$\frac{\xi_{\gamma}\xi_{\rho}}{|\xi|^{5}}=\frac{\xi_{\gamma}\xi_{\rho}}{\langle\xi\rangle^{5}}+O\left(\frac{1}{|\xi|^{5}}\right),$

so the required result follows from Lemma 7.5. Note that the term with
$\delta_{\gamma\rho}$ from (7.10) vanishes upon contraction with
$\,\varepsilon^{\alpha\beta\gamma}\,\nabla_{\alpha}\operatorname{Ric}_{\beta}{}^{\rho}(0)\,$
due to formula (7.4). ∎

Theorem 7.1 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 Local and global trace

The decomposition (7.2), (7.3) depends on the choice of local coordinates.
However, the quantity

$\psi_{\operatorname{curl}}^{\mathrm{loc}}(x):=\mathfrak{a}_{c}(x,x)$ (7.12)

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

$f(x,y):=12\pi^{2}\,\mathfrak{a}_{d}(x,y)\,\operatorname{dist}^{2}(x,y)=E^{\alpha\beta}{}_{\gamma}(x)\,\nabla_{\alpha}\operatorname{Ric}_{\beta\rho}(x)\,(x-y)^{\gamma}(x-y)^{\rho}\,.$
(7.13)

The issue at hand is that the expression (7.13) 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 (7.13) in these new coordinates. It is easy to
see, by writing down the appropriate Jacobians, that

$\widetilde{f}(\widetilde{x}(x),\widetilde{y}(y))-f(x,y)=O(\operatorname{dist}^{3}(x,y)).$
(7.14)

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 (7.2), (7.3), (7.13), (7.14) and the fact that $\mathfrak{a}(x,y)$ is
a true scalar in two variables imply

$\widetilde{\mathfrak{a}}_{c}(\widetilde{x}(x),\widetilde{y}(y))-\mathfrak{a}_{c}(x,y)=O(\operatorname{dist}(x,y)),$

which, in turn,implies
$\widetilde{\mathfrak{a}}_{c}(\widetilde{x}(x),\widetilde{x}(x))=\mathfrak{a}_{c}(x,x)$.

Using formulae (7.2), (7.3) and (7.12) for the calculation of the function
$\psi_{\operatorname{curl}}$ is impractical. The following lemma provides an
alternative equivalent representation.

######  Lemma 7.6.

We have

$\psi_{\operatorname{curl}}^{\mathrm{loc}}(x)=\lim_{r\to 0^{+}}\frac{1}{4\pi
r^{2}}\int_{\mathbb{S}_{r}(x)}\mathfrak{a}(x,y)\,\mathrm{d}S_{y}\,,$ (7.15)

where $\mathfrak{a}(x,y)$ is the integral kernel of the asymmetry operator
(6.1) defined in accordance with (7.1), $\mathbb{S}_{r}(x)=\\{y\in
M|\operatorname{dist}(x,y)=r\\}$ is the sphere of radius $r$ centred at $x$
and $\mathrm{d}S_{y}$ is the surface area element on this sphere.

###### Proof.

Let us fix an $x\in M$ and choose normal coordinates centred at $x$. Theorem
7.1 tells us that

$\mathfrak{a}(x,y)=\mathfrak{a}_{c}(x,y)+c_{\gamma\rho}(x)\,\frac{(x-y)^{\gamma}(x-y)^{\rho}}{|x-y|^{2}}\,,$

where

$c_{\gamma\rho}(x)=\frac{1}{12\pi^{2}}\,E^{\alpha\beta}{}_{\gamma}(x)\,\nabla_{\alpha}\operatorname{Ric}_{\beta\rho}(x)\,.$

We have

$\int_{\mathbb{S}_{r}(x)}(x-y)^{\gamma}(x-y)^{\rho}\,\mathrm{d}S_{y}=\begin{cases}\frac{4\pi
r^{2}}{3}\quad&\textbf{if}\quad\gamma=\rho,\\\ \ \
0\quad&\textbf{if}\quad\gamma\neq\rho,\end{cases}$

hence,

$\frac{1}{4\pi
r^{2}}\int_{\mathbb{S}_{r}(x)}\mathfrak{a}(x,y)\,dS_{y}=\frac{1}{4\pi
r^{2}}\int_{\mathbb{S}_{r}(x)}\mathfrak{a}_{c}(x,y)\,\mathrm{d}S_{y}+\frac{c_{11}(x)+c_{22}(x)+c_{33}(x)}{3}\,.$
(7.16)

But formula (7.4) implies $c_{11}(x)+c_{22}(x)+c_{33}(x)=0$, so the second
term in the right-hand side of (7.16) vanishes. Formulae (7.16) and (7.12) now
imply (7.15) by the continuity of $\mathfrak{a}_{c}(x,y)$. ∎

The advantage of formula (7.15) 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
$\psi_{\operatorname{curl}}^{\mathrm{loc}}\,$.

######  Definition 7.7.

We call the scalar continuous function
$\psi_{\operatorname{curl}}^{\mathrm{loc}}(x)$,
$\psi_{\operatorname{curl}}^{\mathrm{loc}}:M\to\mathbb{R}$, defined in
accordance with formulae (7.15) and (7.1) _the regularised local trace of the
asymmetry operator_.

######  Definition 7.8.

We call the number

$\psi_{\operatorname{curl}}:=\int_{M}\psi_{\operatorname{curl}}^{\mathrm{loc}}(x)\,\rho(x)\,\mathrm{d}x$

_the regularised global trace of the asymmetry operator_.

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 A conjecture

The natural question is how are our geometric invariants
$\psi_{\operatorname{curl}}(x)$ and $\psi_{\operatorname{curl}}$ related to
classical eta invariants.

Recall that we denote by $\lambda_{k}$ the eigenvalues of
$\operatorname{curl}$ written with account of multiplicity and by $u_{k}$ the
corresponding orthonormal eigenforms. Recall also that zero is not in the
spectrum of $\operatorname{curl}$, see Theorem 2.1, so that $\lambda_{k}\neq
0$ for all $k$.

The (global) eta function for the operator $\operatorname{curl}$

$\eta_{\operatorname{curl}}(s):=\sum_{k\in\mathbb{Z}\setminus\\{0\\}}\frac{\operatorname{sgn}\lambda_{k}}{|\lambda_{k}|^{s}}$
(8.1)

was introduced in [2, p. 229], see also [2, p. 229], [3, pp. 44–45], [4, p.
405] and [48, pp. 190–191].

It was shown in [2, 3, 4, 5] that the series (8.1) converges absolutely for
$\operatorname{Re}s>3$, that it defines a meromorphic function in $\mathbb{C}$
by analytic continuation, with possible first order poles at
$s=3,2,1,0,-1,-2,\ldots$, and that $\eta_{\operatorname{curl}}(s)$ is
holomorphic at $s=0$. The (real) number $\eta_{\operatorname{curl}}(0)$ is
called the (global) eta invariant for the operator $\operatorname{curl}$.

######  Definition 8.1.

We define the _local eta function_ for the operator $\operatorname{curl}$ as

$\eta_{\operatorname{curl}}^{\mathrm{loc}}(x;s):=\sum_{k\in\mathbb{Z}\setminus\\{0\\}}\frac{\operatorname{sgn}\lambda_{k}}{|\lambda_{k}|^{s}}\,*(u_{k}\wedge*u_{k})(x)\,,$
(8.2)

where
$*(u_{k}\wedge*u_{k})(x)=g^{\alpha\beta}(x)\,[u_{k}]_{\alpha}(x)\,[u_{k}]_{\beta}(x)$
is the pointwise norm of the eigenform $u_{k}$.

We believe that for any fixed $x\in M$ the local eta function has the same
properties as the global one: the series (8.2) converges absolutely for
$\operatorname{Re}s>3$, that it defines a meromorphic function in $\mathbb{C}$
by analytic continuation, with possible first order poles at
$s=3,2,1,0,-1,-2,\ldots$, and that
$\eta_{\operatorname{curl}}^{\mathrm{loc}}(x;s)$ is holomorphic at $s=0$.

Of course, the local and global eta functions are related by the identity

$\eta_{\operatorname{curl}}(s)=\int_{M}\eta_{\operatorname{curl}}^{\mathrm{loc}}(x;s)\,\rho(x)\,\mathrm{d}x\,.$

######  Definition 8.2.

We call the (real-valued) function
$\eta_{\operatorname{curl}}^{\mathrm{loc}}(x;0)$ the _local eta invariant_ for
the operator $\operatorname{curl}$.

######  Conjecture 8.3.

a

  1. (a)

The function $\eta_{\operatorname{curl}}(s)$ is holomorphic in the half-plane
$\operatorname{Re}s>-2$.

  2. (b)

For every $x\in M$ the function
$s\mapsto\eta_{\operatorname{curl}}^{\mathrm{loc}}(x;s)$ is holomorphic in the
half-plane $\operatorname{Re}s>-2$.

