Self-dual U(1) lattice field theory with a $\theta$-term

Mariia Anosova$\,{}^{a}$, Christof Gattringer$\,{}^{b}$ and Tin
Sulejmanpasic$\,{}^{c}$
$\;{}^{a}$ Universität Graz, Institut für Physik111Member of NAWI Graz.,
Universitätsplatz 5, 8010 Graz, Austria b FWF - Austrian Science Fund222On
leave of absence from Universität Graz., Sensengasse 1, 1090 Vienna, Austria
$\;{}^{c}$ Department of Mathematical Sciences, Durham University, DH1 3LE
Durham, United Kingdom

###### Abstract

We study U(1) gauge theories with a modified Villain action. Such theories can
naturally be coupled to electric and magnetic matter, and display exact
electric-magnetic duality. In their simplest formulation without a
$\theta$-term, such theories are ultra-local. We extend the discussion to U(1)
gauge theories with $\theta$-terms, such that $\theta$ periodicity is exact
for a free theory, and show that imposing electric-magnetic duality results in
a local, but not ultra-local lattice action, which is reminiscent of the
Lüscher construction of axial-symmetry preserving fermions in 4d. We discuss
the coupling to electric and magnetic matter as well as to dyons. For dyonic
matter the electric-magnetic duality and shifts of the $\theta$-angle by
$2\pi$ together generate an SL$(2,\mathbb{Z})$ duality group of
transformations, just like in the continuum. We finally illustrate how the
SL$(2,\mathbb{Z})$ duality may be used to explore theories at finite $\theta$
without a sign problem.

##  1 Introduction

Abelian U(1) gauge theories are at the heart of electrodynamics, which,
together with gravity, explains most of the world we experience. In modern
high-energy physics, they are typically viewed as boring cousins of their non-
abelian counterparts. The reason for this is that in four space-time
dimensions they are infrared free and UV incomplete. In solid-state physics,
however, U(1) gauge theories pop up as effective descriptions of many
condensed matter systems with non-trivial and strongly coupled dynamics.
Moreover they show up as effective descriptions of non-abelian gauge theories
and were long hoped to shed light on quark confinement and mass gap generation
in those theories. While such abelian mechanisms have shown to be relevant in
both supersymmetric [1, 2], and non-supersymmetric settings [3, 4, 5, 6], a
lot of justified skepticism remains about the potential of abelian gauge
theories to give insights into the mysteries of 4d non-abelian gauge theories.

Both, in condensed matter applications as well as in the abelianized non-
abelian gauge theories, monopoles play a crucial role. Monopoles have the
ability to render the theory confining, and can therefore completely change
the IR dynamics of the theory. The details of monopoles (e.g., their charge or
mass) dictate whether or not they are important for the IR physics. Yet, on
the lattice monopoles typically come as artifact rather than a feature, and
are not under direct control. Two of us proposed a formalism of U(1) lattice
gauge theories which allows electric and magnetic matter to be coupled
simultaneously [7], and provides complete control over the matter content of
the theory – both electric and magnetic. Moreover the formalism allows a
duality transformation, mapping magnetic and electric content to each other,
which is a well known feature of continuum theories. Such models were used for
avoiding the complex action problem associated with the $\theta$-term in 2d
abelian gauge theories [8, 9, 10, 11, 12], and for formulating interacting
exactly self-dual gauge theories [13, 14]. These modified Villain theories
also found applications in fracton physics [15] and construction of theories
with non-invertible symmetries [16, 17].

Furthermore 4d Villain theories allow for $\theta$-terms. Indeed
$\theta$-terms in Villain-like models which have the correct continuum limit
have been introduced long ago [18, 19], but generically they do not preserve
exact $\theta$-periodicity and self-duality, when applicable333The
$\theta$-periodicity is lost in the presence of monopoles due to the Witten
effect, since a monopole turns into a dyon which carries electric charge as
well as a magnetic one. However, if the entire spectrum of dyons exists, whose
masses correctly interchange under a $2\pi$ shift, the $\theta$-periodicity
can be valid even in the presence of magnetically charged matter. This is in
fact what happens in abelianized non-abelian theories in 4d.. In [7] a class
of $\theta$-terms which preserve the exact $2\pi$ periodicity of the
$\theta$-angle in the pure gauge theory were introduced, but self-dual
transformations in the presence of the $\theta$-term were not discussed in
detail. Here we find that insisting on exact $\theta$-periodicity and on
ultra-locality ruins self-duality. More precisely, if one performs a duality
transformation with one of the $\theta$-terms proposed in [7], a dual theory
has a local, but not ultra-local444A lattice action is called _ultra-local_
when it couples only fields at a finite number of lattice distances. In a
_local_ lattice theory fields at arbitrary distances may couple, but the pre-
factors decrease (at least) exponentially with the lattice separation. action,
so that the theory is not self-dual for any choice of parameters.

On the other hand, one can restore self-duality by defining a local, but not
ultra-local lattice action, which transforms covariantly under the duality
transformation. We therefore reformulate the original U(1) lattice gauge
theory in a local but non-ultra-local way, such that self-duality becomes
exact. We recover the $\text{SL}(2,\mathbb{Z})$ duality, well known in
continuum U(1) gauge theories [20]555It is worth noting that [19] is the first
proposal of the SL$(2,\mathbb{Z})$ structure for a lattice gauge theory,
which, however, is only approximate.. Exploiting the $\text{SL}(2,\mathbb{Z})$
structure, we discuss interacting theories free from the sign problem. The
situation is reminiscent of exactly massless Dirac fermions on the lattice.
Exactly massless Dirac fermions have axial symmetries in addition to vector
symmetries, which have mixed ’t Hooft anomalies and are subject to the
Nielsen-Ninomiya theorem [21, 22, 23]. This was widely interpreted as as the
inability to preserve axial symmetries on the lattice. However, Lüscher showed
[24],using Neuberger’s solution [25] of the Ginsparg-Wilson relation [26],
that a lattice theory possesing an exact axial symmetry exists, at the expense
of abandoning ultra-locality.

In fact also abelian gauge theories have ‘t Hooft anomalies [27, 28, 29, 30]
(see also [31]), so a straightforward implementation of symmetries is bound to
face an obstructions, much like in the case of fermionic chiral symmetry. What
is perhaps surprising is that the lattice action can still be made ultra-local
when $\theta$ angle is vanishing [7], as we will review. Nevertheless we are
forced to abandon ultra-locality when insisting on the $\theta$-angle
periodicity666The $2\pi$ $\theta$-angle shifts are only an invariance of a
theory without dynamical monopoles, because $2\pi$ shifts cause all monopoles
to become dyons. Alternatively the theory can posses an entire tower of
dynamical dyons of unit magnetic charge and all electric charges, in which
case the $2\pi$ periodicity will be restored. This is precisely what happens
in the abelianized regimes of the $SU(N)$-gauge theory.  and imposing self-
duality in free abelian gauge theory. As is well known, these transformations
form an $\text{SL}(2,\mathbb{Z})$ group.

The paper is organized as follows: In the next section we review the approach
to lattice discretization from [7] that gives rise to the Villain form with an
additional closedness constraint for the 2-form Villain variables. This
closedness constraint removes monopoles and we show that pure U(1) lattice
gauge theory becomes self-dual. In Section 3 we add electrically and
magnetically charged matter and demonstrate that with our formulation self-
duality can be extended to full QED with either bosonic or fermionic matter.
For the case of bosonic matter we show that with a worldline formulation all
sign problems are overcome (for fermionic matter the Grassmann nature of the
matter fields may lead to a remaining sign problem). Based on the closedness
constraint, in Section 4 we define a topological charge in a consistent way
and discuss its properties including the Witten effect. Using this topological
charge we include a $\theta$-term and discuss the generalized duality
transformation that takes into account this term. To obtain self-duality with
the $\theta$-term we generalize the action to a non-ultra-local form which
leads to a fully self-dual lattice discretization of QED with a $\theta$-term.
We discuss the resulting self-duality relations for several observables as
well as the full $\text{SL}(2,\mathds{Z})$ duality. Finally we employ this
duality to discuss theories with non-zero topological angle $\theta$ that can
be simulated without complex action problem.

Some more technical parts are collected in several appendices: Appendix A
summarizes our notation and some results for differential forms on the
lattice. In Appendix B we discuss properties of our definition of the
topological charge. In Appendix C we discuss properties of the action kernels
used in our paper and in Appendix D we provide a short derivation of a
generalized Poisson resummation formula. Appendix E shows how self dual QED
can be mapped to a worldline representation that avoids the complex action
problem and thus makes the theory accessible to Monte Carlo simulations.
Finally, in Appendix F we discuss a further generalization of the topological
charge that implements all lattice symmetries.

##  2 Generalized Villain action for U(1) gauge fields and self-duality

In this section we briefly summarize the discretization strategy outlined in
[7], which leads to a Villain form of the Boltzmann factor [32]. We implement
additional closedness constraints for the Villain variables that suppress the
artificial monopoles that plague the standard lattice discretization of U(1)
gauge fields. We show that the construction leads to a self-dual
discretization of the photon field which will be the starting point for more
general self-dual theories in the subsequent sections.

###  2.1 Generalized Villain discretization with closedness constraints

When coupling U(1) gauge fields to matter in a gauge invariant way one uses
the U(1)-valued compact link variables $U_{x,\mu}\in$ U(1). The coupling
between gauge and matter fields is then implemented via nearest neighbor terms
such as $\phi_{x}^{\,*}\,U_{x,\mu}\,\phi_{x+\hat{\mu}}$. Under a U(1) gauge
transformation the matter fields transform as $\phi_{x}\rightarrow
e^{i\lambda_{x}}\phi_{x}$, such that the transformation $U_{x,\mu}\rightarrow
e^{i\lambda_{x}}U_{x,\mu}\,e^{-\,i\lambda_{x+\hat{\mu}}}$ gives rise to gauge
invariance of $\phi_{x}^{\,*}\,U_{x,\mu}\,\phi_{x+\hat{\mu}}$.

We now parameterize the link variables in the form777To be specific, we
consider a 4-d hypercubic lattice with lattice extents $N_{\mu},\mu=1,2,3,4$
and a total number of sites $V\equiv N_{1}N_{2}N_{3}N_{4}$. The lattice
constant is set to $a=1$ and all fields obey periodic boundary conditions.

$U_{x,\mu}\;\equiv\;e^{iA^{e}_{x,\mu}}\;,$ (1)

with the gauge fields $A^{e}_{x,\mu}\in[-\pi,\pi)$. We added the superscript e
to mark the gauge fields as _electric gauge fields_ , a notation that will
turn out to be useful when studying electric-magnetic duality.

We may define the field strength tensor as the naive discretization of the
continuum form $\partial_{\mu}A^{e}_{\nu}-\partial_{\nu}A^{e}_{\mu}$ and
obtain

$A^{e}_{x+\hat{\mu},\nu}-A^{e}_{x,\nu}-A^{e}_{x+\hat{\nu},\mu}+A^{e}_{x,\mu}\;\equiv\;(dA^{e})_{x,\mu\nu}\;,$
(2)

where in the last step we have defined the exterior lattice derivative $d$,
here acting on the link variables. We will partly use the language of
differential forms on the lattice and in Appendix A briefly summarize our
conventions and some of the results we use (for a more extensive presentation
see the appendix of [7] or the mathematical standard literature such as [33]).

The compact link variables (1) are obviously invariant under the shifts

$A^{e}_{x,\mu}\;\rightarrow\;A^{e}_{x,\mu}\;+\;2\pi\,k_{x,\mu}\;,\;\;k_{x,\mu}\;\in\;\mathds{Z}\;.$
(3)

However, the exterior derivative $(dA^{e})_{x,\mu\nu}$ is not invariant under
shifts, but instead transforms as

$(dA^{e})_{x,\mu\nu}\;\rightarrow\;(dA^{e})_{x,\mu\nu}\;+\;2\pi\;(dk)_{x,\mu\nu}\;,$
(4)

i.e., $(dA^{e})_{x,\mu\nu}$ is shifted by multiples of $2\pi$. One way to
establish invariance under the shift symmetry is to construct the action from
the periodic function $\cos((dA^{e})_{x,\mu\nu})$, which gives rise to the
Wilson gauge action. Another option is to define the field strength as

$F^{e}_{x,\mu\nu}\;\equiv\;(dA^{e})_{x,\mu\nu}\;+\;2\pi\,n_{x,\mu\nu}\;=\;(dA^{e}\;+\;2\pi\,n)_{x,\mu\nu}\;,$
(5)

where $n_{x,\mu\nu}\in\mathds{Z}$ is a 2-form (i.e., plaquette based) variable
which subsequently is summed over and obviously eats up a possible term
$2\pi\,(dk)_{x,\mu\nu}$ generated by the shifts (3). In other words the
Villain variables $n_{x,\mu\nu}$ can be viewed as gauge field of the shift
symmetry [7]. In its simplest form this construction gives rise to a gauge
field Boltzmann factor

$B_{\beta}[A^{e}]\,\equiv\,\prod_{x\in\Lambda}\prod_{\mu<\nu}\;\sum_{n_{x,\mu\nu}\in\mathds{Z}}\\!\\!\\!e^{\,-\frac{\beta}{2}F^{e}_{x,\mu\nu}F^{e}_{x,\mu\nu}}\;=\;\prod_{x\in\Lambda}\prod_{\mu<\nu}\;\sum_{n_{x,\mu\nu}\in\mathds{Z}}\\!\\!\\!e^{\,-\frac{\beta}{2}(dA^{e}\;+\;2\pi\,n)_{x,\mu\nu}(dA^{e}\;+\;2\pi\,n)_{x,\mu\nu}}\;,$
(6)

which is known as the _Villain discretization_ [32]. The first product runs
over our 4-dimensional lattice $\Lambda$ with periodic boundary conditions. We
will refer to the variables $n_{x,\mu\nu}$ as _Villain variables_. As usual
$\beta$ is the inverse gauge coupling $\beta=1/e^{2}$, where $e$ is the
electric charge.

The Boltzmann factor (6) is not only invariant under the shifts (3), (4), but
also under the transformation $A^{e}_{x,\mu}\rightarrow
A^{e}_{x,\mu}-\lambda_{x+\hat{\mu}}+\lambda_{x}$ which, up to a possible re-
projection into the interval $[-\pi,\pi)$, corresponds to the gauge
transformation of the gauge fields $A^{e}_{x,\mu}$.

A key insight is to note that the Villain variables $n_{x,\mu\nu}$ may be
constrained further: They were introduced to implement the invariance of the
Boltzmann factor $B_{\beta}[A^{e}]$ under the shifts (3), (4) which leads to a
shift of the exterior derivative $(dA^{e})_{x,\mu\nu}$ by
$2\pi\,(dk)_{x,\mu\nu}$. Note that this shift obeys $(d(dk))_{x,\mu\nu\rho}=0$
due to the nil-potency of the exterior derivative operator $d$ , i.e.,
$d^{2}=0$ (compare Appendix A). Thus we may constrain also the Villain
variables $n_{x,\mu\nu}$ to obey

$(dn)_{x,\mu\nu\rho}\;=\;0\;\;\;\;\forall x\;,\;\;\mu<\nu<\rho\;.$ (7)

This constraint implies that for all 3-cubes $(x,\mu<\nu<\rho)$ of the lattice
the oriented sum over the Villain variables on the surface of the cube
vanishes (Appendix A). Using again the language of differential forms, the
Villain variables are restricted to be a _closed_ integer-valued 2-form. We
point out that this constraint implies the absence of monopoles as we will
discuss in more detail below.

Taking into account this closedness constraint we may formulate the partition
sum of pure U(1) gauge theory in the form

$Z(\beta)\;\equiv\;\int
D[A^{e}]\;\sum_{\\{n\\}}\;e^{\,-\frac{\beta}{2}\sum_{x,\mu<\nu}(F^{e}_{x,\mu\nu})^{2}}\;\prod_{x}\prod_{\mu<\nu<\rho}\delta\big{(}(dn)_{x,\mu\nu\rho}\big{)}\;.$
(8)

Here we have defined the measure for the gauge fields and the sum over all
configurations of the Villain variables as

$\int
D[A^{e}]\;\equiv\;\prod_{x}\prod_{\mu}\int_{-\pi}^{\pi}\\!\\!\frac{dA^{e}_{x,\mu}}{2\pi}\;\;\;,\;\;\;\;\sum_{\\{n\\}}\;\equiv\;\prod_{x}\prod_{\mu<\nu}\sum_{n_{x,\mu\nu}\in\mathds{Z}}\;.$
(9)

The product in (8) runs over all 3-cubes $(x,\mu<\nu<\rho)$ and implements the
closedness constraint (7) using Kronecker deltas which we here denote with
$\delta(j)\equiv\delta_{j,0}$.

It is useful to write the constraints in (8) by using the Fourier
representation of the Kronecker deltas, such that

$\displaystyle\prod_{x}\\!\prod_{\mu<\nu<\rho}\\!\\!\delta\Big{(}\\!(dn)_{x,\mu\nu\rho}\\!\Big{)}$
$\displaystyle=$
$\displaystyle\prod_{x}\\!\prod_{\mu<\nu<\rho}\int_{-\pi}^{\pi}\\!\frac{dA^{m}_{x,\mu\nu\rho}}{2\pi}\,e^{-iA^{m}_{x,\mu\nu\rho}(dn)_{x,\mu\nu\rho}}$
(10) $\displaystyle=$
$\displaystyle\int\\!\\!D[A^{m}]\,e^{-i\sum_{x}\\!\sum_{\mu<\nu<\rho}A^{m}_{x,\mu\nu\rho}(dn)_{x,\mu\nu\rho}},$

where we have introduced auxiliary fields $A^{m}_{x,\mu\nu\rho}\in\mathds{R}$
assigned to the cubes $(x,\mu\nu\rho)$ of the lattice which, when integrated
over, generate the closedness constraint (7) for the Villain variables. The
notation $A^{m}$ is chosen to reflect the fact that in the electric-magnetic
duality transformations we discuss below, the auxiliary field $A^{m}$ will
take over the role of the vector potential, while in the dual form the vector
field $A^{e}$ will generate the dual closedness constraints. Due to this role
in the duality transformation we will also use the nomenclature _magnetic
gauge field_ for $A^{m}$, while $A^{e}$ is referred to as _electric gauge
field_ (see above).

For notational convenience, in the last step of (10) we have introduced the
integral over all configurations of the magnetic gauge fields
$A^{m}_{x,\mu\nu\rho}$,

$\int\\!\\!D[A^{m}]\;\equiv\,\prod_{x}\\!\prod_{\mu<\nu<\rho}\int_{-\pi}^{\pi}\\!\frac{dA^{m}_{x,\mu\nu\rho}}{2\pi}\;.$
(11)

We thus may write the partition sum of our discretization of the U(1) gauge
field in the form,

$Z(\beta)\;=\;\int\\!\\!D[A^{e}]\int\\!\\!D[A^{m}]\;B_{\beta}[A^{e},A^{m}]\;,$
(12)

where we have introduced the Boltzmann factor

$B_{\beta}[A^{e},A^{m}]\;\equiv\;\sum_{\\{n\\}}\;e^{\,-\frac{\beta}{2}\sum_{x}\\!\sum_{\mu<\nu}(F^{e}_{x,\mu\nu})^{2}}\;e^{\,-i\sum_{x}\\!\sum_{\mu<\nu<\rho}A^{m}_{x,\mu\nu\rho}(dn)_{x,\mu\nu\rho}},$
(13)

that depends on both, the electric gauge field $A^{e}$ that in the Boltzmann
factor $B_{\beta}[A^{e},A^{m}]$ describes the photon dynamics, as well as on
the magnetic gauge field $A^{m}$ that here generates the constraints. As
announced, in the duality transformation we discuss in the next subsection
these two fields will interchange their role.

