# Remarks on the $\mathbb{Z}/p$ Dijkgraaf-Witten invariants of 3D mapping tori

William Chen  Department of Mathematics, Rutgers University<EMAIL_ADDRESS>,  Alex Kontorovich  Department of Mathematics, Rutgers University
<EMAIL_ADDRESS>and  Shehryar Sikander  New High Energy Theory
Center, Rutgers University<EMAIL_ADDRESS>
###### Abstract.

We make some remarks on the $\mathbb{Z}/p$ Dijkgraaf-Witten invariants of 3D
mapping tori and determine the asymptotic behavior of their sum over all
diffeomorphism classes of 3D mapping tori of genus one.

A. Kontorovich supported under NSF grants: DMS-1802119 and BSF 2020119.

S. Sikander supported under US DOE grant: DOE-SC0010008 to Rutgers

###### Contents

  1. 1 Introduction
    1. 1.1 Acknowledgements
  2. 2 Preliminaries: $\mathbb{Z}/p$ Dijkgraaf-Witten invariants of 3D mapping tori and the Smith normal form
  3. 3 $\mathbb{Z}/p$ Dijkgraaf-Witten invariants of 3D mapping tori of genus one
  4. 4 Counting estimates and Proofs of Theorems 1.1 and 1.2
  5. 5 $\mathbb{Z}/2$ Dijkgraaf-Witten invariants of 3D mapping tori of genus one and modular forms
  6. 6 Further remarks: $\mathrm{SU}(2)$ Chern-Simons-Witten invariants of mapping tori of genus one

##  1\. Introduction

Starting with the Euclidean gravitational path integral defined by Hartle and
Hawking, sums over diffeomorphism classes of D dimensional manifolds are by
now ubiquitous in quantum gravity. Recently, Banerjee and Moore [1] considered
sums of TQFT invariants over equivalence classes of one and two dimensional
cobordisms. In this note we study sums of $\mathbb{Z}/p$ Dijkgraaf-Witten
invariants over diffeomorphism classes of 3D mapping tori.

Let $G$ be a finite group. Using gauge theory with gauge group $G$, Dijkgraaf
and Witten [3] constructed invariants of oriented three manifolds which depend
on a choice of a class in $H^{3}(BG,U(1))$. Let $S$ be a closed, connected,
and oriented surface of genus bigger than zero, let $\Gamma(S)$ denote its
mapping class group, and let $[\Gamma(S)]$ be the set of conjugacy classes of
$\Gamma(S)$. For $f$ in $\Gamma(S)$ let $M(f)$ denote the three dimensional
mapping torus of $f$. Since conjugate mapping classes give diffeomorphic
mapping tori, the Dijkgraaf-Witten invariant only depends on the conjugacy
class $[f]$ in $[\Gamma(S)]$. Let $Z(M([f]),G,\omega)$ denote the Dijkgraaf-
Witten invariant for some $\omega$ in $H^{3}(BG,U(1))$, and let
$H:[\Gamma(S)]\to\mathbb{R}$ be a suitable ordering function. We seek the
asymptotics of the sum

$\displaystyle\sum_{[f]\in[\Gamma(S)],\atop H([f])<T}Z(M([f],G,\omega)$ (1)

as $T\to\infty$. Let $S$ be of genus one, then $\Gamma(S)$ is isomorphic to
$\mathrm{SL}(2,\mathbb{Z})$ and the absolute value of the trace provides a
natural ordering function. Let $p$ be an odd prime and for a conjugacy class
$[A]$ in $[\mathrm{SL}(2,\mathbb{Z})]$ let $Z(M([A]),\mathbb{Z}/p)$ denote the
$\mathbb{Z}/p$ Dijkgraaf-Witten invariant, with $\omega$ the identity class in
$H^{3}(B(\mathbb{Z}/p),U(1))$, of the mapping torus $M([A])$. Using the
formula for $Z(M([A],\mathbb{Z}/p)$ derived in Corollary 3.3 and Sarnak’s
Chebotarev density theorem for prime geodesics, we prove the following.

######  Theorem 1.1.

Let $p$ be a fixed odd prime. Then the $\mathbb{Z}/p$ Dijkgraaf-Witten
invariants of genus one mapping tori satisfy

$\displaystyle\sum_{[A]\in[\mathrm{SL}(2,\mathbb{Z})],\atop|Tr([A])|<T}Z(M([A]),\mathbb{Z}/p)=\frac{2p^{3}-2p+1}{p^{3}-p}\operatorname{li}(T^{2})+O(T^{3/2}(\log
T)^{2}),$ (2)

as $T\to\infty$. Here

$\displaystyle\operatorname{li}(x)=\int_{2}^{x}\frac{dt}{\log t},$ (3)

is the logarithmic integral function (as in the prime number theorem).

In the case $p=2$, similar asymptotics can be derived using results of Section
5. In Section 5, we also prove that the numbers $Z(M([A]),\mathbb{Z}/2)+2$ are
Fourier coefficients of a modular form of weight one, see Theorem 5.3 for
details.

Let $S$ be of arbitrary genus $g$ and for a mapping class $f$ in $\Gamma(S)$
denote by $\hat{f}$ the matrix in $\mathrm{Sp}(2g,\mathbb{Z})$ which gives the
action of $f$ on $H_{1}(S,\mathbb{Z})$, with respect to a fixed basis of
$H_{1}(S,\mathbb{Z})$. It is classical that the Smith normal form of the
matrix $(\hat{f}-\operatorname{Id})$ determines $H_{1}(M([f]),\mathbb{Z})$,
and hence the invariants $Z([f],\mathbb{Z}/p,\omega)$ when $\omega$ is the
identity class, completely, see (23), (25), and (28). The duality between the
3D Dijkgraaf-Witten theory and 2D rational conformal field theory (holomorphic
orbifold models in particular), allows one to use methods of rational
conformal field theory to study Smith normal forms of matrices
$(\hat{f}-\operatorname{Id})$ where $\hat{f}$ is in the symplectic group. As
an application of these 2D methods and Sarnak’s Chebotarev density theorem, we
prove the following refinements of Theorem 1.1.

######  Theorem 1.2.

For $[A]$ in $[\mathrm{SL}(2,\mathbb{Z})]$, let $A(1)$ and $A(2)$ denote the
diagonal entries of the Smith normal form of the matrix
$(A-\operatorname{Id})$, see (62). Let $p$ be an odd prime. Then

$\displaystyle\sum_{[A]\in[\mathrm{SL}(2,\mathbb{Z})],|Tr([A])|<T,\atop
p|A(1)\,\text{and}\,p|A(2)}$
$\displaystyle=\frac{1}{p^{3}-p}\operatorname{li}(T^{2})+O(T^{3/2}(\log
T)^{2})$ (4)
$\displaystyle\sum_{[A]\in[\mathrm{SL}(2,\mathbb{Z})],|Tr([A])|<T,\atop p\nmid
A(1)\,\text{and}\,p|A(2)}$
$\displaystyle=\frac{p^{2}-1}{p^{3}-p}\operatorname{li}(T^{2})+O(T^{3/2}(\log
T)^{2})$ (5)
$\displaystyle\sum_{[A]\in[\mathrm{SL}(2,\mathbb{Z})],|Tr([A])|<T,\atop p\nmid
A(1)\,\text{and}\,p\nmid A(2)}$
$\displaystyle=\frac{p^{3}-p^{2}-p-1}{p^{3}-p}\operatorname{li}(T^{2})+O(T^{3/2}(\log
T)^{2})$ (6)

as $T\to\infty$.

These asymptotics should be compared with the nice results in [10]. For an
application of these 2D methods in genus greater than one, see (59). Perhaps
these 2D methods can be leveraged to give faster than the state of the art
algorithms for computing the Smith normal forms of
$(\hat{f}-\operatorname{Id})$; one wonders if and how the topological quantum
computers based on 2D rational conformal fields theories can calculate Smith
normal forms faster than classical computers.

###  1.1. Acknowledgements

Gregory Moore, Daniel Friedan, Henryk Iwaniec, Will Sawin, and Anindya
Banerjee are heartily thanked for discussions and encouragement.

##  2\. Preliminaries: $\mathbb{Z}/p$ Dijkgraaf-Witten invariants of 3D
mapping tori and the Smith normal form

Let $G$ be a finite group and let $M$ be a closed, connected, and oriented
three manifold. Dijkgraaf and Witten introduced an invariant of $M$, which
depends on the choice of a class in $H^{3}(G,U(1))$, as follows: Any $\rho$ in
$\operatorname{Hom}(\pi_{1}(M),G)$ determines a principal $G$-bundle
$E_{\rho}\to M$ as the quotient

$\displaystyle\tilde{M}\times_{\pi_{1}(M)}G:=(\tilde{M}\times G)/\pi_{1}(M)$
(7)

where $\tilde{M}$ is the universal cover of $M$ and $\pi_{1}(X)$ acts on
$\tilde{M}$ as the deck group and acts on $G$ by left multiplication via
$\rho$. The existence of the universal $G$-bundle $EG\to BG$ on the
classifying space $BG$ of the group $G$ implies that there exists a unique (up
to homotopy) continuous map

$\displaystyle\hat{\rho}:M\to BG$ (8)

such that

$\displaystyle E_{\rho}\cong\hat{\rho}^{*}(EG)$ (9)

While isomorphism classes of $G$-bundles are parametrized by the set
$\operatorname{Hom}(\pi_{1}(M),G)/G$, the set
$\operatorname{Hom}(\pi_{1}(M),G)$ parametrizes homotopy classes of pointed
maps; if one chooses a point in $M$ and a point in $BG$, then there is an
equivalence between the set $\operatorname{Hom}(\pi_{1}(M),G)$, where
$\pi_{1}(M)$ and $\pi_{1}(BG)\cong G$ are defined with respect to the chosen
point in $M$ and $BG$, and the set of homotopy equivalence classes of
continuous maps from $M$ to $BG$ which map the chosen point in $M$ to the
chosen point in $BG$.

Let $\omega$ be a class in $H^{3}(BG,U(1))$ and let $\rho$ be a homomorphism
in $\operatorname{Hom}(\pi_{1}(M),G)$. Using the continuous map $\hat{\rho}$
from (8) associated with $\rho$, one pulls back the class $\omega$ to obtain
the class

$\displaystyle\hat{\rho}^{*}(\omega)\in H^{3}(M,U(1))$ (10)

on the three manifold $M$. The pulled back class (10) only depends on the
homotopy class of $\hat{\rho}$ and in this way one associates a well defined
cohomology class in $H^{3}(M,U(1))$ with every $\rho$ in
$\operatorname{Hom}(\pi_{1}(M),G)$. The top degree homology group
$H_{3}(M,\mathbb{Z})$ is infinite cyclic and the orientation of $M$ determines
a generator of this group. Denote this generator by $[M]$ which is also called
the fundamental class of $M$. One can now evaluate the pull-back (10) on the
fundamental class, i.e.

$\displaystyle\langle\hat{\rho}^{*}(\omega),[M]\rangle=\int_{M}\hat{\rho}^{*}(\omega)\in
U(1)$ (11)

The Dijkgraaf-Witten invariant of $M$ with respect to $\omega$ in
$H^{3}(BG,U(1))$ is defined as

$\displaystyle
Z(M,G,\omega):=\frac{1}{|G|}\sum_{\rho\in\operatorname{Hom}(\pi_{1}(M),G)}\langle\hat{\rho}^{*}(\omega),[M]\rangle$
(12)

In case one chooses $\omega$ to be the identity element of the group
$H^{3}(M,U(1))$, the formula (12) specializes to

$\displaystyle Z(M,G):=\frac{|\operatorname{Hom}(\pi_{1}(M),G)|}{|G|}$ (13)

and will be refered to as the untwisted Dijkgraaf-Witten invariant.

######  Example 2.1.

This example is taken from [6] and will be used later. Let $G=\mathbb{Z}/2$,
its classifying space is the infinite dimensional real projective space

$\displaystyle BG=\mathbb{R}\mathbb{P}^{\infty}$ (14)

The authors work with cohomology of $BG$ with $\mathbb{Z}/2$ coeffecients
rather than with $U(1)$ coefficients. We have that

$H^{3}(\mathbb{R}\mathbb{P}^{\infty},\mathbb{Z}/2)=\mathbb{Z}/2$

Let $\alpha$ denote the non-trivial element of
$H^{3}(\mathbb{R}\mathbb{P}^{\infty},\mathbb{Z}/2)$. For any $\rho$ in
$\operatorname{Hom}(\pi_{1}(M),G)$ the evaluation

$\displaystyle\langle\hat{\rho}^{*}(\alpha),[M]\rangle\in\mathbb{Z}/2$ (15)

so that (12) specializes to

$\displaystyle
Z(M,\mathbb{Z}/2,\alpha):=\frac{1}{2}\sum_{\rho\in\operatorname{Hom}(\pi_{1}(M),\mathbb{Z}/2)}(-1)^{\langle\hat{\rho}^{*}(\alpha),[M]\rangle}$
(16)

If $H^{1}(M,\mathbb{Z}/2)=0$ then $Z(M,\mathbb{Z}/2,\alpha)=\frac{1}{2}$,
otherwise $Z(M,\mathbb{Z}/2,\alpha)$ is an integer since an even number of
terms are involved in the sum.

