1 sec:Main32results↩︎

Let \((M,g)\) be a connected closed Riemannian manifold of dimension \(d\ge2\) and let \(\Omega^k(M)\) be the Hilbert space of real-valued \(k\)-forms, \(1\le k\le d-1\). Hodge’s Theorem [1] tells us that \(\Omega^k(M)\) decomposes into a direct sum of three orthogonal closed subspaces \[\label{Hodge32decompostion} \Omega^k(M) = \mathrm{d}\Omega^{k-1}(M) \oplus \delta\Omega^{k+1}(M) \oplus \mathcal{H}^k(M),\tag{1}\] where \(\mathrm{d}\Omega^{k-1}(M)\), \(\delta\Omega^{k+1}(M)\) and \(\mathcal{H}^k(M)\) are the Hilbert subspaces of exact, coexact and harmonic \(k\)-forms, respectively. The overarching idea of our paper is that in the special case when \(d=3\), \(k=1\) and \(M\) is oriented the space \(\delta\Omega^{k+1}(M)\) admits a further decomposition into a distinguished pair of orthogonal Hilbert subspaces \(\delta\Omega_\pm^{k+1}(M)\). Moreover, this further decomposition is effectively described in terms of pseudodifferential projections for which the symbols can be written down explicitly in terms of curvature and its covariant derivatives. Remarkably, this leads to a new approach to the study of spectral asymmetry.

The study of spectral asymmetry, that is, the difference in the distribution of positive and negative eigenvalues of (pseudo)differential operators, has a long and noble history, initiated by the seminal series of papers by Atiyah, Patodi and Singer [2]–[5]. In a nutshell, the classical approach goes as follows. One considers the eta function of the operator at hand — say, curl or Dirac — defined as \[\label{eta32function32intro} \eta(s):=\sum_{\lambda_k\ne 0}\operatorname{sgn}(\lambda_k)\, |\lambda_k|^{-s}, \qquad s\in \mathbb{C},\tag{2}\] where \(\lambda_k\) are the nonzero eigenvalues of the operator. The complex function \(\eta(s)\) can be easily shown to be holomorphic for \(\operatorname{Re} s>d\); one then tries to give a meaning to the quantity \(\eta(0)\), called eta invariant, by examining the meromorphic extension of 2 for \(\operatorname{Re}s<d\) and showing that there is no pole at \(s=0\). The motivation for considering \(\eta(0)\) as a measure of spectral asymmetry is that, when the operator in question is simply a Hermitian matrix, \(\eta(0)\) is precisely the number of positive eigenvalues minus the number of negative eigenvalues. Classically, it has been shown that \(\eta(0)\) is a geometric invariant which can be computed resorting to complex analysis and algebraic topology.

Perhaps surprisingly, there is almost no literature on the spectrum of curl on a 3-manifold with or without boundary. Indeed, if one looks carefully, the overwhelming majority of existing papers deal, effectively, with \(\operatorname{curl}^2\) as opposed to \(\operatorname{curl}\) itself. Many authors [6]–[11] have studied Maxwell’s equations on domains in \(\mathbb{R}^3\) subject to appropriate boundary conditions, which leads to the analysis of spectral problems of the form \[\label{Maxwell32resonator} \begin{pmatrix} 0&i\operatorname{curl}\\ -i\operatorname{curl}&0 \end{pmatrix} \begin{pmatrix} E\\ B \end{pmatrix} = \lambda \begin{pmatrix} E\\ B \end{pmatrix}.\tag{3}\] One would think that the spectral problem 3 reduces to the spectral problem for \(\operatorname{curl}\) by means of a unitary transformation \[\label{unitary32transformation} \begin{pmatrix} E\\ B \end{pmatrix} \mapsto \begin{pmatrix} v_+\\ v_- \end{pmatrix} := \frac{1}{\sqrt{2}} \begin{pmatrix} -i&1\\ i&1 \end{pmatrix} \begin{pmatrix} E\\ B \end{pmatrix}\tag{4}\] but this argument fails because physically meaningful boundary conditions do not agree with 4 . The presence or absence of a boundary is a major issue in the subject.

To the best of our knowledge, one of the few papers dedicated to the study of the spectrum of \(\operatorname{curl}\) on a closed oriented Riemannian manifold — at least, in the way it is understood in the current paper — is [12]. In [12] the author examined the qualitative properties of the spectrum of \(\operatorname{curl}\) depending on the dimension \(d\) and established one-term asymptotic formulae (Weyl law) for the (global) eigenvalue counting functions, alongside a number of explicit examples. Although not entirely aligned in spirit with the aims of the current paper, we refer the reader to [13]–[21] and references therein, examining the spectrum of \(\operatorname{curl}\) (or \(\operatorname{curl}^2\)) from different perspectives.

Let us now formulate our problem and state our main results, postponing proofs and detailed arguments until later sections. In the rest of this paper, unless otherwise stated, the dimension \(d\) is 3 and the manifold is orientable and oriented. Let \(*\) be the Hodge dual (see Appendix 9 for our sign convention), and define \(\operatorname{curl}\) to be the differential expression \[\label{curl32differential32experssion} \operatorname{curl}:=\ast \mathrm{d}\tag{5}\] acting in \(\Omega^1(M)\).

We will show in Section 2 — more precisely, in Theorem 7 — that the differential expression 5 gives rise to a self-adjoint operator (denoted by the same symbol) in the space of coexact 1-forms \(\delta\Omega^2(M)\). The latter has discrete spectrum, accumulating to both \(+\infty\) and \(-\infty\), and not necessarily symmetric about zero. As a way of illustrating that spectral asymmetry actually occurs, in Appendices 11 and 12 we write down explicitly the spectrum of \(\operatorname{curl}\) on a Berger sphere and compute the corresponding (classical) eta invariant.

Note that the definition of \(\operatorname{curl}\) does not involve the concept of connection. In this respect, \(\operatorname{curl}\) is one of the most fundamental operators of mathematical physics, like the Laplace–Beltrami operator. Furthermore, \(\operatorname{curl}\) lies at the heart of (homogeneous) Maxwell’s equations. Namely, seeking solutions harmonic in time reduces Maxwell’s equations on \(M\times\mathbb{R}\) to the spectral problem for \(\operatorname{curl}\). And the solution of the Cauchy problem for Maxwell’s equations can be expressed in terms of eigenvalues and eigenforms of \(\operatorname{curl}\), see formula 275 . Finally, it is worth emphasising that the sign of the eigenvalues of curl has a physical meaning: it coincides with the sign of electromagnetic chirality of time-harmonic polarised solutions of Maxwell’s equations generated by the corresponding \(\operatorname{curl}\) eigenpair (see 272 , 273 ). We refer the reader to Appendix 13 for further details and a self-contained exposition of this fact. These are but few of the reasons why a more detailed examination of the spectrum of curl — and in particular of its spectral asymmetry — is very timely, if not overdue.

As a first step, we introduce the following definitions.