###  2.2 Proof of self-duality

The first step towards establishing self-duality is to rewrite the exponent of
the second term in the Boltzmann factor (13),

$\displaystyle\sum_{x}\\!\sum_{\mu<\nu<\rho}A^{m}_{x,\mu\nu\rho}(dn)_{x,\mu\nu\rho}$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\sum_{x}\\!\sum_{\mu<\nu<\rho}A^{m}_{x,\mu\nu\rho}\big{(}d(dA^{e}+2\pi
n)\big{)}_{x,\mu\nu\rho}$ (14) $\displaystyle=$
$\displaystyle-\frac{1}{2\pi}\sum_{x}\\!\sum_{\mu<\nu}(\partial
A^{m})_{x,\mu\nu}(dA^{e}+2\pi n)_{x,\mu\nu}\;,$

where in the first step we used $d^{\,2}=0$ (see Appendix A) to insert the
term $dA^{e}$. Subsequently we applied the partial integration formula (125).
The Boltzmann factor (13) thus can be written in the form of a product over
all plaquettes (we here inserted
$F^{e}_{x,\mu\nu}=(dA^{e}\,+\,2\pi\,n)_{x,\mu\nu}$),

$B_{\beta}[A^{e},A^{m}]\;=\;\prod_{x}\prod_{\mu<\nu}\sum_{n_{x,\mu\nu}\in\mathds{Z}}e^{\,-\frac{\beta}{2}\big{(}(dA^{e}\,+\,2\pi\,n)_{x,\mu\nu}\big{)}^{2}}\;e^{\,\frac{i}{2\pi}(dA^{e}\,+\,2\pi
n)_{x,\mu\nu}(\partial A^{m})_{x,\mu\nu}}.$ (15)

Each factor in this product is $2\pi$-periodic in the corresponding variable
$dA^{e}$, where the periodicity is generated by the sum over the Villain
variable on the plaquette. Thus we may use Poisson resummation (See Appendix D
for the proof of a more general result that collapses to the usual Poisson
resummation when setting $N=1$.),

$\sum_{n\in\mathds{Z}}e^{\,-\frac{\beta}{2}(dA^{e}\,+\,2\pi\,n)^{2}}\;e^{\,\frac{i}{2\pi}(dA^{e}\,+\,2\pi
n)(\partial
A^{m})}\;=\;\frac{1}{\sqrt{2\pi\beta}}\sum_{p\in\mathds{Z}}e^{\,-\frac{1}{2}\frac{1}{4\pi^{2}\beta}(\partial
A^{m}\,+\,2\pi\,p)^{2}}\;e^{\,-i\,p\,dA^{e}}\;,$ (16)

where we have omitted the plaquette indices for notational convenience. Using
this expression for all factors in the Boltzmann weight (15), we find

$B_{\beta}[A^{e},A^{m}]\;=\;\left(\\!\frac{1}{2\pi\beta}\\!\right)^{\\!3V}\sum_{\\{p\\}}\;e^{\,-\frac{\widetilde{\beta}}{2}\sum_{x,\mu<\nu}\big{(}(\partial
A^{m}\,+\,2\pi\,p)_{x,\mu\nu}\big{)}^{2}}\;e^{\,-\,i\sum_{x,\mu<\nu}(dA^{e})_{x,\mu\nu}\,p_{x,\mu\nu}},$
(17)

where we have defined the dual gauge coupling

$\widetilde{\beta}\;\equiv\;\frac{1}{4\pi^{2}\beta}\;,$ (18)

and denote the sum over all configurations of the newly introduced plaquette
occupation numbers $p_{x,\mu\nu}\in\mathds{Z}$ as

$\sum_{\\{p\\}}\;\equiv\;\prod_{x}\prod_{\mu<\nu}\sum_{p_{x,\mu\nu}\in\mathds{Z}}\;.$
(19)

The next step is to use again the partial integration formula (125) from
Appendix A to rewrite the second exponent in (17) such that the Boltzmann
factor reads

$B_{\beta}[A^{e},A^{m}]\;=\;\left(\\!\frac{1}{2\pi\beta}\\!\right)^{\\!3V}\sum_{\\{p\\}}\;e^{\,-\frac{\widetilde{\beta}}{2}\sum_{x,\mu<\nu}\big{(}(\partial
A^{m}\,+\,2\pi\,p)_{x,\mu\nu}\big{)}^{2}}\;e^{\,-i\sum_{x,\mu}A^{e}_{x,\mu}\,(\partial\,p)_{x,\mu}}.$
(20)

We remark that we will use the form (20) of the Boltzmann factor to show that
the self-dual formulation of scalar electrodynamics that we construct in the
next section is free of any complex action problem. The self-dual formulation
of pure gauge theory is already free of a complex action problem when the form
(8) of the partition sum is used.

To complete the proof of self-duality of pure U(1) gauge theory, we now switch
to the dual lattice. Using (127) we identify the $r$ forms on the original
lattice with $4-r$ forms on the dual lattice (when forms are considered on the
dual lattice they are marked with “$\widetilde{\;\;\;}$” - compare Appendix
A),

$A^{e}_{x,\mu}=\\!\\!\\!\sum_{\nu<\rho<\sigma}\\!\\!\epsilon_{\mu\nu\rho\sigma}\,\widetilde{A}^{e}_{\tilde{x}-\hat{\nu}-\hat{\rho}-\hat{\sigma},\nu\rho\sigma}\,,\;A^{m}_{x,\mu\nu\rho}=\sum_{\sigma}\\!\epsilon_{\mu\nu\rho\sigma}\,\widetilde{A}^{m}_{\tilde{x}-\hat{\sigma},\sigma}\,,\;p_{x,\mu\nu}=\\!\sum_{\rho<\sigma}\\!\epsilon_{\mu\nu\rho\sigma}\,\widetilde{p}_{\tilde{x}-\hat{\rho}-\hat{\sigma},\rho\sigma}\,.$
(21)

The $d$ and $\partial$ operators interchange their role when switching to the
dual lattice (see Equation (128) in Appendix A), such that

$(\partial\,A^{m})_{x,\mu\nu}\;=\,\sum_{\rho<\sigma}\\!\epsilon_{\mu\nu\rho\sigma}\,(d\widetilde{A}^{m})_{\tilde{x}-\hat{\rho}-\hat{\sigma},\rho\sigma}\;,\;(\partial\,p)_{x,\mu}\;=\sum_{\nu<\rho<\sigma}\\!\\!\epsilon_{\mu\nu\rho\sigma}\,(d\,\widetilde{p}\,)_{\tilde{x}-\hat{\nu}-\hat{\rho}-\hat{\sigma},\nu\rho\sigma}\;.$
(22)

Using (21) and (22) in (20) we find the following dual form of the Boltzmann
factor,

$\displaystyle B_{\beta}[A^{e},A^{m}]$ $\displaystyle=$
$\displaystyle\left(\\!\frac{1}{2\pi\beta}\\!\right)^{\\!\\!3V}\\!\\!\sum_{\\{\widetilde{p}\\}}e^{\,-\frac{\widetilde{\beta}}{2}\sum_{\tilde{x},\mu<\nu}\\!\big{(}\\!(d\widetilde{A}^{m}\,+\,2\pi\,\widetilde{p}\,)_{\tilde{x},\mu\nu}\big{)}^{2}}\,e^{\,-i\sum_{\tilde{x},\mu<\nu<\rho}\widetilde{A}^{e}_{\tilde{x},\mu\nu\rho}\,(d\,\widetilde{p}\,)_{\tilde{x},\mu\nu\rho}}$
(23) $\displaystyle=$
$\displaystyle\left(\\!\frac{1}{2\pi\beta}\\!\right)^{\\!\\!3V}\\!\\!\sum_{\\{\widetilde{p}\\}}e^{\,-\frac{\widetilde{\beta}}{2}\sum_{\tilde{x},\mu<\nu}\\!\big{(}\widetilde{F}^{m}_{\tilde{x},\mu\nu}\big{)}^{2}}e^{\,-i\sum_{\tilde{x},\mu<\nu<\rho}\widetilde{A}^{e}_{\tilde{x},\mu\nu\rho}\,(d\,\widetilde{p}\,)_{\tilde{x},\mu\nu\rho}},$

where we defined

$\widetilde{F}^{m}_{\tilde{x},\mu\nu}\;\equiv\;(d\widetilde{A}^{m}\,+\,2\pi\,\widetilde{p}\,)_{\tilde{x},\mu\nu}\;,$
(24)

and the sum over all configurations of the dual plaquette occupation numbers,

$\sum_{\\{\widetilde{p}\,\\}}\;\;\equiv\;\;\prod_{\tilde{x}}\prod_{\mu<\nu}\;\sum_{\widetilde{p}_{\tilde{x},\mu\nu}\in\mathds{Z}}\;=\;\;\sum_{\\{p\\}}\;,$
(25)

where the identity on the right hand side is an obvious consequence of the
last equation in (21). Thus, up to an overall factor the Boltzmann weights
(13) and (23) have the same form.

Comparing (23) with (13) we can summarize the duality relation for the
Boltzmann factor as

$B_{\beta}[A^{e},A^{m}]\;=\;\left(\\!\frac{1}{2\pi\beta}\\!\right)^{\\!\\!3V}\widetilde{B}_{\widetilde{\beta}}\big{[}\widetilde{A}^{m},\widetilde{A}^{e}\big{]}\quad\mbox{with}\quad\widetilde{\beta}\;=\;\frac{1}{4\pi^{2}\beta}\;.$
(26)

Equation (26) constitutes the self-duality relation for the generalized
Boltzmann factor $B_{\beta}[A^{e},A^{m}]$, where in the lhs. form of the
Boltzmann factor the electric gauge field $A^{e}$ describes the dynamics and
the magnetic gauge field $A^{m}$ generates the constraints. Under the duality
transformation the gauge coupling $\beta$ is replaced by the dual coupling
$\widetilde{\beta}$, the Boltzmann factor picks up the overall factor
$(2\pi\beta)^{-3V}$, all fields are replaced by their dual living on the dual
lattice (thus the notation
$\widetilde{B}_{\widetilde{\beta}}[\widetilde{A}^{m},\widetilde{A}^{e}]$), and
finally, the electric and the magnetic field interchange their role. This
means that in the dual form on the rhs. of (26) the dual magnetic field
$\widetilde{A}^{m}$ describes the dynamics, while the dual electric field
$\widetilde{A}^{e}$ now generates the constraints.

To establish full duality of the partition sum $Z(\beta)$ we replace the
integral measures $\int\\!D[A^{e}]$ and $\int\\!D[A^{m}]$ by the corresponding
dual integral measures defined as,

$\displaystyle\int\\!\\!D\big{[}\widetilde{A}^{e}\big{]}\;\equiv\;\prod_{\tilde{x}}\\!\prod_{\mu<\nu<\rho}\int_{-\pi}^{\pi}\\!\frac{d\widetilde{A}^{e}_{\tilde{x},\mu\nu\rho}}{2\pi}\;=\;\int\\!\\!D[A^{e}]\;,$
$\displaystyle\int\\!\\!D\big{[}\widetilde{A}^{m}\big{]}\;\equiv\;\prod_{\tilde{x}}\prod_{\mu}\int_{-\pi}^{\pi}\\!\frac{d\widetilde{A}^{m}_{\tilde{x},\mu}}{2\pi}\;=\;\int\\!\\!D[A^{m}]\;,$
(27)

where again (21) ensures that the dual integration measures are equal to the
ones on the original lattice. Thus we find

$\displaystyle Z(\beta)$ $\displaystyle=$
$\displaystyle\int\\!\\!D[A^{e}]\int\\!\\!D[A^{m}]\;B_{\beta}[A^{e},A^{m}]$
(28) $\displaystyle=$
$\displaystyle\left(\\!\frac{1}{2\pi\beta}\\!\right)^{\\!\\!3V}\\!\\!\int\\!\\!D[\widetilde{A}^{e}]\int\\!\\!D[\widetilde{A}^{m}]\;\widetilde{B}_{\beta}\big{[}\widetilde{A}^{m},\widetilde{A}^{e}\big{]}\;=\left(\\!\frac{1}{2\pi\beta}\\!\right)^{\\!\\!3V}\\!\\!Z(\widetilde{\beta})\;,$

and identify the final form of the self-duality relation for the partition sum

$Z(\beta)\;=\;\left(\\!\frac{1}{2\pi\beta}\\!\right)^{\\!\\!3V}\\!\\!Z(\widetilde{\beta})\qquad\mbox{with}\qquad\widetilde{\beta}\;=\;\frac{1}{4\pi^{2}\beta}\;.$
(29)

Before we discuss properties and consequences of the self-duality relation
(29) we note that iterating the duality relation gives the identity map, i.e.,

$Z(\beta)\;=\;\left(\\!\frac{1}{2\pi\beta}\\!\right)^{\\!\\!3V}\\!\\!Z\big{(}\widetilde{\beta}\big{)}\;=\;\left(\\!\frac{1}{2\pi\beta}\\!\right)^{\\!\\!3V}\left(\frac{1}{2\pi\widetilde{\beta}}\\!\right)^{\\!\\!3V}\\!\\!Z\Big{(}\,\widetilde{\\!\widetilde{\beta}}\,\Big{)}\;=\;Z(\beta)\;,$
(30)

where in the last step we used the obvious properties
$\beta\widetilde{\beta}=1/4\pi^{2}$ and
$\widetilde{\\!\widetilde{\beta}}=\beta$. Equation (30) constitutes an
important consistency check for the self-duality relation we constructed.

The self-duality relation (29) obviously maps the weak- and strong-coupling
regions of the partition sum $Z(\beta)$ onto each other. Suitable derivatives
of $\ln Z(\beta)$ thus will relate observables in the strong and the weak
coupling region. To give an example, we consider the expectation value of the
square of the field strength, which is proportional to the first derivative of
$\ln Z(\beta)$:

$\displaystyle\langle F^{2}\rangle_{\beta}$ $\displaystyle\equiv$
$\displaystyle-\,\frac{1}{3V}\frac{\partial}{\partial\beta}\ln
Z(\beta)\;=\;-\,\frac{1}{3V}\frac{\partial}{\partial\beta}\ln\left((2\pi\beta)^{-3V}Z\big{(}\widetilde{\beta})\right)$
(31) $\displaystyle=$
$\displaystyle\frac{1}{\beta}\,-\,\frac{1}{3V}\left(\frac{\partial}{\partial\widetilde{\beta}}\ln
Z(\widetilde{\beta})\right)\frac{d\widetilde{\beta}}{d\beta}\;=\;\frac{1}{\beta}\;-\;\langle
F^{2}\rangle_{\widetilde{\beta}}\;\frac{1}{4\pi^{2}\beta^{2}}\;.$

Multiplying the equation with $\beta$ we can summarize the duality relation
for $\langle F^{2}\rangle$ in a sum rule that connects the weak and strong
coupling results for $\langle F^{2}\rangle$,

$\beta\,\langle F^{2}\rangle_{\beta}\;+\;\widetilde{\beta}\,\langle
F^{2}\rangle_{\widetilde{\beta}}\;=\;1\;.$ (32)

In a similar way one may relate the weak and strong coupling results of higher
derivatives of $\ln Z(\beta)$, i.e., the susceptibility of the action density
and higher moments. Self-duality relations for correlators of $F^{2}$ can be
obtained by introducing an $x$-dependence of $\beta$ (which leaves the duality
transformation unchanged) and by performing local derivatives that generate
the correlators.

##  3 Self-dual lattice QED

In this section we generalize our self-dual discretization of pure gauge
theory to self-dual lattice QED, where we explicitly discuss the coupling of
electrically and magnetically charged scalar fields. More particularly we
first discuss in detail the case where we couple separate species of matter
fields, one that is electrically charged and a second field with magnetic
charges. Subsequently we briefly address the possibility of coupling dyonic
matter in our formulation.

###  3.1 Coupling separate species for electric and magnetic matter

We here first present the construction for coupling separate species of
electric and magnetic matter fields, then establish self-duality and finally
discuss some of its consequences. We remark that in Appendix E we show that
one may switch to a worldline formulation that for the case of bosonic matter
solves the complex action problem introduced by the closedness constraints,
such that the worldline form can be used for numerical simulations.

We couple electrically charged matter to the electric gauge field $A^{e}$ and
magnetically charged matter to the magnetic gauge field $A^{m}$. Note that we
have defined the magnetic gauge field $A^{m}$ on the cubes of the original
lattice, such that the corresponding dual form $\widetilde{A}^{m}$ corresponds
to fields $\widetilde{A}^{m}_{\tilde{x},\mu}$ that live on the links
$(\tilde{x},\mu)$ of the dual lattice (compare (21)). We now use this dual
form to couple the magnetically charged matter fields.

The partition function with gauge fields coupling to electric and magnetic
matter fields reads

$Z(\beta,M^{e}\\!,\lambda^{e}\\!,q^{e}\\!,\,M^{m}\\!,\lambda^{m}\\!,q^{m})\equiv\int\\!\\!D[A^{e}]\\!\int\\!\\!D[A^{m}]\;B_{\beta}[A^{e},A^{m}]\;Z_{M^{e}\\!,\,\lambda^{e}\\!,\,q^{e}}[A^{e}]\;\widetilde{Z}_{M^{m}\\!,\,\lambda^{m}\\!,\,q^{m}}\big{[}\widetilde{A}^{m}\big{]}\,,$
(33)

where $B_{\beta}[A^{e},A^{m}]$ is the gauge field Boltzmann factor in the form
(13). We have introduced the partition function
$Z_{M^{e}\\!,\,\lambda^{e}\\!,\,q^{e}}[A^{e}]$ for electrically charged matter
$\phi^{e}$ that couples to the electric background gauge field $A^{e}$ as

$\displaystyle Z_{M^{e}\\!,\,\lambda^{e}\\!,\,q^{e}}[A^{e}]$
$\displaystyle\\!\\!\equiv\\!\\!$
$\displaystyle\int\\!\\!D[\phi^{e}]\,e^{-S_{M^{e}\\!,\,\lambda^{e}\\!,\,q^{e}}[\phi^{e},A^{e}]}\;,\;\;\int\\!\\!D[\phi^{e}]\;\equiv\;\prod_{x}\int_{\mathds{C}}\frac{d\phi_{x}^{e}}{2\pi}\;,$
(34) $\displaystyle S_{M^{e}\\!,\,\lambda^{e}\\!,\,q^{e}}[\phi^{e},A^{e}]$
$\displaystyle\\!\\!\equiv\\!\\!$
$\displaystyle\sum_{x}\left[M^{e}|\phi_{x}^{e}|^{2}+\lambda^{e}|\phi_{x}^{e}|^{4}-\sum_{\mu}\Big{[}\phi_{x}^{e\,*}\,e^{\,i\,q^{e}A_{x,\mu}^{e}}\,{\phi_{x+\hat{\mu}}^{e}}+c.c.\
\Big{]}\\!\right]\;,$ (35)

where we imparted the electric matter with a charge $q^{e}\in\mathbb{Z}$. As
already announced, we couple bosonic matter, which for the electric field is a
complex scalar $\phi_{x}^{e}\in\mathds{C}$ that is assigned to the sites $x$
of the original lattice. The path integral measure $\int\\!\\!D[\phi^{e}]$ is
the usual product measure and the action
$S_{M^{e}\\!,\,\lambda^{e}\\!,\,q^{e}}[\phi^{e},A^{e}]$ is the free action
plus a quartic term with coupling $\lambda^{e}$. The mass parameter $M^{e}$ is
related to the tree level mass $m^{e}$ via $M^{e}=8+(m^{e})^{2}$.

The magnetically charged scalar field
$\widetilde{\phi}_{\tilde{x}}^{m}\in\mathds{C}$ lives on the sites $\tilde{x}$
of the dual lattice and couples to the magnetic gauge field
$\widetilde{A}_{\tilde{x},\mu}^{m}$ on the links of the dual lattice. The
corresponding partition sum has the same form as the partition sum (34) for
the electric matter but for the magnetic matter is defined entirely on the
dual lattice,

$\displaystyle\widetilde{Z}_{M^{m}\\!,\,\lambda^{m}\\!,\,q^{m}}\big{[}\widetilde{A}^{m}\big{]}$
$\displaystyle\\!\\!\equiv\\!\\!$
$\displaystyle\int\\!\\!D\big{[}\widetilde{\phi}^{m}\big{]}\,e^{-\widetilde{S}_{M^{m}\\!,\,\lambda^{m}\\!,\,q^{m}}\big{[}\widetilde{\phi}^{m},\widetilde{A}^{m}\big{]}}\;,\;\;\int\\!\\!D\big{[}\widetilde{\phi}^{m}\big{]}\;\equiv\;\prod_{\tilde{x}}\int_{\mathds{C}}\frac{d\widetilde{\phi}_{\tilde{x}}^{m}}{2\pi}\;,$
(36)
$\displaystyle\widetilde{S}_{M^{m}\\!,\,\lambda^{m}\\!,\,q^{m}}\big{[}\widetilde{\phi}^{m},\widetilde{A}^{m}\big{]}$
$\displaystyle\\!\\!\equiv\\!\\!$
$\displaystyle\sum_{\tilde{x}}\left[M^{m}|\widetilde{\phi}_{\tilde{x}}^{m}|^{2}+\lambda^{m}|\widetilde{\phi}_{\tilde{x}}^{m}|^{4}-\sum_{\mu}\Big{[}\widetilde{\phi}_{\tilde{x}}^{m\,*}\,e^{\,i\,q^{m}\widetilde{A}_{\tilde{x},\mu}^{m}}\,\widetilde{\phi}_{\tilde{x}+\hat{\mu}}^{m}+c.c.\
\Big{]}\\!\right].$ (37)

Similarly to the electric matter, we chose to give the magnetic matter a
charge $q^{m}\in\mathbb{Z}$. As before we allow for a quartic self-interaction
term with corresponding coupling $\lambda^{m}$, and the relation of the mass
parameter $M^{m}$ to the tree level mass $m^{m}$ is again given by
$M^{m}=8+(m^{m})^{2}$.

