# Anomalies and gauging of $\boldsymbol{U(1)}$ symmetries

Andrea Antinucci  SISSA and INFN, via Bonomea 265, 34136 Trieste, Italy
Francesco Benini  SISSA and INFN, via Bonomea 265, 34136 Trieste, Italy
International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34151
Trieste, Italy

###### Abstract

We propose the Symmetry TFT for theories with a $U(1)$ symmetry in arbitrary
dimension. The Symmetry TFT describes the structure of the symmetry, its
anomalies, and the possible topological manipulations. It is constructed as a
BF theory of gauge fields for groups $U(1)$ and $\mathbb{R}$, and contains a
continuum of topological operators. We also propose an operation that produces
the Symmetry TFT for the theory obtained by dynamically gauging the $U(1)$
symmetry. We discuss many examples. As an interesting outcome, we obtain the
Symmetry TFT for the non-invertible $\mathbb{Q}/\mathbb{Z}$ chiral symmetry in
four dimensions.

††preprint: SISSA 01/2024/FISI

##  I Introduction

Symmetries and anomalies are some of the most basic, and yet robust and
predictive, properties of a quantum field theory (QFT). Symmetries constrain
spectra and correlation functions. Anomalies are exactly calculable and
renormalization-group invariant, impose constraints on the low-energy dynamics
(a famous example being ’t Hooft anomaly matching [1]), and can explain a
variety of phenomena. By inflow [2, 3], the anomalies of a $d$-dimensional QFT
can be described by a classical topological field theory (TQFT) of background
fields in $d+1$ dimensions (this is also called an SPT phase, or an invertible
TQFT [4]).

An improvement of this description, proposed in [5, 6, 7, 8], captures at the
same time the symmetries and anomalies of all possible variants (or global
forms) of a seed theory, obtained by gauging finite subgroups of the symmetry
group. This description goes under the name of Symmetry TFT. It is a
nontrivial $(d+1)$-dimensional TQFT placed on a slab with two parallel
boundaries. On one boundary it is coupled to the physical QFTd. On the other
boundary one prescribes a topological boundary condition. The claim is that
there is a one-to-one correspondence between topological boundary conditions
of the Symmetry TFT and global forms of the QFT. This proposal has been
developed and verified in a variety of examples, that include the generalized
symmetries of [5], and including cases with non-invertible (or categorical)
symmetries [9, 10, 11]. Moreover the Symmetry TFT proved to be a useful tool
to characterize both representations [12] and anomalies, even of non-
invertible symmetries [13, 14, 15, 16], and has been applied to the
classification of gapped phases [17]. So far it has been limited, though, to
symmetries with a finite number of elements.

In this Letter we propose a construction of the Symmetry TFT for QFTs with
continuous $U(1)$ symmetries. We provide a Lagrangian description of the
Symmetry TFT, written in terms of gauge fields for group $\mathbb{R}$ — as
opposed to $U(1)$. These theories are TQFTs with a continuum of operators,
hence they constitute new mathematical objects with few properties that
deviate from the standard well-known cases. The different boundary conditions
of the TQFT correspond to topological manipulations of the boundary QFT, i.e.,
to gaugings of discrete symmetry subgroups, or alternatively of the whole
$U(1)$ but endowed with the discrete topology 111Gauging a $U(1)$ $p$-form
symmetry with the discrete topology means summing over all flat bundles. It is
a topological manipulation that does not introduce new degrees of freedoms,
and yields a dual $(d{-}p{-}2)$-form symmetry $\mathbb{Z}$.. This is enough to
capture the anomalies of the theory, including the ordinary perturbative
anomalies.

Physically, however, it is more interesting to understand the effect of the
ordinary _dynamical_ gauging of a $U(1)$ symmetry, which involves the
introduction of a new degree of freedom — the photon – coupled to the theory.
In our setup, this is not a topological operation and therefore is not
described by a boundary condition in the original TQFT. Rather, it induces an
operation that maps the Symmetry TFT to another one. We use this map to
explicitly construct the Symmetry TFT for some interesting Abelian gauge
theories in four dimensions. Interestingly, we observe that the various
$(d+1)$-dimensional TQFTs can also be obtained from different boundary
conditions in a $(d+2)$-dimensional TQFT that is a dynamical version of the
anomaly polynomial [2].

We provide a few selected examples of our construction. For instance, we give
the Symmetry TFT description of the chiral anomaly in two and four dimensions,
as well as the Symmetry TFT for a 4d Abelian gauge theory with 2-group
symmetry. We also provide a 3d example, illustrating interesting phenomena
even in the absence of anomalies (that do not exist in odd dimensions). A
particularly interesting outcome of our construction is the Symmetry TFT for
the non-invertible $\mathbb{Q}/\mathbb{Z}$ symmetry that arises from a $U(1)$
chiral symmetry with ABJ (Adler-Bell-Jackiw) anomaly in 4d Abelian gauge
theories [19, *Cordova:2022ieu]. Our theory resembles, but is different from,
the one discussed in [21]. Constructions of $U(1)$ symmetry defects in string
theory have been proposed in [22].

In our notation, the Symmetry TFT lives in $d+1$ spacetime dimensions and has
action $\mathcal{Z}$, while the physical QFT is $d$-dimensional. We use
capital letters $A_{p}$ for $U(1)$ $p$-form gauge fields, and lowercase
letters $a_{p}$ for $\mathbb{R}$ $p$-form gauge fields. Sometimes we indicate
a $p$-form symmetry with a superscript, e.g., $U(1)^{(p)}$.

Note added. When this work was under completion, we received [23], and were
informed that [24] will appear, which address similar questions as ours.

##  II Symmetry TFT for $\boldsymbol{U(1)}$ symmetries

The Symmetry TFT contains all the categorical data of the global symmetry of
the boundary QFTd. For the simple Abelian TQFTs considered in this paper, both
the topological symmetry defects of the boundary QFT that (in the language of
[5]) generate the symmetry, and the charges that the operators can carry, are
described by the bulk operators 222More generally, the Symmetry TFT contains
complete information also on generalized charges, namely representations of
$p$-form symmetries on $q$-dimensional objects, where $q\geq p$ [12].. A
choice of boundary condition corresponds to a maximal set of mutually-
transparent bulk operators (which we call a Lagrangian algebra $\mathscr{L}$)
that can terminate on the boundary. In other words, the boundary condition
sets those operators to be trivial on the boundary. The endpoints (or more
generally end-surfaces) of those operators correspond to the charged operators
in the boundary theory, therefore $\mathscr{L}$ is also the set of charges
that the operators of the boundary theory can have. On the contrary, we can
produce topological operators of QFTd by laying the bulk operators on the
topological boundary. Therefore the symmetry defects that generate the
symmetry of the boundary theory are the operators of the Symmetry TFT modulo
$\mathscr{L}$.