One can extend the definiton of these invariants to manifolds with boundary
and obtain a 3d TQFT where the invariant $Z(M,G,\omega)$ is the partition
function of the theory evaluated on $M$.

Let $S$ be a closed, connected, and oriented surface of genus $g>0$. Let
$\text{Diff}(S)^{+}$ be the group of orientation preserving diffeomorphisms of
$S$ and let $\text{Diff}(S)_{0}^{+}\subset\text{Diff}(S)^{+}$ be the connected
component of the identity, with respect to the compact open topology. The
mapping class group of $S$ is

$\displaystyle\Gamma(S):=\text{Diff}(S)^{+}/\text{Diff}(S)_{0}^{+}$ (17)

It is the group of isotopy classes of orientation preserving diffeomorphims of
$S$. For $f$ in $\Gamma(S)$, the mapping torus of $f$ is the three fold

$\displaystyle M(f):=S\times[0,1]/\sim$ (18)

where the equivalence $\sim$ is given by

$\displaystyle(x,0)\sim(f(x),1)$ (19)

The three fold $M(f)$ is closed, connected, and orientable and its
diffeomorphism class does not depend on the choice of the representative of
the mapping class $f$. It is the total space of a fiber bundle over the circle

$\displaystyle\pi:M(f)\to S^{1}$ (20)

such that for all $x$ in $S^{1}$ the fiber $\pi^{-1}(x)$ is diffeomorphic to
the surface $S$ and the image of the monodromy map
$\pi_{1}(S^{1})\to\Gamma(S)$ is generated by $f$. Two mapping tori $M(f)$ and
$M(g)$ are diffeomorphic to each other as fibre bundles if and only if $f$ is
conjugate to $g$. We note that there exist examples of two mapping tori that
are diffeomorphic to each other as three manifolds, but not diffeomorphic as
fiber bundles.

Let $N(S)\subset M(f)$ be the collar neighbourhood of some fiber of (20) and
let $W=M(f)-\text{Interior}(N(S))$. Since $W$ is diffeomorphic to $S\times I$,
the Mayer-Vietoris sequence associated with the decomposition $M(f)=W\cup
N(S)$ reduces to

$\displaystyle\cdots\to
H_{1}(S,\mathbb{Z})\xrightarrow{\hat{f}-\operatorname{Id}}H_{1}(S,\mathbb{Z})\to
H_{1}(M(f),\mathbb{Z})\to\mathbb{Z}\to 0$ (21)

where

$\displaystyle\hat{f}:H_{1}(S,\mathbb{Z})\to H_{1}(S,\mathbb{Z})$ (22)

is the induced action of $f$ on the homology of $S$, see [7, Fact 2.1]. The
sequence (21) implies

$\displaystyle H_{1}(M(f),\mathbb{Z})\cong\mathbb{Z}\oplus
Coker(\hat{f}-\operatorname{Id})$ (23)

There exist two matrices $P$ and $Q$ in $\mathrm{GL}(2g,\mathbb{Z})$ such that

$\displaystyle P.(\hat{f}-\operatorname{Id}).Q=\begin{bmatrix}A(1)&&&&&&\\\
&A(2)&&&&&\\\ &&\ddots&&&&\\\ &&&A(N)&&&\\\ &&&&0&&\\\ &&&&&\ddots&&\\\
&&&&&&&0\end{bmatrix}$ (24)

where for all $1\leqslant i\leqslant N$ the diagonal entries $A(i)$ are
positive integers such that for all $i>1$, $A(i-1)$ divides $A(i)$. The
diagonal matrix on the RHS is called the Smith normal form of
$(\hat{f}-\operatorname{Id})$ and will be denoted as
$SNF(\hat{f}-\operatorname{Id})$. The $SNF(\hat{f}-\operatorname{Id})$ does
not depend on the pair $P,Q$ and is an invariant of the conjugacy class of
$\hat{f}$ in $\mathrm{Sp}(2g,\mathbb{Z})$. In particular,

$\displaystyle
Coker(\hat{f}-\operatorname{Id})=\mathbb{Z}^{2g-N}\oplus\mathbb{Z}/A(1)\oplus\cdots\oplus\mathbb{Z}/A(N)$
(25)

The diagonal entries $A(i)$ of $SNF(\hat{f}-\operatorname{Id})$ can be
computed from the the entries of the matrix $(\hat{f}-\operatorname{Id})$ as
follows: Let $D(i)$ be the greatest common divisor of all the determinants of
$i\times i$ minors of $(\hat{f}-\operatorname{Id})$, and let $D(0):=0$, then

$\displaystyle A(i)=\frac{D(i)}{D(i-1)}$ (26)

The fastest known algorithm for computing these integers $A(i)$ has runtime
complexity

$\displaystyle
O(\lVert\hat{f}-\operatorname{Id}\rVert\log\lVert\hat{f}-\operatorname{Id}\rVert
16g^{4}\log 2g)$ (27)

Let $G=\mathbb{Z}/p$ for $p$ a prime. From (23) and (25) it follows that the
untwisted Dijkgraaf-Witten invariant is

$\displaystyle Z(M(f),\mathbb{Z}/p)$
$\displaystyle=\frac{|\operatorname{Hom}(H_{1}(M(f),\mathbb{Z}),\mathbb{Z}/p)|}{|\mathbb{Z}/p|}$
(28) $\displaystyle=p^{\big{|}\\{A(i),\,1\leqslant i\leqslant N\,|\,\text{p
divides A(i)}\\}\big{|}+(2g-N)}$ (29)

######  Example 2.2.

Figure 1.

For $1\leqslant i\leqslant 5$ let $D^{n}(t_{i})$ denote the $n^{th}$ power of
the Dehn twist around the loop $t_{i}$ on a genus two surface displayed in
Fig.1, (1). If we let

$\displaystyle f:=D^{39}(t_{1})D^{13}(t_{4})D^{3}(t_{2})D^{19}(t_{3})D(t_{5})$
(30)

then

$\displaystyle\hat{f}=\left(\begin{array}[]{cccc}-116&-1463&39&-2926\\\
0&1&0&1\\\ -3&-38&1&-76\\\ 0&-13&0&-12\\\ \end{array}\right)\quad\quad\quad
SNF(\hat{f}-\operatorname{Id})=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&1&0&0\\\ 0&0&3&0\\\ 0&0&0&507\\\ \end{array}\right)$ (39)

It follows that
$H_{1}(M(f),\mathbb{Z})=\mathbb{Z}\oplus\mathbb{Z}/3\oplus\mathbb{Z}/507$, and
we have

$Z(M(f),\mathbb{Z}/3)=3^{2},\quad
Z(M(f),\mathbb{Z}/13)=13,\;\;\text{and}\;\;Z(M(f),\mathbb{Z}/7)=1$

For another example, let

$\displaystyle f:=D(t_{1})D^{2}(t_{4})$ (40)

then

$\displaystyle\hat{f}=\left(\begin{array}[]{cccc}1&0&1&0\\\ 0&1&0&0\\\
0&0&1&0\\\ 0&-2&0&1\\\ \end{array}\right)\quad\quad\quad
SNF(\hat{f}-\operatorname{Id})=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&2&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\ \end{array}\right)$ (49)

It follows that $H_{1}(M(f),\mathbb{Z})=\mathbb{Z}^{3}\oplus\mathbb{Z}/2$, and

$\displaystyle
Z(M(f),\mathbb{Z}/2)=2^{3},\;\;\text{and}\;\;Z(M(f),\mathbb{Z}/p)=p^{2}\,\,\text{for
all odd primes $p$}$ (50)

The duality between 3D TQFT and 2D RCFT allows one to compute $\mathbb{Z}/p$
Dijkgraaf-Witten invariants without computing the Smith normal form: The long
exact sequence of homotopy groups associated with the fibration (20) reduces
to

$\displaystyle 1\to\pi_{1}(S)\to\pi_{1}(M(f))\to\pi_{1}(S^{1})\to 1$ (51)

The sequence (51) induces an outer action of $\pi_{1}(S^{1})$ on $\pi_{1}(S)$.
This outer action can be refined to an honest action by choosing a section of
the sequence, which is possible since $\pi_{1}(S^{1})\cong\mathbb{Z}$. With
this choice of splitting, $\pi_{1}(M(f))$ becomes the semidirect product

$\displaystyle\pi_{1}(M(f))=\pi_{1}(S)\rtimes\langle f\rangle$ (52)

The action of $\langle f\rangle$ on $\pi_{1}(S)$ induces an action on
$\operatorname{Hom}(\pi_{1}(S),G)$, and

$\displaystyle\operatorname{Hom}(\pi_{1}(M(f)),G)=\\{(\rho,g)\in\operatorname{Hom}(\pi_{1}(S),G)\times
G\,|\,g.\rho.g^{-1}=f^{*}(\rho)\\}$ (53)

In particular, when $G$ is an Abelian group the untwisted Dijkgraaf-Witten
invariant becomes the number of fixed points of the $f$ action on
$\operatorname{Hom}(\pi_{1}(S),G)$;

$\displaystyle Z(M(f),G)$
$\displaystyle=\frac{|\operatorname{Hom}(\pi_{1}(M(f)),G)|}{|G|}$ (54)
$\displaystyle=|\\{\rho\in\operatorname{Hom}(\pi_{1}(S),G)\,|\,f^{*}(\rho)=\rho\\}|$
(55)

In case $G=\mathbb{Z}/p$ for $p$ a prime, we have that
$\operatorname{Hom}(\pi_{1}(S),\mathbb{Z}/p)\cong\operatorname{Hom}((\mathbb{Z}/p)^{2g},\mathbb{Z}/p)$
and the action of $f$ on $\operatorname{Hom}(\pi_{1}(S),\mathbb{Z}/p)$ is
given by the action of the matrix

$\displaystyle(\hat{f}\,\,\text{mod $p$})\in\mathrm{Sp}(2g,\mathbb{Z}/p)$ (56)

on $(\mathbb{Z}/p)^{2g}$. Let
$\mathcal{F}:=L^{2}(\operatorname{Hom}((\mathbb{Z}/p)^{2g},\mathbb{Z}/p),\mathbb{C})$
be the space of complex valued $L^{2}$ functions on
$\operatorname{Hom}((\mathbb{Z}/p)^{2g},\mathbb{Z}/p)$. $\mathcal{F}$ is a
complex vector space of dimension $p^{2g}$, and the action of
$\mathrm{Sp}(2g,\mathbb{Z}/p)$ on
$\operatorname{Hom}((\mathbb{Z}/p)^{2g},\mathbb{Z}/p)$ induces a linear action
on this vector space which we denote by

$\displaystyle\Phi:\mathrm{Sp}(2g,\mathbb{Z}/p)\to\operatorname{Aut}(\mathcal{F})=\mathrm{GL}(p^{2g},\mathbb{C})$
(57)

For every mapping class $f$ in $\Gamma(S)$ this representation has the
property:

$\displaystyle
Trace(\Phi(\hat{f}\,\text{mod}\,\,p))=|\\{\rho\in\operatorname{Hom}((\mathbb{Z}/p)^{2g},\mathbb{Z}/p)\,|\,(\hat{f}\,\text{mod}\,p)(\rho)=\rho\\}|$
(58)

Together with (54) and (28), this implies the following relationship between
the diagonal entries $A(i)$ of the $SNF(\hat{f}-\operatorname{Id})$ and the
representation (57)

$\displaystyle\frac{\log Trace(\Phi(\hat{f}\,\text{mod}\,\,p))}{\log
p}={\big{|}\\{A(i),\,1\leqslant i\leqslant N\,|\,\text{p divides
A(i)}\\}\big{|}+(2g-N)}$ (59)

######  Remark 2.3.

With a rather clever use of two cocycles, it is shown in [3] that for any
finite group $G$ and a class $\omega$ in $H^{3}(BG,U(1))$, there exists a
space of complex valued wave functions on $\operatorname{Hom}(\pi_{1}(S),G)/G$
denoted as $\mathcal{F}^{(\omega)}$, such that the induced representation of
the mapping class group

$\displaystyle\Phi^{(\omega)}:\Gamma(S)\to\operatorname{Aut}(\mathcal{F}^{(\omega)})$
(60)

satisfies

$\displaystyle Trace(\Phi^{(\omega)}(f))=Z(M(f),G,\omega)$ (61)

##  3\. $\mathbb{Z}/p$ Dijkgraaf-Witten invariants of 3D mapping tori of genus
one

Let $S$ be a closed, connected, and oriented surface of genus $g=1$. The
mapping class group $\Gamma(S)$ is isomorphic to $\mathrm{SL}(2,\mathbb{Z})$.
If $A=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$ in $\mathrm{SL}(2,\mathbb{Z})$,
then the entries of the Smith normal form

$\displaystyle SNF(A-\operatorname{Id})=\begin{pmatrix}A(1)&0\\\
0&A(2)\end{pmatrix}$ (62)

can be calculated using (28), and are given as

$\displaystyle A(1)=\mathrm{GCD}(a-1,b,c,d-1)\quad\quad\text{and}\quad\quad
A(2)=\frac{|\operatorname{Tr}(A)-2|}{A(1)}$ (63)

Since $2-\operatorname{Tr}(A)=\operatorname{Det}(A-\operatorname{Id})$, it
follows that $|\operatorname{Tr}(A)-2|$ is divisible by $A(1)^{2}$, confirming
that $A(2)$ is an integer which is divisible by $A(1)$. Denoting by $M(A)$ the
mapping torus of $A$, formula (63) along with (23) and (25) implies the
following trichotomy;

$\displaystyle
H_{1}(M(A),\mathbb{Z})=\begin{cases}\mathbb{Z}\oplus\mathbb{Z}/A(1)\oplus\mathbb{Z}/A(2)&\text{if
$\operatorname{Tr}(A)\neq$ }2\\\ \mathbb{Z}^{2}\oplus\mathbb{Z}/A(1)&\text{if
$\operatorname{Tr}(A)=2$ and $A\neq\operatorname{Id}$ }\\\
\mathbb{Z}^{3}&\text{if $A=\operatorname{Id}$}\end{cases}$ (64)

The following proposition gives a formula for $Z(M(A),\mathbb{Z}/p)$, the
untwisted Dijkgraaf-Witten invariant, defined in (13).