It is obvious that the construction can easily be generalized to coupling
fermionic electric and magnetic matter fields, by simply replacing the
partition sums $Z_{M^{e}\\!,\,\lambda^{e}\\!,\,q^{e}}[A^{e}]$ and
$\widetilde{Z}_{M^{m}\\!,\,\lambda^{m}\\!,\,q^{m}}\big{[}\widetilde{A}^{m}\big{]}$
by the corresponding fermion determinants in the background fields $A^{e}$ and
$\widetilde{A}^{m}$ where for the latter case the corresponding discretized
lattice Dirac operator lives entirely on the dual lattice.

The proof of self-duality is straightforward and is essentially a corollary of
the self-duality relation (26) for the gauge field Boltzmann factor
$B_{\beta}[A^{e},A^{m}]$. Using (26) in the partition sum (33) we find (here
$C=(1/2\pi\beta)^{3V}$)

$\displaystyle
Z(\beta,M^{e}\\!,\lambda^{e}\\!,q^{e}\\!,\,M^{m}\\!,\lambda^{m}\\!,q^{m})\\!\\!\\!$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!\\!\\!$ $\displaystyle
C\\!\\!\int\\!\\!D[A^{e}]\\!\\!\int\\!\\!D[{A}^{m}]B_{\widetilde{\beta}}[\widetilde{A}^{m}\\!,\widetilde{A}^{e}]Z_{M^{e}\\!,\,\lambda^{e}\\!,\,q^{e}\\!}[A^{e}]\widetilde{Z}_{M^{m}\\!,\,\lambda^{m}\\!,\,q^{m}\\!}\big{[}\\!\widetilde{A}^{m}\big{]}$
$\displaystyle\\!\\!\\!\\!=\\!\\!\\!\\!\\!\\!$ $\displaystyle
C\\!\\!\int\\!\\!D[\widetilde{A}^{m}]\\!\\!\int\\!\\!D[\widetilde{A}^{e}]B_{\widetilde{\beta}}[\widetilde{A}^{m}\\!,\widetilde{A}^{e}]\widetilde{Z}_{M^{m}\\!,\,\lambda^{m}\\!,\,q^{m}\\!}\big{[}\\!\widetilde{A}^{m}\big{]}\widetilde{\\!\widetilde{Z}}_{M^{e}\\!,\,\lambda^{e}\\!,-q^{e}}\\!\Big{[}\widetilde{\widetilde{A}^{e}\\!}\Big{]},$

where in the second step we have used (27) to replace the path integral
measures by their dual counterparts, as well as the identity

$Z_{M^{e}\\!,\,\lambda^{e}\\!,\,q^{e}\\!}[A^{e}]\;=\;\widetilde{\\!\widetilde{Z}}_{M^{e}\\!,\,\lambda^{e}\\!,\,-q^{e}\\!}\Big{[}\widetilde{\widetilde{A}^{e}\\!}\,\Big{]}\;.$
(39)

It is important to note that here the iterated duality relation gives rise to
a flip of the sign of the electric charge $q^{e}$: For vector fields Eq. (127)
from Appendix A implies

$A^{e}_{x,\mu}\;=\;\sum_{\nu<\rho<\sigma}\epsilon_{\mu\nu\rho\sigma}\tilde{A}^{e}_{\tilde{x}-\hat{\nu}-\hat{\rho}-\hat{\sigma},\nu\rho\sigma}\;,\;\;\tilde{A}^{e}_{\tilde{x},\mu\nu\rho}\;=\;-\,\sum_{\sigma}\epsilon_{\mu\nu\rho\sigma}A_{x+\hat{s}-\hat{\sigma},\sigma}\;,$
(40)

where $\hat{s}=\hat{1}+\hat{2}+\hat{3}+\hat{4}$. As a consequence we find
$\widetilde{\widetilde{A}^{e}\\!}_{x,\mu}=-A^{e}_{x,\mu}$, such that the link
variables $e^{\,iq^{e}A_{x,\mu}}$ in the electric mater action (35) become
complex conjugate under iterated duality, which in turn is equivalent to
changing the sign of the electric charge $q^{e}$.

Comparing the second line of (LABEL:Zfull_aux) with the definition (33) of the
full partition sum we can identify the self-duality relation for our
discretization of QED

$\displaystyle\beta^{\frac{3V}{2}}Z(\beta,M^{e}\\!,\lambda^{e}\\!,q^{e}\\!,\,M^{m}\\!,\lambda^{m}\\!,q^{m})\;=\;\widetilde{\beta}^{\frac{3V}{2}}\,Z\big{(}\widetilde{\beta},\widetilde{M}^{e}\\!,\widetilde{q}^{\,e}\\!,\widetilde{\lambda}^{e}\\!,\widetilde{M}^{m}\\!,\widetilde{\lambda}^{m}\\!,\widetilde{q}^{\,m}\big{)}\qquad\mbox{with}$
(41)
$\displaystyle\widetilde{\beta}\,=\,\frac{1}{4\pi^{2}\beta}\,,\;\widetilde{M}^{e}\,=\,M^{m}\,,\;\widetilde{\lambda}^{e}\,=\,\lambda^{m}\,,\;\widetilde{q}^{\,e}\,=\,q^{m}\,,\;\widetilde{M}^{m}\,=\,M^{e}\,,\;\widetilde{\lambda}^{m}\,=\,\lambda^{e}\,,\;\widetilde{q}^{\;m}\,=\,-\,q^{e}\;.$

Note that we have split the prefactor $C=(1/2\pi\beta)^{3V}$ of the partition
sums in (LABEL:Zfull_aux) by dividing it into equal powers of $\beta$ and
$\widetilde{\beta}$ to fully display the symmetry. As for the case of pure
gauge theory, the duality transformation generates an overall factor, and the
gauge coupling $\beta$ is replaced by the dual coupling
$\widetilde{\beta}=1/(4\pi^{2}\beta)$, thus interchanging weak and strong
coupling. In addition the parameters $M^{e},\lambda^{e}$ of electric matter
are interchanged with the parameters $M^{m},\lambda^{m}$ of magnetic
matter888When coupling fermionic matter only the bare fermion masses appear as
parameters.. For the charge vector $\bm{q}=(q^{e},q^{m})^{t}$ (the superscript
t denotes transposition) we find the following relation between the dual and
the original charges,

$\widetilde{\bm{q}}\;=\;S\,{\bm{q}}\quad\mbox{with}\quad
S\;=\;\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}\;.$ (42)

The self-duality symmetry described by (41) and (42) will be referred to as
${\cal S}$. However, we remark already now that in the next section we will
extend the symmetry ${\cal S}$ to also include the $\theta$-angle shifts by
$2\pi$, the so called $\cal T$ transformation.

Again we can generate self-duality relations for observables by evaluating
derivatives of $\ln Z$ with respect to the couplings and when the couplings
are considered to be space-time dependent suitable derivatives give rise to
self-duality relations for correlation functions. We discuss two examples for
such self-duality relations, the first being the generalization of the sum
rule (32),

$\beta\,\langle(F^{e})^{2}\rangle_{\beta,M^{e}\\!,\,\lambda^{e}\\!,\,q^{e}\\!,\,M^{m}\\!,\,\lambda^{m}\\!,\,q^{m}}\;+\;\widetilde{\beta}\,\langle(F^{e})^{2}\rangle_{\widetilde{\beta},\widetilde{M}^{e}\\!,\,\widetilde{q}^{\,e}\\!,\,\widetilde{\lambda}^{e}\\!,\,\widetilde{M}^{m}\\!,\,\widetilde{\lambda}^{m}\\!,\,\widetilde{q}^{\;m}}\;=\;1\;.$
(43)

The second term in the sum rule is evaluated with the dual parameters, i.e.,
with gauge coupling $\widetilde{\beta}$ and interchanged electric and magnetic
coupling parameters.

Derivatives with respect to $M^{e}$ and $M^{m}$ generate field expectation
values for the electric and magnetic matter fields. Exploring the duality
relation (41) one finds

$\displaystyle\left\langle|\phi^{e}|^{2}\right\rangle_{\beta,M^{e}\\!,\,\lambda^{e}\\!,\,q^{e}\\!,\,M^{m}\\!,\,\lambda^{m}\\!,\,q^{m}}$
$\displaystyle\equiv$ $\displaystyle-\,\frac{1}{V}\frac{\partial}{\partial
M^{e}}\,\ln
Z(\beta,M^{e}\\!,\lambda^{e}\\!,q^{e}\\!,\,M^{m}\\!,\lambda^{m}\\!,q^{m})$
(44) $\displaystyle=$
$\displaystyle\left\langle|\widetilde{\phi}^{m}|^{2}\right\rangle_{\widetilde{\beta},\widetilde{M}^{e}\\!,\,\widetilde{q}^{\,e}\\!,\,\widetilde{\lambda}^{e}\\!,\,\widetilde{M}^{m}\\!,\,\widetilde{\lambda}^{m}\\!,\,\widetilde{q}^{\;m}}\;,$

with $\left\langle|\widetilde{\phi}^{m}|^{2}\right\rangle=-1/V\partial\ln
Z/\partial M^{m}$. According to the self-duality relation (44), the electric
and magnetic field expectation values are converted into each other when
changing from weak to strong coupling and simultaneously interchanging
electric and magnetic coupling parameters. In a similar way self-duality
relations for various observables and correlation functions can be obtained,
where electric and magnetic fields and their parameters are interchanged, when
switching between the weak and strong coupling domains of the theory. We
stress once more, that the self-duality relations we discuss here hold
identically for fermionic and for bosonic matter.

We remark again that the Boltzmann factor (13) which we use in (33) is
complex, such that we cannot use Monte Carlo simulations for self-dual QED
directly in the form (13), (33). This can be overcome by switching to a
worldline representation, which we discuss in Appendix E.

###  3.2 Comments on coupling dyons

In addition to purely electric and purely magnetic matter, one can also couple
dyonic matter by identifying the lattice and its dual, e.g., by specifying a
map $F:\Lambda\rightarrow\tilde{\Lambda}$ which sends $x\in\Lambda$ to
$x+\frac{\hat{s}}{2}\in\tilde{\Lambda}$. We also define the map
$\tilde{F}:\tilde{\Lambda}\rightarrow\Lambda$ which sends
$\tilde{x}\in\tilde{\Lambda}$ to $\tilde{x}+\frac{\hat{s}}{2}\in\Lambda$. The
maps $F$ and $\tilde{F}$ do not have to necessarily be shifts by the same
amount, and we could have just as well defined
$\tilde{F}:\tilde{x}\rightarrow\tilde{x}-\frac{\hat{s}}{2}$. We now work with
only a single species of matter, the dyonic field $\phi_{x}\in\mathds{C}$,
that lives on the sites $x$ of the lattice and carries both an electric charge
$q^{e}$ and a magnetic charge $q^{m}$. The corresponding action is obtained by
replacing $\phi^{e}_{x}\rightarrow\phi_{x}$ in the mass and quartic terms of
(35) and by replacing the hopping terms as follows,

$\phi_{x}^{e\,*}\,e^{\,i\,q^{e}\,{A^{e}_{x,\mu}}}\,\phi^{e}_{x+\hat{\mu}}\;\rightarrow\;\phi_{x}\,e^{\,i\,q^{e}\,{A^{e}_{x,\mu}}\,+\,i\,q^{m}\,{\tilde{A}^{m}_{F(x),\mu}}}\,\phi_{x+\hat{\mu}}\;.$
(45)

The field $\phi^{m}$ from the previous section is dropped completely999We
remark that we could have placed the dyonic field also on the dual lattice by
coupling the dual of the electric field there.. Under a duality transformation
the dyon field is mapped to the dual lattice, and since in the construction
here we identify original and dual lattice, the duality relations (41) apply
also to the dyonic case, of course with the modification that we have only a
single mass parameter $M$ and only a single quartic coupling $\lambda$. As in
the previous subsection, also here the construction carries over to fermionic
dyon matter in an obvious way.

Notice, however, that coupling dyons makes it difficult to eliminate the sign
problem even if they are bosonic. In the previous subsection and Appendix E
the complex action problem was solved by switching to a worldline formulation
for the magnetic matter, which converts the dependence of the magnetic matter
action on the magnetic gauge field $\widetilde{A}^{m}$ into terms of the form
$e^{i\,q^{m}\,\widetilde{A}^{m}_{\tilde{x},\mu}\,\widetilde{k}_{\tilde{x},\mu}}$
where the $\widetilde{k}_{\tilde{x},\mu}\in\mathds{Z}$ are the flux variables
that describe the dynamics of the magnetic matter in the worldline language
(see Appendix E). In this form the $\widetilde{A}^{m}$ then can be integrated
out together with the constraint factors (10) to generate more general
constraints that couple the Villain and the flux variables. However, when
dyons are considered, the worldline expansion also generates terms of the form
$e^{i\,q^{e}\,A^{e}_{x,\mu}\,{k}_{x,\mu}}$ that can not be integrated out,
since the action for the electric gauge field $A^{e}$ is quadratic. Further
the $\mathcal{S}$-duality transformation does not change the situation, since
it merely interchanges the form of the electric and the magnetic gauge action,
i.e., one of the two is always quadratic. Hence dyons generically have a
complex action problem.

However, it is well known that a $\theta$-term can add or remove an electric
charge from a monopole, which is the famous Witten effect [34], and we will
exploit this effect to relate some dyonic theories to their simpler cousins
discussed in the previous subsection, where indeed a worldline representation
solves the complex action problem. We will review the Witten effect and its
relation to the complex action problem in the next section when we introduce
the $\theta$-term for the Villain action. For the moment let us just note that
when we change $\theta\rightarrow\theta\pm 2\pi$ we can absorb this shift by
changing the dyonic charges $q^{e}\rightarrow q^{e}\mp q^{m}$ and
$q^{m}\rightarrow q^{m}$. Obviously this can be used to relate theories with
sign problem to theories that can be simulated without the sign problem. This
idea becomes more powerful when the shift of $\theta$ is combined with the
duality transformation such that the group of transformations becomes
$\text{SL}(2,\mathbb{Z})$. We will see that this allows us to connect some
theories which have dyons101010As we will see, such dyonic theories in general
have smeared electric and magnetic charges, as the introduction of the
$\theta$-term will force the Witten effect to endow monopoles with electric
charges which are generically spread across multiple lattice links. and a
nonzero $\theta$-term, to theories without the complex action problem. This
will be discussed in detail in Section 4.5.

##  4 QED with a $\theta$-term

We generalize our formulation of self-dual U(1) lattice gauge theory and self-
dual lattice QED further by adding a $\theta$-term. We first construct a
suitable discretization of the topological charge and analyze its properties.
Subsequently we add the $\theta$-term to the gauge field Boltzmann factor and
identify the corresponding duality transformation. In yet another
generalization step we construct a self-dual Boltzmann factor at finite
$\theta$ for the gauge fields, which then is the basis for fully self-dual
U(1) lattice gauge theory and self-dual lattice QED both now containing also
the $\theta$-term.

Before we go through these steps, let us comment on our construction. It is
well known that there are many ways to discretize a $\theta$-term. We will,
however, insist that the $\theta$-term capturea the topological aspecta of the
continuum U(1) theory. In the continuum the $\theta$-term of the U(1) gauge
theory is

$S_{\theta}\;=\;{i\theta}\,Q\;,$ (46)

where

$Q\;=\;\frac{1}{8\pi^{2}}\int\\!d^{4}x\sum_{\mu<\nu\atop\rho<\sigma}F_{\mu\nu}F_{\rho\sigma}\epsilon^{\mu\nu\rho\sigma}\;,$
(47)

is the topological charge. It is well known that the $\theta$-term does not
depend on the local details of the gauge field $A_{\mu}$, and only depends on
the topological class of the gauge bundle. In other words, the variation of
the above action with respect to small local changes of the gauge field
$A_{\mu}$ is zero.

Moreover when $\theta$ is shifted $\theta\rightarrow\theta+2\pi$, the
Euclidean path integral weight is unchanged in the continuum,
because111111This is only true on spin manifolds. On non-spin manifolds the
$\theta$-periodicity is $\theta\rightarrow\theta+4\pi$ because $Q$ can be
half-integer. $Q\in\mathbb{Z}$.

The above features are both ruined in the presence of a dynamical
monopole121212Such an operator is constructed by excising a contour
$\mathcal{C}$ from the space-time manifold, and imposing that on a small
sphere linking the contour $\mathcal{C}$ the flux is $\int F=2\pi q_{m}$,
where $q_{m}\in\mathbb{Z}$ is the charge of the monopole operator.. Indeed the
Witten effect would endow the worldline of the monopole with electric charge
$\theta/2\pi$, hence generating an explicit dependence on the gauge field
$A_{\mu}$ along the monopole worldline, turning the monopole into a dyon. We
will see a manifestation of this in our lattice construction below. The
$\theta$-periodicity can be restored if the theory contains an entire tower of
dyons of all electric charges, but generically it will be destroyed by the
presence of dynamical magnetic matter.

To construct the desired $\theta$-term, we will therefore be guided by a
theory without monopoles being gauge-invariant, independent of the local
details of the gauge field $A_{x,\mu}$ and that has a $2\pi$-periodicity in
$\theta$. To construct such a term it is sufficient to construct the
appropriate topological charge $Q$. The most natural way to construct such a
topological charge is to identify the lattice $\Lambda$ and its dual
$\widetilde{\Lambda}$ by defining a map
$\Lambda\rightarrow\widetilde{\Lambda}$, as was discussed in [7]. Since such
maps are not unique, the definition of the topological charge $Q$ will not be
unique either, and the most general definition of the topological charge will
not be local. All of these definitions will turn out to be equivalent when
monopoles are absent, because of the topological nature of $Q$.

We will see, however, that if such a $\theta$ term is introduced, any attempt
to define an ultra-local kinetic term will fail to satisfy self-duality. In
particular electric-magnetic duality transformation of such an ultra-local
action would produce a local, but not ultra local action, and therefore
violate the self-dual covariance. When we insist on self-duality we will be
forced to consider local, but not ultra-local actions.

It is well known that $\theta$-periodicity and self-duality generate an
$\text{SL}(2,\mathbb{Z})$ duality group. Therefore, insisting on the exact
$\text{SL}(2,\mathbb{Z})$ structure of a lattice theory will force us to
abandon ultra-locality.

###  4.1 Construction of a suitable topological charge

Following [7] we now introduce a topological charge for the Villain
formulation but already generalize the discretization presented in [7] to a
form that will turn out to be useful for the construction of self-dual
theories with a $\theta$-term. More specifically we introduce a whole family
$Q_{T}[F^{e}]$ of lattice discretizations for the topological charge that are
labelled by an integer-valued vector $T$,

$\displaystyle Q_{T}[F^{e}]$ $\displaystyle\equiv$
$\displaystyle\frac{1}{8\pi^{2}}\sum_{x}\sum_{\mu<\nu\atop\rho<\sigma}F^{e}_{x,\mu\nu}\,\epsilon_{\mu\nu\rho\sigma}\,F^{e}_{x-\hat{\rho}-\hat{\sigma}+T,\,\rho\sigma}$
$\displaystyle=$
$\displaystyle\frac{1}{8\pi^{2}}\sum_{x}\sum_{\mu<\nu\atop\rho<\sigma}(dA^{e}+2\pi
n)_{x,\mu\nu}\,\epsilon_{\mu\nu\rho\sigma}\,(dA^{e}+2\pi
n)_{x-\hat{\rho}-\hat{\sigma}+T,\,\rho\sigma}\;,$

where we have introduced $T=\sum_{\mu}t_{\mu}\hat{\mu}$ with
$t_{\mu}\in\mathds{Z}$. Note that $F^{e}_{x,\mu\nu}$ is again given by the
combination (5) of the exterior derivative and the Villain variable, i.e.,
$F^{e}_{x,\mu\nu}=(dA^{e})_{x,\mu\nu}+2\pi n_{x,\mu\nu}$. We now want to show
that the definition has the desired properties of the continuum topological
charge, when magnetic monopoles are not present. Since monopoles are not
present, we will assume that Villain variables obey the closedness constraints
(7). Thus the topological properties of $Q_{T}[F^{e}]$ as we define it in
(4.1) are specific to the case when monopoles are absent.