We propose that the Symmetry TFT for continuous $U(1)$ symmetries, either
0-form or higher $p$-form, is a BF theory of gauge fields for gauge group
$\mathbb{R}$, as opposed to $U(1)$. To explain this point, we will first
review the ordinary BF theory description of $\mathbb{Z}_{N}$ gauge theory.

$\boldsymbol{U(1)}$ gauge fields. The $\mathbb{Z}_{N}$ gauge theory in $d+1$
dimensions can be formulated as a BF theory of standard $U(1)$ gauge fields
using the action [26, 27, 28]

$\mathcal{Z}=\frac{N}{2\pi}\int_{X_{d+1}}B_{d-p}\wedge dA_{p}\,.$ (1)

The gauge fields are not globally-defined forms: they are patched using $U(1)$
gauge transformations $\delta A_{p}=d\lambda_{p-1}$ and $\delta
B_{d-p}=d\lambda_{d-p-1}$. This leads to the Dirac quantization condition

$\frac{1}{2\pi}\int_{\gamma_{p+1}}dA_{p}\in\mathbb{Z}\,,\qquad\frac{1}{2\pi}\int_{\gamma_{d-p+1}}dB_{d-p}\in\mathbb{Z}\,,$
(2)

where $\gamma_{j}$ are closed $j$-cycles. We can define extended operators

$U_{\alpha}[\gamma_{d-p}]=e^{i\alpha\int_{\gamma_{d-p}}B_{d-p}}\,,\qquad
W_{\beta}[\gamma_{p}]=e^{i\beta\int_{\gamma_{p}}A_{p}}\,,$ (3)

and invariance under large gauge transformations requires
$\alpha,\beta\in\mathbb{Z}$.

In order to obtain the EOMs, it is convenient to decompose each gauge field
into a representative of nontrivial $U(1)$ bundles, over which one has to sum,
and a globally-defined form describing fluctuations within a given bundle.
Variations with respect to the fluctuations give $dA_{p}=0$ and $dB_{d-p}=0$,
which guarantee that the operators (3) are topological. The sum over bundles
produces delta functions imposing

$\frac{N}{2\pi}\int_{\gamma_{p}}\\!A_{p}\in\mathbb{Z}\,,\qquad\frac{N}{2\pi}\int_{\gamma_{d-p}}\\!B_{d-p}\in\mathbb{Z}\,.$
(4)

This implies that the operators (3) with $\alpha\to\alpha+N$ or
$\beta\to\beta+N$ are equivalent. Hence the theory has the following
nonequivalent topological operators 333The theory might contain other
topological operators, for instance condensates [52] or theta-defects [53,
*Bhardwaj:2022kot].:

$U_{n}[\gamma_{d-p}]\,,\qquad W_{m}[\gamma_{p}]\,,\qquad\text{with
}n,m\in\mathbb{Z}_{N}\,.$ (5)

The braiding between these operators can be computed by inserting one operator
in (1) as a source, and then evaluating the VEV of the other one with the
EOMs. The result for the braiding is

$\mathcal{B}(\alpha,\beta)=\exp\bigl{(}\tfrac{2\pi
i}{N}\,\alpha\beta\bigr{)}\;.$ (6)

This is the phase picked up by correlation functions when an operator
$U_{\alpha}$ is moved across an operator $W_{\beta}$. The data (5) and (6)
characterizes the $\mathbb{Z}_{N}$ gauge theory.

There are various topological boundary conditions.

  * •

A natural boundary condition is that the operators $W_{m}$ (with
$m\in\mathbb{Z}_{N}$) can terminate on the boundary. The defects $U_{n}$ with
$n\in\mathbb{Z}_{N}$ can lie on the boundary and play the role of symmetry
defects for a

$\text{{\hbox{\mathbb{Z}_N}} {\hbox{(p \,{-}1)}}-form symmetry}\,.$ (7)

We can express the condition as a Dirichlet boundary condition for $A_{p}$
which then plays the role of the background field for the $(p-1)$-form
symmetry. Anomalies are obtained by adding topological terms to $\mathcal{Z}$
written in terms of $A_{p}$. These terms affect the EOMs and in general change
the list of topological operators, the braiding, the possible boundary
conditions, etc.

  * •

Another boundary condition is to let the defects $U_{n}$ (with
$n\in\mathbb{Z}_{N}$) terminate on the boundary. This gives

Z_N (d-p-1)-form symmetry (8)

on the boundary. Indeed, the two symmetries are related by the discrete
gauging of $\mathbb{Z}_{N}$ in the boundary theory.

Other boundary conditions might exist, for instance when $N$ has divisors, or
when $p=d-p$.

$\boldsymbol{\mathbb{R}}$ gauge fields. In [30] it was found that the theory
of a 2d free compact scalar — that has $U(1)^{2}$ global symmetry — admits a
dual holographic description in terms of a certain 3d Chern-Simons theory of
two gauge fields for group $\mathbb{R}$, as opposed to $U(1)$. This suggests
to study the Abelian BF theory with action

$\mathcal{Z}=\frac{1}{2\pi}\int_{X_{d+1}}b_{d-p}\wedge da_{p}\,,$ (9)

where $a_{p}$ and $b_{d-p}$ are gauge fields for the group $\mathbb{R}$. This
means that they are globally-defined forms, subject to small but not large
gauge transformations. $\mathbb{R}$-bundles are necessarily trivial, namely
the Dirac quantization conditions collapse to

$\int_{\gamma_{p+1}}da_{p}=\int_{\gamma_{d-p+1}}db_{d-p}=0\,,$ (10)

which is Stokes’ theorem. We can always rescale $a_{p}$ or $b_{d-p}$ by a real
constant, so that the overall coefficient in $\mathcal{Z}$ is unphysical and
we have fixed it to $1$.

The EOMs simply set $da_{p}=0$ and $db_{d-p}=0$, so that the operators
$U_{\alpha}$ and $W_{\beta}$ are topological. Since there are no large gauge
transformations, those operators are gauge invariant with no restrictions on
$\alpha$ and $\beta$. Since there are no nontrivial bundles to sum over, there
are no restrictions on the holonomies. We conclude that the theory has the
following topological operators:

$U_{\alpha}[\gamma_{d-p}]\,,\qquad W_{\beta}[\gamma_{p}]\,,\qquad\text{with
}\alpha,\beta\in\mathbb{R}\,.$ (11)

The braiding is as in (6) but with $N=1$. Various topological boundary
conditions are possible.