The operator \(P_0\) is the orthogonal projection onto the kernel of \(\operatorname{curl}\) with harmonic 1-forms removed, whereas the operators \(P_\pm\) are the positive (+) and negative (-) spectral projections of \(\operatorname{curl}\).

Further on we use the notation \[\label{norm32of32xi} \|\xi\| :=\sqrt{g^{\mu\nu}(x)\,\xi_\mu\xi_\nu}\tag{7}\] and we denote by \(|\,\cdot\,|\) the Euclidean norm of vectors. Throughout this paper we use Greek letters for tensor indices.

Euclidean versions of formulae 8 and 9 appeared in [22]. On the other hand, the notion of subprincipal symbol for operators acting on 1-forms was never previously defined in full generality. We address this matter in detail, including a discussion of known results, in Section 3.

Remarkably, the pseudodifferential projections \(P_\pm\) can be constructed explicitly. In Proposition 19 we will provide an algorithm for the calculation of their full symbols. The latter proposition and Theorem 1 give a detailed description of the structure of the projection operators \(P_0\) and \(P_\pm\), which is our first main result.

The projections \(P_+\) and \(P_-\) are the key ingredients of our new approach to spectral asymmetry. One starts by observing that, at a formal level, we have \[\label{at32a32formal32level} \#\{\text{positive eigenvalues}\} \,-\, \#\{\text{negative eigenvalues}\} =\operatorname{Tr}(P_+-P_-).\tag{12}\] Here and further on \(\operatorname{Tr}\) stands for operator trace, whereas \(\operatorname{tr}\) stands for matrix trace. Unfortunately, one is immediately presented with two major issues: (i) the LHS of 12 is a difference of two infinities and (ii) the (matrix) operator \(P_+-P_-\) in the RHS of 12 is not of trace class. Our strategy is to assign a meaning to the RHS of 12 by decomposing the procedure of taking operator trace into two steps:

1) take the pointwise matrix trace of the matrix pseudodifferential operator \(P_+-P_-\,\),

2) compute the operator trace of the resulting scalar pseudodifferential operator.

Step 1 above defines a scalar self-adjoint pseudodifferential operator which we call the asymmetry operator and denote by \(A\) — see Definition 6. This operator is defined uniquely, up to the addition of an integral operator with infinitely smooth kernel vanishing in a neighbourhood of the diagonal \(M\times M\), and it is determined by the Riemannian 3-manifold \((M,g)\) and its orientation.

The operator \(A\) is prima facie a pseudodifferential operator of order 0. However, taking the matrix trace brings about unexpected cancellations, so that the operator \(A\) turns out, in fact, to be of order \(-3\). This is our second main result.

A sufficient condition for a self-adjoint pseudodifferential operator to be of trace class and to have continuous integral kernel is for its order to be strictly less than \(-d\), where \(d\) is the dimension of the manifold. See also [23]. In the case at hand, the operator \(A\) has order \(-3\) and the dimension of the manifold is 3, so we are looking at a borderline situation. One would hope that with a mild regularisation one could cross the finish line and be able to define a notion of regularised trace.

Indeed, we will establish in Theorem 28 that the integral kernel \(\mathfrak{a}(x,y)\) of \(A\) decomposes as \(\mathfrak{a}(x,y)=\mathfrak{a}_d(x,y)+\mathfrak{a}_c(x,y)\), where \(\mathfrak{a}_c\) is continuous, whereas \(\mathfrak{a}_d\) is possibly discontinuous but bounded, and integrates to zero over small spheres centred at either \(x\) or \(y\).

This leads us to our third main result.

We call the function \(\psi_{\operatorname{curl}}^\mathrm{loc}(x)\) and the number \(\,\psi_{\operatorname{curl}}:=\int_M\psi_{\operatorname{curl}}^\mathrm{loc}(x) \,\rho(x)\,\mathrm{d}x\,\) the local and global regularised trace of \(A\), see Definitions 7 and 8. These two quantities are geometric invariants, in that they are determined by the Riemannian 3-manifold and its orientation. In particular, the global regularised trace is a real number measuring the asymmetry of the spectrum of \(\operatorname{curl}\).

The natural question is how are our geometric invariants \(\psi_{\operatorname{curl}}(x)\) and \(\psi_{\operatorname{curl}}\) related to classical eta invariants. We comprehensively address this matter in our companion paper [24] where we prove that our \(\psi_{\operatorname{curl}}(x)\) and \(\psi_{\operatorname{curl}}\) are precisely the classical local and global eta invariants for \(\operatorname{curl}\).

All in all, our results offer a new approach to the subject of spectral asymmetry, one that possesses the following elements of novelty.

1. We characterise asymmetry in terms of a pseudodifferential operator of negative order, as opposed to a single number. Here the fact that the order is negative is key as it opens the way to defining the regularised trace.

2. Our approach is not operator-specific. It is versatile and can be deployed in a variety of different scenarios. For example, our strategy can be applied to the Dirac operator⁴, as we will show in a separate paper.

3. Our technique is explicit and based on direct computations. Moreover, it rigorously implements the intuitive notion of spectral asymmetry as difference between numbers of positive and negative eigenvalues, without relying on “black boxes” such as analytic continuation from the very beginning, or using heat-kernel-type arguments (see, e.g., [28]).

There are, of course, many different approaches to the subject of spectral asymmetry. A notable one, for instance, relies on a local invariant known as Wodzicki residue [29], see also [30], [31] for a broader discussion. Since the literature on spectral asymmetry is vast, we decided to refrain from including a detailed literature review here, as it would inevitably be incomplete and would deserve a separate paper (if not a monograph).

Structure of the paper↩︎

Our paper is structured as follows.

In Section 2 we put \(\operatorname{curl}\) on a rigorous operator-theoretic footing, and state its basic properties. Although most of the results from this section are known at least in some form, detailed proofs, often hard to find in the literature, are provided in Appendix 10 in full and in a self-contained fashion, for the convenience of the reader and future reference. In doing so, we shall rely on the auxiliary operator \(\operatorname{curl}_E\), called extended curl, which can be viewed as an elliptic “extension” of \(\operatorname{curl}\).

In Section 3 we temporarily move away from dimension 3 and develop an invariant calculus for pseudodifferential operators acting on 1-forms over a Riemannian closed \(d\)-manifold. In particular we define a notion of subprincipal symbol for these operators, and write down the composition formula.

In Section 4 we prepare the ground for the formulation of our main results and examine the issue of taking the trace of pseudodifferential operators on 1-forms. The highlight of this section is the notion of matrix trace \(\operatorname{\mathfrak{tr}}\,\), defined by 78 and 79 , which addresses the issue of taking the pointwise matrix trace of integral operators whose integral kernel is a two-point tensor.

In Section 5 we study the pseudodifferential projections \(P_0\), \(P_+\) and \(P_-\), and provide an explicit algorithm for the construction of their full symbols. This section contains the proof of Theorem 1.

In Section 6 we define the asymmetry operator \(A\) (subsection 6.1), prove that it is a pseudodifferential operator of order \(-3\) (subsection 6.2), and compute its principal symbol \(A_\mathrm{prin}\) (subsection 6.3). This section establishes Theorem 2.