######  Proposition 3.1.

Let $p$ be a prime and let $A$ be in $\mathrm{SL}(2,\mathbb{Z})$, then

$\displaystyle Z(M(A),\mathbb{Z}/p)=\begin{cases}p^{2}&\text{if
$A\equiv\operatorname{Id}\mod p$}\\\ p&\text{if $A\nequiv\operatorname{Id}\mod
p$ and $\operatorname{Tr}(A)\equiv 2\mod p$}\\\ 1&\text{otherwise}\end{cases}$
(65)

###### Proof.

Assume that $\operatorname{Tr}(A)\neq 2$. Then both $A(1)$ and $A(2)$ are non-
zero and

$\displaystyle\hash\\{\operatorname{Hom}(\mathbb{Z}/A(1)\oplus\mathbb{Z}/A(2),\mathbb{Z}/p)\\}=\begin{cases}p^{2}&\text{if
$p$ divides both $A(1)$ and $A(2)$}\\\ p&\text{if $p$ divides $A(1)$ or $A(2)$
}\\\ 1&\text{if $p$ does not divide $A(1)$ and $A(2)$}\end{cases}$ (66)

_If $A\equiv\operatorname{Id}\mod p$ then $p$ divides both $A(1)$ and $A(2)$_.
This follows since if $A\equiv\operatorname{Id}\mod p$ then all four integers
$a-1,b,c,d-1$ are divisible by $p$. Since A(1) is the GCD of these four
integers it follows that $A(1)$ is divisible by $p$. Since $A(2)$ is always
divisible by $A(1)$, it follows that $A(2)$ is also divisible by $p$.

_If $A\nequiv\operatorname{Id}\mod p$ and $\operatorname{Tr}(A)\equiv 2\mod p$
then $p$ divides $A(2)$ but does not divide $A(1)$_. This follows since if
$A\nequiv\operatorname{Id}\mod p$ then $p$ does not divide $A(1)$. Since $p$
does not divide $A(1)$ and $\operatorname{Tr}(A)\equiv 2\mod p$ then from
definition of A(2) in (63) it is clear that $p$ divides A(2).

In all other cases, $p$ divides neither $A(1)$ nor $A(2)$. This proves (69) in
the case $\operatorname{Tr}(A)\neq 2$.

Assume $\operatorname{Tr}(A)=2$ and $A\neq\operatorname{Id}$. In this case the
conjugacy class of $A$ contains a unique upper triangular matrix
$\begin{pmatrix}1&n\\\ 0&1\end{pmatrix}$. Since $A(1)$ and $A(2)$ are
invariants of the conjugacy class of $A$, it follows that $A(1)=n$ and
$A(2)=0$. This implies

$\displaystyle\hash\\{\operatorname{Hom}(\mathbb{Z}\oplus\mathbb{Z}/A(1),\mathbb{Z}/p)\\}=\begin{cases}p^{2}&\text{if
$n\equiv 0\mod p$}\\\ p&\text{otherwise}\end{cases}$ (67)

But if $n\equiv 0\,\,mod\,p$ then $A\equiv\operatorname{Id}\,mod\,p$. If
$A=\operatorname{Id}$, then $M(A)$ is the 3-torus and it is clear that
$Z(M(A),\mathbb{Z}/p)=p^{2}$. ∎

In light of the 3D/2D correspondence given by (54), the corresponding 2D
statement and proof are as follows:

######  Proposition 3.2.

Let $p$ be a prime and let $A$ be in $\mathrm{SL}(2,\mathbb{Z})$, then

$\displaystyle\hash\\{\rho\in\operatorname{Hom}(\mathbb{Z}^{2},\mathbb{Z}/p)\,|\,A(\rho)=\rho\\}=\begin{cases}p^{2}&\text{if
$A\equiv\operatorname{Id}\mod p$}\\\ p&\text{if $A\nequiv\operatorname{Id}\mod
p$ and $\operatorname{Tr}(A)\equiv 2\mod p$}\\\ 1&\text{otherwise}\end{cases}$
(68)

###### Proof.

The $\mathrm{SL}_{2}(\mathbb{Z})$-action on
$\operatorname{Hom}(\mathbb{Z}^{2},\mathbb{Z}/p)$ factors through the
reduction mod $p$ map
$\mathrm{SL}_{2}(\mathbb{Z})\rightarrow\mathrm{SL}_{2}(\mathbb{F}_{p})$. Since
$\mathbb{Z}/p$ is $p$-torsion, any map $\mathbb{Z}^{2}\rightarrow\mathbb{Z}/p$
factors uniquely through $\mathbb{F}_{p}^{2}$, so this gives a bijection
$\operatorname{Hom}(\mathbb{Z}^{2},\mathbb{Z}/p)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\operatorname{Hom}_{\mathbb{F}_{p}}(\mathbb{F}_{p}^{2},\mathbb{F}_{p})$,
and the induced action of $\mathrm{SL}_{2}(\mathbb{F}_{p})$ on
$X:=\operatorname{Hom}_{\mathbb{F}_{p}}(\mathbb{F}_{p}^{2},\mathbb{F}_{p})$
agrees with the action induced by the standard action on $\mathbb{F}_{p}^{2}$.
There is also a free action of $\mathbb{F}_{p}^{\times}$ on $X$ (by acting on
the codomain), which commutes with the
$\mathrm{SL}_{2}(\mathbb{F}_{p})$-action.

Let $\overline{A}\in\mathrm{SL}_{2}(\mathbb{F}_{p})$ be image of $A$. Since
any map to $\mathbb{F}_{p}$ is either zero or surjective, it suffices to count
the points in $X$ fixed by $\overline{A}$. Given a surjection
$\varphi:\mathbb{F}_{p}^{2}\rightarrow\mathbb{F}_{p}$, $\overline{A}$
preserves the $\mathbb{F}_{p}^{\times}$-orbit of $\varphi$ if and only if it
preserves $\ker\varphi$, which is a line in $\mathbb{F}_{p}^{2}$. Since the
action of $\overline{A}$ commutes with that of $\mathbb{F}_{p}^{\times}$,
$\overline{A}$ fixes $\varphi$ if and only if it fixes every element in the
$\mathbb{F}_{p}^{\times}$ orbit of $\varphi$, which is to say that it induces,
via $\varphi$, the identity on $\mathbb{F}_{p}$. Thus for any
$v\in\varphi^{-1}(1)$, $\varphi\circ\overline{A}=\varphi$ if and only if
$\overline{A}$ preserves the kernel and $\overline{A}v\equiv
v\mod\ker\varphi$, which is to say that $\overline{A}$ is conjugate to a
matrix of the form $\begin{pmatrix}a&b\\\ 0&1\end{pmatrix}$, and since
$\det\overline{A}=1$, we must have $a=1$, or equivalently, that
$\operatorname{tr}\overline{A}=2$. Thus, if such an $\overline{A}$ is not the
identity and has a fixed point, then it preserves at most one line in
$\mathbb{F}_{p}^{2}$, and fixes every point in the corresponding
$\mathbb{F}_{p}^{\times}$-orbit, which is to say it has exactly $p-1$ fixed
points. Since $|X|=p^{2}-1$ and the zero map
$\mathbb{F}_{p}^{2}\rightarrow\mathbb{F}_{p}$ is fixed by everything, this
implies the desired result.

∎

From (58) it follows that $Z(M(A),\mathbb{Z}/p)$ only depends on the conjugacy
class of $A\mod p$ in $\mathrm{SL}(2,\mathbb{Z}/p)$. The conjugacy classes of
the group $\mathrm{SL}(2,\mathbb{Z}/p)$ are described in Table 1, due
essentially to Frobenius. Here $p$ is an odd prime, $x$ is in
$(\mathbb{Z}/p)^{*}$, $x^{-1}$ is the multiplicative inverse of $x$, and
$\epsilon$ is a generator of $(\mathbb{Z}/p)^{*}$.

Representative |  $\hash$ elements in class |  $\hash$ classes | Label
---|---|---|---
$\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}$ | 1 | 1 | $\mathcal{C}1$
$\begin{pmatrix}-1&0\\\ 0&-1\end{pmatrix}$ | 1 | 1 | $\mathcal{C}2$
$\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}$ | $\frac{p^{2}-1}{2}$ | 1 | $\mathcal{C}3$
$\begin{pmatrix}1&\epsilon\\\ 0&1\end{pmatrix}$ | $\frac{p^{2}-1}{2}$ | 1 | $\mathcal{C}4$
$\begin{pmatrix}-1&1\\\ 0&-1\end{pmatrix}$ | $\frac{p^{2}-1}{2}$ | 1 | $\mathcal{C}5$
$\begin{pmatrix}-1&\epsilon\\\ 0&-1\end{pmatrix}$ | $\frac{p^{2}-1}{2}$ | 1 | $\mathcal{C}6$
$\begin{pmatrix}x&0\\\ 0&x^{-1}\end{pmatrix},\,x\neq\pm 1$ | $p(p+1)$ | $\frac{p-3}{2}$ | $\mathcal{C}7$
$\begin{pmatrix}x&y\\\ \epsilon y&x\end{pmatrix},\,x\neq\pm 1$ | $p(p-1)$ | $\frac{p-1}{2}$ | $\mathcal{C}8$
Table 1. The conjugacy classes of $\mathrm{SL}(2,\mathbb{Z}/p)$

Using this table we can prove the following,

######  Corollary 3.3.

Let $p$ be an odd prime and let $[A]$ be a conjugacy class in
$\mathrm{SL}(2,\mathbb{Z})$, then

$\displaystyle Z(M([A]),\mathbb{Z}/p)=\begin{cases}p^{2}&\text{if $[A]\bmod p$
is the identity class $\mathcal{C}1$}\\\ p&\text{if $[A]\bmod p$ a unipotent
class, that is, $\mathcal{C}3$ or $\mathcal{C}4$}\\\
1&\text{otherwise}\end{cases}$ (69)

###### Proof.

Notice that $x+x^{-1}=2$ is only solved by $x=1$; indeed, multiply by $x$ to
obtain

$\displaystyle x^{2}-2x+1=0,$ (70)

and observe that the left hand side factors as $(x-1)^{2}$. It is now easily
checked that only the representatives of $\mathcal{C}3$ and $\mathcal{C}4$
whose trace mod $p$ is 2. The corollary is now implied by Propositions 3.1 and
3.2. ∎

######  Example 3.4.

Let $p=3$. We take $A\in\mathrm{SL}(2,\mathbb{Z})$ such that
$A\equiv\operatorname{Id}\mod 3$ and notice that both $A(1)$ and $A(2)$ are
divisible by $3$

$\displaystyle A=\begin{pmatrix}10&3\\\ 3&1\end{pmatrix}\quad\quad\quad
SNF(A-ID)=\begin{pmatrix}3&0\\\ 0&3\end{pmatrix}$ (71)

The powers of $A$ are also congruent to $\operatorname{Id}$ $\mod 3$, we see
that the Smith normal form invariants of powers are also divisible by $3$

$\displaystyle A^{2}=\begin{pmatrix}109&33\\\
33&10\end{pmatrix}\quad\quad\quad SNF(A^{2}-ID)=\begin{pmatrix}3&0\\\
0&39\end{pmatrix}$ (72)

$\displaystyle A^{3}=\begin{pmatrix}1189&360\\\
360&109\end{pmatrix}\quad\quad\quad SNF(A^{3}-ID)=\begin{pmatrix}36&0\\\
0&36\end{pmatrix}$ (73)

Following is an example of $A\in\mathrm{SL}(2,\mathbb{Z})$ such that
$A\nequiv\operatorname{Id}\mod 3$ and $\operatorname{Tr}(A)\equiv 2\mod 3$. We
see that $A(1)$ is not divisible by $3$ but $A(2)$ is divisible by $3$.