  * •

Let all defects $W_{\beta}[\gamma_{p}]$ terminate on the boundary. They
represent charged operators along $\partial\gamma_{p}$ with generic charges
$\beta\in\mathbb{R}$. All defects $U_{\alpha}[\gamma_{d-p}]$ can lie on the
boundary and play the role of the symmetry defects for an

$\text{{\hbox{\mathbb{R}}} {\hbox{(p\,{-}1)}}-form symmetry}\,.$ (12)

Such a symmetry indeed allows for generic charges. An example of a theory with
this symmetry is a free $\mathbb{R}$ $(p-1)$-form gauge field (for $p=1$ this
is a free noncompact scalar).

  * •

Similarly, let all defects $U_{\alpha}[\gamma_{d-p}]$ terminate on the
boundary. This describes an

R (d-p-1)-form symmetry (13)

on the boundary. It is obtained from (12) by gauging the whole $\mathbb{R}$ on
the boundary (the Pontryagin dual to $\mathbb{R}$ is $\mathbb{R}$). This
gauging, in order to be topological, has to be done with the _discrete
topology_ on $\mathbb{R}$. This means that the field strength of the boundary
gauge field is identically zero and we only sum over flat connections.

  * •

Let the defects $U_{n}$ and $W_{m}$ with $n,m\in\mathbb{Z}$ terminate on the
boundary. From (6), this choice constitutes a maximal set of mutually-
transparent operators and is thus a Lagrangian algebra. The coset classes of
operators that can lie on the boundary are given by $U_{\alpha}$ and
$W_{\beta}$ with $\alpha,\beta\in\mathbb{R}/\mathbb{Z}=U(1)$. Thus there are
two factors, such that charged operators have integer charges while defects
are valued in $U(1)$. This describes a

$\text{{\hbox{U(1)^{(p-1)} \times U(1)^{(d-p-1)}}} symmetry}\,.$ (14)

The two factors have a mixed anomaly. This is obtained from (12) by gauging a
$\mathbb{Z}$ subgroup of $\mathbb{R}$ ($U(1)$ is the Pontryagin dual to
$\mathbb{Z}$) and the mixed anomaly follows from the fact that the exact
sequence $0\to\mathbb{Z}\to\mathbb{R}\to U(1)\to 0$ does not split [31]. An
example of a theory with this symmetry is a free $U(1)$ $(p-1)$-form gauge
field (for $p=1$ this is a free compact scalar, for $p=2$ a free photon), dual
to a free $U(1)$ $(d-p-1)$-form gauge field. We refer to these theories as
generalized Maxwell theories.

In the last case, one could more generally consider the defects $U_{nR}$ and
$W_{mR^{-1}}$ for any real constant $R$. This would amount to rescale the
radii of the two $U(1)$ factors. The Symmetry TFT does not determine the
actual value of the radius, but can compare the radii arising in two different
topological boundaries. For instance, for $p=1$ we get a compact boson whose
radius is $R$ times larger than the one in case (14). Similarly, for $p=2$ the
choice of $R$ corresponds to a rescaling of the electric charge in Maxwell’s
theory. The special case of a rescaling by $R=N\in\mathbb{Z}$ also corresponds
to gauging a $\mathbb{Z}_{N}$ subgroup of $U(1)^{(d-p-1)}$. The cases (12) and
(13) with symmetry $\mathbb{R}$ correspond to the decompactification limits of
the original field or of its dual, respectively.

$\boldsymbol{U(1)/\mathbb{R}}$ gauge fields. Lastly, consider the case

$\mathcal{Z}=\frac{1}{2\pi}\int_{X_{d+1}}\\!b_{d-p}\wedge dA_{p}$ (15)

where $A_{p}$ and $b_{d-p}$ are $U(1)$ and $\mathbb{R}$ gauge fields,
respectively. As before, the overall coefficient of $\mathcal{Z}$ can always
be set to $1$. This time Dirac’s quantization conditions read
$\frac{1}{2\pi}\\!\int\\!dA_{p}\in\mathbb{Z}$ and $\int\\!db_{d-p}=0$. By the
same arguments as before, one concludes that the theory has the following
nonequivalent topological operators:

$U_{\alpha}[\gamma_{d-p}]\text{ with
}\alpha\in\mathbb{R}/\mathbb{Z}=U(1)\,,\quad W_{m}[\gamma_{p}]\text{ with
}m\in\mathbb{Z}\,.$ (16)

Some interesting boundary conditions are the following.

  * •

Let the defects $W_{m}$ (with $m\in\mathbb{Z}$) terminate on the boundary.
Then the defects $U_{\alpha}$ with $\alpha\in U(1)$ can lie on the boundary
and represent a

$\text{{\hbox{U(1)}} {\hbox{(p\,{-}1)}}-form symmetry}\,.$ (17)

  * •

Let all defects $U_{\alpha}$ with $\alpha\in U(1)$ terminate on the boundary.
Then the defects $W_{m}$ with $m\in\mathbb{Z}$ can lie on the boundary and
represent a

$\text{{\hbox{\mathbb{Z}}} {\hbox{(d{-}p{-}1)}}-form symmetry}\,.$ (18)

This is obtained from (17) by gauging the whole $U(1)$ with discrete topology.
For $p=d-1$, an example of a theory with 0-form symmetry $\mathbb{Z}$ is a
scalar field $\phi$ with periodic potential, as in band theory.

  * •

Let the defects $W_{bm}$ with $m\in\mathbb{Z}$ and $b>1$ an integer constant,
as well as $U_{n/b}$ with $n\in\mathbb{Z}_{b}$, terminate on the boundary.
Then the nonequivalent defects that can lie on the boundary are $U_{\alpha}$
with $\alpha\in\mathbb{R}/\bigl{(}\frac{1}{b}\mathbb{Z}\bigr{)}\cong U(1)$ and
$W_{k}$ with $k\in\mathbb{Z}_{b}$. They represent a

$\text{{\hbox{U(1)^{(p-1)} \times\mathbb{Z}_b^{(d-p-1)}}} symmetry}\,.$ (19)

This is obtained from (17) by gauging a $\mathbb{Z}_{b}$ subgroup of $U(1)$.
Indeed the charged operators have integer charges that are multiples of $b$.
There is a mixed anomaly between $U(1)$ and $\mathbb{Z}_{b}$ that follows from
the short exact sequence $0\to\mathbb{Z}_{b}\to U(1)\,\to U(1)\to 0$.