Section 7 is the centrepiece of our paper. After examining the singular structure of the integral kernel \(\mathfrak{a}(x,y)\) of \(A\) near the diagonal (subsection 7.1), we devise a regularisation procedure which allows one to define the local and global trace of the asymmetry operator, geometric invariants measuring spectral asymmetry (subsection 7.2). This yields Theorem 4.

In Section 8 we briefly elaborate on the challenges involved in extending our results to dimensions higher than 3, and why doing so could be interesting.

The paper is complemented by a list of notation given at the end of this section, and by seven appendices. In Appendix 9 we specify our notation and conventions on exterior calculus. In Appendix 10 we provide proofs for the basic properties of the operator curl, summarised in Section 2. Appendices 11 and 12 are concerned with spectral theory on Berger 3-spheres: in Appendix 11 we recall the definition of a Berger sphere and list the eigenvalues of the Laplace–Beltrami operator, whereas in Appendix 12 we list the eigenvalues of \(\operatorname{curl}\) on a Berger sphere as an illustration of spectral asymmetry, and compute the eta invariant by explicitly examining the eta function for curl. In Appendix 13 we examine the relation between the spectral problem for curl and solutions of Maxwell’s equations, and discuss the physical meaning of the sign of the eigenvalues of \(\operatorname{curl}\). In Appendix 14 we derive series expansions for the parallel transport maps. Finally, in Appendix 15 we independently verify formula 13 .

Notation↩︎

2 sec:The32operator32curl↩︎

In this section we put \(\operatorname{curl}\) on a rigorous mathematical foundation, defining it within the framework of the theory of unbounded self-adjoint operators in Hilbert spaces. This would provide the reader with an account of the fundamental properties of \(\operatorname{curl}\), in a self-contained fashion, postponing proofs until Appendix 10.

The main challenge that one faces when dealing with \(\operatorname{curl}\) is that it is not elliptic⁵. Indeed, the formula for the principal symbol of \(\operatorname{curl}\) (which happens to coincide with its full symbol) reads \[\label{principal32symbol32curl} [\operatorname{curl}_\mathrm{prin}]_\alpha{}^\beta(x,\xi)= -i\,E_\alpha{}^{\beta\gamma}(x)\,\xi_\gamma\,,\tag{15}\] where the tensor \(E\) is defined in accordance with 10 . Although \(\xi\) is not a 1-form in its own right, at a formal level the RHS of 15 can be rewritten, concisely, as \(-i*\xi\). A straightforward calculation shows that \[\label{determinant32principal32symbol32curl} \det (\operatorname{curl}_\mathrm{prin})=0.\tag{16}\] The eigenvalues of \(\operatorname{curl}_\mathrm{prin}\) are simple and read \[\label{eigenvalues32principal32symbol32curl} h^{(0)}(x,\xi)=0, \qquad h^{(\pm)}(x,\xi)=\pm\|\xi\|\,,\tag{17}\] for all \((x,\xi)\in T^*M\setminus \{0\}\). Consequently, standard elliptic theory does not apply and care is required in defining the appropriate Hilbert space, domain of the operator and establishing the mapping properties of \(\operatorname{curl}\).

Let \(H^s(M)\), \(s>0\), be the space of differential forms that are square integrable together with their partial derivatives up to order \(s\). We do not carry in our notation for Sobolev spaces the degree of differential forms: this will be clear from the context. Henceforth, to further simplify notation we drop the \(M\) and write \(\Omega^k\) for \(\Omega^k(M)\) and \(H^s\) for \(H^s(M)\).

To begin with, let us observe that \(\operatorname{curl}\) maps smooth coexact 1-forms to smooth coexact 1-forms. Therefore, it is natural to define \(\operatorname{curl}\) as an operator \[\label{definition32curl} \operatorname{curl}=*\mathrm{d}: \delta\Omega^2 \cap H^1\to \delta\Omega^2,\tag{18}\] where \(\delta\Omega^2\) is the Hilbert space of real-valued coexact 1-forms with inner product \[\label{inner32product} \langle u,v \rangle:=\int_M *u \wedge v=\int_M u \wedge * v.\tag{19}\]

The basic properties of the operator 18 are summarised by the following theorem.

We would like to emphasise the significance of property (c) in the above theorem. If we start deforming the metric the eigenvalues of \(\operatorname{curl}\), understood as those of a self-adjoint operator 18 , will never turn to zero or cross zero (change sign). This is in stark contrast with the Dirac operator where zero may be an eigenvalue, depending on the choice of metric [32]. In fact, the study of zero modes of the Dirac operator (harmonic spinors) is an active area of research.

3 sec:Pseudodifferential32operators32acting32on321-forms↩︎

In this section we develop an invariant calculus for pseudodifferential operators acting on 1-forms over a closed Riemannian manifold \((M,g)\) of arbitrary dimension \(d\).

Let \(Q\) be a classical polyhomogeneous pseudodifferential operator of order \(s\) acting on 1-forms, and let \[\label{1132June32202132equation321} Q: u_\alpha(x) \mapsto v_\alpha(x) =(2\pi)^{-d} \int e^{i(x-y)^\gamma\xi_\gamma}\,q_\alpha{}^\beta(x,\xi)\, u_\beta(y)\,\mathrm{d}y\,\mathrm{d}\xi\,,\tag{20}\] \[\label{1132June32202132equation322} q_\alpha{}^\beta(x,\xi) \sim [q_s]_\alpha{}^\beta(x,\xi) + [q_{s-1}]_\alpha{}^\beta(x,\xi) +\dots\tag{21}\] be its representation in local coordinates (the same for \(x\) and \(y\)). Here \(\sim\) stands for asymptotic expansion [23], and ‘polyhomogeneous’ means that \([q_{s-k}]_\alpha{}^\beta(x, \lambda \,\xi)=\lambda^{s-k}\,[q_{s-k}]_\alpha{}^\beta(x, \xi)\) for any positive \(\lambda\) and \(k=0,1,2,\dots\).

Throughout the paper we denote pseudodifferential operators with upper case letters, and their symbols and homogeneous components of symbols — with lower case. For the principal and subprincipal symbol we use upper case letters and denote them by \(Q_\mathrm{prin}\) and \(Q_\mathrm{sub}\) respectively, with the subscript indicating that we are looking at invariant (or covariant) objects ‘living’ on the cotangent bundle as opposed to the operator \(Q\) itself which ‘lives’ (acts) on the base manifold.

Later on we will sometimes write a pseudodifferential operator as an integral operator with distributional integral kernel (Schwartz kernel), see, for example, formula 69 . We will use lower case Fraktur font for the Schwartz kernel.

Returning to the pseudodifferential operator 20 , 21 , it is easy to see that \[\label{formal32definition32of32principal32symbol} [Q_\mathrm{prin}]_\alpha{}^\beta(x,\xi):=[q_s]_\alpha{}^\beta(x,\xi)\tag{22}\] provides an invariant definition for the principal symbol \(Q_\mathrm{prin}\) of \(Q\).