######  Remark 8.4.

It is known [3, Theorem 4.14(iii)] that the function
$\eta_{\operatorname{curl}}(s)$ is holomorphic in the half-plane
$\operatorname{Re}s>-1/2$.

######  Conjecture 8.5.

a

  1. (a)

We have

$\psi_{\operatorname{curl}}=\eta_{\operatorname{curl}}(0),$

where $\psi_{\operatorname{curl}}$ is our geometric invariant introduced in
Definition 7.8.

  2. (b)

Moreover, we have

$\psi_{\operatorname{curl}}^{\mathrm{loc}}(x)=\eta_{\operatorname{curl}}^{\mathrm{loc}}(x;0).$

We will examine the validity of the above conjectures in a separate paper
because doing this requires very extensive and delicate analysis. However, we
give below an indication of why we believe Conjecture 8.3 to be true.

There are two ways of approaching Conjecture 8.3.

First argument. Arguing in the spirit of Seeley [46] (with appropriate
modifications) one expects the residues of the poles of the eta function,
global or local, to be expressed via local geometric invariants: curvature,
its covariant derivatives and the totally antisymmetric tensor. More
precisely, the residues in question should be scalar quantities formed out of
products of these invariants containing exactly one appearance of the totally
antisymmetric tensor. Let us assign to such scalar invariants degrees
according to the following rules:

  * •

the degrees of factors add up,

  * •

curvature carries degree $-2$,

  * •

each covariant differentiation contributes $-1$ to the degree.

After analysing possible scalar invariants of this type one concludes that the
nonvanishing invariant of highest degree is the invariant

$(\nabla_{\alpha}\operatorname{Ric}_{\beta\gamma})\,\operatorname{Ric}^{\beta}{}_{\nu}\,E^{\alpha\gamma\nu}$

which has degree $-5$. An invariant of degree $n$ corresponds to a pole at
$s=3+n$. Hence, we expect the first pole to be at $s=3-5=-2$. This would agree
with similar results obtained for the massless Dirac operator [10, 11].

Second argument. Consider the global counting function for
$\operatorname{curl}$ (2.36) and its local analogue

$N^{\pm}_{\operatorname{curl}}(x;\lambda):=\begin{cases}0\quad&\text{for}\quad\lambda\leq
0,\\\
\sum_{0<\pm\lambda_{k}<\lambda}\,*(u_{k}\wedge*u_{k})(x)\quad&\text{for}\quad\lambda>0.\end{cases}$

Of course, the local and global counting functions are related by the identity

$N^{\pm}_{\operatorname{curl}}(\lambda)=\int_{M}N^{\pm}_{\operatorname{curl}}(x;\lambda)\,\rho(x)\,dx\,.$

Let $\widehat{\mu}:\mathbb{R}\to\mathbb{C}$ be a smooth function such that
$\widehat{\mu}=1$ in some neighbourhood of the origin and
$\operatorname{supp}\widehat{\mu}\subset(-T_{0},T_{0})$, where $T_{0}$ is the
infimum of lengths of all the geodesic loops originating from all the points
of the manifold. Let $\mu$ be the inverse Fourier transform of
$\widehat{\mu}$. Although the operator $\operatorname{curl}$ is not elliptic,
arguing in the spirit of [14, Appendix B] and references therein, it is
natural to expect that the mollified derivative of the local and global
counting functions admits complete asymptotic expansions of the form

$((N^{\pm}_{\operatorname{curl}})^{\prime}*\mu)(x;\lambda)=c^{\pm}_{2}(x)\,\lambda^{2}+c^{\pm}_{1}(x)\,\lambda+c^{\pm}_{0}(x)+c^{\pm}_{-1}(x)\,\lambda^{-1}+\ldots,$
(8.3)
$((N^{\pm}_{\operatorname{curl}})^{\prime}*\mu)(\lambda)=c^{\pm}_{2}\,\lambda^{2}+c^{\pm}_{1}\,\lambda+c^{\pm}_{0}+c^{\pm}_{-1}\,\lambda^{-1}+\ldots,$
(8.4)

as $\lambda\to+\infty$. Here the star stands for convolution in the variable
$\lambda$. We call the coefficients $c^{\pm}_{k}$ appearing in (8.3) and (8.4)
_Weyl coefficients_ , local and global respectively. Of course, the local and
global Weyl coefficients are related by the identity
$c^{\pm}_{k}=\int_{M}c^{\pm}_{k}(x)\,\rho(x)\,\mathrm{d}x\,$. It is known [7]
that $c^{\pm}_{2}(x)=\frac{1}{2\pi^{2}}\,$.

In the remainder of the argument we, for the sake of brevity, deal only with
local functions.

The local eta function is expressed via the local counting functions as

$\eta_{\operatorname{curl}}^{\mathrm{loc}}(x;s)=\int_{0}^{+\infty}\lambda^{-s}\left[(N^{+}_{\operatorname{curl}})^{\prime}(x;\lambda)-(N^{-}_{\operatorname{curl}})^{\prime}(x;\lambda)\right]\mathrm{d}\lambda\,.$

One can show [25, Theorem 3.2.2] that the residues of the local eta function
are given by

$\mathrm{Res}(\eta_{\operatorname{curl}}^{\mathrm{loc}}(x;\,\cdot\,),n)=c_{n-1}^{+}(x)-c_{n-1}^{-}(x)\,,\qquad
n=3,2,1,0,-1,-2,\ldots.$

We are pursuing a separate project aimed at calculating the Weyl coefficients
$c^{\pm}_{k}(x)$, $k=1,0,-1,\ldots$, for the operator $\operatorname{curl}$,
and we are doing this along the lines of [16] where similar analysis was
initiated for the massless Dirac operator. We expect the first five Weyl
coefficients for $N^{+}_{\operatorname{curl}}$ and
$N^{-}_{\operatorname{curl}}$ to coincide,

$c_{k}^{+}(x)=c_{k}^{-}(x),\qquad k=2,1,0,-1,-2,$

which would imply that the local eta function is holomorphic in the half-plane
$\operatorname{Re}s>-2$.

##  9 Challenges in higher dimensions

Let $(M,g)$ be a connected closed Riemannian manifold of dimension

$d=4k+3,\qquad k=1,2,\ldots.$ (9.1)

In this case one can define $\operatorname{curl}$ in the usual way (1.5) 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.

######  Remark 9.1.

One could have also considered the case $d=4k+1$, $k=1,2,\ldots$, with
$\operatorname{curl}$ acting in $\Omega^{2k}(M)$ in accordance with (1.5).
However, in this case $\operatorname{curl}$ is formally anti-self-adjoint, the
spectrum is purely imaginary and there is no spectral asymmetry [7, Theorem
3.2]. As our interest is in spectral asymmetry, we do not discuss the case
$d=4k+1$. Note also that the spectral problem for $\operatorname{curl}$ in
dimension $d=4k+1$ cannot be formulated in terms of the spectral theory of
self-adjoint operators in _real_ Hilbert spaces: one can introduce the factor
$\,i\,$ in the right-hand side of (1.5) to make the operator formally self-
adjoint (as was done in [7, Definition 2.2]), but the resulting operator will
no longer be an operator acting in the vector space of _real_ differential
forms. The bottom line is that the two cases, $d=4k+3$ and $d=4k+1$, are
fundamentally different. We stick with (9.1) because this gives the only
genuine generalisation of the operator $\operatorname{curl}$ to higher
dimensions.

Analysis of higher dimensions (9.1) 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 [17] which apply
to operators whose principal symbols have simple eigenvalues (see also [18]).
In dimension 3 the operator $\operatorname{curl}$ does possess this property —
its principal symbol has three simple eigenvalues (2.3). In higher dimensions
the eigenvalues of the principal symbol of the operator $\operatorname{curl}$
are still given by formulae (2.3), 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 [17] to operators whose principal symbols have multiple eigenvalues. This
will be a difficult task as the construction presented in [17] 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 [40, 34] 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, and by a grant of the Heilbronn Institute for Mathematical Research
(HIMR) via the UKRI/EPSRC Additional Funding Programme for Mathematical
Sciences.

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

##  Appendix A Exterior calculus

In this appendix we set out out conventions on exterior calculus. Henceforth,
we identify differential forms with covariant antisymmetric tensors, and $M$
is an oriented 3-manifold equipped with Riemannian metric $g$ and Levi-Civita
connection $\nabla$.

It is well known that the metric $g$ induces a canonical isomorphism between
the tangent bundle $TM$ and the contangent bundle $T^{*}M$, the so-called
_musical isomorphism_. We denote it by $\flat:TM\to T^{*}M$ (lowering of
indices) and its inverse by $\sharp:T^{*}M\to TM$ (raising of indices).