As a check, consider the case (14) that we derived from the Symmetry TFT (9)
using only $\mathbb{R}$ gauge fields. It should also arise from the Symmetry
TFT of two $U(1)$ symmetries with a mixed anomaly:

$\mathcal{Z}=\frac{1}{2\pi}\\!\int\\!\Bigl{[}b_{d-p}\wedge dB_{p}+a_{p}\wedge
dA_{d-p}-B_{p}\wedge dA_{d-p}\Bigr{]}.$ (20)

Indeed the field $A_{d-p}$ can be integrated out producing a delta function
that enforces $B_{p}=a_{p}$ (restricting $B_{p}$ to be an $\mathbb{R}$ gauge
field), and integrating out $B_{p}$ one reproduces the action (9).
Alternatively, and more precisely, one can list the topological operators and
compute their correlations functions. One realizes that
$e^{in\\!\int\\!B_{p}}$ has identical correlation functions to those of
$e^{i\alpha\\!\int\\!a_{p}}$ for $\alpha=n$, hence the two operators are
identified. Similarly for $e^{im\\!\int\\!A_{d-p}}$ and
$e^{i\beta\\!\int\\!b_{d-p}}$ for $\beta=m$. The alternative presentation (20)
of the Symmetry TFT for a generalized Maxwell field will be important in
Section IV.

##  III Examples

Let us present a few interesting examples. Other examples, which require us to
understand how to dynamically gauge a $U(1)$ symmetry from the point of view
of the Symmetry TFT, are described in Section V.

###  III.1 Chiral anomaly in 2d

The Symmetry TFT for a two-dimensional theory with $U(1)$ 0-form symmetry and
an ’t Hooft anomaly is obtained from the action (15) with $d=2$ and $p=1$ by
adding a term that describes the chiral anomaly:

$\mathcal{Z}=\int_{X_{3}}\biggl{[}\frac{1}{2\pi}\,b_{1}\wedge
dA_{1}+\frac{k}{4\pi}\,A_{1}\wedge dA_{1}\biggr{]}\,,$ (21)

where $k$ is constrained to be integer (when $k$ is odd the theory requires a
spin structure [32]).

The theory has topological line operators given by

$U_{(n,\alpha)}[\gamma_{1}]=e^{i\\!\int_{\gamma_{1}}\\!(nA_{1}+\alpha\,b_{1})}$
(22)

with $n\in\mathbb{Z}$ and $\alpha\in\mathbb{R}$. The braiding between them is

$\mathcal{B}\bigl{[}(n_{1},\alpha_{1}),(n_{2},\alpha_{2})\bigr{]}=\exp\Bigl{[}2\pi
i\bigl{(}n_{1}\alpha_{2}+n_{2}\alpha_{1}-k\alpha_{1}\alpha_{2}\bigr{)}\Bigr{]}$
(23)

and the exponentiated spin is given by a quadratic refinement thereof:

$\theta_{(n,\alpha)}=\exp\Bigl{[}2\pi
i\,\alpha\bigl{(}n-\tfrac{k}{2}\,\alpha\bigr{)}\Bigr{]}\,.$ (24)

The line $(0,1)$ has spin $\theta=(-1)^{k}$ and is a transparent fermion for
$k$ odd. Both spin and braiding are invariant under the following
identifications:

$\begin{cases}(n,\alpha)\,\sim\,(n+k,\,\alpha+1)&\text{for $k$ even,}\\\
(n,\alpha)\,\sim\,(n+2k,\,\alpha+2)&\text{for $k$ odd.}\end{cases}$ (25)

Let us discuss boundary conditions. First, we can let all lines $(n,0)$
terminate on the boundary. For $k$ even, this is a maximal set $\mathscr{L}$
of mutually-transparent lines and they are all bosonic. For $k$ odd, we should
also let the lines $(n,1)$ terminate on the boundary in order to have a
maximal set $\mathscr{L}$, and the extra lines have spin $-1$. Thus, the
Lagrangian algebra $\mathscr{L}$ is bosonic for $k$ even, and spin for $k$
odd. The nonequivalent topological line operators that can lie on the boundary
are labeled by $\alpha\in\mathbb{R}/\mathbb{Z}\cong U(1)$, so this describes a
$U(1)$ 0-form symmetry.

Let us use the lines $(0,\alpha)$ as representatives of the symmetry defect
operators of the boundary theory. For $k$ even, the bosonic lines $(n,0)$ that
end on the 2d boundary represent boundary local operators, and their charge
measured by the braiding (23) is $n$. For $k$ odd, the lines $(n,0)$ and
$(n+k,1)$ that end on the boundary represent local operators with charge $n$
and spin $\pm 1$, respectively. Thus the local operators can have arbitrary
and independent integer charges and statistics.

It is possible to specialize to theories with a spin-charge relation, in which
even-charge operators are bosonic and odd-charge operators are fermionic
(e.g., a free complex Weyl fermion). In this case the Symmetry TFT (21) is
written in terms of a spinc connection $A_{1}$ [33, 34] whose Dirac
quantization condition on 2-cycles is modified according to the second
Stiefel-Whitney class of the manifold:
$\frac{1}{2\pi}dA_{1}=\frac{1}{2}w_{2}\text{ mod }1$ (one also needs to add a
gravitational term to the action). Since $A_{1}$ is not an ordinary connection
but $2A_{1}$ is, gauge invariance restricts the operators $U_{(m,\alpha)}$ to
have even $m$. This in turn implies the spin-charge relation for the boundary
local operators: the endpoints of $(n,0)$ are bosonic for $n$ even, and the
endpoints of $(n+k,1)$ are fermionic for $n$ odd.

When $k\neq 0$, the operators $U_{(0,\alpha)}$ are no longer mutually
transparent and therefore there is no boundary condition corresponding to
(18). This is a manifestation of the chiral anomaly: the $U(1)^{(0)}$ symmetry
of the boundary theory cannot be gauged. Indeed the anomaly of a continuous
group is uniquely determined by the anomaly of all its discrete subgroups
[35]. Therefore, a $U(1)$ symmetry has a perturbative anomaly if and only if
it is anomalous when endowed with the discrete topology.

However, it might still be possible to gauge a discrete subgroup of $U(1)$.
Consider the bosonic case of $k$ even. Given an integer $d$ that divides
$k/2$, consider the set $\mathscr{L}$ of line operators
$\bigl{(}\frac{k}{2d}m+d\,\ell,\frac{m}{d}\bigr{)}$ with $\ell,m\in\mathbb{Z}$
(modulo identifications). Such operators have spin $\theta=1$ and are thus
bosons. From (23), a line that has trivial braiding with all elements of
$\mathscr{L}$ must be in $\mathscr{L}$, thus $\mathscr{L}$ is maximal and is a
Lagrangian algebra. This shows that when $k/2$ is divisible by $d$, the
$\mathbb{Z}_{d}$ subgroup of $U(1)$ is anomaly free and can be gauged in the
boundary theory. The nonequivalent topological lines that can lie on the
boundary are $U_{(n,\alpha)}$ with $n\in\mathbb{Z}_{d}$ and
$\alpha\in\mathbb{R}/\bigl{(}\frac{1}{d}\mathbb{Z}\bigr{)}$. They represent a
symmetry
$U(1)\raisebox{0.0pt}[0.0pt]{${}^{(0)}$}\times\mathbb{Z}_{d}\raisebox{0.0pt}[0.0pt]{${}^{(0)}$}$
with an ’t Hooft anomaly for $U(1)$ and a mixed anomaly between $U(1)$ and
$\mathbb{Z}_{d}$.