$\displaystyle A=\begin{pmatrix}4&3\\\ 5&4\end{pmatrix}\quad\quad\quad
SNF(A-ID)=\begin{pmatrix}1&0\\\ 0&6\end{pmatrix}$ (74)

For the last case we take $A\in\mathrm{SL}(2,\mathbb{Z})$ which does not
satisfy the mod $3$ conditions and see that $3$ does not divide $A(1)$ and
$A(2)$.

$\displaystyle A=\begin{pmatrix}27&1\\\ -1&0\end{pmatrix}\quad\quad\quad
SNF(A-ID)=\begin{pmatrix}1&0\\\ 0&25\end{pmatrix}$ (75)

##  4\. Counting estimates and Proofs of Theorems 1.1 and 1.2

Let $G=\mathrm{SL}(2,\mathbb{Z}/p\mathbb{Z})$ (or any finite group) and let
$\rho:SL(2,\mathbb{Z})\twoheadrightarrow G$ be an epimorphism. A conjugacy
class of $\mathrm{SL}(2,\mathbb{Z})$ is hyperbolic if its trace has absolute
value $>2$, and is primitive if it is not a power any other conjugacy class.
To each primitive hyperbolic conjugacy class $[A]$ of
$\mathrm{SL}(2,\mathbb{Z})$, let $C([A])$ denote the corresponding conjugacy
class in $G$ which is the image under $\rho$ of $[A]$. Then Sarnak’s
Chebotarev Density Theorem for Prime Geodesics [8, Theorem 3.16] states that,
for any fixed conjugacy class $C$ in $G$, we have:

$\displaystyle\sum_{[A]\in[\mathrm{SL}(2,\mathbb{Z})],|\operatorname{Tr}([A])|<T\atop
C([A])=C}1={|C|\over|G|}\operatorname{li}(T^{2})+O(T^{3/2}(\log T)^{2}),$ (76)

as $T\to\infty$. Here $\operatorname{li}(x)$ is the logarithmic integral
function,

$\operatorname{li}(x)=\int_{2}^{x}{dt\over\log t}.$

Note that the above is a general phenomenon not restricted to
$\Gamma=\mathrm{SL}(2,\mathbb{Z})$ and
$G=\mathrm{SL}(2,\mathbb{Z}/p\mathbb{Z})$; in the general case, there may be
further “main” terms, corresponding to Laplace-Beltrami eigenvalues
$\lambda_{j}=s_{j}(1-s_{j})$ with $1/2<s_{j}<1$, see Sarnak op. cit. In our
case of the modular group $\mathrm{SL}(2,\mathbb{Z})$ and its principal
congruence (normal) subgroup
$\Gamma(p)=\ker(\mathrm{SL}(2,\mathbb{Z})\twoheadrightarrow G)$, we have by
Selberg’s 3/16ths theorem [9]111Today even better bounds are known towards the
generalized Ramanujan conjectures [5], which in this setting at infinity
amount to the Selberg’s 1/4 conjecture. But Selberg’s result is already
sufficient for our purposes here. that $\lambda_{j}\geqslant 3/16,$ that is,
$s_{j}\leqslant 3/4$. Therefore any lower order terms, which are of the order
$T^{2s_{j}}$, are already dominated by the error term $T^{3/2}$.

We can now prove the main theorems.

###### Proof of Theorems 1.1 and 1.2.

Theorem 1.1 follows from (76), Corollary 3.3, the values in Table 1 and the
fact that

$\displaystyle|\mathrm{SL}(2,\mathbb{Z}/p)|=p^{3}-p.$ (77)

Theorem 1.2 now similarly follows from Sarnak’s Chebotarev density theorem
(76), together with the previous results on Smith normal forms of matrices
$A-\operatorname{Id}$ for $A$ in $\mathrm{SL}(2,\mathbb{Z})$. ∎

##  5\. $\mathbb{Z}/2$ Dijkgraaf-Witten invariants of 3D mapping tori of genus
one and modular forms

The Table 1, Corollary 3.3, and Theorem 1.1 are all valid for $p$ an odd
prime; we now extend these to $p=2$: Since $(\mathbb{Z}/2)^{2}$ has three
distinct subgroups of order two it follows that

$\displaystyle\hash\operatorname{Hom}((\mathbb{Z}/2)^{2},\mathbb{Z}/2)=4$ (78)

Label these four elements as

$\displaystyle\\{\rho_{0},\rho_{1},\rho_{2},\rho_{3}\\}=\operatorname{Hom}((\mathbb{Z}/2)^{2},\mathbb{Z}/2)$
(79)

where $\rho_{0}$ is the trivial homomorphism, i.e.

$\displaystyle
Ker(\rho_{0}:(\mathbb{Z}/2)^{2}\to\mathbb{Z}/2)=(\mathbb{Z}/2)^{2}$ (80)

and the other three elements are labelled arbitrarily. The action of the group
$\mathrm{SL}(2,\mathbb{Z}/2)$ on
$\operatorname{Hom}((\mathbb{Z}/2)^{2},\mathbb{Z}/2)$ fixes the trivial
homomorphism $\rho_{0}$ and permutes the other three homomorphisms
$\\{\rho_{1},\rho_{2},\rho_{3}\\}$ amongst themselves. The group
$\mathrm{SL}(2,\mathbb{Z}/2)$ has order six and its six elements are

$\displaystyle\mathrm{SL}(2,\mathbb{Z}/2):=\Bigg{\\{}$
$\displaystyle\operatorname{Id}=\begin{pmatrix}1&0\\\
0&1\end{pmatrix},\,T=\begin{pmatrix}1&1\\\
0&1\end{pmatrix},\,S=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix},\,$ (81)
$\displaystyle T.S=\begin{pmatrix}1&-1\\\
1&0\end{pmatrix},\,T.S.T=\begin{pmatrix}1&0\\\
1&1\end{pmatrix},\,-S.T^{-1}=\begin{pmatrix}0&1\\\
-1&1\end{pmatrix}\Bigg{\\}}$ (82)

It has three conjugacy classes which are composed of

$\displaystyle\mathcal{C}1:=\\{\operatorname{Id}\\}$ (83)
$\displaystyle\mathcal{C}2:=\\{T,\,S,\,T.S.T\\}$ (84)
$\displaystyle\mathcal{C}3:=\\{T.S,\,-S.T^{-1}\\}$ (85)

Elements in $\mathcal{C}2$ have order 2 and elements in $\mathcal{C}3$ have
order 3. The group $\mathrm{SL}(2,\mathbb{Z}/2)$ is isomorphic to
$\mathfrak{S}_{3}$, the symmetric group on three letters, and every possible
permutation of $\\{\rho_{1},\rho_{2},\rho_{3}\\}$ occurs under the action of
$\mathrm{SL}(2,\mathbb{Z}/2)$. The following table shows the permutations
assigned to elements of $\mathrm{SL}(2,\mathbb{Z}/2)$ under this isomorphism:

(92)

We see that a non-trivial element $A$ of $\mathrm{SL}(2,\mathbb{Z}/2)$ fixes
at most a single element of the set $\\{\rho_{1},\rho_{2},\rho_{3}\\}$, which
happens if and only if $A$ is in the conjugacy class $\mathcal{C}2$.

######  Corollary 5.1.

Let $[A]$ be a conjugacy class in $\mathrm{SL}(2,\mathbb{Z})$, then

$\displaystyle Z(M([A]),\mathbb{Z}/2)=\begin{cases}4&\text{if $[A]\bmod 2$ is
the identity class $\mathcal{C}1$}\\\ 2&\text{if $[A]\bmod 2$ is the class
$\mathcal{C}2$}\\\ 1&\text{otherwise}\end{cases}$ (93)

###### Proof.

From (54) we have that $Z(M(A),\mathbb{Z}/2)$ is equal to the number of fixed
points of $A\,\bmod\,2$ acting on
$\operatorname{Hom}((\mathbb{Z}/p)^{2},\mathbb{Z}/p)$. From the previous
discussion we know that if $A\bmod 2$ is not the identity, then it has two
fixed points if and only if it belongs to the class $\mathcal{C}2$, and one
fixed point if and only if it belongs to $\mathcal{C}3$. The result follows. ∎

Let

$\displaystyle\Gamma(2):=Ker(\mathrm{SL}(2,\mathbb{Z})\to\mathrm{SL}(2,\mathbb{Z}/2)$
(94)

the principal congruence subgroup of level two, and

$\displaystyle X(2)=\mathbb{H}/\Gamma(2)$ (95)

the modular curve of level two. The curve $X(2)$ has genus zero, zero elliptic
points, and three cusps. Let

$\displaystyle\lambda:\mathbb{H}\to\mathbb{C}$ (96)

denote the modular lambda function where

$\displaystyle\lambda(\tau)=\frac{e_{3}(\tau)-e_{2}(\tau)}{e_{1}(\tau)-e_{2}(\tau)}$
(97)

and

$\displaystyle e_{1}(\tau)=\wp(\tau/2,\tau),\quad
e_{2}(\tau)=\wp(1/2,\tau),\quad\quad e_{3}(\tau)=\wp((1+\tau)/2,\tau)$ (98)

with

$\displaystyle\wp(z,\tau)=\frac{1}{z^{2}}\,+\sum_{(m,n)\in\mathbb{Z}^{2}-\\{0,0\\}}\bigg{(}\frac{1}{(z-m\tau-n)^{2}}-\frac{1}{(m\tau+n)^{2}}\bigg{)}$
(99)

The function (97) is invariant under the $\Gamma(2)$ action on $\mathbb{H}$
and descends to a holomorphic function on the quotient

$\displaystyle\lambda:X(2)\to\mathbb{C}-\\{0,1\\}$ (100)

######  Lemma 5.2.

Let $\zeta_{3}:=e^{2\pi i/3}$ and let $[A]$ be a conjugacy class in
$[\mathrm{SL}(2,\mathbb{Z})]$ such that $[A]\bmod 2$ is not the identity
class. Then

$\displaystyle Z(M(A),\mathbb{Z}/2)=\bigg{|}\frac{3}{2\pi
i}\log\lambda(A(\zeta_{3}))\bigg{|}^{-1}$ (101)

and if $[A]\bmod 2$ is the identity class, then

$\displaystyle Z(M(A),\mathbb{Z}/2)=4$ (102)

The following values, which are calculated using a computer algebra program,
give numerical evidence for Lemma 5.2:

$\displaystyle\lambda(\zeta_{3})$
$\displaystyle=0.5\,-0.866025i\,\,\approx-e^{\frac{2\pi i}{3}}$ (103)
$\displaystyle\lambda(T(\zeta_{3}))$ $\displaystyle=0.5+0.866025i\,\,\approx
e^{\frac{2\pi i}{6}}$ (104) $\displaystyle\lambda(S(\zeta_{3}))$
$\displaystyle=0.5+0.866025i\,\,\approx e^{\frac{2\pi i}{6}}$ (105)
$\displaystyle\lambda(T.S(\zeta_{3}))$
$\displaystyle=0.5-0.866025i\,\,\approx-e^{\frac{2\pi i}{3}}$ (106)
$\displaystyle\lambda(T.S.T(\zeta_{3}))$
$\displaystyle=0.5+0.866025i\,\,\approx e^{\frac{2\pi i}{6}}$ (107)
$\displaystyle\lambda(-S.T^{-1}(\zeta_{3}))$
$\displaystyle=0.5-0.866025i\,\,\approx-e^{\frac{2\pi i}{3}}$ (108)

###### Proof.

The functions $\\{e_{1},e_{2},e_{3}\\}$ defined in (98) while invariant under
the action of $\Gamma(2)$ are transformed into each other under the action of
$\mathrm{SL}(2,\mathbb{Z}/2)$, i.e. for any $A$ in
$\mathrm{SL}(2,\mathbb{Z}/2)$ we have

$\displaystyle e_{i}(A(\tau))=e_{j}(\tau).$ (109)

If we identify $\\{e_{1},e_{2},e_{3}\\}$ with
$\\{\rho_{1},\rho_{2},\rho_{3}\\}$ as a set, then the action of
$\mathrm{SL}(2,\mathbb{Z}/2)$ on $\\{e_{1},e_{2},e_{3}\\}$ is precisely the
action of $\mathrm{SL}(2,\mathbb{Z}/2)$ on $\\{\rho_{1},\rho_{2},\rho_{3}\\}$
as outlined in (92).