However, the subleading term, \([q_{s-1}]_\alpha{}^\beta\,\), is not invariant under change of coordinates. The task at hand is to define and provide a formula for the subprincipal symbol of \(Q\), by determining appropriate correction terms to \([q_{s-1}]_\alpha{}^\beta\).

The original definition of subprincipal symbol goes back to J.J. Duistermaat and L. Hörmander [33]. They defined the subprincipal symbol for operators acting on half-densities.

Duistermaat and Hörmander’s definition of subprincipal symbol extends to operators acting on scalar fields because scalars can be identified with half-densities: it is just a matter of multiplying or dividing by \(\sqrt\rho\,\). This leads to the appearance of an additional correction term. Namely, given a pseudodifferential operator \(Q\) of order \(s\) \[\label{1132June32202132equation32132scalar} Q: f(x) \mapsto (2\pi)^{-d} \int e^{i(x-y)^\gamma\xi_\gamma}\,q(x,\xi)\, f(y)\,\mathrm{d}y\,\mathrm{d}\xi\,,\tag{23}\] \[\label{1132June32202132equation32232scalar} q(x,\xi) \sim q_s(x,\xi) + q_{s-1}(x,\xi) +\dots\tag{24}\] acting on scalar fields, its subprincipal symbol reads \[\label{subprincipal32symbol32of32operator32acting32on32a32scalar32field} Q_\mathrm{sub} := q_{s-1} + \frac{i}{2}\dfrac{\partial^2 q_s}{\partial x^\gamma\partial\xi_\gamma} + \frac{i}{2}\, \frac{\partial\ln\rho}{\partial x^\gamma}\, \frac{\partial q_s}{\partial\xi_\gamma}\,,\tag{25}\] where \(\rho\) is the Riemannian density. In formula 25 one can, in principle, instead of the Riemannian density use any other prescribed positive reference density. However, in this paper we always stick with the Riemannian density. In particular, this implies the identity \[\label{relation32between32Riemannian32density32and32Christoffel32symbols} \frac{\partial\ln\rho}{\partial x^\gamma} = \Gamma^{\alpha}{}_{\gamma\alpha}\,.\tag{26}\]

The definition of subprincipal symbol 25 extends further, without change, to operators acting on \(m\)-columns of scalar fields (sections of the trivial \(\mathbb{R}^m\)- or \(\mathbb{C}^m\)-bundle over \(M\)). By choosing a global framing (an orientable \(3\)-manifold is automatically parallelizable), one can then produce a coordinate-invariant reformulation of the classical construction of subprincipal symbol for operators on 1-forms. However, we should like to emphasise that the approach presented in this manuscript avoids the reliance on a specific global framing. For this reason, we believe it is still of interest and worth presenting.

Note that a notion of subprincipal symbol for operators acting on 1-forms is available in the literature in the special case when the principal symbol is of the form \(fI\), where \(f\) is a scalar function on \(T^*M\setminus\{0\}\) and \(I\) is the \(d\times d\) identity matrix, see, e.g., [34], [35].

We now proceed to the proof of Theorem 9. One way of proving the theorem is by a lengthy explicit calculation establishing 28 . However, we will present an alternative proof which provides insight into the origins of formula 27 . In order to do so we need to introduce first some geometric notions that will be useful here and later on in the paper.

By a frame at a point \(x\in M\) we mean a basis of orthonormal vectors in \(T_xM\). By a framing in an open set \(\mathcal{U}\subset M\) we mean the choice of a frame at each point \(x\) of \(\mathcal{U}\), depending smoothly on the base point \(x\).

Given a framing \(\{e_j{}^\alpha\}_{j=1}^d\), the corresponding dual32framing \(\{e^k{}_\beta\}_{k=1}^d\) is defined as \[\label{dual32framing} e^k{}_\beta(x):=\delta^{jk}\,g_{\alpha\beta}(x)\,e_j{}^\alpha(x),\tag{29}\] \(\delta\) being the Kronecker symbol. Here and further on, \(e_j{}^\alpha\) denotes the \(\alpha\)-th component of the \(j\)-th vector field and \(e^k{}_\beta\) denotes the \(\beta\)-th component of the \(k\)-th covector field.

Let us emphasise that the Levi-Civita framing is uniquely determined in the neighbourhood of every point \(z\in M\) up to a rigid (constant) orthogonal transformation, which reflects the freedom in choosing the frame \(\{V_j\}_{j=1}^d\) at the point \(z\). By definition, the Levi-Civita framing and its dual satisfy \[\label{1432June32202132equation321} \left. \left[ \partial_\beta \widetilde{e}_j{}^\alpha+\Gamma^\alpha{}_{\beta\gamma}\,\widetilde{e}_j{}^\gamma \right] \right|_{x=z} =0\,,\tag{30}\] \[\label{1432June32202132equation322} \left. \left[ \partial_\beta \widetilde{e}\,{}^j{}_\alpha-\Gamma^\gamma{}_{\beta\alpha}\,\widetilde{e}\,{}^j{}_\gamma \right] \right|_{x=z} =0,\tag{31}\] see also [36].

Let us consider the special case when \(M\) is an oriented 3-manifold. In this case we have the following result.

Returning to the analysis of the general case of a Riemannian manifold of arbitrary dimension, let us examine the second order differential operators \(\delta d\) and \(d\delta\) acting on 1-forms. Similarly to 43 , one can show that \[\label{subprincipal32symbol32of32delta32d} (\delta \mathrm{d})_\mathrm{sub}=0,\tag{46}\] \[\label{subprincipal32symbol32of32d32delta} (\mathrm{d}\delta)_\mathrm{sub}=0.\tag{47}\] Of course, formulae 46 and 47 imply that the Hodge Laplacian \(\,\boldsymbol{\Delta}:=-(\delta \mathrm{d}+\mathrm{d}\delta)\,\) acting on 1-forms has zero subprincipal symbol.

The inner product \[\label{general32inner32product32on321-forms} \langle u,v \rangle:=\int_M g^{\alpha\beta}(x)\,\overline{u_\alpha(x)}\,v_\beta(x)\,\rho(x)\,\mathrm{d}x\tag{48}\] allows us to define the formal adjoint \(Q^*\) of a pseudodifferential operator \(Q\) acting on 1-forms.

4 sec:Trace32of32pseudodifferential32operators32acting32on321-forms↩︎

The aim of this section is to extend, in a natural way, the notion of trace of a pseudodifferential operator acting on 1-forms. Given a pseudodifferential operator \(Q\) acting on 1-forms, we will define a scalar operator \(\operatorname{\mathfrak{tr}} Q\) is such a way that, if \(Q\) is of trace class, then the operator-theoretic trace of \(Q\) coincides with the operator-theoretic trace of \(\operatorname{\mathfrak{tr}} Q\). Later in the paper we will use this definition without assuming that \(Q\) is of trace class: it may happen that \(Q\) itself is not of trace class, but \(\operatorname{\mathfrak{tr}} Q\) is — which is the case, up to an additional minor regularisation, for \(Q=\theta(\operatorname{curl})-\theta(-\operatorname{curl})\).