As a check, we can simply restrict to $\mathbb{Z}_{d}$ the anomaly-inflow
action $\mathcal{Z}_{\text{inflow}}=\frac{k}{4\pi}\\!\int\\!A\wedge dA$, where
$A$ is seen as an extension from 2d to 3d of the background gauge field for
the $U(1)$ symmetry [36]. This is achieved by replacing

$A\,\mapsto\,\frac{2\pi}{d}\,\mathcal{A}\,,\qquad\quad\frac{dA}{2\pi}\,\mapsto\,\beta(\mathcal{A})\,,$
(26)

where $\mathcal{A}\in H^{1}(X;\mathbb{Z}_{d})$ is (an extension of) the
background field for $\mathbb{Z}_{d}$, while $\beta:H^{1}(X;\mathbb{Z}_{d})\to
H^{2}(X;\mathbb{Z})$ is the Bockstein homomorphism associated to the exact
sequence $0\to\mathbb{Z}\,\to\mathbb{Z}\to\mathbb{Z}_{d}\to 0$. The inflow
action reduces to
$2\pi\frac{k}{2d}\\!\int\\!\mathcal{A}\cup\beta(\mathcal{A})$, where the
integral is an integer modulo $d$, which is indeed an integer multiple of
$2\pi$ whenever $d$ divides $k/2$ and thus $\mathbb{Z}_{d}$ is anomaly free.
The converse, to determine which $\mathbb{Z}_{d}$ subgroups are actually
anomalous, is a delicate issue [37, 38]: one should determine whether the
reduced anomaly-inflow action is trivial when evaluated on the generator(s) of
the relevant bordism group $\Omega^{SO}_{3}(B\mathbb{Z}_{d})$.

In the fermionic case of $k$ odd one can perform a similar analysis. Given $d$
that divides $k$ (in particular $d$ is odd), the set $\mathscr{L}$ of line
operators $\bigl{(}\frac{k}{d}\frac{d+1}{2}m+d\,\ell,\frac{m}{d}\bigr{)}$
labelled by $\ell,m\in\mathbb{Z}$ (here $\frac{d+1}{2}=2^{-1}$ mod $d$) is a
maximal set of mutually-transparent lines with spins $\theta=(-1)^{m}=\pm 1$,
suggesting that the $\mathbb{Z}_{d}$ subgroup can be gauged. This should be
compared with the evaluation of the reduced inflow action on the generator(s)
of the bordism group $\Omega^{\text{spin}}_{3}(B\mathbb{Z}_{d})$.

We ask whether there can be other more exotic topological boundary conditions.
To be concrete, take $k=1$ and consider the following set:

$\mathcal{C}=\bigl{\\{}\bigl{(}n,n+\ell\sqrt{2}\bigr{)}\bigm{|}n,\ell\in\mathbb{Z}\bigr{\\}}\,.$
(27)

The set is closed under sum. Moreover, mapping the second component to the
interval $[0,2)$ (by shifting also the first component), we obtain a dense
irrational subset. The operators in $\mathcal{C}$ have $\theta=(-1)^{n}$ and
are all mutually transparent, however the set is not maximal (hence it does
not describe a topological boundary condition) and in fact it cannot be made
into a maximal set. A line mutually transparent with $\mathcal{C}$ is of the
form $\bigl{(}m,m+\frac{h}{\sqrt{2}}\bigr{)}$ for some $m,h\in\mathbb{Z}$. For
$h$ odd, these lines are not contained in $\mathcal{C}$, however they cannot
be included in $\mathcal{C}$ because they have nontrivial spin
$\theta=i\,(-1)^{m+1}$. We have thus found an example of an Abelian algebra
that cannot be completed into a Lagrangian algebra. Interestingly, this is a
peculiarity of TQFTs with a continuum of lines, as can be intuitively
understood from the necessity of taking the square root. Indeed a finite semi-
simple modular tensor category admitting Lagrangian algebras is a Drinfeld
center, and the condensation of an algebra of a Drinfeld center yields another
Drinfeld center [39]. As we showed, this result is no longer true in the
presence of a continuum of lines.

###  III.2 $\boldsymbol{U(1)}$ symmetry in 3d

In odd dimensions there are no anomalies for $U(1)$ symmetries. Nevertheless
there are useful pieces of information encoded in the symmetry TFT. Here we
consider the 3d case, hence the basic 4d Symmetry TFT describing a $U(1)$
symmetry is (15) with $d=3$, $p=1$. There are two more terms that can be
added: a theta-term $\frac{\theta}{8\pi^{2}}\\!\int_{X_{4}}\\!dA_{1}\wedge
dA_{1}$ and a phi-term $\frac{\phi}{8\pi^{2}}\\!\int_{X_{4}}\\!b_{2}\wedge
b_{2}$. In the absence of the latter, the theta-term is unphysical since it
can be reabsorbed by a shift of $b_{2}$. Here we study the effect of the phi-
term, while we leave the analysis of the most general action with both terms
for the future.

The 4d TQFT we consider is

$\mathcal{Z}=\frac{1}{2\pi}\int_{X_{4}}b_{2}\wedge
dA_{1}+\frac{\phi}{8\pi^{2}}\int_{X_{4}}b_{2}\wedge b_{2}\,.$ (28)

This can be thought of as the non-compact version of the theory studied in
[28]. The gauge transformations are

$\delta b_{2}=d\lambda_{1}\,,\qquad\delta
A_{1}=d\rho_{0}-\tfrac{\phi}{2\pi}\,\lambda_{1}\,.$ (29)

The gauge-invariant operators are the surfaces

$U_{\alpha}[\gamma_{2}]=e^{i\alpha\\!\int_{\gamma_{2}}b_{2}}$ (30)

and the (generically) non-genuine lines

$W_{n}[\gamma_{1},D_{2}]=\exp\biggl{(}in\int_{\gamma_{1}}A_{1}+i\,\frac{n\phi}{2\pi}\int_{D_{2}}b_{2}\biggr{)},$
(31)

both topological. Here $n\in\mathbb{Z}$ whilst $D_{2}$ is a disk with
$\partial D_{2}=\gamma_{1}$. A non-trivial correlator on the sphere is the
braiding of $U_{\alpha}[\gamma_{2}]$ and $W_{n}[\gamma_{1},D_{2}]$ in a
configuration in which $\gamma_{1}$ and $\gamma_{2}$ link, hence $\gamma_{2}$
intersects $D_{2}$ at a point:

$\bigl{\langle}U_{\alpha}[\gamma_{2}]\,W_{n}[\gamma_{1},D_{2}]\bigr{\rangle}=e^{2\pi
i\alpha n}\,.$ (32)

We read off that $\alpha\sim\alpha+1$. This implies that if
$\frac{n\phi}{2\pi}$ is an integer in (31), there is no dependence of $W_{n}$
on the disk $D_{2}$: choosing a different disk would give an operator which
differs by $U_{\alpha}[\gamma_{2}]$ for some integer $\alpha$ and where
$\gamma_{2}$ is the union (with opposite orientations) of the two disks, hence
that difference is trivial. In particular we read off that $\phi$ is a
periodic parameter:

$\phi\sim\phi+2\pi\,.$ (33)

An interesting observation is that, while for irrational values of
$\frac{\phi}{2\pi}$ the theory has no genuine lines, hence it is essentially
trivial in the bulk, when $\phi=2\pi p/q$ with $p,q\in\mathbb{Z}$ and
$\gcd(p,q)=1$, the lines $W_{mq}$ are genuine.

Let us study topological boundary conditions. First, we can let all surfaces
$U_{\alpha}$ terminate on the boundary. This corresponds to a Dirichlet
boundary condition for $b_{2}$. Since the gauge transformations of $b_{2}$ are
forced to vanish at the boundary, there we can construct the genuine line
operators $\mathcal{W}_{n}=e^{in\\!\int\\!A_{1}}$ (equivalently, since $b_{2}$
is a background field at the boundary, the dependence of $W_{n}$ on it can be
removed by a counterterm). This boundary condition thus describes a
$\mathbb{Z}$ 1-form symmetry, whose topological operators are the lines
$\mathcal{W}_{n}$ and whose charged operators are the endlines of $U_{\alpha}$
on the boundary. The phi-term represents an anomaly for this 1-form symmetry,
and if $\frac{\phi}{2\pi}$ is irrational there are no non-anomalous subgroups
of $\mathbb{Z}$. Indeed in this case the bulk does not have other genuine
topological operators besides $U_{\alpha}$, hence there are no other
topological boundary conditions. In particular, there is no boundary with a
$U(1)$ symmetry.

On the contrary, consider the case $\frac{\phi}{2\pi}=p/q\in\mathbb{Q}$. The
lines $W_{qm}$ with $m\in\mathbb{Z}$ are genuine, as we noticed, and we can
let them terminate on the boundary. In order to have a maximal set of
mutually-transparent objects, we should let the surfaces $U_{\alpha}$ with
$\alpha=l/q$ and $l\in\mathbb{Z}$ terminate on the boundary as well. This
boundary condition describes the symmetry $\mathbb{Z}_{q}{}^{(1)}\times
U(1)^{(0)}$ with a mixed anomaly and a pure anomaly for
$\mathbb{Z}_{q}{}^{(1)}$, and is obtained from the previous one by gauging the
subgroup $q\mathbb{Z}\subset\mathbb{Z}$ of the 1-form symmetry, which is non-
anomalous in this case. The symmetry $U(1)^{(0)}$ is generated by $U_{\alpha}$
with identification $\alpha\sim\alpha+1/q$ (the reduced range is due to the
boundary condition). The local operators $\mathcal{M}_{m}$ charged under
$U(1)^{(0)}$ are the endpoints of the genuine lines $W_{qm}$.

To get some intuition on the nature of the QFTs described by this boundary
condition, let us present a Lagrangian example. On a manifold with boundary,
the variation of the action (28) under a gauge transformation (29) generates a
boundary term:

$\delta\mathcal{Z}=\frac{1}{2\pi}\\!\int_{\partial
X_{4}}\biggl{[}\lambda_{1}\wedge dA_{1}-\frac{\phi}{4\pi}\,\lambda_{1}\wedge
d\lambda_{1}\biggr{]}$ (34)

where the boundary value of $A_{1}$ is fixed. This can be cancelled by edge
modes. Since the local operators $\mathcal{M}_{m}$ charged under $U(1)^{(0)}$
are the endpoints of $W_{qm}$, the corresponding background field is $qA_{1}$
at the boundary. Since the line operators charged under
$\mathbb{Z}_{q}{}^{(1)}$ are the endlines of $U_{l/q}$, the background field
is $b_{2}/q$. At the boundary we can place the Chern-Simons theory
$U(1)_{pq}$:

$S_{\partial}=-\int_{\partial
X_{4}}\biggl{[}\frac{pq}{4\pi}\,\mathcal{B}\wedge
d\mathcal{B}+\frac{1}{2\pi}(qA_{1})\wedge d\mathcal{B}\biggr{]}\,.$ (35)

This theory has 1-form symmetry
$\mathbb{Z}_{pq}=\mathbb{Z}_{p}\times\mathbb{Z}_{q}$, but we only consider the
$\mathbb{Z}_{q}$ subgroup which has anomaly $p$ [40]. The generator of
$\mathbb{Z}_{q}$ acts as $\delta\mathcal{B}=\frac{1}{q}\lambda_{1}$ so that
the variation of $S_{\partial}$ cancels (34). The Chern-Simons theory also has
a magnetic $U(1)$ 0-form symmetry with current $*d\mathcal{B}$, which is
coupled to the background field $(qA_{1})$. The local operators
$\mathcal{M}_{m}$ are monopoles. In this example the current is trivial at
separated points, but it still has an interesting contact term [41] (to make
the current nontrivial at separated points, one could consider Maxwell-Chern-
Simons theory instead). Because of the 4d EOM, we can identify
$dA_{1}=-p\bigl{(}\frac{1}{q}b_{2}\bigr{)}$ with the background for the 1-form
symmetry, hence the theory is coupled to the latter as well, in a subtle way.

Another possibility is to use the topological theory
$\mathcal{A}^{q,p}\bigl{[}\frac{1}{q}dA_{1}\bigr{]}$ [40] as boundary theory.
Because of the identifications among those theories, this in particular shows
that $p\sim p+q$ in accord with (33).