For $T=0$ the definition of $Q_{0}[F^{e}]$ is essentially a direct lattice
discretization of the continuum topological charge $Q\propto\int
d^{4}xF_{\mu\nu}(x)\epsilon_{\mu\nu\rho\sigma}F_{\rho\sigma}(x)$ with an
additional shift of the second factor $F^{e}_{x,\rho\sigma}$ by one lattice
unit in the negative $\rho$ and $\sigma$ directions131313Without this shift we
would not be able to show the topological properties of $Q_{T}$, and the
$\theta$-term in the absence of monopoles would lose the independence on
$A_{x,\mu}^{e}$ and the $\theta$-periodicity.. This, in the absence of
monopoles, will have as a consequence that the $\theta$-term depends only on
the villain variables $n_{x,\mu\nu}$ and not on gauge fields, and will also
imply $\theta$ periodicity by $2\pi$. Following [7], this $\theta$-term may be
interpreted as defining a map from a lattice to a dual lattice, and then
making a natural product of $F^{e}_{x,\mu\nu}$ on a plaquette $(x,\mu\nu)$
with the corresponding dual field strength on the plaquette dual to
$(x,\mu\nu)$. For $T\neq 0$ this definition from [7] is generalized to a
product of $F^{e}_{x,\mu\nu}$ with the dual field strength on the dual
plaquette shifted by $T$, which we implement by adding $T$ in the argument of
the second factor in (4.1). Note that when $T$ becomes larger than the extent
of the lattice the periodic boundary conditions will identify the shifted site
$x+T$ with some site of the lattice, such that only a finite number of vectors
$T$ correspond to distinct definitions of $Q_{T}[F^{e}]$.

We will now show, that $Q_{T}[F^{e}]$ is topological in nature and that
$Q_{T}[F^{e}]$ gives the same result for all vectors $T$ as long as the
Villain variables are closed. We proceed towards this goal in several steps
where we discuss and use properties of $Q_{T}[F^{e}]$ that are partly proven
in Appendix B.

The first property shown in Appendix B is the fact that $Q_{T}[F^{e}]$ is
invariant under adding an arbitrary exterior derivative $(dB)_{x,\mu\nu}$ to
the field strength $F^{e}_{x,\mu\nu}$, i.e.,

$Q_{T}[F^{e}+dB]\;=\;Q_{T}[F^{e}]\;,$ (49)

where $B_{x,\mu}$ is an arbitrary 1-form, i.e., an arbitrary set of link-based
fields. We stress again that this property only holds when the Villain
variables $n_{x,\mu\nu}$ in
$F^{e}_{x,\mu\nu}=(dA^{e})_{x,\mu\nu}+2\pi\,n_{x,\mu\nu}$ obey the closedness
condition (7).

The property (49) immediately implies that $Q_{T}[F^{e}]$ is independent of
the exterior derivative $(dA^{e})_{x,\mu\nu}$ in
$F^{e}_{x,\mu\nu}=(dA^{e})_{x,\mu\nu}+2\pi\,n_{x,\mu\nu}$. Thus the
topological charge depends only on the Villain variables $n_{x,\mu\nu}$ and we
may write (compare the definition (4.1))

$Q_{T}[F^{e}]\;=\;\frac{1}{2}\sum_{x}\sum_{\mu<\nu\atop\rho<\sigma}n_{x,\mu\nu}\,\epsilon_{\mu\nu\rho\sigma}\,n_{x-\hat{\rho}-\hat{\sigma}+T,\,\rho\sigma}\;\;\mbox{with}\;\;(dn)_{x,\mu\nu\rho}=0\;\;\forall
x,\mu\\!<\\!\nu\\!<\\!\rho\,.$ (50)

The fact that the Villain variables obey the closedness condition (7) can be
used together with the Hodge decomposition (see Eq. (126) of Appendix A) to
write the Villain variables in the form

$n_{x,\mu\nu}\;=\;(d\,l)_{x,\mu\nu}\;+\;h_{x,\mu\nu}\;,$ (51)

where the condition $(d\,n)_{x,\,\mu\nu\rho}=0$ implies that we do not have a
contribution $(\partial\,c)_{x,\,\mu\nu\rho}$ in the Hodge decomposition (126)
of $n_{x,\mu\nu}$. In (51) $l_{x,\mu}\in\mathds{Z}$ is an integer-valued
1-form which due to $d^{2}l=0$ does not contribute to $dn$. The second term in
the Hodge decomposition (51) are the closed integer-valued harmonic
contributions $h_{x,\mu\nu}$ which obey $(dh)_{x,\mu\nu\rho}=0$ and cannot be
written as exterior derivatives. We may parameterize them in the form
($\mu<\nu$)

$h_{x,\mu\nu}\;=\;\omega_{\mu\nu}\,\sum_{i=1}^{N_{\rho}}\sum_{j=1}^{N_{\sigma}}\delta_{x,i\hat{\rho}+j\hat{\sigma}}^{(4)}\;\;\;\;\;\mbox{with}\;\;\;\;\;\rho\neq\mu,\nu\,;\,\sigma\neq\mu,\nu\,;\,\rho\neq\sigma\;,$
(52)

where $N_{\rho}$ and $N_{\sigma}$ denote the lattice extents in the $\rho\,$\-
and $\sigma$-directions and $\delta_{x,i\hat{\rho}+j\hat{\sigma}}^{(4)}$ is
the 4-dimensional Kronecker delta. In other words $h_{x,\mu\nu}$ is the
constant $\omega_{\mu\nu}\in\mathds{Z}$ for $\mu$-$\nu$ plaquettes
$(x,\mu\nu)$ that have their root site $x$ in the $\rho$-$\sigma$ plane that
is orthogonal to the $\mu$-$\nu$ plane and contains the origin. For all other
plaquettes $h_{x,\mu\nu}=0$. It is easy to see that the harmonics
$h_{x,\mu\nu}$ are closed forms, i.e., they obey $(dh)_{x,\mu\nu\rho}=0$. We
stress that our choice for the parameterization of the $h_{x,\mu\nu}$ is not
unique, since they can be deformed by adding arbitrary exterior derivatives.

In Appendix B we show the following result for the topological charge141414In
the notation for the final expression on the rhs. of (53) we assume
antisymmetry of the $\omega_{\mu\nu}$.

$Q_{T}[2\pi n]=Q_{T}[2\pi
h]=\omega_{12}\,\omega_{34}-\omega_{13}\omega_{24}+\omega_{14}\omega_{23}=\frac{1}{8}\epsilon_{\mu\nu\rho\sigma}\omega_{\mu\nu}\omega_{\rho\sigma}\in\mathds{Z}\;,\;\;\forall\;T\;.$
(53)

The first identity follows already from (49) and shows that the topological
charge depends only on the harmonic contributions to the Villain variables.
The second step is the explicit evaluation of $Q_{T}[2\pi n]=Q_{T}[2\pi h]$
for the parameterization (52) of the harmonics, which, as shown in Appendix B,
turns out to be independent of the vector $T$ that appears as a parameter in
the definitions (4.1) and (50). This completes the proof of our statement that
for closed Villain variables $Q_{T}[F^{e}]=Q_{T}[2\pi n]=Q_{T}[2\pi h]$ is
independent of $T$ due to its topological nature. In addition the definition
of the topological charge is integer-valued and can be computed uniquely from
the harmonics in the Hodge decomposition of the Villain variables.

###  4.2 The Witten effect

Before we come to performing the duality transformation in the next section
let us discuss an interesting physical aspect of the abelian $\theta$-term,
the Witten effect, which constitutes an important consistency check of our
formulation. The Witten effect states that the $\theta$-term endows a magnetic
monopole with the minimal possible magnetic charge $m=2\pi$ with an electric
charge $q=\theta/2\pi$.

We here consider a general definition of the topological charge given by the
superposition of different discretizations $Q_{T}$,

$Q\;=\;\sum_{T}\gamma_{T}\,Q_{T}[dA^{e}+2\pi n]\;,$ (54)

where the $Q_{T}$ are given by (4.1) and we require $\sum_{T}\gamma_{T}=1$.
Now consider a magnetic Wilson loop defined by

$\prod_{(\tilde{x},\mu)\in\widetilde{C}}e^{\,i\,\widetilde{A}^{m}_{\tilde{x},\mu}}\;,$
(55)

along some contour $\widetilde{C}$ on the dual lattice. Combining the Wilson
loop with the closedness factor (10) and integrating over the
$\widetilde{A}^{m}$ we find

$\displaystyle\int\\!\\!D[\widetilde{A}^{m}]\,e^{-i\sum_{x}\\!\sum_{\mu<\nu<\rho}A^{m}_{x,\mu\nu\rho}(dn)_{x,\mu\nu\rho}}\\!\\!\prod_{(\tilde{x},\mu)\in\widetilde{C}}\\!\\!e^{\,i\,\widetilde{A}^{m}_{\tilde{x},\mu}}$
(56)
$\displaystyle\qquad=\;\int\\!\\!D[\widetilde{A}^{m}]\\!\\!\prod_{(\tilde{x},\mu)\in\widetilde{C}}\\!\\!e^{\,i\,\widetilde{A}^{m}_{\tilde{x},\mu}[1+\sum_{\nu<\rho<\sigma}\epsilon_{\mu\nu\rho\sigma}(dn)_{x+\hat{\mu},\nu\rho\sigma}]}\\!\\!\\!\prod_{(\tilde{x},\mu)\not\in\widetilde{C}}\\!\\!e^{\,i\,\widetilde{A}^{m}_{\tilde{x},\mu}\sum_{\nu<\rho<\sigma}\epsilon_{\mu\nu\rho\sigma}(dn)_{x+\hat{\mu},\nu\rho\sigma}},$

where in the second step we rewrote the $A^{m}$ in terms of the dual fields
$\widetilde{A}^{m}$. Obviously this integral imposes the new constraint

$\sum_{\nu<\rho<\sigma}\epsilon_{\mu\nu\rho\sigma}(dn)_{x+\hat{\mu},\nu\rho\sigma}\;=\;\begin{cases}-1&\text{if
$(\tilde{x},\mu)\in\widetilde{C}$}\;\;,\\\ 0&\text{if
$(\tilde{x},\mu)\notin\widetilde{C}$ \; .}\end{cases}$ (57)

This constraint modifies the condition (7), which in its original form ensures
the complete absence of monopoles, while now along the contour $\widetilde{C}$
of the magnetic Wilson loop (55) monopole charges are inserted.

Let us now inspect $Q_{T}$ in the presence of the modified constraint (57)
induced by the magnetic Wilson loop. We find (use $Q_{T}[dA^{e}]=0$),

$Q_{T}[dA^{e}+2\pi
n]=\frac{1}{4\pi}\sum_{x}\sum_{\mu<\nu\atop\rho<\sigma}n_{x,\mu\nu}\epsilon_{\mu\nu\rho\sigma}(dA^{e})_{x+T-\hat{\rho}-\hat{\sigma},\,\rho\sigma}\\\
+\frac{1}{4\pi}\sum_{x}\sum_{\mu<\nu\atop\rho<\sigma}(dA^{e})_{x,\mu\nu}\epsilon_{\mu\nu\rho\sigma}n_{x+T-\hat{\rho}-\hat{\sigma},\,\rho\sigma}+Q_{T}[2\pi
n]\;.$ (58)

Reorganizing the sums one finds

$\displaystyle\sum_{x}\sum_{\mu<\nu\atop\rho<\sigma}n_{x,\mu\nu}\,\epsilon_{\mu\nu\rho\sigma}(dA^{e})_{x+T-\hat{\rho}-\hat{\sigma},\,\rho\sigma}=\sum_{x}\sum_{\mu<\nu\atop\rho<\sigma}n_{x-T+\hat{\rho}+\hat{\sigma},\mu\nu}\,\epsilon_{\mu\nu\rho\sigma}(dA^{e})_{x,\,\rho\sigma}$
(59)
$\displaystyle\quad=\sum_{x}\sum_{\rho<\sigma}N_{x,\rho\sigma}(dA^{e})_{x,\,\rho\sigma}=\sum_{x}\sum_{\mu}A^{e}_{x,\mu}(\partial
N)_{x,\mu}=\sum_{x}\sum_{\mu}\\!A_{x+T,\mu}\sum_{\nu<\rho<\sigma}\\!\\!\epsilon_{\mu\nu\rho\sigma}(dn)_{x+\hat{\mu},\nu\rho\sigma}\;,$

where

$N_{x,\,\rho\sigma}=\sum_{\mu<\nu}n_{x-T+\hat{\rho}+\hat{\sigma},\mu\nu}\,\epsilon_{\mu\nu\rho\sigma}\;.$
(60)

In the third step of (59) we used the partial integration formula (125), and
again some reordering of terms to get to the final form. In a similar way we
find for the second sum in (58),

$\sum_{x}\sum_{\mu<\nu\atop\rho<\sigma}(dA^{e})_{x,\mu\nu}\,\epsilon_{\mu\nu\rho\sigma}\,n_{x+T-\hat{\rho}-\hat{\sigma},\rho\sigma}\;=\;\sum_{x}\sum_{\mu}A_{x-T+\hat{s},\mu}\sum_{\nu<\rho<\sigma}\\!\\!\epsilon_{\mu\nu\rho\sigma}(dn)_{x+\hat{\mu},\nu\rho\sigma}\;,$
(61)

so that inserting (58), (59) and (61) in (54) we find

$Q[dA^{e}+2\pi
n]=\frac{1}{2\pi}\sum_{x}\sum_{\mu}\sum_{T}\gamma_{T}\frac{A_{x+T,\mu}+A_{x-T+\hat{s},\mu}}{2}\sum_{\nu<\rho<\sigma}\\!\\!\epsilon_{\mu\nu\rho\sigma}(dn)_{x+\hat{\mu},\nu\rho\sigma}+Q[2\pi
n]\;.$ (62)

Since in the presence of the magnetic Wilson loop
$\prod_{(\tilde{x},\mu)\in\widetilde{C}}e^{\,i\,\widetilde{A}^{m}_{\tilde{x},\mu}}$
upon integrating over the $\widetilde{A}^{m}$ we have
$\sum_{\nu<\rho<\sigma}\epsilon_{\mu\nu\rho\sigma}(dn)_{x,\nu\rho\sigma}=-1$
whenever $(\tilde{x},\mu)\in\widetilde{C}$, and since $Q$ comes with a weight
$e^{-i{\theta}Q}$ in the partition function, a magnetic Wilson loop generates
the contribution

$\prod_{(\tilde{x},\mu)\in\widetilde{C}}e^{\,i\frac{\theta}{2\pi}\sum_{T}\gamma_{T}\frac{A^{e}_{x+T,\mu}+A^{e}_{x-T+\hat{s},\mu}}{2}}\;,$
(63)

which is the $A^{e}$-dependent part of $Q[dA^{e}+2\pi n]$ that gets added to
the topological part $Q[2\pi n]$ given by the harmonic contributions as stated
in (53). The interpretation of the above formula is that the monopole gets an
electric charge $q^{e}=\frac{\theta}{2\pi}$, although smeared to the
neighborhood of the dual link $(\tilde{x},\mu)$. This is the famous Witten
effect [34].

When we discussed the coupling of dyonic matter in Subsection 3.2, we already
announced, that in this context the Witten effect gives rise to a new duality
${\cal T}$ which relates shifts of the $\theta$ angle by $2\pi$ to a shift of
the magnetic charge. The remainder of this subsection is devoted to discussing
the duality ${\cal T}$.

Similar to the derivation of the worldline expansion of magnetic matter
discussed in Appendix E we may expand the hopping terms of the dyon action
(compare (45)) to bring the dependence of the dyon partition sum on the two
gauge fields into the form

$\prod_{(x,\mu)}e^{\,i\,\big{[}q^{e}\,{A}^{e}_{x,\mu}\,+\,q^{m}\,\widetilde{A}^{m}_{F(x),\mu}\big{]}\,k_{x,\mu}}\;=\;\prod_{(x,\mu)}e^{\,i\,q^{e}\,{A}^{e}_{x,\mu}\,k_{x,\mu}}\;\prod_{(x,\mu)}e^{\,i\,q^{m}\,\widetilde{A}^{m}_{F(x),\mu}\,k_{x,\mu}}\;,$
(64)

where $k_{x,\mu}\in\mathds{Z}$ are the flux variables for dyon matter (compare
Appendix E). Obviously the second factor on the rhs. generalizes the insertion
of the magnetic Wilson loop considered in (55). Upon integrating out the
magnetic gauge field this generates the modified constraint

$\sum_{\nu<\rho<\sigma}\epsilon_{\mu\nu\rho\sigma}(dn)_{x+\hat{\mu},\nu\rho\sigma}\;=\;-\,q^{m}\,k_{x,\mu}\quad\forall\;(x,\mu)\;,$
(65)

which now replaces (57). However, the steps discussed in the previous
paragraphs go through essentially unchanged also with the new constraint, and
the additional term that is generated by the topological charge is a
generalization of (63) given by

$\prod_{(x,\mu)}e^{\,i\frac{\theta}{2\pi}\,q_{m}\,\sum_{T}\gamma_{T}\frac{A^{e}_{x+T,\mu}+A^{e}_{x-T+\hat{s},\mu}}{2}\,k_{x,\mu}}\prod_{(x,\mu)}\delta\left(\sum_{\nu<\rho<\sigma}\epsilon_{\mu\nu\rho\sigma}(dn)_{x+\hat{\mu},\nu\rho\sigma}\,+\,q^{m}\,k_{x,\mu}\right)\;e^{-\,i\,\theta\,Q[2\pi
n]}\;,$ (66)

where we also wrote explicitly the constraint (65), using the already familiar
product over Kronecker deltas. We also display the remaining topological
contribution. The first factor in (63) may be combined with the flux terms
(64) and we replace

$\prod_{(x,\mu)}e^{\,i\,\big{[}q^{e}\,{A}^{e}_{x,\mu}\,+\,q^{m}\,\widetilde{A}^{m}_{F(x),\mu}\big{]}\,k_{x,\mu}}\;\rightarrow\;\prod_{(x,\mu)}e^{\,i\,\big{[}q^{e}\,{A}^{e}_{x,\mu}\,+\,q^{m}\,\sum_{T}\gamma_{T}\frac{A^{e}_{x+T,\mu}+A^{e}_{x-T+\hat{s},\mu}}{2}\,+\,q^{m}\,\widetilde{A}^{m}_{F(x),\mu}\big{]}\,k_{x,\mu}}\;,$
(67)

where we have reinstated the terms with $\widetilde{A}^{m}$ that upon
integration generate the constraint in (66), given that the topological term
is kept in the form $Q[2\pi n]$.

Note that the required 1-form gauge invariance [7]

$\begin{split}&A^{e}_{x,\mu}\rightarrow A^{e}_{x,\mu}+2\pi k_{x,\mu}\;,\\\
&n_{x,\mu\nu}\rightarrow n_{x,\mu\nu}-(dk)_{x,\mu\nu}\;,\end{split}$ (68)

can be achieved by a shift of $A^{m}_{\tilde{x},\mu}$ ,

$\widetilde{A}^{m}_{F(x),\mu}\rightarrow\widetilde{A}^{m}_{F(x),\mu}-\frac{\theta}{2}\sum_{T}\gamma_{T}\left(k_{x+T-\hat{\mu},\mu}+k_{x-T+s-\hat{\mu},\mu}\right)\;.$
(69)

Indeed the remainder of the action, has phase terms

$i\theta Q[2\pi
n]+i\sum_{x}\\!\sum_{\mu<\nu<\rho}\epsilon^{\mu\nu\rho\sigma}(dn)_{x,\mu\nu\rho}\tilde{A}^{m}_{F(x)-\hat{\sigma},\sigma}\;,$
(70)

which are invariant under the transformations (68) and (69). Note that $Q[2\pi
n]$ is now no longer necessarily an integer, because it is a sum of integers
weighted by $\gamma_{T}$, and must be kept in the action even if $\theta\in
2\pi\mathbb{Z}$. In fact it is crucial that this term is not dropped, as then
the gauge symmetry (69) would be ruined.