In this section, as in the previous one, we work on a closed Riemannian manifold \((M,g)\) of arbitrary dimension \(d\).

Let us start with the simpler case when \(Q\) is a pseudodifferential operator of order \(s\) acting on \(m\)-columns of scalar fields. Suppose that \[\label{s32less32than32d} s<-d.\tag{64}\] Then [23] \(Q\) is an integral operator with continuous integral kernel, \[\label{Q32acting32on32column32of32scalars321} Q: u_j(x)\mapsto\int_M\mathfrak{q}_j{}^k(x,y)\,u_k(y)\,\rho(y)\,\mathrm{d}y\,.\tag{65}\] We introduced the factor \(\rho(y)\) in the above integral in order to make the integral kernel \(\mathfrak{q}_j{}^k(x,y)\) a (matrix-valued) scalar function on \(M\times M\). Furthermore, if \(Q\) is self-adjoint, then it is of trace class and we have \[\label{Q32acting32on32column32of32scalars322} \operatorname{Tr}Q=\int_M(\operatorname{tr}\mathfrak{q})(x,x)\,\rho(x)\,\mathrm{d}x\,,\tag{66}\] where \((\operatorname{tr}\mathfrak{q})(x,y):=\mathfrak{q}_j{}^j(x,y)\) is the trace of the matrix-function \(\mathfrak{q}_j{}^k(x,y)\). We define the scalar operator \[\label{Q32acting32on32column32of32scalars323} \operatorname{tr}Q: f(x)\mapsto\int_M (\operatorname{tr}\mathfrak{q})(x,y)\,f(y)\,\rho(y)\,\mathrm{d}y\tag{67}\] and call it the matrix trace of the operator \(Q\). It is easy to see that \[\label{Q32acting32on32column32of32scalars324} \operatorname{Tr}Q=\operatorname{Tr}(\operatorname{tr}Q).\tag{68}\]

Note that 67 is well defined for any \(s\in\mathbb{R}\), not necessarily satisfying condition 64 . More precisely, if \(Q\) is of order \(s\), it is easy to see that

(i) \(\operatorname{tr}Q\) is also an operator of order \(s\),

(ii) \((\operatorname{tr}Q)^*=\operatorname{tr}(Q^*)\), where the star refers to formal adjoints with respect to the natural inner products,

(iii) \((\operatorname{tr}Q)_\mathrm{prin}=\operatorname{tr}(Q_\mathrm{prin})\),

(iv) \((\operatorname{tr}Q)_\mathrm{sub}=\operatorname{tr}(Q_\mathrm{sub})\).

Of course, when condition 64 is not satisfied the integral kernel \(\mathfrak{q}_j{}^k(x,y)\) appearing in formula 65 should be understood in the distributional sense (Schwartz kernel), but this does not prevent us from taking its matrix trace. The distribution \((\operatorname{tr}\mathfrak{q})(x,y)\) will be the Schwartz kernel of the scalar operator 67 .

We will now adapt the above construction to the case of a pseudodifferential operator \[\label{Q32acting32on321-forms321} Q: u_\alpha(x)\mapsto\int_M \mathfrak{q}_\alpha{}^\beta(x,y)\,u_\beta(y)\,\rho(y)\,\mathrm{d}y\tag{69}\] acting on 1-forms. Here we encounter a problem: the quantity \(\mathfrak{q}_\alpha{}^\alpha(x,y)\) is not a scalar on \(M\times M\) because the tensor indices \(\alpha\) and \(\beta\) in \(q_\alpha{}^\beta(x,y)\) ‘live’ at different points, \(x\) and \(y\) respectively. More precisely, the problem here is that the Schwartz kernel \(\mathfrak{q}_\alpha{}^\beta(x,y)\) of the operator 69 is a two-point tensor: it is a covector at \(x\) in the tensor index \(\alpha\) and a vector at \(y\) in the tensor index \(\beta\). We overcome this impediment as follows.

Let \(x\) and \(y\) be two points on the manifold \(M\) which are sufficiently close. Then there is a unique shortest geodesic \(\gamma(x,y)\) connecting \(x\) and \(y\). Given a point \(z\in\gamma(x,y)\), put \[\label{Q32acting32on321-forms322} t=\frac{\operatorname{dist}(z,x)}{\operatorname{dist}(x,y)}\,,\tag{70}\] which is the variable arc length of the geodesic normalised by its total length. The variable \(t\) provides a convenient parameterisation of our geodesic \(\gamma(x,y;\,\cdot\,):[0,1]\to M\), so that \(\gamma(x,y;0)=x\) and \(\gamma(x,y;1)=y\).

We denote by \[\label{Q32acting32on321-forms323} Z: T_xM\ni u^\alpha\mapsto u^\alpha\,Z_\alpha{}^\beta(x,y)\in T_yM\tag{71}\] the linear map realising the parallel transport of vectors from \(x\) to \(y\) along the unique shortest geodesic connecting \(x\) and \(y\). Note that the result of parallel transport is independent of the choice of the particular parameterisation of the curve along which it takes place.

In what follows we will be raising and lowering indices in the 2-point tensor \(Z_\alpha{}^\beta(x,y)\) using the Riemannian metric \(g(x)\) in the first index and \(g(y)\) in the second. Of course, \[\label{Q32acting32on321-forms323a32extra} Z_\alpha{}^\kappa(x,y)\,Z_\kappa{}^\beta(y,x) \,=\, \delta_\alpha{}^\beta\,, \qquad Z^\alpha{}_\kappa(x,y)\,Z^\kappa{}_\beta(y,x) \,=\, \delta^\alpha{}_\beta\,,\tag{72}\] \[\label{Q32acting32on321-forms323a} Z_\alpha{}^\kappa(x,y)\,Z^\beta{}_\kappa(x,y) \,=\, \delta_\alpha{}^\beta\,, \qquad Z_\kappa{}^\alpha(x,y)\,Z^\kappa{}_\beta(x,y) \,=\, \delta^\alpha{}_\beta\,.\tag{73}\] Formula 72 expresses the fact that if we parallel transport a vector/covector from \(x\) to \(y\) along the shortest geodesic connecting these two points, and then parallel transport it back from \(y\) to \(x\) along the same geodesic, we get the original vector/covector. Formula 73 is a consequence of the fact that the Levi-Civita connection is metric compatible.

Let us introduce a one-parameter family of scalar distributions defined by \[\label{Q32acting32on321-forms325} Z^\alpha{}_\kappa(x,\gamma(x,y;\tau)) \, \mathfrak{q}_{\alpha}{}^\beta(x,y) \, Z_\beta{}^\kappa(y,\gamma(x,y;\tau))\,,\tag{75}\] where \(\mathfrak{q}_{\alpha}{}^\beta(x,y)\) is the Schwartz kernel from 69 and \(\tau\in[0,1]\). What happens here is that tensor indices are parallel transported to the common point \(z=\gamma(x,y;\tau)\) on the geodesic \(\gamma(x,y)\) connecting \(x\) and \(y\). Thus, the quantity 75 is a genuine scalar.