Given a scalar field $f\in C^{\infty}(M)$, its exterior derivative
$\mathrm{d}f$ is defined as the gradient. Given a 1-form $u\in\Omega^{1}(M)$,
its exterior derivative $\mathrm{d}u\in\Omega^{2}(M)$ is defined,
componentwise, as

$(\mathrm{d}u)_{\alpha\beta}=\partial_{x^{\alpha}}u_{\beta}-\partial_{x^{\beta}}u_{\alpha}\,.$

We define the action of the Hodge star $*:\Omega^{k}(M)\to\Omega^{3-k}(M)$ on
a rank $k$ antisymmetric tensor as

$(*v)_{\mu_{k+1}\ldots\mu_{3}}\\!:=\frac{1}{k!}\,\sqrt{\det g_{\alpha\beta}}\
v^{\mu_{1}\ldots\mu_{k}}\,\varepsilon_{\mu_{1}\mu_{2}\mu_{3}}\,,$ (A.1)

where $\varepsilon$ is the totally antisymmetric symbol,
$\varepsilon_{123}:=+1$.

With regards to the totally antisymmetric symbol, we adopt the convention of
raising indices using the Kronecker symbol, so that, for example,

$\varepsilon^{\alpha_{1}\alpha_{2}\alpha_{3}}:=\varepsilon_{\alpha_{1}\alpha_{2}\alpha_{3}}.$

Let $\wedge$ denote the exterior product. Given a pair of real-valued rank $k$
covariant antisymmetric tensors $u$ and $v$ we define their $L^{2}$ inner
product as

$\begin{split}\langle
u,v\rangle:&=\int_{M}\frac{1}{k!}\,u_{\alpha_{1}\dots\alpha_{k}}\,v_{\beta_{1}\dots\beta_{k}}\,g^{\alpha_{1}\beta_{1}}\cdots
g^{\alpha_{k}\beta_{k}}\,\sqrt{\det g_{\mu\nu}}\ \mathrm{d}x\\\
&=\int_{M}u\wedge*v=\int_{M}*u\wedge v\,.\end{split}$

For the sake of clarity, let us mention that the exterior product of 1-forms
reads

$(u\wedge v)_{\alpha\beta}=u_{\alpha}v_{\beta}-u_{\beta}v_{\alpha}\,.$

Given $w\in\Omega^{k}(M)$ and $u\in\Omega^{k-1}(M)$ we define the action of
the codifferential $\delta:\Omega^{k}(M)\to\Omega^{k-1}(M)$ in accordance with

$\langle w,\mathrm{d}u\rangle=\langle\delta w,u\rangle.$

In particular, when $u\in\Omega^{1}(M)$ and $w\in\Omega^{2}(M)$, we get in
local coordinates

$\delta u=-\nabla^{\alpha}u_{\alpha},$ $(\delta
w)_{\alpha}=\nabla^{\beta}w_{\alpha\beta}.$

##  Appendix B The spectrum of the Laplacian on a Berger sphere

In preparation for the calculations that will appear in Appendix C, we list
here, in the form of a theorem, the eigenvalues of the Laplacian on a Berger
sphere. These were first obtained in [50, Lemma 4.1] and reproduced in [38,
Theorem 5.5] and [35, Proposition 3.9].

###  B.1 Definitions and notation

In Euclidean space $\mathbb{R}^{4}$ equipped with Cartesian coordinates
$\mathbf{x}^{\alpha}$, $\alpha=1,2,3,4$, let

$\mathbb{S}^{3}:=\\{\mathbf{x}\in\mathbb{R}^{4}\ |\ \|\mathbf{x}\|=1\\}$

be the $3$-sphere. We use bold script for 4-dimensional objects and normal
script for 3-dimensional objects.

We prescribe orientation of $\mathbb{S}^{3}$ as follows. We define local
coordinates $y=(y^{1},y^{2},y^{3})$ on $\mathbb{S}^{3}$ to be positively
oriented if

$\det\begin{pmatrix}\frac{\partial\mathbf{x}^{1}}{\partial
y^{1}}(y)&\frac{\partial\mathbf{x}^{1}}{\partial
y^{2}}(y)&\frac{\partial\mathbf{x}^{1}}{\partial
y^{3}}(y)&\mathbf{x}^{1}(y)\\\ \frac{\partial\mathbf{x}^{2}}{\partial
y^{1}}(y)&\frac{\partial\mathbf{x}^{2}}{\partial
y^{2}}(y)&\frac{\partial\mathbf{x}^{2}}{\partial
y^{3}}(y)&\mathbf{x}^{2}(y)\\\ \frac{\partial\mathbf{x}^{3}}{\partial
y^{1}}(y)&\frac{\partial\mathbf{x}^{3}}{\partial
y^{2}}(y)&\frac{\partial\mathbf{x}^{3}}{\partial
y^{3}}(y)&\mathbf{x}^{3}(y)\\\ \frac{\partial\mathbf{x}^{4}}{\partial
y^{1}}(y)&\frac{\partial\mathbf{x}^{4}}{\partial
y^{2}}(y)&\frac{\partial\mathbf{x}^{4}}{\partial
y^{3}}(y)&\mathbf{x}^{4}(y)\end{pmatrix}<0\,.$

The above definition of orientation agrees with that from [24, Appendix A].
Spherical coordinates

$\begin{pmatrix}\mathbf{x}^{1}\\\ \mathbf{x}^{2}\\\ \mathbf{x}^{3}\\\
\mathbf{x}^{4}\end{pmatrix}=\begin{pmatrix}\cos y^{1}\\\ \sin y^{1}\cos
y^{2}\\\ \sin y^{1}\sin y^{2}\cos y^{3}\\\ \sin y^{1}\sin y^{2}\sin y^{3}\\\
\end{pmatrix},\quad y^{1},y^{2}\in(0,\pi),\quad y^{3}\in[0,2\pi),$

are an example of positively oriented local coordinates on $\mathbb{S}^{3}$.

Of course, if one is only interested in the study of the Laplace–Beltrami
operator, orientation is irrelevant. However, with $\operatorname{curl}$ or
the Dirac operator [16, subsection 9.1] prescribing orientation of the
manifold is essential.

Consider the vector fields

$\displaystyle\mathbf{V}_{1}:=-\textbf{x}^{4}\dfrac{\partial}{\partial\textbf{x}^{1}}-\textbf{x}^{3}\frac{\partial}{\partial\textbf{x}^{2}}+\textbf{x}^{2}\dfrac{\partial}{\partial\textbf{x}^{3}}+\textbf{x}^{1}\dfrac{\partial}{\partial\textbf{x}^{4}}\,,$
(B.1)
$\displaystyle\mathbf{V}_{2}:=\phantom{-}\textbf{x}^{3}\dfrac{\partial}{\partial\textbf{x}^{1}}-\textbf{x}^{4}\frac{\partial}{\partial\textbf{x}^{2}}-\textbf{x}^{1}\dfrac{\partial}{\partial\textbf{x}^{3}}+\textbf{x}^{2}\dfrac{\partial}{\partial\textbf{x}^{4}}\,,$
$\displaystyle\mathbf{V}_{3}:=-\textbf{x}^{2}\dfrac{\partial}{\partial\textbf{x}^{1}}+\textbf{x}^{1}\frac{\partial}{\partial\textbf{x}^{2}}-\textbf{x}^{4}\dfrac{\partial}{\partial\textbf{x}^{3}}+\textbf{x}^{3}\dfrac{\partial}{\partial\textbf{x}^{4}}\,.$

The vector fields (B.1) are tangent to $\mathbb{S}^{3}$, i.e., they are
orthogonal to the gradient of $\|\textbf{x}\|$, or, equivalently,
$\mathbf{V}_{j}(\|\textbf{x}\|^{2})=0$, $j=1,2,3$. For $j=1,2,3$ let $V_{j}$
be the restriction of $\mathbf{V}_{j}$ to $\mathbb{S}^{3}$.

Let $a\in\mathbb{R}$ be a positive parameter. Consider the differential
operator

$B:=V_{1}^{2}+V_{2}^{2}+\frac{1}{a^{2}}V_{3}^{2}\,.$ (B.2)

The _Berger metric_ $g$ on $\mathbb{S}^{3}$ is defined via the identity

$B_{\mathrm{prin}}(y,\eta)=-g^{\alpha\beta}(y)\eta_{\alpha}\eta_{\beta}$ (B.3)

or, equivalently,

$g^{\alpha\beta}(y):=-\frac{\partial^{2}B_{\mathrm{prin}}}{\partial\eta_{\alpha}\partial\eta_{\beta}}\,.$

We call $(\mathbb{S}^{3},g)$ the _Berger sphere_.