The rhs. of (67) shows that in the presence of magnetic matter, the
topological term generates an additional contribution to the electric charge,
such that the combined electric charge is given by $q^{e}+q^{m}\theta/2\pi$,
although this charge is generally smeared across multiple lattice links,
weighted by $\gamma_{T}$. In fact, no choice of the coefficients $\gamma_{T}$
in the definition (54) allows for an ultra-local Witten effect. Indeed, if
even only a single $\gamma_{T}$ is nonzero, i.e., $\gamma_{T}=1$ with
$\gamma_{T^{\prime}}=0\;\forall\;T^{\prime}\neq T$, this will produce the
spread of the electric charge. However, independent of the details of the
definition of $Q$ we find that shifting $\theta\rightarrow\theta+2\pi$ is
equivalent to shifting the electric charge $q^{e}\rightarrow q^{m}+q^{e}$.
Thus, if we shift $\theta\rightarrow\theta-2\pi$ and simultaneously
$q^{e}\rightarrow q^{m}+q^{e}$ this becomes an invariance. We will refer to
this invariance as ${\cal T}$ duality, as is conventional, and summarize it as
follows,

$\theta\,\rightarrow\,\overline{\theta}\,=\,\theta-2\pi\;,\;\;{\bm{q}}\,\rightarrow\,\overline{\bm{q}}\,=\,T\,{\bm{q}}\;\;\mbox{with}\;\;T=\begin{pmatrix}1&1\\\
0&1\end{pmatrix}\;,$ (71)

where again we use the vector of charges ${\bm{q}}=(q^{e},q^{m})^{t}$. We will
later see that the ${\cal S}$ duality defined in (42), i.e., the self-dual
transformation, and the duality ${\cal T}$ together generate the group
SL$(2,\mathbb{Z})$.

In the next two subsections we will see that a generalized form of the ${\cal
S}$ duality holds also for the theory with a $\theta$-term, but that the
discretization we discussed so far will not map the action into the same form
under a duality transformation. We will then go on to construct lattice
actions which enjoy an exact self-duality, at the price of abandoning the
ultra-local structure we so far employed. That such a non-ultra-local
structure may be expected is already suggested by the discussion of the Witten
effect in this subsection.

###  4.3 $\theta$-term and duality transformation

We now generalize the partition sum (33) of self-dual lattice QED further by
adding a $\theta$-term and discuss the form of the duality transformation in
the presence of such a term. This will not yet give rise to a self-dual
theory, which will be identified only in the next subsection after another
necessary generalization of the action.

We may write the partition sum that now also contains the $\theta$-term in the
form

$Z(\beta,\theta,M^{e}\\!,\lambda^{e}\\!,q^{e}\\!,\,M^{m}\\!,\lambda^{m}\\!,q^{e})\equiv\\!\int\\!\\!D[A^{e}]\\!\int\\!\\!D[A^{m}]B_{\beta,\theta}[A^{e},A^{m}]Z_{M^{e}\\!,\,\lambda^{e}\\!,\,q^{e}}[A^{e}]\widetilde{Z}_{M^{m}\\!,\,\lambda^{m}\\!,\,q^{m}}\big{[}\widetilde{A}^{m}\big{]},$
(72)

where we have generalized the Boltzmann factor (13) by adding the
$\theta$-term,

$B_{\beta,\theta}[A^{e},A^{m}]\;\equiv\;\sum_{\\{n\\}}\;e^{\,-\frac{\beta}{2}\sum_{x}\\!\sum_{\mu<\nu}\big{(}F^{e}_{x,\mu\nu}\big{)}^{2}}\;e^{\,-i\,\theta\,Q_{0}[F^{e}]}\;\;e^{\,-\,i\sum_{x}\\!\sum_{\mu<\nu<\rho}A^{m}_{x,\mu\nu\rho}(dn)_{x,\mu\nu\rho}}.$
(73)

Note that for now we use the $T=0$ discretization $Q_{0}[F^{e}]$ from the
family $Q_{T}[F^{e}]$ of possible equivalent lattice forms of the topological
charge we have introduced in Subsection 4.1 in Eq. (4.1). We will generalize
the $\theta$-term further in the next subsection when we construct the self-
dual form of the Boltzmann factor.

Both, the gauge field action and the topological charge $Q_{0}[F^{e}]$ are
quadratic in $F^{e}=dA^{e}\,+\,2\pi\,n$, such that we can combine them in a
quadratic form. The Boltzmann factor thus reads

$B_{\beta,\theta}[A^{e},A^{m}]\;=\,\sum_{\\{n\\}}e^{\,-\frac{\beta}{2}\sum_{x,\mu<\nu\atop
y,\rho<\sigma}F^{e}_{x,\mu\nu}\,M_{x,\mu\nu|y,\rho\sigma}\,F^{e}_{y,\rho\sigma}}\;e^{\,i\sum_{x,\mu<\nu}(\partial
A^{m})_{x,\mu\nu}\,n_{x,\mu\nu}},$ (74)

where we used the partial integration formula (125) from Appendix A to also
rewrite the exponent of the last exponential that upon integration over
$A^{m}$ generates the constraints, which in the absence of magnetically
charged matter are the closedness constraints (7) or, when magnetic matter is
coupled, the modified constraints (57). The kernel $M_{x,\mu\nu|y,\rho\sigma}$
of the quadratic form (74) is defined as (note that in our notation $\mu<\nu$
and $\rho<\sigma$)

$M_{x,\mu\nu|y,\rho\sigma}\;=\;\delta_{\mu\rho}\,\delta_{\nu\sigma}\,\delta_{x,y}^{\;(4)}\;+\;i\,\xi\;\epsilon_{\mu\nu\rho\sigma}\,\delta_{x-\hat{\rho}-\hat{\sigma},y}^{\;(4)}\qquad\mbox{with}\quad\xi\;\equiv\;\frac{\theta}{4\pi^{2}\beta}\;.$
(75)

Using Fourier transformation (see Appendix C for our conventions) we may
diagonalize the lattice site dependence of $M$. Using (137) we find for the
Fourier transform of $M$,

$\widehat{M}(p)_{\mu\nu|\rho\sigma}\;=\;\sum_{x}e^{ip\cdot(x-y)}M_{x,\mu\nu|y,\rho\sigma}=\delta_{\mu\rho}\,\delta_{\nu\sigma}\;+\;i\,\xi\;\epsilon_{\mu\nu\rho\sigma}\;e^{\,ip_{\rho}\,+\,ip_{\sigma}}\;.$
(76)

It is straightforward to see that for $|\xi|<1$ (this is a sufficient
condition) we have $\det\widehat{M}\neq 0$, such that the matrix $\widehat{M}$
in Fourier space is invertible and thus also the real space matrix $M$. One
finds (see Appendix C),

$\displaystyle\widehat{M}(p)_{\mu\nu|\rho\sigma}^{\;-1}$ $\displaystyle=$
$\displaystyle\frac{\delta_{\mu\rho}\,\delta_{\nu\sigma}\,-\,i\,\xi\,\epsilon_{\mu\nu\rho\sigma}\;e^{\,ip_{\rho}\,+\,ip_{\sigma}}}{1\;+\;\xi^{2}\;e^{\,ip\cdot\hat{s}}}$
(77) $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}(-\xi^{2})^{k}\left[\delta_{\mu\rho}\,\delta_{\nu\sigma}\,e^{\,ip\cdot
k\hat{s}}\,-\,i\,\xi\,\epsilon_{\mu\nu\rho\sigma}\;e^{\,ip_{\rho}\,+\,ip_{\sigma}\,+\,ip\cdot
k\hat{s}}\right],$

where $\hat{s}=\hat{1}+\hat{2}+\hat{3}+\hat{4}$ and in the second step we have
expanded the denominator using the geometric series which converges
exponentially for $|\xi|<1$. Using this second form and (138) we can evaluate
$M^{-1}_{\,x,\mu\nu|y,\rho\sigma}$ and obtain,

$M^{-1}_{\,x,\mu\nu|y,\rho\sigma}\;=\;\sum_{k=0}^{\infty}(-\xi^{2})^{k}\left[\delta_{\mu\rho}\,\delta_{\nu\sigma}\,\delta_{x-k\hat{s},y}^{\;(4)}\;-\;i\,\xi\,\epsilon_{\mu\nu\rho\sigma}\;\delta_{x-\hat{\rho}-\hat{\sigma}-k\hat{s},y}^{\;(4)}\right]\;.$
(78)

We now apply the generalized Poisson resummation formula proven in Appendix D
for $N=6V$ and find151515We remark that it is easy to show that for $|\xi|<1$
all eigenvalues of $M$ have positive real parts, which guarantees the
existence of the Gaussian integral in the Poisson resummation. (use again Eq.
(14) to first rewrite the second exponent in (74))

$B_{\beta,\theta}[A^{e},A^{m}]=C_{M}(\beta)\\!\sum_{\\{p\\}}\\!e^{\\!-\frac{\widetilde{\beta}}{2}\sum_{x,\mu<\nu\atop
y,\rho<\sigma}\\!(\partial A^{m}+2\pi
p)_{x,\mu\nu}\,M^{-1}_{\,x,\mu\nu|y,\rho\sigma}\,(\partial A^{m}+2\pi
p)_{y,\rho\sigma}}e^{-i\sum_{x,\mu<\nu}(dA^{e})_{x,\mu\nu}\,p_{x,\mu\nu}}\\!,$
(79)

with

$\widetilde{\beta}\;\equiv\;\frac{1}{4\pi^{2}\beta}\quad,\qquad\sum_{\\{p\\}}\;\equiv\;\prod_{x,\mu<\nu}\;\sum_{p_{x,\mu\nu}\in\mathds{Z}}\quad,\qquad
C_{M}(\beta)\;\equiv\;\frac{1}{(\sqrt{2\pi\beta})^{6V}\sqrt{\det M}}\;.$ (80)

$\sum_{\\{p\\}}$ denotes the sum over all configurations of plaquette
occupation numbers $p_{x,\mu\nu}\in\mathds{Z}$.

Note that the first exponent in (79) with $M^{-1}_{\,x,\mu\nu|y,\rho\sigma}$
given by (78) gives rise to a lattice action that combines terms at arbitrary
distances shifted relative to each other by $k\hat{s}$ with
$k\in\mathds{N}_{0}$, such that this action is not ultra-local, i.e., it
connects terms at arbitrary distances $k\hat{s}$. However, the corresponding
terms are suppressed exponentially with $k$, due to the condition $|\xi|<1$.
Such types of non-ultra local lattice actions with exponential suppression of
shifted terms are widely used, with prominent examples being fixed point
actions [35] or the overlap operator [25], and it is known that the
exponential suppression of distant terms guarantees a local continuum limit
[36]. As already mentioned, such lattice actions are referred to as local but
non-ultra-local actions.

Let us now complete the duality transformation. The final step of the duality
transformation is to express the electric and the magnetic gauge fields, as
well as the plaquette occupation numbers in terms of their counterparts on the
dual lattice, i.e., we again use Eqs. (21) and (22). We find

$B_{\beta,\theta}[A^{e},A^{m}]\;=\;C_{M}(\beta)\\!\sum_{\\{p\\}}e^{\,-\frac{\widetilde{\beta}}{2}\sum_{\tilde{x},\mu<\nu\atop\tilde{y},\rho<\sigma}\widetilde{F}^{m}_{\tilde{x},\mu\nu}\,\widetilde{M}_{\tilde{x},\mu\nu|\tilde{y},\rho\sigma}\,\widetilde{F}^{m}_{\tilde{y},\rho\sigma}}\;e^{\,-\,i\sum_{\tilde{x},\mu<\nu<\rho}\widetilde{A}^{e}_{\tilde{x},\mu\nu\rho}\,(d\widetilde{p})_{\tilde{x},\mu\nu\rho}},$
(81)

where
$\widetilde{F}^{m}_{\tilde{x},\mu\nu}=(d\widetilde{A}^{m}+2\pi\widetilde{p})_{\tilde{x},\mu\nu}$.
The kernel $\widetilde{M}_{\,\tilde{x},\mu\nu|\tilde{y},\rho\sigma}$ is
identified from $M^{-1}_{\,x,\mu\nu|y,\rho\sigma}$ when replacing $(\partial
A^{m}\,+\,2\pi p)_{x,\mu\nu}$ by the dual expression
$\sum_{\mu^{\prime}<\nu^{\prime}}\epsilon_{\mu\nu\mu^{\prime}\nu^{\prime}}(d\widetilde{A}^{m}\,+\,2\pi\tilde{p})_{\tilde{x}-\hat{\mu}^{\prime}-\hat{\nu}^{\prime},\mu^{\prime}\nu^{\prime}}$
in the exponent of (79) using (22), i.e.,

$\widetilde{M}_{\,\tilde{x},\mu\nu|\tilde{y},\rho\sigma}\;\equiv\;\sum_{\mu^{\prime}<\nu^{\prime}\atop\rho^{\prime}<\sigma\prime}\epsilon_{\mu\nu\mu^{\prime}\nu^{\prime}}\,M^{-1}_{\,x+\hat{\mu}^{\prime}+\hat{\nu}^{\prime},\mu^{\prime}\nu^{\prime}|y+\hat{\rho}^{\prime}+\hat{\sigma}^{\prime},\rho^{\prime}\sigma^{\prime}}\,\epsilon_{\rho^{\prime}\sigma^{\prime}\rho\sigma}\;\bigg{|}_{x\rightarrow\tilde{x}\atop
y\rightarrow\tilde{y}}\;.$ (82)

Inserting the explicit form (78) of $M^{-1}_{\,x,\mu\nu|y,\rho\sigma}$ and
summing over the indices of the epsilon tensors that appear in (82) we obtain
in a few lines of algebra

$\widetilde{M}_{\,\tilde{x},\mu\nu|\tilde{y},\rho\sigma}\;=\;\sum_{k=0}^{\infty}(-\xi^{2})^{k}\left[\delta_{\mu\rho}\,\delta_{\nu\sigma}\,\delta_{\tilde{x}-k\hat{s},\tilde{y}}^{\;(4)}\;-\;i\,\xi\,\epsilon_{\mu\nu\rho\sigma}\;\delta_{\tilde{x}-\hat{\rho}-\hat{\sigma}-k\hat{s},\tilde{y}}^{\;(4)}\right]\;.$
(83)

It is important to note that as for the case without $\theta$-term, the
Boltzmann factor in its dual form (81) is structurally similar to the original
form (74) we started from. The dual form lives on the dual lattice and the
Villain variables $n$ were replaced by the dual plaquette occupation numbers
$\widetilde{p}$. The electric and magnetic gauge fields interchanged their
role, such that $\widetilde{A}^{m}$ now appears in the quadratic form, while
$\widetilde{A}^{e}$ generates the constraints for the dual plaquette
occupation numbers.

However, the original kernel $M_{\,x,\mu\nu|y,\rho\sigma}$ Eq. (75) and the
dual kernel $\widetilde{M}_{\,\tilde{x},\mu\nu|\tilde{y},\rho\sigma}$ Eq. (83)
differ. More specifically, the original kernel $M_{\,x,\mu\nu|y,\rho\sigma}$
is ultra-local, while the dual kernel
$\widetilde{M}_{\,\tilde{x},\mu\nu|\tilde{y},\rho\sigma}$ is a sum over terms
that are shifted relative to each other by $k\hat{s}$ with
$k\in\mathds{N}_{0}$. Comparing this form with our definition of $Q_{T}$ in
Eq. (4.1) we find that for a fixed $k$ the second term in (83) gives rise to
the topological charge $Q_{T}[\widetilde{F}^{m}]$ with $T=-k\hat{s}$, i.e.,
$Q_{-k\hat{s}}[\widetilde{F}^{m}]$, with
$\widetilde{F}^{m}_{\tilde{x},\mu\nu}=(d\widetilde{A}^{m})_{\tilde{x},\mu\nu}+2\pi\widetilde{p}_{\tilde{x},\mu\nu}$.
Since integrating over $\widetilde{A}^{e}$ in (81) generates the closedness
constraints for the dual plaquette occupation numbers
$\widetilde{p}_{\tilde{x},\mu\nu}$, the topological charge
$Q_{-k\hat{s}}[\widetilde{F}^{m}]$ obeys all properties we showed for the
original definition (4.1). In particular it is independent of $T=-k\hat{s}$,
such that $Q_{-k\hat{s}}[\widetilde{F}^{m}]=Q_{0}[\widetilde{F}^{m}]\;\forall
k\in\mathds{Z}$. Thus we may sum up $k$ in the term generated by the second
factor of (83) such that the corresponding term in the exponent reads

$-\,\frac{\widetilde{\beta}}{2}\,8\pi^{2}\,Q_{0}[\widetilde{F}^{m}]\,(-i\xi)\sum_{k=0}^{\infty}(-\xi^{2})^{k}\;=\;i\,\frac{\theta}{4\pi^{2}\beta^{2}+\theta^{2}/4\pi^{2}}\,Q_{0}[\widetilde{F}^{m}]\;=\;-\,i\,\theta^{\prime}\,Q_{0}[\widetilde{F}^{m}]\;,$
(84)

where
$\theta^{\prime}\equiv-\,\theta/(4\pi^{2}\beta^{2}+\theta^{2}/4\pi^{2})$. Thus
we may rewrite the dual form (81) as

$B_{\beta,\theta}[A^{e},A^{m}]\;=\;C_{M}\\!\sum_{\\{p\\}}e^{-\frac{\widetilde{\beta}}{2}\sum_{k=0}^{\infty}(-\xi^{2})^{k}\sum_{\tilde{x},\mu<\nu}\widetilde{F}^{m}_{\tilde{x},\mu\nu}\,\widetilde{F}^{m}_{\tilde{x}-k\hat{s},\mu\nu}\;-\;i\theta^{\prime}\,Q_{0}[\widetilde{F}^{m}]}\;e^{\,-\,i\sum_{\tilde{x},\mu<\nu<\rho}\widetilde{A}^{e}_{\tilde{x},\mu\nu\rho}\,(d\widetilde{p}\,)_{\tilde{x},\mu\nu\rho}},$
(85)

where we have again used the partial integration formula to re-express the
exponent in the last factor. The dual form (85) has to be compared to the
original form of the Boltzmann factor (73). One sees that after interchanging
the gauge fields with their dual counterparts, multiplying the trivial overall
factor, and replacing $\beta$ by $\widetilde{\beta}$ and $\theta$ by
$\theta^{\prime}$, the two expressions are almost identical. The only
remaining difference is that the gauge field action has the form

$\frac{\tilde{\beta}}{2}\\!\sum_{k=0}^{\infty}\\!(-\xi^{2})^{k}\\!\\!\\!\sum_{\tilde{x},\mu<\nu}\widetilde{F}^{m}_{\tilde{x},\mu\nu}\,\widetilde{F}^{m}_{\tilde{x}-k\hat{s},\mu\nu}=\frac{\tilde{\beta}}{2}\\!\sum_{k=0}^{\infty}\\!(-\xi^{2})^{k}\\!\\!\\!\sum_{\tilde{x},\mu<\nu}\\!\\!(d\widetilde{A}^{m}\\!+\\!2\pi\widetilde{p}\,)_{\tilde{x},\mu\nu}(d\widetilde{A}^{m}\\!+\\!2\pi\widetilde{p}\,)_{\tilde{x}-k\hat{s},\mu\nu}\;,$
(86)

i.e., the gauge field action is a superposition of terms where the two factors
of the field strength $\widetilde{F}^{m}$ are shifted relative to each other
by $k\hat{s}$ with $k\in\mathds{N}_{0}$. The $k=0$ term corresponds to the
ultra-local action we have started from, but the duality transformation has
generated the non-ultra-local extension, which for $|\xi|<1$ still gives rise
to a local continuum limit as we discussed above. We remark, however, that
also the restriction $|\xi|<1$, which corresponds to
$\beta>|\theta|/4\pi^{2}$, is an unphysical restriction for a proper self-dual
lattice version of U(1) lattice gauge theory with a $\theta$-term we are
aiming at.