As we are dealing with distributions we need to clarify the precise meaning of formula 75 .

Consider a map from smooth scalar functions of two variables to smooth \((1,1)\) two-point tensors defined as \[\label{Q32acting32on321-forms32532bis} C^\infty(M\times M)\ni f(x,y) \mapsto Z^\alpha{}_\kappa(x,\gamma(x,y;\tau)) \, Z_\beta{}^\kappa(y,\gamma(x,y;\tau)) \, f(x,y)\,.\tag{76}\] The quantity \(\mathfrak{q}_{\alpha}{}^\beta(x,y)\) in 75 is a distribution (continuous complex-valued linear functional) acting on smooth \((1,1)\) two-point tensors. Formula 75 should be understood as a composition of \(\mathfrak{q}\) and 76 .

Since the definition 75 is independent of the choice of \(\tau\), henceforth we will set, for convenience, \(\tau=0\), in which case formula 75 simplifies and reads \[\label{Q32acting32on321-forms32532with32tau32zero} \mathfrak{q}_{\alpha}{}^\beta(x,y) \, Z_\beta{}^\alpha(y,x)\,.\tag{77}\]

There is still one problem with formula 77 : the linear operator \(Z\) appearing in this formula is defined only for \(x\) and \(y\) sufficiently close. In order to view 77 as a well-defined distribution we need a cut-off. Let \(\chi:[0,+\infty)\to \mathbb{R}\) be a compactly supported infinitely smooth scalar function such that \(\chi=1\) in a neighbourhood of zero. We modify formula 75 to read \[\label{Q32acting32on321-forms326} \mathfrak{q}_{\alpha}{}^\beta(x,y) \, Z_\beta{}^\alpha(y,x) \,\chi(\operatorname{dist}(x,y)/\epsilon)\,,\tag{78}\] where \(\epsilon>0\) is a small parameter which ensures that the quantity 78 vanishes when \(x\) and \(y\) are not sufficiently close.

Thus, given a pseudodifferential operator \(Q\) acting on 1-forms we defined the scalar pseudodifferential operator \(\operatorname{\mathfrak{tr}} Q\,\). The latter depends on the small parameter \(\epsilon>0\) and the cut-off \(\chi\), but, as we will see, the choice of this parameter and the cut-off does not affect the main results of our paper. Let us emphasise that the matrix trace 79 is defined uniquely, modulo the addition of a scalar integral operator whose integral kernel is infinitely smooth and vanishes in a neighbourhood of the diagonal.

Proposition 15 part (b) immediately implies

5 The projection operators \(P_0\) and \(P_\pm\)↩︎

This section is devoted to the study of the operators \(P_0\) and \(P_\pm\). We will show that they are pseudodifferential operators whose full symbols can be constructed via an explicit algorithm. Furthermore, we will compute their principal and subprincipal symbols, thus proving Theorem 1.

By \(\Psi^s\) we denote the space of classical pseudodifferential operators of order \(s\) with polyhomogeneous symbols acting on 1-forms, recall 20 , 21 . We define \[\label{Psi32minus32infinity} \Psi^{-\infty}:=\bigcap_s \Psi^s\tag{88}\] and we write \(Q=R \mod \Psi^{-\infty}\) if \(Q-R\) is an integral operator with (infinitely) smooth integral kernel.

Let us begin by proving parts (a) and (b) of Theorem 1. The matter of subprincipal symbols (part (c) of Theorem 1) and the construction of our operators will be addressed afterwards.

Recall that, as explained in Section 2, \(\operatorname{curl}_\mathrm{prin}\) has three simple eigenvalues \(h^{(\aleph)}\), \(\aleph\in\{0,+,-\}\), given by 17 . Let us denote by \(P^{(\aleph)}\), \(\aleph\in\{0,+,-\}\), the corresponding eigenprojections. It is easy to see that these are exactly the principal symbols 8 , 9 of our pseudodifferential operators \(P_\aleph\), \(\aleph \in\{0,+,-\}\).

Observe that the pseudodifferential operators \(P_\aleph\), \(\aleph \in\{0,+,-\}\), satisfy \[\label{theorem32about32projections32equation322} P_\aleph^*= P_\aleph, \qquad P_\aleph P_\beth= \delta_{\aleph\beth} P_\aleph\,, \qquad \sum_\aleph \;P_\aleph=\mathrm{Id}\textcolor{black}{\mod \Psi^{-\infty}}\,, \qquad [\operatorname{curl}, P_\aleph]=0\,,\tag{104}\] where \(\delta\) is the Kronecker delta.

Next, let us examine what happens if we relax 104 , asking that they are satisfied not exactly, but only modulo \(\Psi^{-\infty}\), cf. 96 and 97 . Namely, let us seek operators \(\widetilde{P}_\aleph\in\Psi^0\), \(\aleph\in\{0,+,-\}\), satisfying \[\tag{105} \begin{equation} \tag{106} (\widetilde{P}_\aleph)^*=\widetilde{P}_\aleph \mod \Psi^{-\infty}, \end{equation} \begin{equation} \tag{107} \widetilde{P}_\aleph \widetilde{P}_\beth= \delta_{\aleph\beth} \widetilde{P}_\aleph\mod \Psi^{-\infty}\,, \end{equation} \begin{equation} \tag{108} \sum_\aleph \widetilde{P}_\aleph=\mathrm{Id}\mod \Psi^{-\infty}\,, \end{equation} \begin{equation} \tag{109} [\operatorname{curl}, \widetilde{P}_\aleph]=0 \mod \Psi^{-\infty}\,. \end{equation}\]

In fact, we have a stronger version of Theorem 17.

What happens here is that in Theorem 17 assumptions 106 and 108 are redundant, and 107 can be relaxed to 112 .

The overall strategy of our paper hinges on the key observation 111 . Indeed, this gives us an effective way of constructing the full symbols of our projections \(P_\aleph\), \(\aleph \in\{0,+,-\}\), without resorting explicitly to the representations 92 and 101 .

Let introduce refined notation for the principal symbol. Namely, we denote by \((\,\cdot\,)_{\mathrm{prin},s}\) the principal symbol of the expression within brackets, regarded as an operator in \(\Psi^{-s}\). To appreciate the need for such notation, consider the following example. Let \(Q\) and \(R\) be pseudodifferential operators in \(\Psi^{-s}\) with the same principal symbol. Then, as an operator in \(\Psi^{-s}\), \(Q-R\) has vanishing principal symbol: \((Q-R)_{\mathrm{prin},s}=0\). But this tells us that \(Q-R\) is, effectively, an operator in \(\Psi^{-s-1}\) and, as such, it may have nonvanishing principal symbol \((Q-R)_{\mathrm{prin},s+1}\). This refined notation will be used whenever there is risk of confusion.

We are now in a position to prove part (c) of Theorem 1, which will be quite useful in the analysis of the asymmetry operator.