The notion of a Berger sphere is well established in differential geometry,
nevertheless, for the benefit of a wider audience, we provide here an explicit
description in the simplest possible coordinate system. Working in the
southern hemisphere $\mathbf{x}^{4}<0$, consider the positively oriented local
coordinates

$y^{\alpha}=\mathbf{x}^{\alpha},\qquad\alpha=1,2,3,$

so that

$\mathbf{x}^{4}=-\sqrt{1-(\mathbf{x}^{1})^{2}-(\mathbf{x}^{2})^{2}-(\mathbf{x}^{3})^{2}}=-\sqrt{1-|y|^{2}}.$

In these coordinates the vector fields $V_{j}$ read

$\begin{split}V_{1}&=\sqrt{1-|y|^{2}}\dfrac{\partial}{\partial
y^{1}}-y^{3}\dfrac{\partial}{\partial y^{2}}+y^{2}\dfrac{\partial}{\partial
y^{3}},\\\ V_{2}&=y^{3}\dfrac{\partial}{\partial
y^{1}}+\sqrt{1-|y|^{2}}\dfrac{\partial}{\partial
y^{2}}-y^{1}\dfrac{\partial}{\partial y^{3}},\\\
V_{3}&=-y^{2}\dfrac{\partial}{\partial y^{1}}+y^{1}\dfrac{\partial}{\partial
y^{2}}+\sqrt{1-|y|^{2}}\dfrac{\partial}{\partial y^{3}}.\end{split}$ (B.4)

Formulae (B.2), (B.3) and (B.4) imply

$\footnotesize
g^{\alpha\beta}(y)=\begin{pmatrix}1-(y^{1})^{2}-\frac{a^{2}-1}{a^{2}}(y^{2})^{2}&-\frac{y^{1}y^{2}}{a^{2}}&-y^{1}y^{3}+\frac{a^{2}-1}{a^{2}}y^{2}\sqrt{1-|y|^{2}}\\\
-\frac{y^{1}y^{2}}{a^{2}}&1-(y^{2})^{2}-\frac{a^{2}-1}{a^{2}}(y^{1})^{2}&-y^{2}y^{3}-\frac{a^{2}-1}{a^{2}}y^{1}\sqrt{1-|y|^{2}}\\\
-y^{1}y^{3}+\frac{a^{2}-1}{a^{2}}y^{2}\sqrt{1-|y|^{2}}&-y^{2}y^{3}-\frac{a^{2}-1}{a^{2}}y^{1}\sqrt{1-|y|^{2}}&\frac{a^{2}-1}{a^{2}}(y^{2})^{2}+\frac{a^{2}-1}{a^{2}}(y^{2})^{2}-\frac{1-(y^{3})^{2}}{a^{2}}\\\
\end{pmatrix}.$ (B.5)

We summarise the properties of the vector fields $V_{j}$, $j=1,2,3$, and the
Berger metric $g$ in the following Lemma. The proof amounts to straightforward
calculations and is omitted.

######  Lemma B.1.

We have the following properties.

  1. (a)

$[V_{j},V_{k}]=-2\varepsilon_{jkl}\,V_{l}\,$, $j,k=1,2,3$.

  2. (b)

The vector fields $V_{j}$ are positively oriented, i.e. $\det
V_{j}{}^{\alpha}>0$, $j,\alpha=1,2,3$. Here the $V_{j}{}^{\alpha}$ are the
components of the vector fields $V_{j}$ with respect to the basis
$\frac{\partial}{\partial y^{\alpha}}\,$.

  3. (c)

The vector fields

$\widetilde{V}_{1}:=V_{1}\,,\qquad\widetilde{V}_{2}:=V_{2}\,,\qquad\widetilde{V}_{3}:=\frac{1}{a}V_{3}$
(B.6)

are orthonormal with respect to the metric $g$ defined by (B.5).

  4. (d)

The Riemannian density $\rho(y)$ generated by the Berger metric $g$ reads

$\rho(y)=a\,\rho_{0}(y),$

where $\rho_{0}(y)$ is the standard Riemannian density of the round
$\mathbb{S}^{3}$.

  5. (e)

The vector fields $V_{j}$, $j=1,2,3$, viewed as differential operators are
skew-Hermitian.

  6. (f)

The vector fields $V_{j}$, $j=1,2,3$, are Killing vector fields for the
standard round metric $g_{0}$ on $\mathbb{S}^{3}$.

  7. (g)

The vector field $V_{3}$ is a Killing vector field for the Berger metric $g$.

  8. (h)

The operator $B$ defined in accordance with (B.2) is the Laplace–Beltrami
operator (henceforth denoted by $\Delta$) on the Berger sphere.

  9. (i)

The operator $\operatorname{curl}$ on the Berger sphere can be represented as

$\operatorname{curl}_{s}=\begin{pmatrix}\frac{2}{a}&-\frac{1}{a}V_{3}&V_{2}\\\
\frac{1}{a}V_{3}&\frac{2}{a}&-V_{1}\\\ -V_{2}&V_{1}&2a\\\ \end{pmatrix}.$
(B.7)

Here the subscript $\,s\,$ indicates that the operator appearing in (B.7) acts
on 3-columns of scalar functions. The latter can be obtained by projecting
1-forms onto the orthonormal framing (B.6). The operators
$\operatorname{curl}$ and $\operatorname{curl}_{s}$ are related as

$\operatorname{curl}_{\alpha}{}^{\beta}=\widetilde{V}^{j}{}_{\alpha}\,[\operatorname{curl}_{s}]_{j}{}^{k}\,\widetilde{V}_{k}{}^{\beta},\qquad\widetilde{V}^{j}{}_{\alpha}:=\delta^{jl}\,\widetilde{V}_{l}{}^{\gamma}\,g_{\alpha\gamma}\,.$

See also the projection operator $\mathbf{S}$ in [15, Section 5.1].

###  B.2 The spectrum

######  Theorem B.2.

The eigenvalues of the operator $-\Delta$ on the Berger 3-sphere are

$n(n+2)+\left(a^{-2}-1\right)\left(n-2l\right)^{2}\,,\quad
n=0,1,2,3,\dots,\quad 0\leq l\leq\left\lfloor\frac{n}{2}\right\rfloor$ (B.8)

with

  1. (i)

multiplicity

$1$

if $n=0$,

  2. (ii)

multiplicity

$2n+2$

if $n$ is odd,

  3. (iii)

multiplicity

$2n+2$

if $n$ is even and $0\leq l<\frac{n}{2}$, and

  4. (iv)

multiplicity

$n+1$

if $n$ is even and $l=\frac{n}{2}$.

For a proof of the above theorem we refer the reader to [38, Theorem 5.5].
Somewhat more involuted proofs may also be found in [50, Lemma 4.1] and [35,
Proposition 3.9].

######  Remark B.3.

When $a=1$ Theorem B.2 reduces to the usual spectrum of $\,-\Delta\,$ on the
round 3-sphere: eigenvalues $n(n+2)$, $n=0,1,2,\dots$, with multiplicity
$(n+1)^{2}\,$.

##  Appendix C The spectrum of curl on a Berger sphere

In this appendix we will write down explicitly the eigenvalues and the
$\eta$-invariant for the operator curl on a Berger sphere. This will be done
using classical (as opposed to pseudodifferential) techniques and will serve
as an illustration of the spectral asymmetry of curl.

Throughout this appendix we use the notation from Appendix B.1.

###  C.1 The spectrum

The positive spectrum of curl (B.7) was first computed by Lotay in [38,
Proposition 5.6]. Explicit formulae for the negative spectrum do not appear to
be available in the literature. We give below, for future reference and in the
form of a theorem, explicit formulae for the _full_ spectrum of curl: Lotay’s
result (slightly reformulated) as well as negative eigenvalues. The proof is
omitted.

######  Theorem C.1.

The spectrum of the operator $\operatorname{curl}$ on the Berger 3-sphere is
the (disjoint) union of the following four series of eigenvalues:

  1. I.

Eigenvalues

$\frac{n}{a}\,,\qquad n=2,3,\dots,$ (C.1)

with multiplicity

$2n-2.$ (C.2)

  2. II.

Eigenvalues

$\frac{n+2(a^{2}-1)}{a}\,,\qquad n=2,3,\dots.$ (C.3)

Here the multiplicity is as follows.

    1. (a)

If $n=2$ the multiplicity is

$1.$ (C.4)

    2. (b)

If $n=3,4,\dots$ the multiplicity is

$2n-2.$ (C.5)

  3. III.

Eigenvalues

$a+\sqrt{a^{2}+n(n+2)+\left(a^{-2}-1\right)\left(n-2l\right)^{2}}\ ,\quad
n=2,3,\dots,\quad 1\leq l\leq\left\lfloor\frac{n}{2}\right\rfloor.$ (C.6)

Here the multiplicity is as follows.

    1. (a)

If $n$ is odd the multiplicity is

$2n+2.$ (C.7)

    2. (b)

If $n$ is even and $l<\frac{n}{2}$ the multiplicity is

$2n+2.$ (C.8)

    3. (c)

If $n$ is even and $l=\frac{n}{2}$ the multiplicity is

$n+1.$ (C.9)

  4. IV.

Eigenvalues

$a-\sqrt{a^{2}+n(n+2)+\left(a^{-2}-1\right)\left(n-2l\right)^{2}}\ ,\quad
n=2,3,\dots,\quad 1\leq l\leq\left\lfloor\frac{n}{2}\right\rfloor,$ (C.10)

with multiplicities as in (C.7)–(C.9).