###  4.4 Identification of the exactly self-dual theory

Having developed the duality transformation in the presence of a $\theta$-term
we are now ready to identify the fully self-dual discretization that includes
the $\theta$-term. We have seen that the duality transformation has converted
the ultra-local lattice action of the original theory into a non-ultra-local
lattice action for the dual theory. The key insight is that in order to obtain
self-duality, we must not start with an ultra-local discretization for the
original theory, but with a more general ansatz that allows terms at arbitrary
distances with exponentially decreasing coefficients. We write the Boltzmann
factor in the form (compare (74))

$B_{\beta,\theta}[A^{e},A^{m}]\;\equiv\,\sum_{\\{n\\}}e^{\,-\frac{\beta}{2}\sum_{x,\mu<\nu\atop
y,\rho<\sigma}F^{e}_{x,\mu\nu}\,K_{x,\mu\nu|y,\rho\sigma}\,F^{e}_{y,\rho\sigma}}\;e^{\,i\sum_{x,\mu<\nu}(\partial
A^{m})_{x,\mu\nu}\,n_{x,\mu\nu}},$ (87)

where $K$ is a new kernel that we will specify below. Again the electric gauge
field $A^{e}$ describes the dynamics and the magnetic gauge field $A^{m}$
generates the constraints.

The key step towards self-duality is to identify a kernel $K$ that is form-
invariant under inversion. This can be done directly in momentum space, and
from inspection of (76) and (77) one finds that an ansatz of the form

$\widehat{K}(p)_{\mu\nu|\rho\sigma}\;=\;\frac{\delta_{\mu\rho}\,\delta_{\nu\sigma}\,+\,i\,\frac{\gamma}{2}\,\epsilon_{\mu\nu\rho\sigma}\,[e^{\,ip_{\rho}\,+\,ip_{\sigma}}+e^{\,-ip_{\mu}\,-\,ip_{\nu}}]}{\sqrt{1\;+\;\frac{\gamma^{2}}{2}\,[1+\cos(p\cdot\hat{s})]}}\;,$
(88)

with

$\widehat{K}(p)_{\mu\nu|\rho\sigma}^{\,-1}\;=\;\frac{\delta_{\mu\rho}\,\delta_{\nu\sigma}\,-\,i\,\frac{\gamma}{2}\,\epsilon_{\mu\nu\rho\sigma}\,[e^{\,ip_{\rho}\,+\,ip_{\sigma}}+e^{\,-ip_{\mu}\,-\,ip_{\nu}}]}{\sqrt{1\;+\;\frac{\gamma^{2}}{2}\,[1+\cos(p\cdot\hat{s})]}}\;,$
(89)

defines a gauge action and a topological term where the original kernel and
the inverse kernel that is used in the dual theory have the same momentum
dependence, and thus the same structure also in real space161616Note that (88)
is not the only choice of a kernel that keeps the electric-magnetic duality of
the theory, however, no ultra-local choice is possible. We remark that the
choice (88) violates hypercubic symmetries. This can indeed be corrected, as
we discuss in Appendix (F).. Here $\gamma$ is a real parameter that we do not
need to specify here and later will relate to the parameters $\beta$ and
$\theta$. Note that we have symmetrized the part that generates the
topological charge in $K$ to
$\epsilon_{\mu\nu\rho\sigma}\,[e^{\,ip_{\rho}\,+\,ip_{\sigma}}+e^{\,-ip_{\mu}\,-\,ip_{\nu}}]$,
which is related to a combination of $Q_{0}[F^{e}]$ and $Q_{-\hat{s}}[F^{e}]$.
With this choice the normalization factor in the denominator is regular for
all $\gamma$, i.e., the argument of the square root is always positive, such
that we have solved the problem of an unphysical restriction of $\beta$ and
$\theta$ we faced in the naive attempt in the previous subsection.

It is obvious, that the non-trivial denominator
$\sqrt{1\;+\;\frac{\gamma^{2}}{2}\,[1+\cos(p\cdot\hat{s})]}$ that appears in
the momentum space kernels (88) and (89) gives rise to non-ultra-local local
real space kernels $K$ and $K^{-1}$, which we now discuss. It is
straightforward to identify the real space equivalent of
$1\;+\;\frac{\gamma^{2}}{2}\,[1+\cos(p\cdot\hat{s})]$ which is the Helmholtz-
type lattice operator

$H_{x,y}\;=\;\delta^{\;(4)}_{x,y}\;+\;\frac{\gamma^{2}}{4}\,\left[\delta^{\;(4)}_{x+\hat{s},y}+2\delta^{(4)}_{x,y}+\delta^{\;(4)}_{x-\hat{s},y}\right]\;.$
(90)

Thus we find in real space,

$\displaystyle K_{x,\mu\nu|z,\rho\sigma}$ $\displaystyle=$
$\displaystyle\sum_{z}H^{-\frac{1}{2}}_{x,y}\,\bigg{[}\delta_{\mu\rho}\,\delta_{\nu\sigma}\,\delta_{y,z}^{\;(4)}\;+\;i\,\frac{\gamma}{2}\;\epsilon_{\mu\nu\rho\sigma}\,\left[\delta_{y-\hat{\rho}-\hat{\sigma},z}^{\;(4)}+\delta_{y+\hat{s}-\hat{\rho}-\hat{\sigma},z}^{\;(4)}\right]\bigg{]}\;,$
(91) $\displaystyle K_{x,\mu\nu|y,\rho\sigma}^{\;-1}$ $\displaystyle=$
$\displaystyle\sum_{z}H^{-\frac{1}{2}}_{x,y}\,\bigg{[}\delta_{\mu\rho}\,\delta_{\nu\sigma}\,\delta_{y,z}^{\;(4)}\;-\;i\,\frac{\gamma}{2}\;\epsilon_{\mu\nu\rho\sigma}\,\left[\delta_{y-\hat{\rho}-\hat{\sigma},z}^{\;(4)}+\delta_{y+\hat{s}-\hat{\rho}-\hat{\sigma},z}^{\;(4)}\right]\bigg{]}\;,$
(92)

where in real space the inverse square root $H^{-\frac{1}{2}}$ of the
Helmholtz operator may be implemented with the spectral theorem or a series
expansion. Since the denominator
$\sqrt{1\;+\;\frac{\gamma^{2}}{2}\,[1+\cos(p\cdot\hat{s})]}$ in the momentum
space kernels $\widehat{K}(p)$ and $\widehat{K}(p)^{\,-1}$ is regular for all
values of $\gamma$, the inverse Fourier transforms, i.e., the real space
kernels $K$ and $K^{\,-1}$ will have entries that decrease exponentially with
increasing lattice distance, i.e., they are local (but not ultra-local of
course).

It is important to note that the operator $H^{-\frac{1}{2}}_{x,y}$ is composed
from simple operators $\delta_{x+n\hat{s},y}$ for integers $n$. When applying
these shifts on the terms with the epsilon tensor in (91) and (92) this
generates terms proportional to $Q_{-\hat{s}k}[F^{e}]$ for different values of
$k$. If there are no dynamical monopoles then we can replace all
$Q_{-k\hat{s}}[F^{e}]$ by $Q_{0}[F^{e}]$.

Thus for the terms with the epsilon tensor in (91) and (92) we may replace the
action of $H^{-\frac{1}{2}}_{x,y}$ simply by multiplication with
$1/\sqrt{1+\gamma^{2}}$. We now use this fact to simplify the topological part
of the quadratic form of the Boltzmann factor.

We may write the Boltzmann factor (87) in the form

$B_{\beta,\theta}[A^{e},A^{m}]\;\equiv\,\sum_{\\{n\\}}e^{\,-\beta
S_{g}[F^{e}]\;-\;i\,\theta Q_{0}[F^{e}]}\;e^{\,i\sum_{x,\mu<\nu}(\partial
A^{m})_{x,\mu\nu}\,n_{x,\mu\nu}},$ (93)

where the gauge field action is given by

$S_{g}[F^{e}]\;=\;\frac{1}{2}\sum_{x,\mu<\nu\atop
y,\rho<\sigma}F^{e}_{x,\mu\nu}\,H^{-\frac{1}{2}}_{x,\mu\nu|y,\rho\sigma}\,F^{e}_{y,\rho\sigma}\;,$
(94)

and using the simplification of $H^{-\frac{1}{2}}_{x,y}$ for the topological
part discussed above, we identify the topological angle $\theta$ as

$\theta\;=\;\beta\,4\pi^{2}\,\frac{\gamma}{\sqrt{1+\gamma^{2}}}\quad\Rightarrow\quad\gamma\;=\;\frac{\theta}{\sqrt{(4\pi^{2}\beta)^{2}-\theta^{2}}}\;.$
(95)

Note that while we obtained this for a model without dynamical magnetic
matter, we can take the above formulas as the definition of the $\theta$ term
even in this case.

In a similar way we may identify the bare electric charge parameter $e$
through the pre-factor of the $F^{e}_{\mu\nu}F^{e}_{\mu\nu}$ term which gives
rise to the relation (again replacing $H^{-\frac{1}{2}}_{x,y}$ by
$1/\sqrt{1+\gamma^{2}}$)

$\frac{1}{e^{2}}\;=\frac{\beta}{\sqrt{1+\gamma^{2}}}\;.$ (96)

Using (95) and (96) we may express the two auxiliary parameters $\beta$ and
$\gamma$ in terms of the bare charge parameter $e$ and the topological angle
$\theta$,

$\beta\;=\;\frac{1}{2\pi}\sqrt{\left(\frac{2\pi}{e^{2}}\right)+\left(\frac{\theta}{2\pi}\right)^{2}}\quad,\qquad\gamma\;=\;\frac{\theta\,e^{2}}{4\pi^{2}}\;.$
(97)

Repeating the steps of the duality transformation from the previous section we
use the generalized Poisson resummation formula and find the dual form of the
Boltzmann factor

$B_{\beta,\theta}[A^{e},A^{m}]\;=\;C_{K}(\beta)\\!\sum_{\\{p\\}}e^{\,-\frac{\widetilde{\beta}}{2}\sum_{\tilde{x},\mu<\nu\atop\tilde{y},\rho<\sigma}\widetilde{F}^{m}_{\tilde{x},\mu\nu}\,\widetilde{K}_{\tilde{x},\mu\nu|\tilde{y},\rho\sigma}\,\widetilde{F}^{m}_{\tilde{y},\rho\sigma}}\;e^{\,i\sum_{\tilde{x},\mu<\nu}(\partial\widetilde{A}^{e})_{\tilde{x},\mu\nu}\,\widetilde{p}_{\tilde{x},\mu\nu}},$
(98)

where (it is straightforward to establish the result $\det K=1$ we use here)

$\tilde{\beta}\;\equiv\;\frac{1}{4\pi^{2}\beta}\qquad\mbox{and}\qquad
C_{K}(\beta)\;\equiv\;\frac{1}{(\sqrt{2\pi\beta})^{6V}\sqrt{\det
K}}\;=\;\frac{1}{(2\pi\beta)^{3V}}\;.$ (99)

The dual kernel $\widetilde{K}_{\tilde{x},\mu\nu|\tilde{y}\rho\sigma}$ is
identified in analogy to (82) and we find

$\displaystyle\widetilde{K}_{\,\tilde{x},\mu\nu|\tilde{y},\rho\sigma}$
$\displaystyle\equiv$
$\displaystyle\sum_{\mu^{\prime}<\nu^{\prime}\atop\rho^{\prime}<\sigma\prime}\epsilon_{\mu\nu\mu^{\prime}\nu^{\prime}}\,K^{-1}_{\,x+\hat{\mu}^{\prime}+\hat{\nu}^{\prime},\mu^{\prime}\nu^{\prime}|y+\hat{\rho}^{\prime}+\hat{\sigma}^{\prime},\rho^{\prime}\sigma^{\prime}}\,\epsilon_{\rho^{\prime}\sigma^{\prime}\rho\sigma}\;\bigg{|}_{x\rightarrow\tilde{x}\atop
y\rightarrow\tilde{y}}$ (100) $\displaystyle=$
$\displaystyle\sum_{z}H^{-\frac{1}{2}}_{x,y}\,\bigg{[}\delta_{\mu\rho}\,\delta_{\nu\sigma}\,\delta_{y,z}^{\;(4)}\;-\;i\,\frac{\gamma}{2}\;\epsilon_{\mu\nu\rho\sigma}\,\left[\delta_{y-\hat{\rho}-\hat{\sigma},z}^{\;(4)}+\delta_{y+\hat{s}-\hat{\rho}-\hat{\sigma},z}^{\;(4)}\right]\bigg{]}\;.$

Obviously, up to the opposite sign of the imaginary part, also the dual kernel
$\widetilde{K}_{\tilde{x},\mu\nu|\tilde{y},\rho\sigma}$ has the structure of
the original kernel $K_{x,\mu\nu|y,\rho\sigma}$ in Eq. (91), and comparing
(87) and (98) we find that we have indeed constructed a self-dual Boltzmann
factor that also includes the $\theta$-term. Writing the dual form of the
kernel as

$B_{\beta,\theta}[A^{e},A^{m}]\;=\;C_{K}(\beta)\\!\sum_{\\{p\\}}e^{\,-\widetilde{\beta}S_{g}[\widetilde{F}^{m}]\;-\;i\,\widetilde{\theta}\,Q_{0}[\widetilde{F}^{e}]}\;e^{\,i\sum_{\tilde{x},\mu<\nu}(\partial\widetilde{A}^{e})_{\tilde{x},\mu\nu}\,\widetilde{p}_{\tilde{x},\mu\nu}}\;,$
(101)

we here identify the dual topological angle as

$\widetilde{\theta}\;=\;-\,4\pi^{2}\widetilde{\beta}\frac{\gamma}{\sqrt{1+\gamma^{2}}}\;.$
(102)

We now may summarize the self-duality relation of the Boltzmann factor with
$\theta$-term as

$\beta^{\,\frac{3V}{2}}\,B_{\beta,\theta}[A^{e},A^{m}]\;=\;\widetilde{\beta}^{\,\frac{3V}{2}}\,\widetilde{B}_{\widetilde{\beta},\widetilde{\theta}}\big{[}\widetilde{A}^{m},\widetilde{A}^{e}\big{]}\;,$
(103)

where we have again distributed the pre-factors symmetrically and use

$\widetilde{\beta}\;=\;\beta\,f\quad,\qquad\widetilde{\theta}\;=\;-\,\theta\,f\qquad\mbox{where}\qquad
f\;\equiv\;\frac{1}{(2\pi\beta)^{2}}\;.$ (104)

Alternatively we may write the transformation of the couplings completely in
terms of the more physical parameters $e$ and $\theta$ (use (95) and (96))

$\frac{1}{\widetilde{e}^{\;2}}\;=\;\frac{1}{e^{2}}\,f\quad,\qquad\widetilde{\theta}\;=\;-\,\theta\,f\qquad\mbox{with}\qquad
f\;\equiv\;\frac{1}{\left(\frac{2\pi}{e^{2}}\right)^{2}+\left(\frac{\theta}{2\pi}\right)^{2}}\;.$
(105)

These equations constitute the generalization of the duality relation Eq. (26)
to the case of a Boltzmann factor that also includes the topological term. We
remark at this point that the self-duality of QED with a $\theta$-term can be
extended to an even more general definition of the topological charge, that
also fully implements all lattice symmetries. For the corresponding discussion
see Appendix F.

As before we check that repeating the duality transformation provides the
identity map. This property follows from
$\;\widetilde{\\!\\!\widetilde{A}}^{\atop e}=-A^{e}$,
$\;\widetilde{\\!\\!\widetilde{A}}^{\atop m}=-A^{m}$ and the trivial
identities

$\widetilde{\\!\widetilde{\beta}}\;=\;\beta\quad,\quad\widetilde{\\!\widetilde{\theta}}\;=\;\theta\qquad\mbox{and}\qquad
C_{K}(\beta)\,C_{K}(\widetilde{\beta})\;=\;1\;.$ (106)

We may now use the Boltzmann factor (101) in the partition sum (72) and based
on (103) obtain the self-duality relation for full QED with a $\theta$-term,

$\displaystyle\beta^{\frac{3V}{2}}\,Z(\beta,\theta,M^{e}\\!,\lambda^{e}\\!,q^{e}\\!,\,M^{m}\\!,\lambda^{m}\\!,q^{m})\;=\;\widetilde{\beta}^{\frac{3V}{2}}\,Z\big{(}\widetilde{\beta},\widetilde{\theta},\widetilde{M}^{e}\\!,\widetilde{\lambda}^{e}\\!,\widetilde{q}^{\,e}\\!,\widetilde{M}^{m}\\!,\widetilde{\lambda}^{m}\\!,\widetilde{q}^{\;m}\big{)}\;,$
(107)
$\displaystyle\qquad\mbox{with}\quad\;\widetilde{\beta}\,=\,\beta\,f\;,\;\widetilde{\theta}\,=\,-\,\theta\,f\;,\;f\,=\,\frac{1}{4\pi^{2}\beta^{2}}\;,$
$\displaystyle\hskip
51.21495pt\widetilde{M}^{e}\,=\,M^{m}\;,\;\widetilde{\lambda}^{e}\,=\,\lambda^{m}\;,\;\widetilde{M}^{m}\,=\,M^{e}\;,\;\widetilde{\lambda}^{m}\,=\,\lambda^{e}\;,\;$
$\displaystyle\hskip 51.21495pt\widetilde{\bf q}\,=\,S\,{\bf
q}\;\;,\;\;S=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}\;,$

where again we use the vector of charges ${\bf q}=(q^{e},q^{m})^{t}$.

As before we can generate self-duality relations for observables by evaluating
derivatives of $\ln Z$ with respect to the couplings. When considering
observables for the matter fields, which are obtained from derivatives with
respect to $M^{e}\\!,\,\lambda^{e}\\!,\,M^{m}$ or $\lambda^{m}$, the self-
duality relations generalize in a straightforward way. For example the
relation (44) now reads,

$\left\langle|\phi^{e}|^{2}\right\rangle_{\beta,\theta,M^{e}\\!,\,\lambda^{e}\\!,\,q^{\,e}\\!,\,M^{m}\\!,\,\lambda^{m}\\!,\,q^{\,m}}\;=\;\left\langle|\phi^{m}|^{2}\right\rangle_{\widetilde{\beta},\widetilde{\theta},\widetilde{M}^{e}\\!,\,\widetilde{\lambda}^{e}\\!,\widetilde{q}^{\,e}\\!,\,\widetilde{M}^{m}\\!,\,\widetilde{\lambda}^{m}\\!,\,\widetilde{q}^{\,m}}\;.$
(108)

However, self-duality relations for observables that contain the gauge fields,
i.e., observables that are generated by derivatives with respect to $\beta$ or
$\theta$ require a little more work, since via $\gamma$ as given in (95) the
two parameters mix, and additional terms appear in the self-duality relations.
We derive two such self-duality relations which for notational convenience we
discuss without matter fields, i.e., we derive them for pure gauge theory with
a $\theta$-term (adding matter fields is trivial).