6 sec:The32asymmetry32operator↩︎

6.1 Definition of the asymmetry operator↩︎

The remainder of this paper is devoted to the study of the pseudodifferential operator \(P_+-P_-\) which encodes information about the asymmetry of the Riemannian manifold \((M,g)\) under change of orientation. In particular, one expects this information to be carried by the trace of \(P_+-P_-\,\), either in its pointwise or global (operator-theoretic) versions.

Unfortunately, one encounters a problem: the pseudodifferential operator \(P_+-P_-\) is of order zero and its spectrum is the set of three points \(\{-1,0,+1\}\), all three being eigenvalues of infinite multiplicity. Hence, it is not trace class. In order to circumvent this problem, acting in the spirit of Section 4, we introduce the following definition.

The above definition warrants a number of remarks.

6.2 The order of the asymmetry operator↩︎

Definition 6 implies that, a priori, our asymmetry operator \(A\) is a pseudodifferential operator of order 0. The goal of this subsection is to show that, as a result of cancellations in the symbol, the order of \(A\) is in fact \(-3\). This will be done in three steps.

1. First, we will show that \(A\) is of order \(-2\). This will follow easily from the facts already established earlier in the paper.

2. Next, we will show that, in normal coordinates, parallel transport does not contribute to homogeneous components of the symbol of \(A\) of degree \(-2\) and \(-3\).

3. Finally, we will show that \(A\) is of order \(-3\). This is the subject of Theorem 24 which will come in the end of the subsection and is one of the main results of this paper.

Let us now fix an arbitrary point \(z\in M\) and work in normal coordinates centred at \(z\). In our chosen coordinate system the operator \(A\) (modulo \(\Psi^{-\infty}\)) reads \[\label{A32in32normal32coordinates} A: f(x) \mapsto \frac{1}{(2\pi)^3} \int e^{i(x-y)^\mu\xi_\mu}\, [p_+-p_-]_{\alpha}{}^\beta(x,\xi) \, Z_\beta{}^\alpha(y,x) \, f(y) \,\mathrm{d}y\,\mathrm{d}\xi\,,\tag{121}\] where \[[p_\pm]_\alpha{}^\beta(x,\xi)\sim \sum_{j=0}^{+\infty} [(p_\pm)_{-j}]_\alpha{}^\beta(x,\xi), \qquad [(p_\pm)_{-k}]_\alpha{}^\beta(x,\lambda\xi)=\lambda^{-k}[(p_\pm)_{-k}]_\alpha{}^\beta(x,\xi) \quad \forall\lambda>0,\] is the full symbol of \(P_\pm\) in the chosen coordinate system.

Next, we observe that 121 can be recast as \[\label{A32in32normal32coordinates32v2} A=A_\mathrm{diag}+A_\mathrm{pt}\,,\tag{122}\] where \[\label{A32diag} A_\mathrm{diag}:=\frac{1}{(2\pi)^3} \int e^{i(x-y)^\mu\xi_\mu}\, [p_+-p_-]_{\alpha}{}^\alpha(x,\xi) \, f(y) \,\mathrm{d}y\,\mathrm{d}\xi\tag{123}\] and \[\label{A32pt} A_\mathrm{pt}:=A-A_\mathrm{diag}.\tag{124}\] Here the subscript “pt” stands for “parallel transport”. The operator \(A_\mathrm{pt}\) describes the contribution to the asymmetry operator arising from the fact that our original operator \(\operatorname{curl}\) acts on 1-forms and not on 3-columns of scalar fields,

The decomposition 122 is not invariant, it relies on our particular choice of local coordinates but it is convenient for our purposes. Namely, in what follows we will show that \(A_\mathrm{pt}\) will not contribute to the principal symbol of order \(-3\) of the asymmetry operator \(A\) at the point \(z\).

Let us denote by \(a_\mathrm{pt}(x,\xi)\sim \sum_{j=0}^{+\infty} (a_\mathrm{pt})_{-j}(x,\xi)\) the full symbol of \(A_\mathrm{pt}\,\). Our task is to show that \[\label{components32of32apt32are32zero} (a_\mathrm{pt})_{-j}(z,\xi)=0 \qquad \text{for} \;j=0,1,2,3.\tag{125}\] In what follows, we will be using the expansion 283 for \(Z_{\alpha}{}^\beta\).

The claim 125 for \(j=0\) and \(j=1\) follows immediately from 283 .

Next, we observe that formulae 11 , 27 and 9 imply \[\label{ppm32minus32132is32zero} [(p_+-p_-)_{-1}]_\alpha{}^\beta(z,\xi)=0.\tag{131}\] This fact is needed in establishing the following lemma.

Lemmata 5 and 6 give us 125 .

We are now in a position to prove the main result of this subsection.

6.3 sec:The32principal32symbol32of32the32asymmetry32operator↩︎

This subsection is concerned with the proof of Theorem 2. As in the previous subsection, in what follows we also assume to have chosen normal coordinates centred at \(z\).

Since formula 13 is one of the key results of this paper, we provide in Appendix 15 an independent verification relying on formula 100 .

7 The regularised trace of the asymmetry operator↩︎

7.1 Structure of the integral kernel near the diagonal↩︎

The asymmetry operator \(A\) can be written as an integral operator \[\label{structure32frac32a32equation321} (Af)(x)=\int_M \mathfrak{a}(x,y)\,f(y)\,\rho(y)\,\mathrm{d}y \,.\tag{171}\]

Now, the operator \(A\) is not guaranteed to be of trace class, so one cannot apply formula ?? to it. The issue here is that in dimension 3 a sufficient condition for a pseudodifferential operator to be of trace class is that its order be strictly less than \(-3\), whereas according to Theorem 2 for a generic Riemannian 3-manifold the order of \(A\) is exactly \(-3\). This means that we need to perform some sort of very basic regularisation in order to get over the finish line and define, in an invariant and geometrically meaningful manner, the trace of \(A\) .

In order to define an appropriate regularisation, we need to understand the structure of the integral kernel \(\mathfrak{a}(x,y)\). More precisely, we need to characterise explicitly the way in which the latter fails to be continuous.

In order to prove Theorem 28 we need to establish first a number of preparatory lemmata. Until further notice we will be working in Euclidean space \(\mathbb{R}^3\) equipped with Cartesian coordinates. In what follows we use the Japanese32bracket notation \[\label{Japanese32bracket} \langle\xi\rangle := (1+|\xi|^2)^{1/2}.\tag{175}\]

Theorem 28 tells us that the singularity of the integral kernel \(\mathfrak{a}(x,y)\) of our asymmetry operator \(A\) is very weak. We are looking at a bounded function of two variables with a discontinuity on the diagonal. The discontinuity of \(\mathfrak{a}(x,y)\) on the diagonal manifests itself in the fact that limit of \(\mathfrak{a}(x,y)\) as \(y\) tends to \(x\) depends on the direction along which \(y\) tends to \(x\).

The bottom line is that working with the asymmetry operator \(A\) does not require the use of the theory of distributions or microlocal analysis. It is an integral operator whose integral kernel has a mild discontinuity on the diagonal.