######  Remark C.2.

?

  1. (i)

Series I, II and III form the positive spectrum, whereas series IV is the
negative spectrum.

  2. (ii)

For $a=1$ the Berger sphere reduces to the standard 3-sphere. It is not hard
to check that for $a=1$ Theorem C.1 gives us eigenvalues

$\pm n,\qquad n=2,3,\dots,$ (C.11)

with multiplicity

$n^{2}-1.$ (C.12)

In particular, the spectrum is symmetric about $0$.

  3. (iii)

Formulae (C.11) and (C.12) agree with [7, Theorem 5.2].

  4. (iv)

Note that in (C.6) the lowest value the integer $l$ can take is $1$. The case
$l=0$ is special: in this case the square root becomes superfluous and the
expression (C.6) turns into (C.3). We have highlighted the special status of
the case $l=0$ by separating out the series of eigenvalues (C.3).

  5. (v)

Observe that the expression (B.8) appears under the square root in formulae
(C.6) and (C.10).

###  C.2 The eta invariant

In this appendix we will compute the eta invariant for the operator
$\operatorname{curl}$ on a Berger 3-sphere directly, using the explicit
formulae for the eigenvalues established above in Theorem C.1. Recall that the
eta-function is defined, in general, in accordance with (1.2).

######  Theorem C.3.

Consider the operator $\operatorname{curl}$ on a Berger 3-sphere, defined as
in Appendix B.1. We have

$\eta_{\operatorname{curl}}(0)=\frac{2}{3}(a^{2}-1)^{2}.$ (C.13)

######  Remark C.4.

The above theorem warrants a number of remarks.

  1. (i)

The eta invariant for the Dirac operator on a Berger 3-sphere was computed by
Hitchin [29, p. 34] and reads

$\eta_{\mathrm{Dirac}}(0)=-\frac{1}{6}(a^{2}-1)^{2}\,.$ (C.14)

Note that the parameter $\lambda$ in [29] corresponds to our parameter $a$,
and that Hitchin adopts the same (or rather, equivalent) conventions as we do
when defining the Dirac operator. The latter can be seen if one compares the
spectra of the Dirac operator in [29] and [24]: Hitchin’s [29, Proposition
3.2] with $p=q=1$ and minus sign in front of the square root agrees with [24,
formulae (6.5) and (6.8)].

The result (C.14) 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 2.1).

  2. (ii)

For a general connected oriented closed Riemannian 3-manifold the invariants
$\eta_{\operatorname{curl}}(0)$ and $\eta_{\mathrm{Dirac}}(0)$ are related as

$\eta_{\operatorname{curl}}(0)=-4\eta_{\mathrm{Dirac}}(0)+k$ (C.15)

for some $k\in\mathbb{Z}$. This follows from abstract algebraic topology
arguments involving Hirzebruch polynomials, see [2, Section 1] and [3,
Theorems 4.2 and 4.14]. Theorem C.3 establishes that, in the case of a Berger
3-sphere, we have (C.15) with $k=0$ for $a<4$. The significance of the value
$a=4$ is that for this particular value of the parameter $a$ the Dirac
operator has a zero mode (harmonic spinor).

  3. (iii)

The properties of the Dirac operator on a Berger 3-sphere, including an
alternative (to Hitchin’s [29]) derivation of the spectrum, were further
studied by Bär in [6]. Note that Bär adopts the opposite convention when
defining the Dirac operator, so that the eigenvalues of the Dirac operator in
[6] differ from those in [29, 24] by sign .

In preparation for the proof of Theorem C.3, let us make the following
observation.

######  Lemma C.5.

The function $\eta_{\operatorname{curl}}$ admits the representation

$\eta_{\operatorname{curl}}(s)=\theta(s)+(2a)^{-s}+4a^{s}\zeta(s-1)\,,$ (C.16)

where

$\theta(s):=\sum_{j=1}^{\infty}\left[\left(\sqrt{a^{2}+\mu_{j}}\,+\,a\right)^{-s}-\left(\sqrt{a^{2}+\mu_{j}}\,-\,a\right)^{-s}\right],$
(C.17)

the $\mu_{j}$ are the eigenvalues (2.17) of $\,-\Delta\,$ enumerated in
increasing order with account of multiplicities, and
$\,\zeta(s):=\sum_{n=1}^{\infty}n^{-s}\,$ is the Riemann zeta function.

###### Proof.

Formulae (C.16) and (C.17) are a straightforward consequence of Theorems B.2
and C.1. ∎

######  Definition C.6.

We define

$\zeta_{-\Delta}(s):=\sum_{j=1}^{\infty}\mu_{j}^{-s}$

to be the zeta function444The function $s\mapsto\zeta_{-\Delta}(s)$ is often
called the _Minakshisundaram–Pleijel zeta function_ , see [41, equation (6)].
associated with (minus) the Laplace–Beltrami operator $-\Delta$. Similarly, we
define

$\zeta_{\sqrt{-\Delta}}(s):=\sum_{j=1}^{\infty}\mu_{j}^{-s/2}=\zeta_{-\Delta}(s/2)\,.$

It is known [41, p. 243] that the function $\zeta_{\sqrt{-\Delta}}(s)$ is
meromorphic in $\mathbb{C}$ with simple poles at $s=3,1,-1,\dots$.

######  Lemma C.7.

For a closed Riemannian 3-manifold $(M,g)$ we have

$\zeta_{\sqrt{-\Delta}}(s)=\frac{1}{2\pi^{2}}\left[\frac{\operatorname{Vol}(M)}{s-3}+\frac{\int_{M}\operatorname{Sc}(x)\,\rho(x)\,\mathrm{d}x}{12(s-1)}+\ldots\right].$
(C.18)

###### Proof.

Let $N_{\sqrt{-\Delta}}(\lambda)$ be the counting function for the positive
eigenvalues of $\sqrt{-\Delta}$, see, e.g., [14, Appendix B]. The functions
$\zeta_{\sqrt{-\Delta}}$ and $N_{\sqrt{-\Delta}}$ are related as

$\zeta_{\sqrt{-\Delta}}(s)=\int_{0}^{+\infty}\lambda^{-s}\,N^{\prime}_{\sqrt{-\Delta}}(\lambda)\,\mathrm{d}\lambda\,,$
(C.19)

where the prime stands for differentiation with respect to $\lambda$. By
combining [14, equation (B.1) and Theorem B.2] with formula (C.19), we obtain
(C.18). ∎

Since for the Berger 3-sphere $\operatorname{Vol}(M)=2\pi^{2}a$ and
$\operatorname{Sc}=8-2a^{2}$, we have the following.

######  Corollary C.8.

For the Berger 3-sphere

$\zeta_{\sqrt{-\Delta}}(s)=a\left[\frac{1}{s-3}+\frac{4-a^{2}}{6(s-1)}+\ldots\right].$
(C.20)

We are now in a position to prove Theorem C.3.

###### Proof of Theorem C.3.

Since $\,\zeta(-1)=-\frac{1}{12}\,$, formula (C.16) implies that the proof of
(C.13) reduces to the proof of

$\theta(0)=\frac{2}{3}\,a^{2}(a^{2}-2)\,,$ (C.21)

where $\theta$ is the function (C.17). Formulae (C.21) and (C.17) reduce the
analysis of the operator $\,\operatorname{curl}\,$ to the analysis of the
Laplace–Beltrami operator.