Note that the function (97) is a cross-ratio of the three functions
$\\{e_{1},e_{2},e_{3}\\}$ and under the action by any $A$ in
$\mathrm{SL}(2,\mathbb{Z}/2)$ we obtain a new cross-ratio of the same three
functions. It is well known that there are in total six different cross-ratios
and they are related to each other under the action of the anharmonic group
which is isomorphic to $\mathrm{SL}(2,\mathbb{Z}/2)$. In other words, the
group $\mathrm{SL}(2,\mathbb{Z}/2)$ also acts on the codomain of the map

$\displaystyle\lambda:X(2)\to\mathbb{C}-\\{0,1\\}$ (110)

by the following assignment

$\displaystyle
T\quad\quad\quad\quad\lambda(\tau)\mapsto\frac{\lambda(\tau)}{\lambda(\tau)-1}$
(111) $\displaystyle S\quad\quad\quad\quad\lambda(\tau)\mapsto
1-\lambda(\tau)$ (112) $\displaystyle
T.S\quad\quad\quad\quad\lambda(\tau)\mapsto\frac{\lambda(\tau)-1}{\lambda(\tau)}$
(113) $\displaystyle
T.S.T\quad\quad\quad\quad\lambda(\tau)\mapsto\frac{1}{\lambda(\tau)}$ (114)
$\displaystyle-$ $\displaystyle
S.T^{-1}\quad\quad\quad\quad\lambda(\tau)\mapsto\frac{1}{1-\lambda(\tau)}$
(115)

It is easy to see that the point $-e^{\frac{2\pi i}{3}}$ in the codomain of
the map (110) is fixed under the action of the elements of order three and is
mapped to $e^{\frac{2\pi i}{6}}$ under the action of elements of order two.
This proves Lemma 5.2. ∎

######  Lemma 5.3.

Let $d>1$ be a cube free integer, let $N=3\prod_{p|d}p$, and let $\psi$ denote
the quadratic character of conductor three. There exists a modular form of
weight one and level $3N^{2}$ twisted by $\psi$,

$\displaystyle f_{\psi}(\tau)\in M_{1}(3N^{2},\psi)$ (117)

such that its $p^{th}$ Fourier coeffecient $a_{p}(f_{\psi})$ satisfies

$\displaystyle a_{p}(f_{\psi})=Z(M(A),\mathbb{Z}/2)+2$ (118)

where

if $p\equiv 1\,\,mod\,3$ and $d$ is a cube $mod\,p$ then $[A]\bmod 2$ is the
identity class (119) if $p\equiv 1\,\,mod\,3$ and $d$ is not a cube $mod\,p$
then $[A]\bmod 2$ is in the class $\mathcal{C}2$ (120) if $p\equiv
2\,\,mod\,3$ then $[A]\bmod 2$ is in the class $\mathcal{C}3$ (121)

For $d=2$, the first hundred Fourier coeffecients, taken from LMFDB, are as
follows:

$\displaystyle
f_{\psi}(\tau)=q-q^{7}-q^{13}-q^{19}+q^{25}+2q^{31}-q^{37}+2q^{43}-q^{61}-q^{67}-q^{73}-q^{79}+q^{91}-q^{97}+....$
(122)

###### Proof.

For $d>1$ a cube free integer, the Galois group of the field extension

$\displaystyle\mathbb{Q}(d^{1/3},\zeta_{3})$ (123)

over $\mathbb{Q}$ is

$\displaystyle\text{Gal}(\mathbb{Q}(d^{1/3},\zeta_{3})|\mathbb{Q})\cong\mathfrak{S}_{3}$
(124)

and there is a representation

$\displaystyle\Psi:\text{Gal}(\mathbb{Q}(d^{1/3},\zeta_{3})|\mathbb{Q})\to\mathrm{GL}(2,\mathbb{Z})$
(125)

given by the assignment

$\displaystyle(123)\mapsto$ $\displaystyle\begin{pmatrix}0&1\\\
-1&-1\end{pmatrix}$ (126) $\displaystyle(23)\mapsto$
$\displaystyle\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$ (127)

such that for any rational prime $p\in\mathbb{Q}$ and a maximal ideal
$\mathfrak{p}$ of the ring of integers of $\mathbb{Q}(d^{1/3},\zeta_{3})$
lying over $p$, we have that

$\displaystyle Trace(\Psi(Frob_{\mathfrak{p}}))=\begin{cases}2&\text{if
$p\equiv 1\,\,mod\,3$ and d is a cube mod $p$},\\\ -1&\text{if $p\equiv
1\,\,mod\,3$ and d is not a cube mod $p$}\\\ 0&\text{if $p\equiv
2\,\,mod\,3$}\end{cases}$ (128)

It is well known that the representation $\Psi$ is modular with the modular
form as assigned in the statement, see for example, [2, p. 371]. This proves
(5.3). ∎

The geometric reasoning behind lemma (5.3) is as follows: Let
$\mathcal{O}_{\mathbb{Q}(d^{1/3},\zeta_{3})}$ denote the ring of integers of
$\mathbb{Q}(d^{1/3},\zeta_{3})$, then for any rational prime $p$ one has that

$\displaystyle
p\mathcal{O}_{\mathbb{Q}(d^{1/3},\zeta_{3})}=\begin{cases}\mathfrak{p}_{1}\cdots\mathfrak{p}_{6}\quad\quad\quad\quad&\text{if
$p\equiv 1\,mod\,\,3$ and $d$ is a cube mod $p$}\\\
\mathfrak{p}_{1}\mathfrak{p}_{2}\quad\quad\quad\quad\quad&\text{if $p\equiv
1\,mod\,\,3$ and $d$ is not a cube mod $p$}\\\
\mathfrak{p}_{1}\mathfrak{p}_{2}\mathfrak{p}_{3}\quad\quad\quad\quad&\text{if
$p\equiv 2\,mod\,\,3$}\end{cases}$ (129)

A closed geodesic $\gamma$ in $X(1)$ can be pulled back by the covering map

$\displaystyle\pi:X(2)\to X(1)$ (130)

The pullback $\pi^{*}(\gamma)$ is a disjoint union of closed geodesics in
$X(2)$ which we shall denote as

$\displaystyle\pi^{*}(\gamma)=\\{\tilde{\gamma}_{1},\dots,\tilde{\gamma}_{k}\\}$
(131)

With every closed geodesic $\gamma$ in $X(1)$ is associated a unique conjugacy
class $[A_{\gamma}]$ in $[\mathrm{SL}(2,\mathbb{Z})]$. From [8, Prop 3.11], we
have the following

$\displaystyle\pi^{*}(\gamma)=\begin{cases}\\{\tilde{\gamma}_{1},\dots,\tilde{\gamma}_{6}\\}\quad\quad\quad&\text{if
$[A_{\gamma}]\bmod 2$ is the identity class $\mathcal{C}1$}\\\
\\{\tilde{\gamma}_{1},\tilde{\gamma}_{2}\\}\quad\quad\quad&\text{if
$[A_{\gamma}]\bmod 2$ is in conjugacy class $\mathcal{C}3$}\\\
\\{\tilde{\gamma}_{1},\tilde{\gamma}_{2},\tilde{\gamma}_{3}\\}\quad\quad\quad&\text{if
$[A_{\gamma}]\bmod 2$ is in conjugacy class $\mathcal{C}2$}\end{cases}$ (132)

We know not how to generalize lemmas (5.2) and (5.3) to odd primes, but we
remark: The function field of the Riemann surface $X(2)$ is

$\displaystyle\mathbb{C}(X(2))=\mathbb{C}(j,e_{1},e_{2},e_{3})$ (133)

where $j:\mathbb{H}\to\mathbb{C}$ is the modular $j$ function which can be
given as

$\displaystyle
j(\tau)=\frac{1}{54}\frac{\big{(}(e_{1}-e_{2})^{2}+(e_{2}-e_{3})^{2}+(e_{3}-e_{1})^{2}\big{)}^{3}}{\big{(}(e_{1}-e_{2})(e_{2}-e_{3})(e_{3}-e_{1})\big{)}^{2}}$
(134)

We have a Galois covering of Riemann surfaces

$\displaystyle\pi:X(2)\to X(1):=\mathbb{H}/\mathrm{SL}(2,\mathbb{Z})$ (135)

where the function field $\mathbb{C}(X(1))=\mathbb{C}(j)$. It follows that
$\mathbb{C}(j,e_{1},e_{2},e_{3})|\mathbb{C}(j)$ is a Galois field extension
with the Galois group

$\displaystyle\text{Gal}\big{(}\mathbb{C}(j,e_{1},e_{2},e_{3})|\mathbb{C}(j)\big{)}\cong\mathrm{SL}(2,\mathbb{Z}/2)$
(136)

Moreover,
$\\{\rho_{0},\rho_{1},\rho_{2},\rho_{3}\\}=\operatorname{Hom}((\mathbb{Z}/2)^{2},\mathbb{Z}/2)$
and $\\{j,e_{1},e_{2},e_{3}\\}$ are naturally isomorphic as sets with
$\mathrm{SL}(2,\mathbb{Z}/2)$ action. This isomorphism generalizes to odd $p$,
i.e. the action of the Galois group $\mathrm{SL}(2,\mathbb{Z}/p)$ on the
extension of the function field $\mathbb{C}(j)$ to the function field of
$X(p):=\mathbb{H}/\Gamma(p)$ associated with the Galois covering

$\displaystyle\pi(p):X(p)\to X(1)$ (137)

can be identified with the action of $\mathrm{SL}(2,\mathbb{Z}/p)$ on
$\operatorname{Hom}((\mathbb{Z}/p)^{2},\mathbb{Z}/p)$.

##  6\. Further remarks: $\mathrm{SU}(2)$ Chern-Simons-Witten invariants of
mapping tori of genus one

Let $A$ be in $\mathrm{SL}(2,\mathbb{Z})$ such that
$|\operatorname{Tr}(A)|>2$. One maps $A$ to an indefinite binary quadratic
form of discriminant $\operatorname{Tr}(A)^{2}-4$ as follows

$\displaystyle A=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\mapsto bx^{2}+(a-d)xy-
cy^{2}=:Q_{A}(x,y)$ (138)

This map is equivariant with respect to the $\mathrm{SL}(2,\mathbb{Z})$ action
on the left by conjugation and on the right by the standard action on binary
quardatic forms, and one gets an isomorphism between conjugacy classes of
hyperbolic elements of $\mathrm{SL}(2,\mathbb{Z})$ with trace $t$ and
isomorphism classes of indefinite binary quadratic forms of discriminant
$t^{2}-4$. Under this isomorphism, if $mx^{2}+lxy+ky^{2}$ is a BQF of
discriminant $t^{2}-4$, then its inverse is

$mx^{2}+lxy+ky^{2}\mapsto\begin{pmatrix}\frac{t-l}{2}&k\\\
-m&\frac{t+l}{2}\end{pmatrix}$ (139)

which has trace equal to $t$. Let $[A]$ be in $[\mathrm{SL}(2,\mathbb{Z})]$
such that $|\operatorname{Tr}([A])|>2$ and let $k$ be a positive integer.
Define

$\displaystyle
Z(M([A]),\mathrm{SU}(2),k):=\frac{\text{sign}(\operatorname{Tr}(A))}{2}\Bigg{(}$
$\displaystyle\frac{1}{\sqrt{|\operatorname{Tr}(A)-2|}}\sum_{\\{(x,y)\in\mathbb{Z}^{2}/(A-\operatorname{Id})\mathbb{Z}^{2}\\}}e^{2\pi
ik\frac{Q_{A}(x,y)}{\operatorname{Tr}(A)-2}}\,\,+\,\,$ (140)
$\displaystyle\frac{1}{\sqrt{|\operatorname{Tr}(A)+2|}}\sum_{\\{(x,y)\in\mathbb{Z}^{2}/(-A-\operatorname{Id})\mathbb{Z}^{2}\\}}e^{2\pi
ik\frac{Q_{A}(x,y)}{\operatorname{Tr}(A)+2}}\Bigg{)}$ (141)

This is the $\mathrm{SU}(2)$ Chern-Simons-Witten invariant of $M([A])$ at
level $k$, see [4, Equation (5.20)]. Let $N(k):=8(k+2)$, the invariant
$Z(M([A]),\mathrm{SU}(2),k)$ only depends on the conjugacy class of $[A]\bmod
N(k)$ in $\mathrm{SL}(2,\mathbb{Z}/N(k))$. This follows from the duality
between 3D Chern-Simons-Witten theory and 2D rational conformal field theory
which constructs a representation

$\displaystyle\Phi^{(k)}:\mathrm{SL}(2,\mathbb{Z})\to\operatorname{Aut}(\mathbb{C}^{k+1})$
(142)

which factors through $\mathrm{SL}(2,\mathbb{Z}/N(k))$ and has the property
that

$\displaystyle Trace(\Phi^{(k)}(A))=Z(M([A]),\mathrm{SU}(2),k)$ (143)

For a fixed positive integer $k$ one can thus apply Sarnak’s Chebotarev
density theorem to obtain asymptotics of sums of (140) over all hyperbolic
conjugacy classes $[A]$ in $[\mathrm{SL}(2,\mathbb{Z})]$. It would be
interesting to also consider the asymptotics of these sums as $k\to\infty$.

