of a map , which is a homomorphism Using the above theorem we can state as a simple corollary, Corollary 4.4.The standard equivariant spectral triple over A( ) is spectrally closed and, in particular, the Dirac operator (4.9) is torsion-free.
Proof.We compute, where we used (4.10).Then, from the definition of traces we see that:

F
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 , 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.

Lemma 3 . 3 ([ 11 ,
Lemma 5.6]).The classical spectral triple over a closed spin-c manifold of dimension = 2 is spectrally closed in the above sense and hence is torsion-free.