To establish (C.21) we use a trick due to Hitchin [29, p. 34]. We expand the
terms in the RHS of (C.17) in inverse powers of $\mu_{j}$ as

$\Bigl{(}\sqrt{a^{2}+\mu_{j}}\,\pm\,a\Bigr{)}^{-s}=\mu_{j}^{-s/2}\Bigl{(}\sqrt{1+a^{2}\mu_{j}^{-1}}\,\pm\,a\mu_{j}^{-1/2}\Bigr{)}^{-s}\\\
=\mu_{j}^{-s/2}\Bigl{(}1\pm
a\mu_{j}^{-1/2}+\frac{1}{2}a^{2}\mu_{j}^{-1}+O\left(\mu_{j}^{-2}\right)\Bigr{)}^{-s}\\\
=\mu_{j}^{-s/2}\Bigl{(}1-s\Bigl{[}\pm
a\mu_{j}^{-1/2}+\frac{1}{2}a^{2}\mu_{j}^{-1}\Bigr{]}+\frac{s(s+1)}{2}\Bigl{[}a^{2}\mu_{j}^{-1}\pm
a^{3}\mu_{j}^{-3/2}\Bigr{]}\mp\frac{s(s+1)(s+2)}{6}a^{3}\mu_{j}^{-3/2}+O\left(\mu_{j}^{-2}\right)\Bigr{)}\\\
=\mu_{j}^{-s/2}\Bigl{(}1\mp
sa\mu_{j}^{-1/2}+\frac{s^{2}}{2}a^{2}\mu_{j}^{-1}\pm\frac{s(1-s^{2})}{6}a^{3}\mu_{j}^{-3/2}+O\left(\mu_{j}^{-2}\right)\Bigr{)}.$
(C.22)

Substituting (C.22) into (C.17) we get

$\theta(s)=-\,2sa\,\zeta_{\sqrt{-\Delta}}(s+1)\,+\,\frac{s(1-s^{2})}{3}\,a^{3}\,\zeta_{\sqrt{-\Delta}}(s+3)\,+\,h(s)\,,$
(C.23)

where $h$ is analytic at $s=0$ and

$h(0)=0\,.$ (C.24)

Formulae (C.23), (C.20) and (C.24) imply

$\lim_{s\to
0}\theta(s)=-\,2a^{2}\,\frac{4-a^{2}}{6}\,+\,\frac{1}{3}\,a^{4}=\frac{2}{3}\,a^{2}(a^{2}-2)\,,$

which proves (C.21). ∎

##  Appendix D Maxwell’s equations and electromagnetic chirality

The focus of our paper is the study of asymmetry between positive and negative
eigenvalues of the operator $\operatorname{curl}$. It is natural to ask the
question: does the sign of an eigenvalue of $\operatorname{curl}$ have a
physical meaning? We show in this appendix that it does. The key here is the
notion of _electromagnetic chirality_ , see Definition D.3.

Let $(M,g)$ be a connected oriented closed Riemannian 3-manifold. Consider
homogeneous vacuum Maxwell equations on $M\times\mathbb{R}$

$\begin{pmatrix}0&\operatorname{curl}\\\
-\operatorname{curl}&0\end{pmatrix}\begin{pmatrix}E\\\
B\end{pmatrix}=\frac{\partial}{\partial t}\begin{pmatrix}E\\\ B\end{pmatrix},$
(D.1) $\delta E=\delta B=0,$ (D.2)

where $E$ (electric field) and $B$ (magnetic field) are the unknown
quantities, time-dependent real-valued 1-forms.

We list below two basic symmetries of Maxwell’s equations (D.1), (D.2). All
the results in this appendix are given without proof because the corresponding
proofs are elementary.

######  Definition D.1.

Given a pair of real-valued 1-forms $E$ and $B$, we define the _duality
transform_ as the linear map

$\operatorname{dual}:\begin{pmatrix}E\\\
B\end{pmatrix}\mapsto\begin{pmatrix}-B\\\ E\end{pmatrix}.$ (D.3)

The duality transform can be interpreted as the action of the 4-dimensional
Lorentzian Hodge star on the electromagnetic tensor

$F_{\alpha\beta}=\begin{pmatrix}0&E_{1}&E_{2}&E_{3}\\\
-E_{1}&0&-B_{3}&B_{2}\\\ -E_{2}&B_{3}&0&-B_{1}\\\
-E_{3}&-B_{2}&B_{1}&0\end{pmatrix}.$ (D.4)

Observe that the duality transform is a skew-involution, i.e. applying it
twice gives minus identity.

######  Lemma D.2.

Maxwell’s equations (D.1), (D.2) are invariant under the action of the duality
transform (D.3), i.e. the duality transform maps solutions to solutions.

######  Definition D.3 ([49, 37]).

Given a pair of real-valued 1-forms $E$ and $B$, we define _electromagnetic
chirality_ as the quadratic functional

$\operatorname{chir}(E,B):=\int_{M}(E\wedge\mathrm{d}E+B\wedge\mathrm{d}B).$
(D.5)

######  Lemma D.4.

Chirality (D.5) is a conserved quantity for Maxwell’s equations (D.1), (D.2),
i.e. if $(E,B)$ is a solution then $\operatorname{chir}(E,B)$ does not depend
on time $t$.

We will be seeking harmonic (in the time variable $t$) solutions of Maxwell’s
equations.

######  Definition D.5.

A harmonic solution $(E,B)$ of Maxwell’s equations (D.1), (D.2) is said to be
_polarised_ if $\,(\dot{E},\dot{B})=\pm\lambda\operatorname{dual}(E,B)\,$.
Here the dot stands for derivative in $t$ and $\lambda$ is the angular
frequency.

Recall that $(\lambda_{j},u_{j})$, $j=\pm 1,\pm 2,\ldots$, is the eigensystem
for $\operatorname{curl}$. It is easy to see that

$\begin{pmatrix}E\\\ B\end{pmatrix}=\begin{pmatrix}u_{j}\cos(\lambda_{j}t)\\\
-u_{j}\sin(\lambda_{j}t)\end{pmatrix}$ (D.6)

and

$\begin{pmatrix}E\\\ B\end{pmatrix}=\begin{pmatrix}u_{j}\sin(\lambda_{j}t)\\\
u_{j}\cos(\lambda_{j}t)\end{pmatrix}$ (D.7)

are harmonic polarised solutions of Maxwell’s equations (D.1), (D.2). The two
solutions (D.6) and (D.7) are related via the duality transform (D.3) and
describe waves with opposite polarisations. We see that the correspondence
between eigenvalues and eigenforms of $\operatorname{curl}$ on the one hand
and harmonic polarised solutions of Maxwell’s equations on the other is one-
to-two.

Let $E_{0}$ and $B_{0}$ be a pair of infinitely smooth real-valued 1-forms
which do not depend on $t$ and satisfy the condition $\delta E_{0}=\delta
B_{0}=0$. Consider the Cauchy problem

$\left.E\right|_{t=0}=E_{0},\qquad\left.B\right|_{t=0}=B_{0}$ (D.8)

for Maxwell’s equations (D.1), (D.2). It is easy to see that the solution can
be written as a series

$\begin{pmatrix}E\\\
B\end{pmatrix}=\sum_{j\in\mathbb{Z}\setminus\\{0\\}}\left[\begin{pmatrix}u_{j}\cos(\lambda_{j}t)\\\
-u_{j}\sin(\lambda_{j}t)\end{pmatrix}\langle
E_{0}\,,u_{j}\rangle+\begin{pmatrix}u_{j}\sin(\lambda_{j}t)\\\
u_{j}\cos(\lambda_{j}t)\end{pmatrix}\langle
B_{0}\,,u_{j}\rangle\right]+\begin{pmatrix}P_{\mathcal{H}^{1}}E_{0}\\\
P_{\mathcal{H}^{1}}B_{0}\end{pmatrix},$ (D.9)

where $P_{\mathcal{H}^{1}}$ is the orthogonal projection onto the space of
harmonic 1-forms and $\langle\,\cdot\ ,\,\cdot\,\rangle$ is the inner product
(2.5). We see that our harmonic solutions (D.6) and (D.7) provide a basis for
the Cauchy problem (D.8) for Maxwell’s equations (D.1), (D.2).

The following proposition is the main result of this appendix.

######  Proposition D.6.

For both solutions (D.6) and (D.7) we have

$\operatorname{chir}(E,B)=\lambda_{j}\,.$ (D.10)

Formula (D.10) tells us that the sign of an eigenvalue of
$\operatorname{curl}$ has the physical meaning of the sign of chirality of the
corresponding harmonic polarised solution of Maxwell’s equations. And here it
does not matter which polarisation one chooses when writing down the harmonic
solution of Maxwell’s equations.

##  Appendix E Taylor expansions for the operator of parallel transport

This appendix is concerned with local expansions of parallel transport maps in
normal coordinates. These can, in principle, be found in many guises and with
varied degrees of accuracy in the literature. Nevertheless, for the
convenience of the reader and for future reference, we provide here a concise
self-contained derivation, one which agrees with the definitions and sign
conventions set out in this paper.

Let us fix a point $x\in M$ and let us choose normal coordinates centred at
$x$. Let $y\in M$ be a point in a (small) neighbourhood of $x$.

Recall that $Z_{\alpha}{}^{\beta}(x,y)$ is the map realising the parallel
transport of vectors along the unique geodesic connecting $x$ to $y$, whereas
$Z^{\alpha}{}_{\beta}(x,y)$ is the map realising the parallel transport of
covectors along the same geodesic, see (4.6)–(4.8).