###  III.3 Chiral anomaly in 4d

For a four-dimensional theory with a $U(1)$ 0-form symmetry and an ’t Hooft
anomaly, the Symmetry TFT is

$\mathcal{Z}=\int_{X_{5}}\biggl{[}\frac{1}{2\pi}\,b_{3}\wedge
dA_{1}+\frac{k}{24\pi^{2}}\,A_{1}\wedge dA_{1}\wedge dA_{1}\biggr{]}$ (36)

where $A_{1}$ is a $U(1)$ gauge field while $b_{3}$ is an $\mathbb{R}$ gauge
field. The parameter $k$ is an integer in fermionic theories, while in general
bosonic theories it should be a multiple of 6.

The theory has topological line and surface operators

$\displaystyle W_{n}[\gamma_{1}]$
$\displaystyle=e^{in\\!\int_{\gamma_{1}}\\!A_{1}}\,,$ (37) $\displaystyle
U_{(\beta,m)}[\gamma_{3}]$
$\displaystyle=\exp\biggl{[}i\\!\int_{\gamma_{3}}\\!\Bigl{(}\beta\,b_{3}+\frac{m}{4\pi}\,A_{1}\wedge
dA_{1}\Bigr{)}\biggr{]}$

with $n,m\in\mathbb{Z}$. The quantization of $m$ corresponds to spin-Chern-
Simons theories. The perturbative EOMs $dA_{1}=db_{3}=0$ guarantee that these
operators are topological. The observables of the theory include the linking
of $W_{n}$ with $U_{(\beta,m)}$, and the triple-linking between three
operators $U_{(\beta_{i},m_{i})}$ on surfaces $\gamma_{3}{}^{(i)}$. The latter
probes the linking between the intersection
$\widetilde{\gamma}_{1}=\gamma_{3}{}^{(1)}\cap\gamma_{3}{}^{(2)}$ and
$\gamma_{3}{}^{(3)}$ (one can show that this is symmetric in the three
surfaces). The braiding can be defined as the following expectation value on
the sphere:

$\bigl{\langle}W_{n}[\gamma_{1}]\,U_{(\beta,m)}[\gamma_{3}]\bigr{\rangle}=\exp\bigl{(}2\pi
i\,n\beta\operatorname{Lk}(\gamma_{1},\gamma_{3})\bigr{)}$ (38)

where $\operatorname{Lk}$ is the geometric linking number. Alternatively, it
is the phase picked up by generic correlation functions when $W_{n}$ is moved
across $U_{(\beta,m)}$. The triple-linking number on the sphere is the
following expectation value:

$\displaystyle\bigl{\langle}U_{(\beta_{1},m_{1})}\bigl{[}\gamma_{3}^{(1)}\bigr{]}\,U_{(\beta_{2},m_{2})}\bigl{[}\gamma_{3}^{(2)}\bigr{]}\,U_{(\beta_{3},m_{3})}\bigl{[}\gamma_{3}^{(3)}\bigr{]}\bigr{\rangle}$
(39) $\displaystyle=\exp\bigl{[}2\pi
i\bigl{(}m_{1}\beta_{2}\beta_{3}+m_{2}\beta_{1}\beta_{3}+m_{3}\beta_{1}\beta_{2}-k\beta_{1}\beta_{2}\beta_{3}\bigr{)}\L\bigr{]}$

where
$\L=\operatorname{Lk}\bigl{(}\widetilde{\gamma}_{1},\gamma_{3}{}^{(3)}\bigr{)}$.
Note that, differently from the 2d case, there is no operator $U_{(\beta,m)}$
with trivial triple-linking with all other pairs of operators. Hence there are
no identifications of labels in this case.

Using (38) and (39) we can look for topological boundary conditions
corresponding to the condensation of a “Lagrangian algebra”. By this we mean a
set of line and surface operators which are: (i) closed under fusion, (ii)
mutually transparent with respect to both braiding and triple linking, and
(iii) maximal in the sense that any operator transparent with the set belongs
itself to the set. These conditions guarantee that the Lagrangian algebra can
be condensed, and the result is the trivial TQFT. We find the following
possibilities.

First, the lines $W_{n}$ and the surfaces $U_{(\beta=j,m)}$ with
$n,j,m\in\mathbb{Z}$ are mutually transparent and maximal. Condensing all of
them we obtain the boundary condition for a theory with $U(1)$ 0-form
symmetry. Its nonequivalent topological symmetry operators are
$U_{(\beta,0)}[\gamma_{3}]$ where $\beta\in\mathbb{R}/\mathbb{Z}$ with
$\beta\sim\beta+1$ because of the condensation.

Second, if $k=0$, another Lagrangian algebra is given by all the operators
$U_{(\beta,0)}$. This correspond to a 2-form symmetry $\mathbb{Z}$ whose
topological symmetry operators are the lines $W_{n}$ with $n\in\mathbb{Z}$, as
in the general case (18) 444The boundary theory also inherits the topological
surfaces $U_{(0,m)}[\gamma_{3}]$ which are theta-defects [53,
*Bhardwaj:2022kot] constructed out of the lines $W_{n}[\gamma_{1}]$.. Such a
boundary condition is obtained from the previous one by gauging the $U(1)$
symmetry (with discrete topology) on the boundary. If $k\neq 0$, however, this
algebra does not exist, consistently with the statement that the Symmetry TFT
describes an anomalous symmetry.

Lastly, given an integer $d$ such that $3d$ divides $k$, one can construct a
Lagrangian algebra made of

$W_{d\rho}\,,\quad U_{(\beta,m)}\text{ with
}(\beta,m)=\biggl{(}\frac{\nu}{d},\,\frac{k}{3d}\,\nu+d^{2}\mu\biggr{)}$ (40)

and labelled by $\rho,\nu,\mu\in\mathbb{Z}$. The endpoints of the lines are
the charged objects. Since the charges are multiples of $d$, we conclude that
this boundary corresponds to gauging the non-anomalous $\mathbb{Z}_{d}$
subgroup of $U(1)^{(0)}$. The symmetry on this boundary is
$U(1)^{(0)}\times\mathbb{Z}_{\raisebox{0.0pt}[0.0pt][0.0pt]{\scriptsize$d$}}^{(2)}$
where the first factor is the $\mathbb{Z}_{d}$ quotient of the original
symmetry, while the second factor is the dual 2-form symmetry generated by the
lines $W_{p}$ with $p\in\mathbb{Z}_{d}$.