Applying a derivative with respect to $\beta$ on the Boltzmann factor in the
form of Eq. (101) generates the insertion of the action $S_{g}$, such that we
find (the matter couplings were omitted as arguments in all expressions since
we discuss the pure gauge case),

$-\frac{\partial\ln
Z(\beta,\theta)}{\partial\beta}\,=\,\big{\langle}S_{g}\big{\rangle}_{\\!\beta,\theta}\,+\,\beta\left\langle\\!\frac{\partial
S_{g}}{\partial\beta}\\!\right\rangle_{\\!\\!\\!\beta,\theta}\\!\\!\\!=\,\big{\langle}S_{g}\big{\rangle}_{\\!\beta,\theta}\,+\,\beta\frac{\partial\gamma}{\partial\beta}\big{\langle}S_{g}\big{\rangle}_{\\!\beta,\theta}\,=\,\big{\langle}S_{g}\big{\rangle}_{\\!\beta,\theta}\,-\,\frac{\Gamma\,\theta}{\beta}\big{\langle}S_{g}\big{\rangle}_{\\!\beta,\theta}.$
(109)

The second terms on the right hand sides come from the $\beta$-dependence of
the action $S_{g}$ via the $\beta$-dependence of $\gamma$ and we use the
notation $S_{g}^{\prime}=d\,S_{g}/d\gamma$. It is straightforward to compute
the derivative $\frac{\partial\gamma}{\partial\beta}$ that appears after the
second step from the explicit expression (95), and we write the result in the
form

$\frac{\partial\gamma}{\partial\beta}\;=\;-\frac{\Gamma\,\theta}{\beta}\qquad\mbox{with}\qquad\Gamma\;\equiv\;\frac{\beta^{3}(4\pi^{2})^{2}}{\big{(}(4\pi^{2}\beta)^{2}-\theta^{2}\big{)}^{3/2}}\;=\;\frac{\widetilde{\beta}^{3}(4\pi^{2})^{2}}{\big{(}(4\pi^{2}\widetilde{\beta})^{2}-\widetilde{\theta}^{2}\big{)}^{3/2}}\;\equiv\;\widetilde{\Gamma}\;,$
(110)

where we have factored out the combination $\Gamma$ which is invariant under
the duality transformation as follows immediately from the transformation
properties (104). Using the self-duality relation (107) for the partition sum
we can apply the derivative with respect to $\beta$ also on the dual form,

$\displaystyle-\frac{\partial\ln Z(\beta,\theta)}{\partial\beta}$
$\displaystyle=$
$\displaystyle-\frac{\partial}{\partial\beta}\ln\left((2\pi\beta)^{-3V}Z(\widetilde{\beta},\widetilde{\theta})\right)\;=\;\frac{3V}{\beta}-\frac{d\widetilde{\beta}}{d\beta}\frac{\partial\ln
Z(\widetilde{\beta},\widetilde{\theta})}{\partial\widetilde{\beta}}$ (111)
$\displaystyle=$
$\displaystyle\frac{3V}{\beta}-\frac{1}{4\pi^{2}\beta^{2}}\left[\big{\langle}S_{g}\big{\rangle}_{\widetilde{\beta},\widetilde{\theta}}\,-\,\frac{\widetilde{\Gamma}\,\widetilde{\theta}}{\widetilde{\beta}}\big{\langle}S_{g}^{\prime}\big{\rangle}_{\widetilde{\beta},\widetilde{\theta}}\right]\;.$

Setting equal the right hand sides of (109) and (111) one obtains after a few
steps of trivial reordering of terms (use also
$\widetilde{\beta}=1/4\pi^{2}\beta$ and $\widetilde{\Gamma}=\Gamma$) the final
form of the self-duality relation

$\beta\,\big{\langle}S_{g}\big{\rangle}_{\\!\beta,\theta}\;-\;\Gamma\,\theta\,\big{\langle}S_{g}^{\prime}\big{\rangle}_{\\!\beta,\theta}\;=\;3V\;-\;\widetilde{\beta}\,\big{\langle}S_{g}\big{\rangle}_{\widetilde{\beta},\widetilde{\theta}}\;+\;\Gamma\,\widetilde{\theta}\,\big{\langle}S_{g}^{\prime}\big{\rangle}_{\widetilde{\beta},\widetilde{\theta}}\;.$
(112)

It is easy to check that for $\theta=0$ this self-duality relation reduces to
the self-duality relation (32) which we derived for pure gauge theory without
$\theta$-term (use $\theta=0\Rightarrow\gamma=0$, the fact that $S_{g}$ at
$\gamma=0$ reduces to the ultra-local action we used initially, and $\langle
F^{2}\rangle=\langle S_{g}\\!\\!\mid_{\gamma=0}\rangle/3V$). The
generalization to the non-ultralocal action that is needed for $\theta\neq 0$
then generates the additional terms with $S_{g}^{\prime}$.

In exactly the same way we may also study derivatives
$i\partial/\partial\theta$ that generate expectation values of the topological
charge $Q_{0}$, and following the same steps as above find the corresponding
self-duality relation

$\beta\,\big{\langle}Q_{0}\big{\rangle}_{\\!\beta,\theta}\;-\;i\,\Gamma\,\beta\,\big{\langle}S_{g}^{\prime}\big{\rangle}_{\\!\beta,\theta}\;=\;-\;\widetilde{\beta}\,\big{\langle}Q_{0}\big{\rangle}_{\widetilde{\beta},\widetilde{\theta}}\;+\;i\,\Gamma\,\widetilde{\beta}\,\big{\langle}S_{g}^{\prime}\big{\rangle}_{\widetilde{\beta},\widetilde{\theta}}\;.$
(113)

This self-duality relation vanishes for $\theta=0$ (as expected), since
$\langle Q_{0}\rangle\big{|}_{\theta=0}=0$ and due to
$\theta=0\Rightarrow\gamma=0$ also the second term disappears because of
$S_{g}^{\prime}\big{|}_{\gamma=0}=0$.

###  4.5 The $\text{SL}(2,\mathbb{Z})$ structure of dyonic lattice QED with a
$\theta$-term

In the course of this paper we have identified two transformations of our
self-dual lattice version of QED with a $\theta$-term. The first one is the
duality transformation ${\cal S}$ itself, which we here write in terms of the
bare charge parameter $e$, the topological angle $\theta$ and the vector of
electric and magnetic charges $\bm{q}\equiv(q^{e},q^{m})^{t}$ (compare (105)),

$\displaystyle\frac{1}{e^{2}}\;\rightarrow\;\frac{1}{\widetilde{e}^{\;2}}\;=\;\frac{1}{e^{2}}\,f\quad,\qquad\theta\;\rightarrow\widetilde{\theta}\;=\;-\,\theta\,f\qquad\mbox{where}\qquad
f\;\equiv\;\frac{1}{\left(\frac{2\pi}{e^{2}}\right)^{2}+\left(\frac{\theta}{2\pi}\right)^{2}}\;,$
$\displaystyle\bm{q}\;\rightarrow\;\widetilde{\bm{q}}\;=\;S\,\bm{q}\qquad\mbox{with}\qquad
S\;=\;\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}\;.$ (114)

The second transformation we identified is ${\cal T}$ which shifts $\theta$ by
$2\pi$ that was discussed in Subsection 4.2 in the context of the Witten
effect. It acts only on $\theta$ and the charge vector $\bm{q}$ and is defined
as follows (see (71))

$\displaystyle\theta\;\rightarrow\;\overline{\theta}\;=\;\theta-2\pi\;,$
$\displaystyle\bm{q}\;\rightarrow\;\overline{\bm{q}}\;=\;T\,\bm{q}\qquad\mbox{with}\qquad
T\;=\;\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}\;.$ (115)

We begin the discussion of the overall duality structure by noting that the
matrices $S$ and $T$ are the generators171717See for example [37] for an
elementary introduction to the group $\text{SL}(2,\mathds{Z})$. of the group
$\text{SL}(2,\mathbb{Z})$, which is the group of $2\times 2$ matrices $M$ with
integer-valued elements and determinant 1, i.e.,

$M\;=\;\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\quad\mbox{with}\quad
a,b,c,d\,\in\,\mathds{Z}\quad\mbox{and}\quad ad-bc\,=\,1\;.$ (116)

And since they are generators, combinations of the matrices $S$ and $T$
implement arbitrary $\text{SL}(2,\mathbb{Z})$ transformations on the charge
vector $\bm{q}$.

Furthermore, also the action of the dualities ${\cal S}$ and ${\cal T}$ on
$1/e^{2}$ and $\theta$ can be identified with the group
$\text{SL}(2,\mathbb{Z})$. In order to see that, we combine the gauge coupling
$e$ and the topological angle $\theta$ into a joint complex-valued coupling
$\tau$, the so-called modular parameter of U(1) gauge theory (see, e.g., [19,
38, 39]), defined as

$\tau\;\equiv\;i\,\frac{2\pi}{e^{2}}-\frac{\theta}{2\pi}\;.$ (117)

It is straightforward to see from (114) and (115) that our two generators
${\cal S}$ and ${\cal T}$ act as

${\cal
S}:\;\tau\,\rightarrow\,\widetilde{\tau}\;=\;-\frac{1}{\tau}\quad\mbox{and}\quad{\cal
T}:\;\tau\,\rightarrow\,\overline{\tau}\;=\;\tau\,+\,1\;.$ (118)

The action of a general $\text{SL}(2,\mathbb{Z})$ transformation on a complex
number $\tau$ is defined as181818Strictly speaking this implements the
projected group $\text{PSL}(2,\mathbb{Z})$, which is $\text{SL}(2,\mathbb{Z})$
with matrices $M$ and $-M$ identified.

$\tau\;\rightarrow\;M\,\tau\;\equiv\;\frac{a\tau+b}{c\tau+d}\;,$ (119)

where $a,b,c,d$ are the entries of an $\text{SL}(2,\mathbb{Z})$ matrix $M$ as
given in (116). It is straightforward to see that choosing the generators
$M=S$ and $M=T$ gives rise to the transformations $\tau\rightarrow-1/\tau$ and
$\tau\rightarrow\tau+1$, i.e., the action of our symmetries ${\cal S}$ and
${\cal T}$ as stated in (118). Thus our transformations ${\cal S}$ and ${\cal
T}$ generate the full set of group transformations (119). We conclude that
combining our two transformations ${\cal S}$ and ${\cal T}$ as stated in (114)
and (115) gives rise to a full $\text{SL}(2,\mathbb{Z})$ invariance of our
self-dual lattice formulation of QCD with a $\theta$-term.

We now ignore the other couplings of the dyonic matter fields, such as $M$ and
$\lambda$, and only focus on the couplings $1/e^{2}$ and $\theta$ combined
into the complex coupling $\tau$, as well as the charge vector $\bm{q}$. Only
those couplings are now listed as arguments of the partition function.
Furthermore it is convenient to redefine the partition function as
$\overline{Z}(\tau,\bm{q})\equiv\beta^{\frac{3V}{2}}Z(\tau,\bm{q})$. Then the
${\cal S}$ and ${\cal T}$ symmetry relations are written as

$\overline{Z}(\tau,\bm{q})\;=\;\overline{Z}(S\tau,S\bm{q})\qquad\mbox{and}\qquad\overline{Z}(\tau,\bm{q})\;=\;\overline{Z}(T\tau,T\bm{q})\;,$
(120)

and according to the discussion above these two transformations generate the
full $\text{SL}(2,\mathbb{Z})$ invariance given by

$\displaystyle\overline{Z}(\tau,\bm{q})\;=\;\overline{Z}(M\tau,M\bm{q})\qquad\mbox{with}\quad$
(121) $\displaystyle M\;=\;\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\;,\;\quad
a,b,c,d\,\in\,\mathds{Z}\;,\;ad-
bc\,=\,1\;,\;M\,\tau\;=\;\frac{a\tau+b}{c\tau+d}\;.$

These equations describe the symmetry content of dyonic self-dual QED with a
$\theta$-term.

Figure 1: Mapping of theories in the complex $\tau$ plane that are related by
different elements of $\text{SL}(2,\mathbb{Z})$. On the vertical axis we plot
Im $\\!\tau=2\pi/e^{2}$ and on the horizontal axis Re $\\!\tau=-\theta/2\pi$.
Theories that have $\theta=0$ and thus can be directly simulated are located
on the positive vertical axis (marked with a thick black line). Different
elements of $\text{SL}(2,\mathbb{Z})$ map this family of theories to
equivalent theories with non-zero $\theta$ which are represented by the semi-
circles (see the legend for which element $M$ of $\text{SL}(2,\mathbb{Z})$ was
used).

Figure 2: The same as Fig. 2, but now plotted in the $(\theta,e^{2})$ plane.

As we have seen in Sec. 3.2, dyonic theories generically have a sign problem.
In addition, also theories with a $\theta$-term have a sign problem. However,
the $\text{SL}(2,\mathbb{Z})$ structure we identified above allows us to map
theories which can be recast in terms of wordlines without the sign-problem
(i.e., with the purely electric and/or purely magnetic matter, at least one of
which is bosonic and considered at $\theta=0$) onto other theories that
naively do have a sign problem. This map can be rather nontrivial. To
illustrate the idea, let us discuss an example. We consider a matrix
$M\in\text{SL}(2,\mathbb{Z})$ given by

$M=\begin{pmatrix}1&0\\\ 1&1\end{pmatrix}\;.$ (122)

We have seen that a theory described by $(\tau,\bm{q})$ is equivalent to a
theory with parameters $(M\tau,M\bm{q})$. Furthermore we can simulate the
family of theories with $\tau=iy,y\in\mathbb{R}_{+}$ and, say a matter field
with charge vector $\bm{q}=(n,0)^{t}$ with $n\in\mathbb{Z}$. This family of
theories is dual to theories with $\bar{\tau}=M\tau=\frac{iy}{iy+1}$ and
$\bar{\bm{q}}=M\bm{q}=(n,n)^{t}$. Identifying
$\text{Re}(\bar{\tau})=\frac{\bar{\theta}}{2\pi}$ and
$\text{Im}(\bar{\tau})=\frac{2\pi}{\bar{e}^{2}}$, we find that the dual
theories have parameters and matter charges charges given by
($y\in\mathds{R}_{+}$)

$\bar{e}^{2}=2\pi\frac{1+y^{2}}{y}\;,\qquad\bar{\theta}=2\pi\frac{y^{2}}{1+y^{2}}\;,\qquad{\text{
and }}\qquad\bar{\bm{q}}=\begin{pmatrix}n\\\ n\end{pmatrix}\;.$ (123)

Obviously this is a family of theories that has a complex action problem,
which, using our $\text{SL}(2,\mathbb{Z})$ mapping we may study via simulating
the corresponding sign free set of theories. The above family of theories is
shown as an orange semi-circle in the $\tau$ plane in Fig. 2, or equivalently,
in the $(\theta,e^{2})$ plane in Fig. 2 (again the orange curve). Note that in
particular the self-dual point of the original theory $y=1$ is mapped to
$e^{2}=4\pi$ and $\bar{\theta}=\pi$ of the dual theory.

Figures 2 and 2 show several $\text{SL}(2,\mathbb{Z})$ families of theories
that are dual to the family of theories which obeys $\text{Im}(\tau)=0$, i.e.,
the $\theta=0$ theories, that can be simulated as long as the matter field is
not dyonic, i.e., the matter fields are purely electric or purely magnetic.
The legend shows the matrices $M$ that need to be applied to the matter fields
in the exactly same manner as in the example above. Note that all the
interesting theories which are related to the sign-problem free theory are at
strong electric coupling $\bar{e}^{2}\geq 4\pi$, as is evident from Fig. 2.

##  5 Concluding remarks

In this work we discussed the construction of U(1) gauge theories in 4d on the
lattice with $\theta$-terms. We showed that modified Villain actions allow for
a construction which admits the inclusion of $\theta$-terms in a natural way.
However, a naive ultra-local $\theta$-term ruins the self-dual nature of the
theory, mapping the ultra-local theory to a merely local one. We showed that
by generalizing the modified Villain actions to include non-ultra-local terms,
we can restore the exactly self-dual nature of the model. The construction is
reminiscent of the inability to regulate a lattice theory in an ultra-local
way, while preserving the axial symmetry in 4d lattice gauge theories. Indeed,
the Nielsen-Ninomiya theorem [21, 22, 23] prohibits a lattice regularization
for which the matrix $\gamma^{5}$ anti-commutes with the Dirac operator, which
is essential for the axial symmetry. However, using the Ginsparg-Wilson
relation [26] and its solution by Neuberger [25], Lüscher showed [24] that an
axial symmetry can be defined. But such constructions, while local, are not
ultra-local. In fact 4d U(1) gauge theories, both free and interacting, have
anomalies [27, 28, 29, 30] and it is not a priory a big surprise that for
maintaining the correct symmetry and anomaly structure on the lattice one must
resort to a local, but not ultra-local form of the action. Yet many examples
exist of lattice discretization of theories with anomalies, which maintain
their ultra-local form. One example is a $\theta=\pi$ $2d$ abelian gauge
theory, whose lattice action was discussed in [8, 9, 7, 10, 11, 12], and may
be chosen ultra-local, even though such models may have ‘t Hooft anomalies
involving their internal symmetries. Such models are also connected to half-
integral spin-chains, which furnish a regulator of the Hilbert space.

It is well known, that the free abelian gauge theory has an
$\text{SL}(2,\mathbb{Z})$ structure in the continuum, generated by self-
duality and shifts of the $\theta$-angle by $2\pi$. As Witten showed in [40],
the free U(1) gauge theory partition function transforms as a modular form,
with a modular weight determined by the Euler characteristic $\chi$ and the
signature of the spin-for manifold $\sigma$. In [31] this was interpreted as a
‘t Hooft anomaly (see Section 4.2.2 of [31]) at special points of the modular
parameter $\tau$, where the theory enjoys a $\mathbb{Z}_{6}$ symmetry, which
is a subgroup of $\text{SL}(2,\mathbb{Z})$. As explained by Honda and Tanizaki
in [31], this $\mathbb{Z}_{6}$ symmetry has a mixed anomaly with gravity. On
the other hand, the mixed axial-gravitational anomaly of fermions is also well
known (see, e.g., [41, 42, 43]). Furthermore, the role of gravitational
anomalies in lattice discretization was noted in [44], where it was proven
(under some mild assumptions) that there exists no lattice
discretization191919The proof was for a lattice discretization of Hilbert
space, where time is viewed as continuous. in two dimensions unless its pure
gravitational anomaly vanishes. One may then naturally wonder if the inability
to write ultra-local actions is related to the involvement of mixed-
gravitational anomalies.

Acknowledgments: We thank Simeon Hellerman, Nabil Iqbal, Masataka Watanabe for
discussions related to this work. We also thank Simon Hands and Simon
Catterall who initially encouraged us to write this work for a collection they
are editing. TS is supported by the Royal Society University Research
Fellowship.

Appendices

##  Appendix A Notation and results for lattice differential forms

In this appendix we summarize the notation and some basic results for
differential forms on the lattice, which we use in our paper. For a more
general presentation see, e.g., [33, 7]. In the main part of the paper we work
in $d=4$ dimensions, but this appendix is kept general with arbitrary $d$.

We consider a $d$-dimensional hypercubic lattice $\Lambda$ and set the lattice
spacing to $a=1$. The spatial extents of the lattice are denoted as
$N_{\mu},\,\mu=1,2\,...\,d$ and we use periodic boundary conditions. Sites are
denoted as $x$. Links are denoted as $(x,\mu)$, plaquettes as $(x,\mu\nu)$
with $\mu<\nu$, 3-cubes as $(x,\mu\nu\rho)$ with $\mu<\nu<\rho$ et cetera.

We can identify these elements with $r$-cells: Sites are 0-cells. A link
$(x,\mu)$ is a 1-cell and contains the 0-cells (sites) $x$ and $x+\hat{\mu}$
where by $\hat{\mu}$ we denote the unit vector in direction $\mu$. A plaquette
$(x,\mu\nu)$ is a 2-cell and contains the sites $x$, $x+\hat{\mu}$,
$x+\hat{\nu}$ and $x+\hat{\mu}+\hat{\nu}$, as well as the links $(x,\mu)$,
$(x+\hat{\mu},\nu)$, $(x+\hat{\nu},\mu)$ and $(x,\nu)$. In an equivalent way
3-cells are the 3-cubes that contain 8 sites, 12 links and 6 plaquettes and
similarly for higher $r$-cells. The site $x$ in the definition of an $r$-cell
we will sometimes refer to as _root site_ or _root_.

We assign integers or real numbers to the $r$-cells and refer to these as
$r$-forms. We denote $r$-forms as
$f_{x,\,\mu_{1}\mu_{2}\,...\,\mu_{r}}\in\mathds{R}$ or $\mathds{Z}$ with
$\mu_{1}<\mu_{2}<\,...\,<\mu_{r}$, i.e., the $r$-forms are labelled by the
root site and the ordered indices $\mu_{j},\,j=1,2,\,...\,r$ that label the
corresponding $r$-cell. It is convenient to not only label the $r$-forms with
the ordered indices $\mu_{1}<\mu_{2}<\,...\,<\mu_{r}$ but allow for arbitrary
ordering with the convention that
$f_{x,\,\mu_{1}\,...\,\mu_{j}\,...\,\mu_{k}\,...\,\mu_{r}}=-f_{x,\,\mu_{1}\,...\,\mu_{k}\,...\,\mu_{j}\,...\,\mu_{r}}$,
i.e., $r$-forms are antisymmetric in their indices. This also implies that an
$r$-form vanishes when two or more of the indices are equal. One may use this
generalized labelling with non-ordered indices also for the cells and the sign
of the permutation of the indices defines an orientation for the cells.