######  Proposition E.1.

We have

$Z_{\alpha}{}^{\beta}(0,y)=\delta_{\alpha}{}^{\beta}+\frac{1}{6}\operatorname{Riem}^{\beta}{}_{\mu\alpha\nu}(0)\,y^{\mu}y^{\nu}-\frac{1}{6}\frac{\partial^{2}\Gamma^{\beta}{}_{\mu\alpha}}{\partial
y^{\nu}\partial y^{\rho}}(0)\,y^{\mu}y^{\nu}y^{\rho}+O(|y|^{4})\,$ (E.1)

and

$Z^{\alpha}{}_{\beta}(0,y)=\delta^{\alpha}{}_{\beta}-\frac{1}{6}\operatorname{Riem}^{\alpha}{}_{\mu\beta\nu}(0)\,y^{\mu}y^{\nu}+\frac{1}{6}\frac{\partial^{2}\Gamma^{\alpha}{}_{\mu\beta}}{\partial
y^{\nu}\partial y^{\rho}}(0)\,y^{\mu}y^{\nu}y^{\rho}+O(|y|^{4})\,.$ (E.2)

Furthermore, for all $\tau\in[0,1]$ we have

$Z_{\alpha}{}^{\beta}(y,\tau
y)=\delta_{\alpha}{}^{\beta}+\frac{\tau^{2}-1}{6}\operatorname{Riem}^{\beta}{}_{\mu\alpha\nu}(0)\,y^{\mu}y^{\nu}-\frac{\tau^{3}-1}{6}\frac{\partial^{2}\Gamma^{\beta}{}_{\mu\alpha}}{\partial
y^{\nu}\partial y^{\rho}}(0)\,y^{\mu}y^{\nu}y^{\rho}+O(|y|^{4})\,$ (E.3)

and

$Z^{\alpha}{}_{\beta}(y,\tau
y)=\delta^{\alpha}{}_{\beta}-\frac{\tau^{2}-1}{6}\operatorname{Riem}^{\alpha}{}_{\mu\beta\nu}(0)\,y^{\mu}y^{\nu}+\frac{\tau^{3}-1}{6}\frac{\partial^{2}\Gamma^{\alpha}{}_{\mu\beta}}{\partial
y^{\nu}\partial y^{\rho}}(0)\,y^{\mu}y^{\nu}y^{\rho}+O(|y|^{4})\,.$ (E.4)

Note that formula (E.1) agrees with [16, formula (7.1)].

###### Proof of Proposition E.1.

Let us start with a vector $v^{\alpha}$ at the origin and let us parallel
transport it along a straight line to $y$ — the unique shortest geodesic in
the chosen coordinate system. This means imposing the condition

$y^{\mu}\nabla_{\mu}v^{\beta}=y^{\mu}\left(\partial_{\mu}v^{\beta}+\Gamma^{\beta}{}_{\mu\alpha}v^{\alpha}\right)=0$
(E.5)

along the straight line. Let us now parameterise our straight line with
parameter $t\in[0,1]$ as $z(t)=ty$. Then

$y^{\mu}\partial_{\mu}v^{\beta}=\dot{v}^{\beta}.$ (E.6)

Formulae (E.5) and (E.6) imply

$\left.v^{\beta}\right|_{t=1}=\left.v^{\beta}\right|_{t=0}-\int_{0}^{1}y^{\mu}\Gamma^{\beta}{}_{\mu\gamma}(ty)\,v^{\gamma}(ty)\,dt\\\
=\left.v^{\beta}\right|_{t=0}-\int_{0}^{1}y^{\mu}\left[\frac{\partial\Gamma^{\beta}{}_{\mu\alpha}}{\partial
y^{\nu}}(0)\,y^{\nu}\,t+\frac{1}{2}\frac{\partial^{2}\Gamma^{\beta}{}_{\mu\alpha}}{\partial
y^{\nu}\partial
y^{\rho}}(0)\,y^{\nu}y^{\rho}\,t^{2}+O(|y|^{3})\right]\left[\left.v^{\alpha}\right|_{t=0}+O(|y|^{2})\right]\,dt\,.$
(E.7)

In the above equation we used the fact that

$\left.\dot{v}^{\alpha}\right|_{t=0}=0\,,$

which follows from (E.5), (E.6) and the fact that Christoffel symbols vanish
at the origin.

Using the elementary differential-geometric identity

$\frac{\partial\Gamma^{\beta}{}_{\mu\alpha}}{\partial
y^{\nu}}(0)=-\frac{1}{3}\left(\operatorname{Riem}^{\beta}{}_{\mu\alpha\nu}(0)+\operatorname{Riem}^{\beta}{}_{\alpha\mu\nu}(0)\right)$

and performing integration in (E.7), we obtain

$\left.v^{\beta}\right|_{t=1}=\left.v^{\beta}\right|_{t=0}+\frac{1}{6}\operatorname{Riem}^{\beta}{}_{\mu\alpha\nu}(0)\,y^{\mu}y^{\nu}\left.v^{\alpha}\right|_{t=0}-\frac{1}{6}\frac{\partial^{2}\Gamma^{\beta}{}_{\mu\alpha}}{\partial
y^{\nu}\partial
y^{\rho}}(0)\,y^{\mu}y^{\nu}y^{\rho}\left.v^{\alpha}\right|_{t=0}+O(|y|^{4})\,.$

The latter implies (E.1).

Combining (E.1) with (4.8) one immediately obtains (E.2).

In order to obtain (E.3) let us parallel transport along our straight line
from $y$ to $\tau y$. Arguing as above, we get

$\left.v^{\beta}\right|_{t=\tau}=\left.v^{\beta}\right|_{t=1}-\int_{1}^{\tau}y^{\mu}\,\Gamma^{\beta}{}_{\mu\gamma}(ty)\,v^{\gamma}(ty)\,ds\\\
=\left.v^{\beta}\right|_{t=1}-\int_{1}^{\tau}y^{\mu}\left[-\frac{1}{3}\operatorname{Riem}^{\beta}{}_{\mu\alpha\nu}(0)\,y^{\nu}\,t+\frac{1}{2}\frac{\partial^{2}\Gamma^{\beta}{}_{\mu\alpha}}{\partial
y^{\nu}\partial
y^{\rho}}(0)\,y^{\nu}y^{\rho}\,t^{2}+O(|y|^{3})\right]\left[\left.v^{\alpha}\right|_{t=1}+O(|y|^{2})\right]\,ds\\\
=\left.v^{\beta}\right|_{t=1}+\frac{\tau^{2}-1}{6}\operatorname{Riem}^{\beta}{}_{\mu\alpha\nu}(0)\,y^{\mu}y^{\nu}\left.v^{\alpha}\right|_{t=1}-\frac{\tau^{3}-1}{6}\frac{\partial^{2}\Gamma^{\beta}{}_{\mu\alpha}}{\partial
y^{\nu}\partial
y^{\rho}}(0)\,y^{\mu}y^{\nu}y^{\rho}\left.v^{\alpha}\right|_{t=1}+O(|y|^{4})\,.$

The latter implies (E.3) which, in turn, combined with (4.8) immediately gives
us (E.4). ∎

######  Remark E.2.

For $\tau=0$ expansions (E.3) and (E.4) simplify to read

$Z_{\alpha}{}^{\beta}(y,0)=\delta_{\alpha}{}^{\beta}-\frac{1}{6}\operatorname{Riem}^{\beta}{}_{\mu\alpha\nu}(0)\,y^{\mu}y^{\nu}+\frac{1}{6}\frac{\partial^{2}\Gamma^{\beta}{}_{\mu\alpha}}{\partial
y^{\nu}\partial y^{\rho}}(0)\,y^{\mu}y^{\nu}y^{\rho}+O(|y|^{4})\,$ (E.8)

and

$Z^{\alpha}{}_{\beta}(y,0)=\delta^{\alpha}{}_{\beta}+\frac{1}{6}\operatorname{Riem}^{\alpha}{}_{\mu\beta\nu}(0)\,y^{\mu}y^{\nu}-\frac{1}{6}\frac{\partial^{2}\Gamma^{\alpha}{}_{\mu\beta}}{\partial
y^{\nu}\partial y^{\rho}}(0)\,y^{\mu}y^{\nu}y^{\rho}+O(|y|^{4})\,.$ (E.9)

Formulae (E.1), (E.2), (E.8) and (E.9) agree with (4.7) and (4.8).

##  Appendix F An alternative derivation of formula (1.13)

In this appendix we verify, using an alternative method, formula (1.13), one
of the main results of our paper.

In what follows we work in normal coordinates and we assume that curvature
(but not its covariant derivative) vanishes at the origin, so that

$g_{\alpha\beta}(x)=\delta_{\alpha\beta}-\frac{1}{6}(\nabla_{\sigma}\operatorname{Riem}_{\alpha\mu\beta\nu})(0)\,x^{\sigma}\,x^{\mu}\,x^{\nu}+O(|x|^{4}).$
(F.1)

We continue using the notation
$\bm{\Delta}=-(\mathrm{d}\delta+\delta\mathrm{d})$ for the Hodge Laplacian on
1-forms (5.8); $|\xi|$ stands for the Euclidean norm of $\xi$, whereas
$\|\xi\|$ stands for the Riemannian norm thereof.