## References

  * [1] Anindya Banerjee, Gregory Moore: Comments on summing over bordisms in TQFT, Journal of High Energy Physics, 2022, Article number: 171 (2022).
  * [2] Fred Diamond, Jerry Shurman: A first course in modular forms, Springer, Graduate text in mathematics 228, 2005.
  * [3] Robbert Dijkgraaf, Edward Witten: Topological gauge theory and group cohomology. Communications in Mathematical Physics, 129, 393 – 429, 1990.
  * [4] Lisa Jeffrey: Chern-Simons-Witten invariants of Lens spaces and Torus bundles, and semi-classical approximation. Communications in Mathematical Physics, 147, 563–604, 1992.
  * [5] Henry Kim, Peter Sarnak: Refined estimates towards Ramanujan and Selberg conjectures, J. Amer. Math. Soc. 16 (2003), 139-183.
  * [6] Sergey Matveev, Valadimir Truaev: Doklady Mathematics 91(1):9-11
  * [7] Igor Rivin: Statistics of Random 3-Manifolds occasionally fibering over the circle, arXiv:1401.5736, 2014.
  * [8] Peter Sarnak: Prime geodesic theorems, Thesis Stanford University, 1980.
  * [9] Atle Selberg: On the estimation of the Fourier coeffecients of modular forms, Proc. Sympos. Pure Math., Vol. VIII, pp 1–15. Amer. Math. Soc., Providence, R.I., 1965.
  * [10] Richard Stanley, Yinghui Wang: Smith normal form distribution of random integer matrices, SIAM Journal on Discrete Mathematics, 31, 2247-2268, 2017.


# Radiation backreaction in axion electrodynamics

A. Patkós
Institute of Physics, Eötvös University
H-1117 Pázmány Péter sétány 1/A, Budapest, Hungary

###### Abstract

Energy-momentum conservation of classical axion-electrodynamics is carefully
analyzed in the Hamiltonian formulation of the theory. The term responsible
for the energy transfer between the electromagnetic and the axion sectors is
identified. As a special application the axion-to-light Primakoff-process in
the background of a static magnetic field is worked out and the radiative
self-damping of the axion oscillations is characterized quantitatively. The
damping time turns out comparable to the age of the Universe in the preferred
axion mass range.

## Homage to Professor Zhengdao Li

50 years ago Professor Li visited Hungary among the distinguished speakers at
the First International Conference on Neutrino Physics and Astrophysics held
in Balatonfüred from 11th to 17th June 1972 (see Fig.1). His friendly but very
sharp dispute with R. Feynman on the interpretation of the scaling phenomenon
then freshly discovered in deep inelastic electron-proton scattering left
lifelong impression in the audience, including also the present author. The
concurrence of the Friedberg-Lee model and the quark parton model of Feynman
has been resolved soon by the discovery of asymptotic freedom of quantum
chromodynamics. It is a great honor for us to contribute to the special volume
dedicated to the 95th anniversary of Professor Li.

Figure 1:  Zhengdao Li with Bruno Pontecorvo at the Balatonfüred Neutrino
Conference in 1972

##  1 Aims

Axion electrodynamics, the Maxwell theory modified by coupling the axion field
to the electromagnetic topological density dates back to more than four
decades of intense theoretical and phenomenological discussion [1].

Following the unexpected discovery of the violation of parity conservation in
weak $\beta$ decays suggested in 1956 by Lee and Yang, the violation of the
combined CP symmetry was soon established in 1964. With the advent of QCD, one
might have also expected the breakdown of these symmetries in strong
interactions. The coefficient of the allowed CP-violating term in the QCD
Lagrangian is proportional to an angle which turned out to have an upper bound
$10^{-10}$, derived from the observed upper bound on the electric dipole
moment of neutron. This somewhat unnatural feature requires explanation. By
far the most convincing interpretation has been proposed by Peccei and Quinn
[2], who introduced a pseudoscalar field, the axion replacing the CP-violating
angle and suggesting a mechanism relaxing its expectation value to zero.
Coherent oscillations of the axion field around this average offer an
attractive alternative for the cosmologically evidenced dark matter.

The axion field also couples to electrodynamical degrees of freedom. Questions
of modified light propagation through this medium are mostly in the focus of
this research. The first studies of axion propagation performed in quantum
mechanical framework investigated photon–axion conversion via the Primakoff
process [3, 4]. These investigations were improved recently using the
formalism of quantum field theory [5], including propagation in curved space
time [6]. Plasmon–axion interactions receive similar attention [7, 8, 9, 10].
Increasing accuracy in estimating the intensity of the solar axion flux [11]
as well as axion effects arising in the strong magnetic field of neutron stars
[12] carry important discovery potential. A specific chapter of such
investigations is represented by the detection techniques of axions, e.g., the
theory of haloscopes [13].

It is somewhat surprising that a sufficiently detailed general analysis of the
issue of energy–momentum conservation in axion electrodynamics, a most
important consequence of the space–time translation symmetry is mostly
bypassed, despite some recent (partly controversial) publications [14, 15, 16,
17]. In particular, characterization of the energy transfer between axions and
the electromagnetic field in the presence of a strong static magnetic field
has a practical interest seeing that the haloscopes rely on the Primakoff
process of photon–axion conversion. Although the invariance of the complete
photon–axion system to space–time translations implies energy–momentum
conservation, from the point of view of the electromagnetism of electrically
charged objects, the existence of axions defines a specific dissipation
channel. Energy–momentum exchange through this channel might be significant
for an indirect observation of axions.

The aim of the present paper is not to put forward some fundamentally new
approach. It represents rather a pedagogically detailed discussion of the
issue of energy–momentum conservation in axion electrodynamics, closely
following the treatment of electrodynamical conservation laws and radiation
backreaction presented in the classic textbook of J.D. Jackson [18]. Our
complete analysis is classical! First, I recall the modified Maxwell equations
(presented by many authors and many times). Then, I proceed with extending the
electromechanical energy conservation to the axion–electromechanical system.
An exact result is established (see (10) below), which is valid beyond the
linear approximation in the strength of the axion–photon coupling. In the
practically relevant case of extremely weak axion–photon coupling, it is
demonstrated that the energy conservation naturally expressed in canonical
variables is easily translated into an expression constructed of electric and
magnetic field strengths. As an application of the general equations, I
discuss the energy balance in presence of a spatially compact static magnetic
field. An interesting point is the appearance of an effective axion equation
arising from the elimination of the electromagnetic variables. This equation
displays full-fledged radiation backreaction effect, damping axion
oscillations. The relaxation time is carefully estimated for the
phenomenologically preferred axion mass range.

##  2 Canonical Equations of Axion Electrodynamics and the Conservation of the
Energy

The Lagrangean density of the coupled axion ($a(x)$) + photon theory is the
following:

$\displaystyle\displaystyle L=\frac{1}{2}\left(\epsilon{\bf
E}^{2}(x)-\frac{1}{\mu}{\bf B}^{2}(x)\right)-ga(x){\bf E}(x)\cdot{\bf B}(x)$
$\displaystyle\displaystyle+\frac{1}{2}\left[\frac{1}{c^{2}}(\partial_{t}a(x))^{2}-(\partial_{i}a(x))^{2}-\frac{m_{a}^{2}c^{2}}{\hbar^{2}}a^{2}\right]-j_{0}A_{0}(x)+{\bf
j}(x)\cdot{\bf A}(x),$ (1)

with

${\bf B}=\nabla\times{\bf A},\qquad{\bf E}=-\dot{\bf A}-\nabla A_{0}.$ (2)

The homogeneous Maxwell equations follow from these relations:

$\nabla{\bf B}=0,\qquad\nabla\times{\bf E}=-\dot{\bf B}.$ (3)

The canonical momenta are

$\Pi_{a}=\frac{1}{c^{2}}\dot{a}\equiv\frac{p_{a}}{c},\qquad\Pi_{A_{0}}=0,\qquad{\bf\Pi}_{A}=-\epsilon{\bf
E}+ga{\bf B}\equiv-{\bf D}.$ (4)

The Hamiltonian density is constructed by the usual Legendre transform:

$\displaystyle\displaystyle{\cal H}=-\dot{\bf A}\cdot{\bf D}+\dot{a}\Pi_{a}-L$
$\displaystyle\displaystyle=\frac{1}{2}\left[p_{a}^{2}+(\nabla
a)^{2}+\frac{m_{a}^{2}c^{2}}{\hbar^{2}}a^{2}\right]+\frac{1}{\epsilon}{\bf
D}\cdot({\bf D}+ga{\bf B})-\frac{1}{2\epsilon}({\bf D}+ga{\bf
B})^{2}+\frac{1}{2\mu}{\bf B}^{2}$
$\displaystyle\displaystyle+\frac{ga}{\epsilon}{\bf B}\cdot({\bf D}+ga{\bf
B})+j_{0}A_{0}-{\bf j}\cdot{\bf A}$
$\displaystyle\displaystyle=\frac{1}{2}\left[p_{a}^{2}+(\nabla
a)^{2}+\frac{m_{a}^{2}c^{2}}{\hbar^{2}}a^{2}\right]+\frac{1}{2}\left[\frac{1}{\epsilon}{\bf
D}^{2}+\frac{1}{\mu}\left(1+\frac{\mu(ga)^{2}}{\epsilon}\right){\bf
B}^{2}\right]+\frac{ga}{\epsilon}{\bf B}\cdot{\bf D}$
$\displaystyle\displaystyle+j_{0}A_{0}-{\bf j}\cdot{\bf A}.$ (5)

The inhomogeneous equations are derived from the Hamiltonian formulation of
the theory. Hamilton’s equations can be written in the conventional form with

${\bf H}=\frac{1}{\mu^{\prime}}{\bf B}+\frac{1}{\epsilon}ga{\bf
D},\qquad\frac{1}{\mu^{\prime}}=\frac{1}{\mu}\left(1+\frac{\mu(ga)^{2}}{\epsilon}\right)$
(6) $\nabla{\bf D}=j_{0},\qquad\nabla\times{\bf H}={\bf j}+\dot{\bf
D},\qquad\ddot{a}-\Delta
a+\frac{m_{a}^{2}c^{2}}{\hbar^{2}}a=-\frac{1}{\epsilon}g{\bf B}\cdot({\bf
D}+ga{\bf B})=-g{\bf E}\cdot{\bf B}.$ (7)

The rate of change in the energy in the axion sector is given as

$\frac{dE_{a}}{dt}=\int d^{3}x\dot{a}\left(\ddot{a}-\Delta
a+\frac{m_{a}^{2}c^{2}}{\hbar^{2}}a\right)=-\int d^{3}xg\dot{a}{\bf
E}\cdot{\bf B}\equiv\int d^{3}x{\bf j}_{axion}\cdot{\bf E}.$ (8)

The extension of the mechanical energy conservation to axion electrodynamics
starts by computing the rate of change in the mechanical energy through the
work on an electromagnetic probe current ${\bf j}_{e}$ (which leads in the
quasi-stationary approximation to Joule heat generation):

$\displaystyle\displaystyle\frac{dE_{mech}}{dt}=\int d^{3}x{\bf
j}_{e}\cdot{\bf E}=\int d^{3}x\left[(\nabla\times{\bf H})\cdot{\bf E}-{\bf
E}\cdot\dot{\bf D}\right]$ $\displaystyle\displaystyle=\int
d^{3}x\left[-\nabla({\bf E}\times{\bf H})-\dot{\bf B}\cdot{\bf H}-\dot{\bf
D}\cdot{\bf E}\right]$ $\displaystyle\displaystyle=\int
d^{3}x\left[-\nabla({\bf E\times H})-\frac{1}{2\mu^{\prime}}\frac{d{\bf
B}^{2}}{dt}-\frac{1}{2\epsilon}\frac{d{\bf
D}^{2}}{dt}-\frac{ga}{\epsilon}\frac{d({\bf B}\cdot{\bf D})}{dt}\right],$
$\displaystyle\displaystyle=-\int d^{3}x\left[\nabla({\bf E}\times{\bf
H})+\frac{d}{dt}\frac{1}{2}\left(\frac{1}{\epsilon}{\bf
D}^{2}+\frac{1}{\mu^{\prime}}{\bf
B}^{2}\right)+\frac{d}{dt}\frac{g}{\epsilon}(a{\bf B}\cdot{\bf D})\right]$
$\displaystyle\displaystyle+\frac{g}{\epsilon}\int d^{3}x\dot{a}{\bf
B}\cdot({\bf D}+ga{\bf B})$ $\displaystyle\displaystyle=-\int d{\bf
F}\cdot{\bf S}_{Poynting}-\frac{dE_{em+int}}{dt}+\int d^{3}xg\dot{a}{\bf
B}\cdot{\bf E}.$ (9)