We define two discrete differential operators that act on $r$-forms, the
_exterior derivative_ $d$ that takes an $r$-form into an $(r\\!+\\!1)$-form
with the definition that $d$ acting on a $d$-form gives zero. The other
differential operator is the _boundary operator_ $\partial$ that takes an
$r$-form into an $(r\\!-\\!1)$-form. When acting on a $0$-form $\partial$
gives zero. The two differential operators are defined as

$\displaystyle(d\,f)_{x,\,\mu_{1}\mu_{2}\,...\,\mu_{r}}$ $\displaystyle\equiv$
$\displaystyle\sum_{j=1}^{r}(-1)^{j+1}\,\Big{[}\,f_{x+\hat{\mu}_{j},\,\mu_{1}\,...\,{\mu}^{\\!\\!\\!{}^{o}}_{j}\,...\,\mu_{r}}\,-\,f_{x,\,\mu_{1}\,...\,{\mu}^{\\!\\!\\!{}^{o}}_{j}\,...\,\mu_{r}}\,\Big{]}\;,$
$\displaystyle(\partial f)_{x,\,\mu_{1}\mu_{2}\,...\,\mu_{r}}$
$\displaystyle\equiv$
$\displaystyle\sum_{\nu=1}^{d}\,\Big{[}\,f_{x,\,\mu_{1}\,...\,\mu_{r}\,\nu}\,-\,f_{x-\hat{\nu},\,\mu_{1}\,...\,\mu_{r}\,\nu}\,\Big{]}\;,$
(124)

where in the first equation ${\mu}^{\\!\\!\\!{}^{o}}_{j}$ indicates that
$\mu_{j}$ is dropped from the list of indices. We remark that due to the
antisymmetry of the $r$-forms with respect to interchange of their indices, in
the second line of (124) only those $(r\\!+\\!1)$-forms contribute to the sum
where $\nu$ is different from all indices $\mu_{1},\mu_{2}\,...\,\mu_{r}$. The
ordering of these indices and $\nu$ determines the sign of the contribution.

Both differential operators are nilpotent, i.e., $d^{\,2}=0$ and
$\partial^{\,2}=0$. Furthermore one may show the following partial integration
formula, where $f$ is an $r$-form and $g$ an $(r\\!-\\!1)$-form,

$\sum_{x\atop\mu_{1}<\,...\,<\,\mu_{r}}\\!\\!\\!\\!\\!f_{\,x,\,\mu_{1}\,...\,\mu_{r}}\;(d\,g)_{\,x,\,\mu_{1}\,...\,\mu_{r}}\quad=\quad(-1)^{r}\\!\\!\\!\\!\\!\\!\\!\sum_{x\atop\mu_{1}<\,...\,<\,\mu_{r-1}}\\!\\!\\!\\!\\!(\partial
f)_{\,x,\,\mu_{1}\,...\,\mu_{r-1}}\;g_{\,x,\,\mu_{1}\,...\,\mu_{r-1}}\;.$
(125)

The _Hodge decomposition_ states that an arbitrary $r$-form $f$ can be written
as the sum of the boundary operator $\partial$ acting on an $(r\\!+\\!1)$-form
$p$, the exterior derivative operator $d$ acting on an $(r\\!-\\!1)$-form $q$
and a _harmonic_ or _defect_ $r$-form $h$ that obeys $d\,h=0$ and
$\partial\,h=0$,

$f_{\,x,\,\mu_{1}\,...\,\mu_{r}}\;=\;(\partial\,p)_{\,x,\,\mu_{1}\,...\,\mu_{r}}\;+\;(d\,q)_{\,x,\,\mu_{1}\,...\,\mu_{r}}\;+\;h_{\,x,\,\mu_{1}\,...\,\mu_{r}}\;.$
(126)

The _dual lattice_ $\tilde{\Lambda}$ is defined by a one-to-one identification
of the $r$-cells of the original lattice with the $(d\\!-\\!r)$-cells of the
dual lattice. The sites $\tilde{x}$ of $\tilde{\Lambda}$ are at the centers of
the $d$-cells of $\Lambda$, i.e.,
$\tilde{x}=x+\frac{1}{2}(\hat{1}+\hat{2}+...+\hat{d}\,)$. There is a natural
identification of the $r$-forms $f_{x,\,\mu_{1}\,...\,\mu_{r}}$ on $\Lambda$
with the $(d\\!-\\!r)$-forms on $\tilde{\Lambda}$. We denote these as
$\tilde{f}_{\tilde{x},\,\nu_{r+1}\,...\,\nu_{d}}$ but stress that the
numerical value is of course the same as the $r$-form
$f_{x,\,\mu_{1}\,...\,\mu_{r}}$ we identify it with. The identification is
given by

$f_{x,\,\mu_{1}\,...\,\mu_{r}}\;=\;\sum_{\nu_{r+1}<\,\nu_{r+2}<\,...\,<\,\nu_{d}}\epsilon_{\mu_{1}\,...\,\mu_{r}\,\nu_{r+1}\,...\,\nu_{d}}\;\tilde{f}_{\tilde{x}-\hat{\nu}_{r+1}-\hat{\nu}_{r+2}\,...\,-\hat{\nu}_{d},\,\nu_{r+1}\,...\,\nu_{d}}\;.$
(127)

Finally, when switching between the original lattice $\Lambda$ and the dual
lattice $\tilde{\Lambda}$ the exterior derivative $d$ and the boundary
operator are converted into each other,

$(\partial\,f)_{x,\,\mu_{1}\,...\,\mu_{r}}\;=\;\sum_{\nu_{r+1}<\,\nu_{r+2}<\,...\,<\,\nu_{d}}\epsilon_{\mu_{1}\,...\,\mu_{r}\,\nu_{r+1}\,...\,\nu_{d}}\;(d\,\tilde{f})_{\tilde{x}-\hat{\nu}_{r+1}-\hat{\nu}_{r+2}\,...\,-\hat{\nu}_{d},\,\nu_{r+1}\,...\,\nu_{d}}\;.$
(128)

##  Appendix B Auxiliary results for the topological charge

In this appendix we collect some results for the topological charge $Q_{T}[F]$
which can be obtained with elementary although lengthy algebra. As a first
result we show (49), i.e., the fact that the topological charge $Q_{T}[F]$ is
invariant when adding the exterior derivative $(dB)_{x,\mu\nu}$ of an
arbitrary 1-form $B_{x,\mu}$ to the field strength
$F_{x,\mu\nu}=(dA^{e})_{x,\mu\nu}+2\pi\,n_{x,\mu\nu}$, as long as the Villain
variables $n_{x,\mu\nu}$ obey the closedness condition (7), i.e.,
$(dn)_{x,\mu\nu\rho}=0$. The expression $Q_{T}[F\\!+\\!dB]$ can be trivially
decomposed into three contributions,

$\displaystyle Q_{T}[F\\!+\\!dB]$ $\displaystyle\\!\\!\\!=\\!\\!\\!$
$\displaystyle\frac{1}{8\pi^{2}}\sum_{x}\sum_{\mu<\nu\atop\rho<\sigma}(F+dB)_{x,\mu\nu}\,\epsilon_{\mu\nu\rho\sigma}\,(F+dB)_{x-\hat{\rho}-\hat{\sigma}+T,\rho\sigma}\;=\;Q_{T}[F]\,+\,Q_{T}[(dB)]$
(129) $\displaystyle\\!\\!\\!+\\!\\!\\!$
$\displaystyle\frac{1}{8\pi^{2}}\sum_{x}\sum_{\mu<\nu\atop\rho<\sigma}\Big{[}F_{x,\mu\nu}(dB)_{x-\hat{\rho}-\hat{\sigma}+T,\rho\sigma}\,+\,F_{x-\hat{\rho}-\hat{\sigma}+T,\rho\sigma}(dB)_{x,\mu\nu}\,\Big{]}\,\epsilon_{\mu\nu\rho\sigma}\;.$

For proving the claimed independence of $dB$ we need to show that the second
and third terms on the right hand side vanish. We begin with the third term:
After inserting the explicit definition of the exterior derivatives
$(dB)_{x,\mu\nu}$ and reshuffling the summation indices the third term can be
written in the form (a factor $1/8\pi^{2}$ was dropped)

$\displaystyle\sum_{x}\sum_{\mu<\nu\atop\rho<\sigma}\epsilon_{\mu\nu\rho\sigma}\Big{[}B_{x,{\rho}}\;(F_{x+\hat{\rho}+\hat{\sigma}-T,\mu\nu}-F_{x+\hat{\rho}-T,\mu\nu})\;-\;B_{x,{\sigma}}\;(F_{x+\hat{\rho}+\hat{\sigma}-T,\mu\nu}-F_{x+\hat{\sigma}-T,\mu\nu})\;+$
$\displaystyle
B_{x,{\mu}}\;(F_{x-\hat{\rho}-\hat{\sigma}+T,\rho\sigma}-F_{x-\hat{\rho}-\hat{\sigma}-\hat{\nu}+T,\rho\sigma})\;-\;B_{x,{\nu}}\;(F_{x-\hat{\rho}-\hat{\sigma}+T,\rho\sigma}-F_{x-\hat{\rho}-\hat{\sigma}-\hat{\mu}+T,\rho\sigma}\;)\Big{]}\;.$
(130)

For a fixed site $x$ we now may identify all the contributions that multiply
$B_{x,\mu}$ for a fixed value of $\mu$. For example, the terms that multiply
$B_{x,1}$ are given by

$\displaystyle
F_{x+T-\hat{3}-\hat{4},34}-F_{x+T-\hat{2}-\hat{3}-\hat{4},34}-F_{x+T-\hat{2}-\hat{4},24}+F_{x+T-\hat{2}-\hat{3}-\hat{4},24}+F_{x+T-\hat{2}-\hat{3},23}$
$\displaystyle-\,F_{x+T-\hat{2}-\hat{3}-\hat{4},23}+F_{x-T+\hat{1}+\hat{4},23}-F_{x-T+\hat{1},23}-F_{x-T+\hat{1}+\hat{3},24}$
(131)
$\displaystyle+\,F_{x-T+\hat{1},24}+F_{x-T+\hat{1}+\hat{2},34}-F_{x-T+\hat{1},34}\,=\,(dF)_{x+T-\hat{2}-\hat{3}-\hat{4},234}\,-\,(dF)_{x-T+\hat{1},234}\;=\;0,$

where in the last line we have combined the contributions into the exterior
derivatives
$(dF)_{x,\mu\nu\rho}=(d^{2}A)_{x,\mu\nu\rho}+2\pi(dn)_{x,\mu\nu\rho}=0$, which
vanish due to $d^{2}=0$ and the closedness condition $(dn)_{x,\mu\nu\rho}=0$
we implemented for the Villain variables. The same steps can be repeated for
the factors that multiply the other $B_{x,\mu}$ thus establishing that the
third term on the rhs. of (129) vanishes.

Let us now explore the second term on the right hand side of (129), i.e., the
term $Q[(dB)]$. Dropping again a factor of $1/8\pi^{2}$ the term reads

$\displaystyle\sum_{x}\sum_{\mu<\nu\atop\rho<\sigma}(dB)_{x,\mu\nu}\,\epsilon_{\mu\nu\rho\sigma}\,(dB)_{x-\hat{\rho}-\hat{\sigma}+T,\rho\sigma}\;=\;\sum_{x}\sum_{\mu<\nu\atop\rho<\sigma}(dB)_{x+\hat{\rho}+\hat{\sigma}-T,\mu\nu}\,\epsilon_{\mu\nu\rho\sigma}\,(dB)_{x,\rho\sigma}\;=$
$\displaystyle\sum_{x}\sum_{\mu<\nu\atop\rho<\sigma}\\!\epsilon_{\mu\nu\rho\sigma}\Big{[}B_{x+\hat{\rho},\sigma}-B_{x,\sigma}-B_{x+\hat{\sigma},\rho}+B_{x,\rho}\Big{]}\,(dB)_{x+\hat{\rho}+\hat{\sigma}-T,\mu\nu}\;=$
$\displaystyle\sum_{x}\sum_{\mu<\nu\atop\rho<\sigma}\\!\epsilon_{\mu\nu\rho\sigma}\Big{[}B_{x,\rho}\Big{(}(dB)_{x+\hat{\rho}+\hat{\sigma}-T,\mu\nu}-(dB)_{x+\hat{\rho}-T,\mu\nu}\Big{)}$
$\displaystyle\hskip
113.81102pt-\,B_{x,\sigma}\Big{(}(dB)_{x+\hat{\rho}+\hat{\sigma}-T,\mu\nu}-(dB)_{x+\hat{\sigma}-T,\mu\nu}\Big{)}\Big{]},$
(132)

where we performed trivial shifts of site indices, explicitly wrote out one of
the exterior derivatives, and organized the terms such that we identify the
factors multiplying $B_{x,\rho}$ and $B_{x,\sigma}$. As with the third term we
now collect all factors that multiply $B_{x,\mu}$ for a fixed $\mu$. For the
case of $B_{x,1}$ this factor is given by

$\displaystyle(dB)_{x-T+\hat{1}+\hat{4},23}\;-\;(dB)_{x-T+\hat{1},23}\;-\;(dB)_{x-T+\hat{1}+\hat{3},24}\;+\;(dB)_{x+T+\hat{1},24}\qquad\;$
(133) $\displaystyle\hskip
56.9055pt+\;(dB)_{x-T+\hat{1}+\hat{2},34}\;-\;(dB)_{x-T+\hat{1},34}\;=\;(d^{2}B)_{x-T+\hat{1},234}\;=\;0\;.\qquad$

In the second line we have identified the terms with
$(d^{2}B)_{x-T+\hat{1},234}$, which vanishes due to the nilpotency of the
exterior derivative. In a similar way one can treat the terms that multiply
$B_{x,\mu}$ for all values of $\mu$ and show that also the second term on the
rhs. of (129) vanishes. Thus we have proven (49), i.e.,
$Q_{T}[F\\!+\\!dB]=Q_{T}[F]$ for configurations of the Villain variables that
obey the closedness condition $(dn)_{x,\mu\nu\rho}=0$.

The second result we show in this appendix is the fact that when using the
Hodge decomposition (51) for the Villain variables the result for the
topological charge $Q_{T}[F]$ at arbitrary $T$ is given by (53) when the
harmonic contribution is parameterized as in (52). Obviously

$Q_{T}[F]=Q_{T}[2\pi n]=Q_{T}[2\pi dl+2\pi h]=Q_{T}[2\pi
h]=\frac{1}{2}\sum_{x}\sum_{\mu<\nu\atop\rho<\sigma}\epsilon_{\mu\nu\rho\sigma}h_{x,\mu\nu}h_{x-\hat{\rho}-\hat{\sigma}+T,\rho\sigma}\;,$
(134)

where in the first step we used (50), then inserted the Hodge decomposition
(51) and subsequently used (49). In the final step $Q_{T}[2\pi h]$ was written
explicitly. Inserting the parametrization (52) for the harmonic contribution
$h$ we find

$\displaystyle Q_{T}[2\pi h]$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{x}\sum_{\mu<\nu\atop\rho<\sigma}\epsilon_{\mu\nu\rho\sigma}\,\omega_{\mu\nu}\,\omega_{\rho\sigma}\,\sum_{i=1}^{N_{\rho}}\sum_{j=1}^{N_{\sigma}}\,\delta_{x,\,i\hat{\rho}+j\hat{\sigma}}\sum_{n=1}^{N_{\mu}}\sum_{l=1}^{N_{\nu}}\,\delta_{x-\hat{\rho}-\hat{\sigma}+T,\,n\hat{\mu}+l\hat{\nu}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\mu<\nu\atop\rho<\sigma}\epsilon_{\mu\nu\rho\sigma}\,\omega_{\mu\nu}\,\omega_{\rho\sigma}\,\sum_{n=1}^{N_{\mu}}\sum_{l=1}^{N_{\nu}}\sum_{i=1}^{N_{\rho}}\sum_{j=1}^{N_{\sigma}}\,\delta_{i\hat{\rho}+j\hat{\sigma}-\hat{\rho}-\hat{\sigma}+T,n\hat{\mu}+l\hat{\nu}}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{\mu<\nu\atop\rho<\sigma}\epsilon_{\mu\nu\rho\sigma}\,\omega_{\mu\nu}\,\omega_{\rho\sigma}\,\sum_{n=1}^{N_{\mu}}\sum_{l=1}^{N_{\nu}}\sum_{i=1}^{N_{\rho}}\sum_{j=1}^{N_{\sigma}}\,\delta_{0,(n-t_{\mu})\hat{\mu}+(l-t_{\nu})\hat{\nu}+(1-i)\hat{\rho}+(1-j)\hat{\sigma}}\;.$

In the first step we summed over $x$ to remove the first Kronecker delta. In
the second step we used that
$T=t_{\mu}\hat{\mu}+t_{\nu}\hat{\nu}+t_{\rho}\hat{\rho}+t_{\sigma}\hat{\sigma}$
for mutually distinct $\mu,\nu,\rho,\sigma$. Then it is obvious that only a
single term remains where the Kronecker deltas give 1, such that the
topological charge reduces to

$Q_{T}[2\pi
h]\;=\;\frac{1}{2}\sum_{\mu<\nu\atop\rho<\sigma}\epsilon_{\mu\nu\rho\sigma}\,\omega_{\mu\nu}\,\omega_{\rho\sigma}\;=\;\omega_{12}\omega_{34}-\omega_{13}\omega_{24}+\omega_{14}\omega_{23}\;\in\;\mathds{Z}\;,$
(136)

which is the result (53).

##  Appendix C Properties of the kernels $M$ and $K$ from Fourier
transformation

In Sections 4.2 and 4.3 we combine the gauge field action and the
$\theta$-term into quadratic forms with kernels $M_{x,\mu\nu|y,\rho\sigma}$
defined in Eq. (75) and similarly in Section 4.3 with a new kernel
$K_{x,\mu\nu|y,\rho\sigma}$ defined in Eq. (91). In order to diagonalize the
space-time dependence of these kernels, which we here generically denote as
$A_{x,\mu\nu|y,\rho\sigma}$, we use Fourier transformation, i.e., a similarity
transformation with the unitary matrices $U_{p,x}\equiv V^{-1/2}e^{-ip\cdot
x}$, where the momenta are given by $p_{\mu}=2\pi
n_{\mu}/N_{\mu}\;,\;n_{\mu}=0,1,2\;...\;N_{\mu}-1$ (we use periodic boundary
conditions). We find

$\sum_{x,y}U_{p,x}^{\star}\,A_{\,x,\mu\nu|y,\rho\sigma}\,U_{q,y}\;=\;\delta^{\;(4)}_{p,q}\;\widehat{A}(p)_{\mu\nu|\rho\sigma}\;,$
(137)

such that

$A_{\,x,\mu\nu|y,\rho\sigma}^{\;-1}\;=\;\frac{1}{V}\sum_{p}e^{ip(y-x)}\,\widehat{A}(p)_{\mu\nu|\rho\sigma}^{\,-1}\;.$
(138)

The Fourier transform $\widehat{A}(p)_{\mu\nu|\rho\sigma}$ is a $6\times 6$
matrix (we order the indices $\mu<\nu$, $\rho<\sigma$). Both, the Fourier
transforms of $M$ and of $K$ have a similar structure given by202020For $K$ we
actually use a symmetrized version of the term with the $\epsilon$-tensor, but
the inversion strategy is identical to the one outlined here. (again we use
the generic notation $\widehat{A}$ for both kernels),

$\widehat{A}(p)_{\mu\nu|\rho\sigma}\;=\;f(p)[\delta_{\mu\rho}\,\delta_{\nu\sigma}\,+\,i\,a\,\epsilon_{\mu\nu\rho\sigma}\;e^{\,ip_{\rho}\,+\,ip_{\sigma}}]\;,$
(139)

where $f(p)$ is some function of the momenta and $a\in\mathds{R}$ some real-
valued parameter. It is straightforward to show that

$\sum_{\rho<\sigma}[\delta_{\mu\rho}\,\delta_{\nu\sigma}\,+\,i\,a\,\epsilon_{\mu\nu\rho\sigma}\;e^{\,ip_{\rho}\,+\,ip_{\sigma}}][\delta_{\rho\tau}\,\delta_{\sigma\omega}\,-\,i\,a\,\epsilon_{\rho\sigma\tau\omega}\;e^{\,ip_{\tau}\,+\,ip_{\omega}}]\;=\;\delta_{\mu\tau}\,\delta_{\nu\omega}(1+a^{2}e^{ip\cdot\hat{s}})\;,$
(140)

where again we use $\hat{s}\equiv\hat{1}+\hat{2}+\hat{3}+\hat{4}$. As a
consequence, the inverse momentum space kernel is given by

$\widehat{A}(p)_{\mu\nu|\rho\sigma}^{\;-1}\;=\;\frac{\delta_{\mu\rho}\,\delta_{\nu\sigma}\,-\,i\,a\,\epsilon_{\mu\nu\rho\sigma}\;e^{\,ip_{\rho}\,+\,ip_{\sigma}}}{f(p)(1+a^{2}e^{ip\cdot\hat{s}})}\;,$
(141)

and with (138) one finds the inverse kernel in real space.

##  Appendix D Generalized Poisson resummation

In this appendix we present a short derivation of the generalized Poisson
resummation formula. We consider a function $b(x_{1},x_{2},\,...\;x_{N})$ of
$N$ real variables $x_{j}$ that has the form