######  Proposition F.1.

Let

$[s]_{\alpha}{}^{\beta}(x,\xi)\sim\|\xi\|^{-1}\delta_{\alpha}{}^{\beta}+[s_{-2}]_{\alpha}{}^{\beta}(x,\xi)+[s_{-3}]_{\alpha}{}^{\beta}(x,\xi)+[s_{-4}]_{\alpha}{}^{\beta}(x,\xi)+\dots$

be the (left) symbol of the operator $(-\bm{\Delta})^{-1/2}$ (recall (5.9)).
Then

$A_{\mathrm{prin}}(x,\xi)=-\varepsilon_{\beta}{}^{\alpha\gamma}\left(\,([s_{-3}]_{\alpha}{}^{\beta})_{x^{\gamma}}+i\,\xi_{\gamma}\,[s_{-4}]_{\alpha}{}^{\beta}\right)+O(|\xi|^{-3}|x|)\,.$
(F.2)

###### Proof.

The claim follows from (5.10), (6.9), (6.30) and the fact that the full symbol
of $\operatorname{curl}$ is (2.1). ∎

In the remainder of this appendix, we will derive explicit formulae for
$[s_{-3}]_{\alpha}{}^{\beta}$ and $[s_{-3}]_{\alpha}{}^{\beta}$. The proofs
are straightforward, hence omitted.

######  Lemma F.2.

Let

$[q]_{\alpha}{}^{\beta}(x,\xi)\sim\|\xi\|^{2}\delta_{\alpha}{}^{\beta}+[q_{1}]_{\alpha}{}^{\beta}(x,\xi)+[q_{0}]_{\alpha}{}^{\beta}(x,\xi)$

be the (left) symbol of the operator $-\bm{\Delta}$. We have

$[q_{1}]_{\alpha}{}^{\beta}(x,\xi)=i\,a_{\alpha}{}^{\beta\gamma}{}_{\mu\nu}\,\xi_{\gamma}\,x^{\mu}\,x^{\nu}+O(|\xi|\,|x|^{3})\,,$
(F.3)
$[q_{0}]_{\alpha}{}^{\beta}(x)=b_{\alpha}{}^{\beta}{}_{\nu}\,x^{\nu}+O(|x|^{2})\,,$
(F.4)

where

$a_{\alpha}{}^{\beta\gamma}{}_{\mu\nu}:=\left[\frac{1}{2}\nabla_{\mu}\operatorname{Ric}^{\gamma}{}_{\nu}-\frac{1}{12}\nabla^{\gamma}\operatorname{Ric}_{\mu\nu}\right]\delta_{\alpha}{}^{\beta}\\\
-\frac{1}{6}\left[\nabla_{\alpha}\operatorname{Riem}^{\gamma}{}_{\mu}{}^{\beta}{}_{\nu}-3\nabla_{\mu}\operatorname{Riem}^{\gamma}{}_{\alpha}{}^{\beta}{}_{\nu}+5\nabla_{\nu}\operatorname{Riem}^{\gamma}{}_{\mu}{}^{\beta}{}_{\alpha}\right],$
(F.5)
$b_{\alpha}{}^{\beta}{}_{\nu}:=-\frac{1}{6}\nabla^{\beta}\operatorname{Ric}_{\alpha\nu}+\frac{1}{2}\nabla_{\alpha}\operatorname{Ric}^{\beta}{}_{\nu}+\frac{1}{2}\nabla_{\nu}\operatorname{Ric}_{\alpha}{}^{\beta}\,.$
(F.6)

######  Lemma F.3.

The symbols

$[r]_{\alpha}{}^{\beta}(x,\xi)\sim\|\xi\|\delta_{\alpha}{}^{\beta}+[r_{0}]_{\alpha}{}^{\beta}(x,\xi)+[r_{-1}]_{\alpha}{}^{\beta}(x,\xi)+[r_{-2}]_{\alpha}{}^{\beta}(x,\xi)+\dots$

and $s_{\alpha}{}^{\beta}$ of $(-\bm{\Delta})^{1/2}$ and
$(-\bm{\Delta})^{-1/2}$ are expressed in terms of (F.3) and (F.4) via the
following hierarchy of identities:

$[r_{0}]_{\alpha}{}^{\beta}=\frac{1}{2|\xi|}[q_{1}]_{\alpha}{}^{\beta}-\frac{1}{2i|\xi|^{2}}\xi^{\mu}(\|\xi\|)_{x^{\mu}}\,\delta_{\alpha}{}^{\beta}+O(|x|^{3})\,,$
$[r_{-1}]_{\alpha}{}^{\beta}\\\
=\frac{1}{2|\xi|}[q_{0}]_{\alpha}{}^{\beta}-\frac{1}{2i|\xi|^{2}}\xi^{\mu}([r_{0}]_{\alpha}{}^{\beta})_{x^{\mu}}\\\
+\frac{1}{4|\xi|}(|\xi|)_{\xi_{\mu}\xi_{\nu}}(\|\xi\|)_{x^{\mu}x^{\nu}}\,\delta_{\alpha}{}^{\beta}\\\
+O(|\xi|^{-1}|x|^{2})\,,$
$[r_{-2}]_{\alpha}{}^{\beta}=-\frac{1}{2i|\xi|^{2}}\xi^{\mu}([r_{-1}]_{\alpha}{}^{\beta})_{x^{\mu}}\\\
+\frac{1}{4|\xi|}(|\xi|)_{\xi_{\mu}\xi_{\nu}}([r_{0}]_{\alpha}{}^{\beta})_{x^{\alpha}x^{\beta}}+\frac{1}{12i|\xi|}(|\xi|)_{\xi_{\mu}\xi_{\nu}\xi_{\rho}}(\|\xi\|)_{x^{\mu}x^{\nu}x^{\rho}}\,\delta_{\alpha}{}^{\beta}\\\
+O(|\xi|^{-2}|x|)\,,$
$[s_{-2}]_{\alpha}{}^{\beta}=-|\xi|^{-2}[r_{0}]_{\alpha}{}^{\beta}-\frac{1}{i|\xi|^{2}}\,\xi^{\mu}\,(\|\xi\|^{-1})_{x^{\mu}}\,\delta_{\alpha}{}^{\beta}+O(|\xi|^{-2}|x|^{3})\,,$
$[s_{-3}]_{\alpha}{}^{\beta}=-|\xi|^{-2}[r_{-1}]_{\alpha}{}^{\beta}-\frac{1}{i|\xi|^{2}}\xi^{\mu}\,([s_{-2}]_{\alpha}{}^{\beta})_{x^{\mu}}\\\
+\frac{1}{2|\xi|}(|\xi|)_{\xi_{\mu}\xi_{\nu}}(\|\xi\|^{-1})_{x^{\mu}x^{\nu}}\,\delta_{\alpha}{}^{\beta}+O(|\xi|^{-3}|x|^{2})\,,$
(F.7)
$[s_{-4}]_{\alpha}{}^{\beta}=-|\xi|^{-2}[r_{-2}]_{\alpha}{}^{\beta}-\frac{1}{i|\xi|^{2}}\,\xi^{\mu}\,([s_{-3}]_{\alpha}{}^{\beta})_{x^{\mu}}\\\
+\frac{1}{2|\xi|}(|\xi|)_{\xi_{\mu}\xi_{\nu}}([s_{-2}]_{\alpha}{}^{\beta})_{x^{\mu}x^{\nu}}+\frac{1}{6i|\xi|}(|\xi|)_{\xi_{\mu}\xi_{\nu}\xi_{\rho}}(\|\xi\|^{-1})_{x^{\mu}x^{\nu}x^{\rho}}\,\delta_{\alpha}{}^{\beta}\\\
+O(|\xi|^{-4}|x|)\,.$ (F.8)

######  Remark F.4.

Lemmata F.2 and F.3 are true in any dimension $d$, not only in dimension
$d=3$.

In order to verify our formula (1.13) for $A_{\mathrm{prin}}\,$, it only
remains to compute (for example, with the help of computer algebra) the right-
hand sides of (F.7) and (F.8), and substitute the resulting expressions into
(F.2).

For instance, for the particular choice of metric determined by

$\operatorname{Ric}(x)=\begin{pmatrix}0&0&0\\\ 0&0&x^{1}\\\ 0&x^{1}&0\\\
\end{pmatrix}+O(|x|^{2})\ $

(recall (6.24) and (F.1)) we get

$\left.s_{-3}\right|_{(x,\xi_{0})}=-\frac{1}{12}\begin{pmatrix}0&3x^{3}&3x^{2}\\\
-x^{3}&0&3x^{1}\\\
-x^{2}&3x^{1}&0\end{pmatrix}+O(|x|^{2}),\quad\left.s_{-4}\right|_{(x,\xi_{0})}=-\frac{i}{2}\begin{pmatrix}0&0&0\\\
1&0&0\\\ 0&0&0\end{pmatrix}+O(|x|)\,,$ (F.9)

where $\xi_{0}$ is given by (6.26). Substituting (F.9) into (F.2) we obtain

$A_{\mathrm{prin}}(0,\xi_{0})=-\frac{1}{2}\,,$

which agrees with (1.13) — see also (6.36).

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