One might be tempted to make exclusive use of the ”vacuum” field strengths:
${\bf E},{\bf B}$. Then, the mixed term disappears from the exact energy
density of the electromagnetic field [14], and one finds

$\frac{dE_{mech}}{dt}=-\int d{\bf F}\cdot{\bf
S}_{Poynting}-\frac{d}{dt}\frac{1}{2}\int d^{3}x\left(\frac{1}{\mu}{\bf
B}^{2}+\epsilon{\bf E}^{2}\right)+\int d^{3}xg\dot{a}{\bf B}\cdot{\bf E}.$
(10)

Adding any form of this rate to the rate of change in the energy in the axion
sector, one recognizes that the power of the energy transfer between the two
sectors is determined by

$W_{transfer}=\int d^{3}xg\dot{a}{\bf B}\cdot{\bf E}=-\int d^{3}x{\bf
j}_{axion}\cdot{\bf E}$ (11)

and the energy balance is exactly fulfilled:

$\displaystyle\displaystyle{\frac{d}{dt}\left[E_{mech}+\int
d^{3}x\left(\frac{1}{\mu}{\bf B}^{2}+\epsilon{\bf E}^{2}\right)+\int
d^{3}x\frac{1}{2}\left(p_{a}^{2}+(\nabla
a)^{2}+\frac{m_{a}^{2}c^{2}}{\hbar^{2}}a^{2}\right)\right]+\int d{\bf F}{\bf
S}}$
$\displaystyle\displaystyle=\frac{d}{dt}\left(E_{mech}+E_{e.m.}+E_{axion}\right)+\int
d{\bf F}{\bf S}_{Poynting}=0.$ (12)

This expression displays a general feature of the energy–momentum tensor when
expressed in terms of ${\bf E}$ and ${\bf B}$, namely the axion–photon
interaction does not appear in it [14] since the term (11) responsible for the
energy transfer between the two sectors mutually cancel between (8) and (10).
One notes that with the present definitions of ${\bf D}$ and ${\bf H}$

${\bf E}\times{\bf H}=\frac{1}{\epsilon\mu}{\bf D}\times{\bf
B},\qquad\epsilon{\bf E}^{2}+\mu{\bf H}^{2}=\frac{1}{\epsilon}{\bf
D}^{2}+\frac{1}{\mu}{\bf B}^{2}+{\cal O}(g^{2});$ (13)

therefore, the above expression with ${\cal O}(g)$ accuracy can be rewritten
equally well in terms of the variables ${\bf E}$ and ${\bf H}$:

$\frac{dE_{mech}}{dt}=\int d^{3}x\left[-\nabla({\bf E}\times{\bf
H})-\frac{\partial}{\partial t}\left[\frac{1}{2}\left(\epsilon{\bf
E}^{2}+\mu{\bf H}^{2}\right)-\mu ga{\bf E}\cdot{\bf H}\right]-{\bf
j}_{axion}\cdot{\bf E}\right]+{\cal O}(g^{2}).$ (14)

The Poynting vector (that is the integrand of the surface integral) is
uniquely defined irrespective of the use of different field variables e.g.,
(${\bf E},{\bf H}$) or (${\bf D},{\bf B}$). The apparent difference between
the rate equation for ${\bf S}_{Abraham}={\bf D}\times{\bf B}$ and ${\bf
S}_{Minkowski}={\bf E}\times{\bf H}$ observed in [16] is a consequence of
restricting all their quantities to ${\cal O}(g)$ accuracy.

##  3 Energy Transfer from Axions in External Magnetic Field

Consider a finite region of volume $V_{H}$ where a strong static magnetic
induction field ${\bf B}_{0}({\bf x})$ is present. One computes the
electromagnetic field (${\bf E}_{1},{\bf B}_{1}$) generated by the temporal
variation of the axion field oscillation in this region, for one linearizes
the dynamical Equation (7) around the static magnetic field to find:

$\frac{1}{\mu}\nabla\times{\bf B}_{1}=\dot{\bf D}=\epsilon(-\ddot{\bf
A}_{1}-\nabla\dot{A}_{01})-g\dot{a}{\bf B}_{0}.$ (15)

Choosing the Lorentz gauge ($\nabla{\bf A}+\dot{A}_{0}=0$), an inhomogeneous
wave equation follows:

$\epsilon\mu\ddot{\bf A}_{1}-\Delta{\bf A}_{1}=-\mu g\dot{a}{\bf
B}_{0}\equiv\mu{\bf j}_{axion}.$ (16)

With the help of the retarded solution to this equation

${\bf A}_{1}({\bf x},t)=\frac{\mu}{4\pi}\int d^{3}x^{\prime}\frac{1}{R}{\bf
j}_{axion}({\bf x},t-R/c),\qquad R=|{\bf x}-{\bf x}^{\prime}|$ (17)

we proceed to calculate the rate of change in the axion energy due to the
electromagnetically mediated self-action of the axion field. This idea is
analogous to the computation of the radiation backreaction on the motion of
electrically charged objects. Its interest has been emphasized recently in
[17], though the attention of that paper is focused on Joule heat production
by the electric field produced by (17) in dielectrics in the quasi-stationary
approximation.

Our interest lies in estimating the size of the radiative backreaction on the
axion field. The steps of the calculation might remind us the computation of
the electromagnetic self-energy of the ”Abraham-electron”. The rate of change
is given as

$\frac{dE_{a}}{dt}=\int d^{3}x{\bf j}_{axion}\cdot{\bf E}_{1}=\int d^{3}x{\bf
j}_{axion}\cdot\left(-\dot{\bf A}_{1}-\nabla A_{01}\right).$ (18)

The contribution from the second term vanishes assuming homogeneity of the
axion field, which can be seen by performing partial integration and
recognizing that $\nabla{\bf j}_{axion}=-g\dot{a}\nabla{\bf B}_{0}=0$. Next,
one substitutes (17) for the first term and arrives for the time-averaged
energy loss at

$\displaystyle\displaystyle\frac{dE_{a}}{dt}=-\frac{\mu}{16\pi}\int d^{3}x\int
d^{3}x^{\prime}\frac{1}{R}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\displaystyle\times\left[{\bf j}_{axion}({\bf
x},t)\cdot\frac{\partial}{\partial t}{\bf j}_{axion}^{*}({\bf
x}^{\prime},t-R/c)+{\bf j}_{axion}^{*}({\bf x},t)\cdot\frac{\partial}{\partial
t}{\bf j}_{axion}({\bf x}^{\prime},t-R/c)\right]$
$\displaystyle\displaystyle=-\frac{\mu g^{2}}{16\pi}\int d^{3}x\int
d^{3}x^{\prime}\frac{1}{R}{\bf B}_{0}({\bf x})\cdot{\bf B}_{0}({\bf
x}^{\prime})\left[\dot{a}(t)\ddot{a}^{*}(t-R/c)+\dot{a}(t)^{*}\ddot{a}(t-R/c)\right].$
(19)

Assuming homogeneous harmonic oscillation of the axion field,
$a(t)=a_{0}\exp(-i\omega_{a}t)$, one arrives at

$-|a_{0}|^{2}\omega_{a}^{3}\frac{\mu g^{2}}{8\pi}\int d^{3}x\int
d^{3}x^{\prime}\frac{\sin\left(\omega_{a}R/c\right)}{R}{\bf B}_{0}({\bf
x})\cdot{\bf B}_{0}({\bf x}^{\prime}).$ (20)

If in the volume where ${\bf B}_{0}\neq 0$ the retardation is small, e.g.,
$\omega_{a}R/c<<1$, then the sine can be approximated by the first term of its
Taylor series:

$\frac{dE_{a}}{dt}\approx-|a_{0}|^{2}\omega_{a}^{4}\frac{\mu g^{2}}{8\pi
c}\left(\int d^{3}x{\bf B}_{0}({\bf x})\right)^{2}.$ (21)

This result can also be obtained working with the starting expression without
assuming the strict harmonic time dependence. In Equation (19), terms
depending on the retarded moment can be expanded in Taylor series:

$\ddot{a}(t-R/c)=\sum_{n=0}^{\infty}\frac{1}{n\\!}\left(-\frac{R}{c}\right)^{n}\frac{\partial^{n}}{\partial
t^{n}}\ddot{a}(t).$ (22)

One can average the power of the self-work of the axion field over the period
of the nearly harmonic time dependence of $a(t)$. One promptly recognizes that
the average of the $n=0$ term is zero. The leading nonzero contribution comes
from the $n=1$ term. The time average is written after performing a partial
integration as

$\frac{1}{T}\int_{0}^{T}dt\frac{dE_{a}}{dt}=-\frac{\mu g^{2}}{8\pi
c}\left(\int_{V_{H}}d^{3}x{\bf B}_{0}({\bf
x})\right)^{2}\frac{1}{T}\int_{0}^{T}|\ddot{a}(t)|^{2}.$ (23)

The relative magnitude of the higher terms is controlled by the combination
$R_{H}\omega_{a}/c$, where $R_{H}$ is the size of the region ${\bf B}_{0}\neq
0$ and $\omega_{a}$ the characteristic frequency of the axion field. If the
wavelength of the axion is much larger than this size, then these higher terms
are negligible. Below, for numerical estimates, we use the following
quantities (which appear the most frequently when discussing haloscopes):

$R_{H}=10{\textrm{cm}},\qquad
10^{-16}{\textrm{GeV}}{/c^{2}}<m_{a}=\frac{\hbar\omega_{a}}{c^{2}}<10^{-14}{\textrm{GeV}}{/c^{2}},$
(24)

using the following conversion from the MKSA to natural units:

$1{\textrm{cm}}=5.06\cdot 10^{13}{\textrm{GeV}}^{-1},$ (25)

one finds for this range of the axion mass

$0.05<\frac{\omega_{a}}{c}R_{H}<5.06.$ (26)

This means that the expansion in retardation is legitimate for $m_{a}\leq
10^{-16}{\textrm{GeV}}$ but cannot be used in the higher mass range. Then, one
has to return to (20) and perform the symmetric double integral over the
volume of the nonzero ${\bf B}_{0}$ region.

##  4 Axion Oscillation in Presence of a Magnetic Field

The equation describing the homogeneous axion oscillation can be written after
substituting the expression of ${\bf E}_{1}({\bf x},t)$:

$\frac{1}{c^{2}}\ddot{a}+\frac{m_{a}^{2}c^{2}}{\hbar^{2}}a=-\frac{\mu
g^{2}}{4\pi}\int d^{3}x^{\prime}\frac{{\bf B}_{0}({\bf x})\cdot{\bf
B}_{0}({\bf x}^{\prime})}{R}\ddot{a}\left(t-\frac{R}{c}\right).$ (27)

For a semiquantitative estimate of the decay rate, we assume that the magnetic
induction is homogeneous in the volume $V_{H}$ and average the right hand side
to erase the apparent dependence on $\bf x$:

$\ddot{a}(t)+\frac{m_{a}^{2}c^{4}}{\hbar^{2}}a(t)=-\frac{\mu
g^{2}}{2\pi}\left(\frac{1}{2}V_{H}{\bf
B}_{0}^{2}\right)\frac{1}{V_{H}^{2}}\int_{V_{H}}d^{3}x\int_{V_{H}}d^{3}x^{\prime}\frac{1}{R}\ddot{a}\left(t-\frac{R}{c}\right).$
(28)

First, we assume the expansion in retardation is justified. The leading
contribution arises by neglecting retardation effects and is just proportional
to $\ddot{a}$. Putting together this piece with the original kinetic term, a
new propagation velocity (smaller than the light velocity in a vacuum) can be
defined after averaging this expression over the volume of the ${\bf
B}_{0}\neq 0$ region:

$\frac{1}{v_{f}^{2}}=\frac{1}{c^{2}}+\frac{\mu g^{2}}{4\pi}\frac{1}{V_{H}}\int
d^{3}x\int d^{3}x^{\prime}\frac{1}{R}{\bf B}_{0}({\bf x}^{\prime})\cdot{\bf
B}_{0}({\bf x})$ (29)

This would produce harmonic oscillation of the axion field with frequency
$\omega_{a}=m_{a}cv_{f}/\hbar$.