7.2 sec:Local32and32global32trace↩︎

The decomposition 172 , 173 depends on the choice of local coordinates. However, the quantity \[\label{local32trace32definitiion321} \psi_{\operatorname{curl}}^\mathrm{loc}(x):=\mathfrak{a}_c(x,x)\tag{186}\] does not depend on the choice of local coordinates, i.e. is a true scalar function \(\psi_{\operatorname{curl}}:M\to\mathbb{R}\). Let us elaborate. Denote \[\label{elaborate321} f(x,y) := 12\pi^2 \,\mathfrak{a}_d(x,y) \,\operatorname{dist}^2(x,y) =E^{\alpha\beta}{}_\gamma(x)\, \nabla_\alpha \operatorname{Ric}_{\beta\rho}(x)\,(x-y)^\gamma (x-y)^\rho\,.\tag{187}\] The issue at hand is that the expression 187 is not a scalar because the quantities \((x-y)^\gamma\) and \((x-y)^\rho\) are not vectors based at the point \(x\). Let us switch from local coordinates \(x\) to local coordinates \(\widetilde{x}\) and let \(\widetilde{f}(\widetilde{x},\widetilde{y})\) be the representation of the quantity 187 in these new coordinates. It is easy to see, by writing down the appropriate Jacobians, that \[\label{elaborate322} \widetilde{f}(\widetilde{x}(x),\widetilde{y}(y))-f(x,y) = O(\operatorname{dist}^3(x,y)).\tag{188}\] Now, let \(\widetilde{\mathfrak{a}}_c(\widetilde{x},\widetilde{y})\) be the representation of the quantity \(\mathfrak{a}_c(x,y)\) in the new coordinates. Formulae 172 , 173 , 187 , 188 and the fact that \(\mathfrak{a}(x,y)\) is a true scalar in two variables imply \[\label{elaborate323} \widetilde{\mathfrak{a}}_c(\widetilde{x}(x),\widetilde{y}(y))-\mathfrak{a}_c(x,y) = O(\operatorname{dist}(x,y)),\tag{189}\] which, in turn,implies \(\widetilde{\mathfrak{a}}_c(\widetilde{x}(x),\widetilde{x}(x))=\mathfrak{a}_c(x,x)\).

Using formulae 172 , 173 and 186 for the calculation of the function \(\psi_{\operatorname{curl}}\) is impractical. The following lemma provides an alternative equivalent representation.

The advantage of formula 190 is that it does not involve the use of local coordinates. Hence, it can be viewed as a more convenient and natural definition of the scalar function \(\psi_{\operatorname{curl}}^\mathrm{loc}\,\).

The function \(\psi_{\operatorname{curl}}^\mathrm{loc}(x)\) and the number \(\psi_{\operatorname{curl}}\) are determined by the Riemannian 3-manifold and its orientation. This means that we are looking at geometric invariants.

8 sec:Challenges32in32higher32dimensions↩︎

Let \((M,g)\) be a connected closed Riemannian manifold of dimension \[\label{higher32dimensions} d=4k+3,\qquad k=1,2,\ldots.\tag{196}\] In this case one can define \(\operatorname{curl}\) in the usual way 5 as an operator acting in \(\Omega^{2k+1}(M)\). It is easy to see that \(\operatorname{curl}\) remains formally self-adjoint and at a formal level everything appears to be similar to what happens in dimension 3.

Analysis of higher dimensions 196 presents two major challenges.

First challenge. We expect that in higher dimensions the asymmetry operator will still be a pseudodifferential operator of order \(-3\), as it is in the 3-dimensional case. However, a sufficient condition for a pseudodifferential operator to be of trace class is that its order be less than \(-d\). This means that defining the regularised trace of the asymmetry operator becomes challenging: one would need to analyse singularities of the integral kernel and perform regularisation reducing the order of the singularity by \(4k+1\). In other words, one would need to perform \(4k+1\) steps of regularisation. Doing this in a geometrically invariant fashion will not be an easy task.

Second challenge. In the current paper we used the results of [37] which apply to operators whose principal symbols have simple eigenvalues (see also [41]). In dimension 3 the operator \(\operatorname{curl}\) does possess this property — its principal symbol has three simple eigenvalues 17 . In higher dimensions the eigenvalues of the principal symbol of the operator \(\operatorname{curl}\) are still given by formulae 17 , only now they have multiplicity: zero is an eigenvalue of multiplicity \(\binom{4k+2}{2k}\), whereas each of the two nonzero eigenvalues has multiplicity \(\frac{1}{2}\binom{4k+2}{2k+1}\). This means that addressing the issue of higher dimensions would require extending the results of [37] to operators whose principal symbols have multiple eigenvalues. This will be a difficult task as the construction presented in [37] relied heavily on the assumption that the eigenvalues of the principal symbol of the operator are simple. Overall, partial differential operators whose principal symbols have multiple eigenvalues are known to present a major challenge in micrololcal analysis.

Despite the challenges involved, examining the asymmetry operator in higher dimension could be an piece of research worth pursuing in the future. For example, since the seminal works by Milnor and Kervaire [42], [43] the study of exotic smooth structures on 7-spheres has been an active area of research in differential geometry. Our asymmetry operator, being very sensitive to the underlying geometry, might help to make some progress in this area.

Acknowledgements↩︎

MC was partially supported by a Leverhulme Trust Research Project Grant RPG-2019-240, by a Research Grant (Scheme 4) of the London Mathematical Society, by a grant of the Heilbronn Institute for Mathematical Research (HIMR) via the UKRI/EPSRC Additional Funding Programme for Mathematical Sciences, and by EPSRC Fellowship EP/X01021X/1.

We are grateful to Christian Bär, Bruno Colbois, Stuart Dowker, Michal Kwasigroch, Jason Lotay and Nikolai Saveliev for useful comments and suggestions. We also thank Michael Levitin for kindly sharing his Mathematica scripts for the Berger sphere.

(a) If $n=2$ the multiplicity is
    $$\label{2032August32202132equation324}
    1.$$ {#eq:2032August32202132equation324} 

(b) If $n=3,4,\dots$ the multiplicity is
    $$\label{2032August32202132equation325}
    2n-2.$$ {#eq:2032August32202132equation325} 

 (a) If $n$ is odd the multiplicity is
     $$\label{2032August32202132equation327}
     2n+2.$$ {#eq:2032August32202132equation327} 

 (b) If $n$ is even and $l<\frac{n}{2}$ the multiplicity is
     $$\label{2032August32202132equation328}
     2n+2.$$ {#eq:2032August32202132equation328} 

 (c) If $n$ is even and $l=\frac{n}{2}$ the multiplicity is
     $$\label{2032August32202132equation329}
     n+1.$$ {#eq:2032August32202132equation329} 

The result @eq:eta32invariant32Dirac was obtained under the
assumption $a<4$. We do not need to impose any restriction on $a$
because, unlike for Dirac operator, where the dimension of the space
of harmonic spinors is highly dependent on the Riemannian metric,
zero is never an eigenvalue of curl (see
Theorem [7](#theorem32properties32of32curl){reference-type="ref"
reference="theorem32properties32of32curl"}).

References↩︎