Radiative damping of the axion oscillations arises when going further in
retardation effects. A force term $\sim\dddot{a}$ comes from the next ($n=1$)
term of the expansion:

$\frac{\mu g^{2}}{4\pi c}\int d^{3}x^{\prime}{\bf B}_{0}({\bf
x}^{\prime})\cdot{\bf B}_{0}({\bf x})\dddot{a}(t)\approx\frac{\mu g^{2}}{4\pi
c}V_{H}\left(\overline{{\bf B}_{0}}^{V_{H}}\right)^{2}\dddot{a}$ (30)

This kind of self-force is completely analogous to the case of the ”self-
accelerating Abraham-electron”, which leads to the following equation of
motion for the axion amplitude in the long-wavelength limit:

$\ddot{a}+\frac{m_{a}^{2}c^{2}v_{f}^{2}}{\hbar^{2}}a=\frac{\mu
g^{2}v_{f}^{2}}{4\pi c}V_{H}\left(\overline{{\bf
B}_{0}}^{V_{H}}\right)^{2}\dddot{a}.$ (31)

This equation coincides formally with the equation of the radiatively damped
charged harmonic oscillator:

$\ddot{x}-\tau\dddot{x}=-\omega_{0}^{2}x,$ (32)

with the correspondence

$\tau=\frac{\mu g^{2}v_{f}^{2}}{4\pi c}V_{H}\left(\overline{{\bf
B}_{0}}^{V_{H}}\right)^{2},\qquad\omega_{0}^{2}=\frac{m_{a}^{2}c^{2}v_{f}^{2}}{\hbar^{2}}.$
(33)

The causal solution (excluding the axion amplitude runaway) of (31) can be
constructed following Jackson’s treatment, which gives with ${\cal O}(g^{2})$
accuracy

$a(t)=a_{0}e^{-\alpha
t},\qquad\alpha=\frac{\Gamma}{2}+i\omega_{0},\qquad\Gamma=\tau\omega_{0}^{2}\approx\frac{\mu
g^{2}c}{4\pi}V_{H}\left(\overline{{\bf
B}_{0}}^{V_{H}}\right)^{2}\left(\frac{m_{a}c^{2}}{\hbar}\right)^{2}.$ (34)

In natural ($\hbar=c=1$) units

$\Gamma=\frac{m_{a}^{2}}{2\pi}g^{2}U_{B}$ (35)

where $U_{B}$ is the magnetic energy of the ${\bf B}_{0}\neq 0$ region. In
addition to those listed at the end of the previous section, we choose

$m_{a}=10^{-16}{\textrm{GeV}}{/c^{2}},\qquad|{\bf
B}_{0}|=10{\textrm{T}},\qquad g=10^{-12}{\textrm{GeV}}^{-1}.$ (36)

One finds $U_{B}\approx 2.5\cdot 10^{14}{\textrm{GeV}}$, which provides for
the radiative relaxation time of the coherent axion oscillations, an estimate
comparable to the age of the Universe:

$\Gamma^{-1}\approx 1.7\cdot 10^{18}{\textrm{ s}}\approx 5.3\cdot
10^{10}{\textrm{yr}}.$ (37)

For an estimate valid at the other edge of the mass range
($m_{a}=10^{-14}{\textrm{GeV}}{/c^{2}}$), we have to return to the equation
describing the homogeneous axion oscillation without relying on the expansion
in the retardation:

$\frac{1}{c^{2}}\ddot{a}+\frac{m_{a}^{2}c^{2}}{\hbar^{2}}a=-\frac{\mu
g^{2}}{4\pi}\int d^{3}x^{\prime}\frac{{\bf B}_{0}({\bf x})\cdot{\bf
B}_{0}({\bf x}^{\prime})}{R}\ddot{a}\left(t-\frac{R}{c}\right).$ (38)

For a semiquantitative estimate of the decay rate, we assume that the magnetic
induction is homogeneous in the volume $V_{H}$ and average the right-hand side
to suppress the apparent dependence on $\bf x$. After substituting the
approximate solution in the form introduced in (34), one has

$\alpha^{2}a(t)+\frac{m_{a}^{2}c^{4}}{\hbar^{2}}a(t)=-\frac{\mu
g^{2}\alpha^{2}}{2\pi}\left(\frac{1}{2}V_{H}{\bf
B}_{0}^{2}\right)\frac{1}{V_{H}^{2}}\int_{V_{H}}d^{3}x\int_{V_{H}}d^{3}x^{\prime}\frac{e^{\alpha
R/c}}{R}\cdot a(t).$ (39)

From this, the dispersion relation is extracted. Its real part can be used to
find $\omega_{0}$, which is slightly shifted from the frequency $\omega_{a}$.
Still, $\omega_{0}R_{H}/c$ is larger than unity, therefore one cannot expand
into Taylor series the exponential $\exp(-i\omega_{0}R/c)$. However, one can
assume $\Gamma<<\omega_{0}$ and linearize $\exp(\Gamma R/2c)$ in $\Gamma$.
Then, the imaginary part of the resulting approximate equation can be used for
the determination of the decay rate:

$\omega_{0}\Gamma=-\frac{\mu g^{2}c^{2}}{2\pi}\left(\frac{1}{2}V_{H}{\bf
B}_{0}^{2}\right)\frac{1}{V_{H}^{2}}\int_{V_{H}}d^{3}x\int_{V_{H}}d^{3}x^{\prime}\left[-\omega_{0}^{2}\frac{\sin(\omega_{0}R/c)}{R}\left(1+\frac{R}{2c}\right)+\omega_{0}\Gamma\frac{\cos(\omega_{0}R)}{R}\right].$
(40)

The integral of the quickly oscillating integrand $\sin(\omega_{0}R/c)$ is
very close to zero. Rearranging the terms one finds

$\displaystyle\displaystyle\Gamma\left[1+\frac{\mu
g^{2}c^{2}}{2\pi}\left(\frac{1}{2}V_{H}{\bf
B}_{0}^{2}\right)\frac{1}{V_{H}^{2}}\int_{V_{H}}d^{3}x\int_{V_{H}}d^{3}x^{\prime}\frac{\cos(\omega_{0}R/c)}{R}\right]$
$\displaystyle\displaystyle=\frac{\omega_{0}\mu
g^{2}c^{2}}{2\pi}\left(\frac{1}{2}V_{H}{\bf
B}_{0}^{2}\right)\frac{1}{V_{H}^{2}}\int_{V_{H}}d^{3}x\int_{V_{H}}d^{3}x^{\prime}\frac{\sin(\omega_{0}R/c)}{R}.$
(41)

The quantities appearing in this equation are related to the real and
imaginary part of the integral:

$I(k_{0},R_{H})=\frac{1}{V_{H}^{2}}\int_{V_{H}}d^{3}x\int_{V_{H}}d^{3}x^{\prime}\frac{e^{ik_{0}R}}{4\pi
R},\qquad R=|{\bf x}-{\bf x}^{\prime}|,\quad k_{0}=\frac{\omega_{0}}{c}.$ (42)

Exploiting the expansion of the integrand in terms of spherical harmonics

$\frac{e^{ik_{0}R}}{4\pi
R}=ik_{0}\sum_{l}j_{l}(k_{0}r_{<})h_{l}^{(1)}(k_{0}r_{>})\sum_{m=-l}^{l}Y^{*}_{lm}(\Omega)Y_{lm}(\Omega^{\prime})$
(43)

the angular integrals can be performed. We choose a sphere of radius $R_{H}$
since it is very convenient for a fully analytic treatment. In addition,
exploiting the symmetry of the range of integration one finds

$I(k_{0},R_{H})=\frac{8\pi
ik_{0}}{V_{H}^{2}}\int_{0}^{R_{H}}r_{>}^{2}dr_{>}\int_{0}^{r_{>}}r_{<}^{2}dr_{<}j_{0}(k_{0}r_{<})h_{0}^{(1)}(k_{0}r_{>}).$
(44)

Introducing the notation $X_{H}=k_{0}R_{H}$, one easily finds

$\displaystyle\displaystyle{\textrm{Im}}I(k_{0},R_{H})=\frac{9k_{0}}{4\pi
X_{H}^{6}}\left[X_{H}^{2}+1+(X_{H}-1)\cos(2X_{H})-2X_{H}\sin(2X_{H})\right]$
$\displaystyle\displaystyle{\textrm{Re}}I(k_{0},R_{H})=\frac{9k_{0}}{4\pi
X_{H}^{6}}\left[-\frac{2}{3}X_{H}^{3}-(X_{H}^{2}-1)\sin(2X_{H})-2X_{H}\cos(2X_{H})\right].$
(45)

With the actual parameters, the second term in the bracket on the left-hand
side of (41) is negligible relative to the unity. Second, on the right-hand
side, the quantity multiplying $\omega_{0}$ is very small, ${\cal
O}(10^{-27})$, which confirms the linearization in $\Gamma$. Due to the larger
axion mass, a somewhat faster decay rate is obtained:

$\Gamma^{-1}\approx 5\cdot 10^{9}{\textrm{yr}}.$ (46)

This is still comparable to the age of the Universe.

##  5 Momentum Conservation with Axions

Again, one starts with the rate of change in the mechanical momentum of the
electrically charged medium:

$\displaystyle\displaystyle\frac{d{\bf P}_{mech}}{dt}=\int d^{3}x(j_{0}{\bf
E}+{\bf j}\times{\bf B})=-\frac{d}{dt}\int d^{3}x({\bf D}\times{\bf B})$
$\displaystyle\displaystyle+\int d^{3}x\left[{\bf E}(\nabla{\bf
D})+(\nabla\times{\bf H})\times{\bf B}-{\bf D}\times(\nabla\times{\bf E})+{\bf
H}(\nabla{\bf B})\right].$ (47)

Changing from (${\bf D},{\bf B})$) to the (${\bf E},{\bf H}$) pair of
variables, one notes the exact relation

${\bf D}\times{\bf B}=(\epsilon{\bf E}-ga{\bf B})\times{\bf B}=\epsilon{\bf
E}\times(\mu{\bf H}-\mu ga{\bf E})=\epsilon\mu{\bf E}\times{\bf H},$ (48)

which gives an unchanged relation of the electromagnetic momentum density to
the Poynting vector.

The second volume integral of the second expression of (47) is the simplest
when using the pair of variables (${\bf E},{\bf B}$). Then, $i$-th component
of its integrand reads (summation is understood over repeated indices):

$\displaystyle\displaystyle{E_{i}(\nabla{\bf D})+[(\nabla\times{\bf
H})\times{\bf B}]_{i}-[{\bf D}\times(\nabla\times{\bf
E})]_{i}+H_{i}(\nabla{\bf B})}$
$\displaystyle\displaystyle=E_{i}\partial_{j}D_{j}+B_{j}\partial_{j}H_{i}-B_{j}\partial_{i}H_{j}+D_{j}\partial_{j}E_{i}-D_{j}\partial_{i}E_{j}+H_{i}\partial_{j}B_{j}$
$\displaystyle\displaystyle=\epsilon\left(\partial_{j}(E_{i}E_{j})-E_{j}\partial_{i}E_{j}\right)-g\partial_{j}(E_{i}B_{j}a)+gaB_{j}\partial_{i}E_{j}$
$\displaystyle\displaystyle+\frac{1}{\mu}\left(\partial_{j}(B_{i}B_{j})-B_{j}\partial_{i}B_{j}\right)+g\partial_{j}(E_{i}B_{j}a)-gB_{j}\partial_{i}(aE_{j})$
(49)

By the cancellations between the last two lines one arrives at

$\displaystyle\displaystyle\frac{d{\bf
P}_{mech,i}}{dt}=-\epsilon\mu\frac{d}{dt}\int d^{3}x({\bf E}\times{\bf
H})_{i}-\int d^{3}x\partial_{j}\Pi_{ij}-\int d^{3}x\partial_{i}a({\bf
E}\cdot{\bf B}),$
$\displaystyle\displaystyle\Pi_{ij}=\left[\frac{1}{2}\left(\epsilon{\bf
E}^{2}+\frac{1}{\mu}{\bf B}^{2}\right)\delta_{ij}-\epsilon
E_{i}E_{j}-\frac{1}{\mu}B_{i}B_{j}\right].$ (50)

Momentum leaks to the axion sector through inhomogeneity of the axion field.
In the long-wavelength limit there is no recoil from the axion sector. This is
consistent with the vanishing Lorentz self-force, when the spatial variation
of the axion is neglected, e.g., ${\bf j}_{a}\times{\bf B}=-g\dot{a}{\bf
B}\times{\bf B}=0$. A spatial gradient of the axion field splits the
electromagnetic eigenmodes between parallel uncharged material plates, though
the summary contribution to the Casimir force [19].

##  6 Summary

A pedagogical derivation of the classical laws of energy–momentum conservation
in Maxwell’s electrodynamics completed by the interaction of the axion field
with the topological charge density of the electromagnetic fields was
presented. This derivation allows the unique identification of the term
responsible for the energy transfer between the electromagnetic and the axion
sectors of the theory. Its power density ${\bf j}_{axion}{\bf E}$ is
completely analogous to the energy dissipation power density of electrically
charged objects. It allowed the computation of the backreaction of the
electromagnetic radiation emitted by the axion field oscillations on these
oscillations themselves. An explicit example is presented for the case of
radiation in the presence of a strong static magnetic field in a finite
macroscopic region. The characteristic damping time, estimated here for a
first time, is comparable to the age of the Universe for typical values of
axion masses and haloscope size.

## Acknowledgements

This research received support of the Hungarian National Research, Development
and Innovation Fund under Project No. K123815. Discussions with Tamás Biró and
Gergely Fejős are gratefully acknowledged.

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