##  IV Non-topological manipulations

The Symmetry TFT for $U(1)$ symmetries that we discussed is the
straightforward generalization of the discrete case. Its topological
boundaries correspond to _topological_ manipulations that use the $U(1)$
symmetry. These include gauging discrete subgroups, possibly with discrete
torsion, as well as gauging the whole $U(1)$ but understood with the discrete
topology. However, differently from the discrete ones, continuous $U(1)$
symmetries also allow for non-topological, _dynamical_ manipulations — such as
coupling to a dynamical photon — that introduce new degrees of freedom. For
instance, a 4d free complex scalar field has a $U(1)^{(0)}$ symmetry and its
Symmetry TFT is (15) with $p=1$, $d=4$. By gauging dynamically the $U(1)$ we
obtain scalar QED which has a $U(1)^{(1)}$ symmetry, hence its Symmetry TFT is
again (15) but with $p=2$, $d=4$. We see that dynamical manipulations are not
described by different topological boundary conditions of the same Symmetry
TFT, but rather by a map between two different Symmetry TFTs. As we will
argue, this is a controlled operation.

###  IV.1 Gauging dynamically a $\boldsymbol{U(1)}$

For concreteness we are going to focus on 0-form symmetries, but the
generalization to higher forms is straightforward. The initial Symmetry TFT is
(15) with $p=1$. The idea is to add to it the Symmetry TFT of a
$d$-dimensional photon, and to couple the two TFTs in the bulk in a way that
reproduces the coupling of the current to the gauge field on the boundary. It
is convenient to use the alternative formulation (20) for the Symmetry TFT of
the photon, which we report here for convenience:

$\mathcal{Z}=\frac{1}{2\pi}\\!\int\Bigl{[}g_{d-2}\wedge dG_{2}+f_{2}\wedge
dF_{d-2}-G_{2}\wedge dF_{d-2}\Bigr{]}.$ (41)

The coupling to the fields appearing in the “matter” part
$\frac{1}{2\pi}\\!\int b_{d-1}\wedge dA_{1}$ must be such that, on the
boundary, the Wilson lines of the Maxwell field are endable on the local
operators charged under $U(1)^{(0)}$. The Wilson lines are the endlines of the
surfaces of $G_{2}$, while the above-mentioned local operators are the
endpoints of the lines of $A_{1}$. Hence, the coupling must allow the surfaces
of $G_{2}$ to end on the lines of $A_{1}$. We are led to the total action:

$\displaystyle\mathcal{Z}$
$\displaystyle=\frac{1}{2\pi}\\!\int_{X_{d+1}}\Bigl{[}b_{d-1}\wedge
dA_{1}+b_{d-1}\wedge G_{2}$ (42) $\displaystyle\qquad\quad+g_{d-2}\wedge
dG_{2}+f_{2}\wedge dF_{d-2}-G_{2}\wedge dF_{d-2}\Bigl{]}.$

Because of the added coupling $b_{d-1}\wedge G_{2}$, the standard gauge
transformation of $G_{2}$ must also act on $A_{1}$:

$\delta G_{2}=d\lambda_{1}\,,\qquad\delta A_{1}=-\lambda_{1}\,.$ (43)

Hence the lines of $A_{1}$ are no longer gauge invariant, but we can construct
the non-genuine line operators

$L_{n}[\gamma_{1},D_{2}]=e^{in\\!\int_{\gamma_{1}}\\!A_{1}\,+\,in\\!\int_{D_{2}}\\!G_{2}}$
(44)

where $\partial D_{2}=\gamma_{1}$. This implies that all the surfaces of
$G_{2}$ can end, and on the boundary all Wilson lines can be cut open,
achieving the coupling of the photon with matter.

Similarly, the gauge transformations of $b_{d-1}$ must act on $g_{d-2}$:

$\delta b_{d-1}=d\eta_{d-2}\,,\qquad\delta g_{d-2}=(-1)^{d}\,\eta_{d-2}\,.$
(45)

As a consequence the surfaces of $g_{d-2}$ are not gauge invariant, and must
be attached to a disk where $b_{d-1}$ is integrated:

$U_{\alpha}[\gamma_{d-2},D_{d-1}]=e^{i\alpha\\!\int_{\gamma_{d-2}}g_{d-2}\,-\,(-1)^{d}i\alpha\\!\int_{D_{d-1}}\\!b_{d-1}}.$
(46)

This also has a clear physical interpretation. The operators
$\exp\bigl{(}i\alpha\\!\int b_{d-1}\bigr{)}$ that used to generate on the
boundary the global $U(1)$ 0-form symmetry we are gauging, can now be opened
and trivialized. Gauging a symmetry trivializes the topological operators that
generate it.

By inspection we notice that the only genuine operators that cannot be opened
are the surfaces of $f_{2}$ and $F_{d-2}$. All the other ones become trivial
because of the coupling $b_{d-1}\wedge G_{2}$. Therefore the theory is
equivalent to

$\mathcal{Z}=\frac{1}{2\pi}\\!\int_{X_{d+1}}\\!f_{2}\wedge dF_{d-2}\,.$ (47)

This is nothing but the action (15) with $p=2$, and it describes the dual
magnetic symmetry appearing after the dynamical gauging.

We also notice that going from $\frac{1}{2\pi}\\!\int b_{d-1}\wedge dA_{1}$ to
(47) can be understood as the replacement:

$dA_{1}\,\mapsto\,f_{2}\,,\qquad\quad b_{d-1}\,\mapsto\,dF_{d-2}\,.$ (48)

The first substitution means that $A_{1}$, which was previously flat, can now
describe a curved background with field strength $f_{2}$, which is free —
hence dynamical — at the boundary. The second substitution means that the
previous (Hodge dual) current $b_{d-1}$ of the $U(1)^{(0)}$ symmetry is
trivialized in terms of a form of lower degree.

The prescription (48) is the basic ingredient to construct maps between
different Symmetry TFTs, implementing the dynamical manipulations. In Section
V we will apply it to more complicated examples in order to construct the
Symmetry TFTs of various symmetry structures.

###  IV.2 The Anomaly Polynomial TFT

It is important to emphasize that, while for discrete symmetries the Symmetry
TFT is unique and its topological boundaries describe all possible
manipulations, here we need a step further. Indeed we find distinct Symmetry
TFTs, related by the map (48), that describe the dynamical manipulations,
while each Symmetry TFT admits various topological boundaries that describe
the topological manipulations. We observe, however, that one can construct a
(unique) $(d+2)$-dimensional TQFT whose $(d+1)$-dimensional topological
boundaries correspond to the distinct Symmetry TFTs. We dub this the _Anomaly
Polynomial_ TFT. It is written entirely in terms of $\mathbb{R}$ gauge fields,
which can be identified with the field strengths of the gauged symmetries in
the various theories related by dynamical manipulations.

We illustrate the idea in the simplest example of a 2d theory with a single
$U(1)^{(0)}$ symmetry. Denoting by $k\in\mathbb{Z}$ the anomaly, the Anomaly
Polynomial TFT is a 4d TQFT with action:

$\mathcal{P}=\frac{1}{2\pi}\\!\int_{X_{4}}\biggl{[}g_{1}\wedge
df_{2}+\frac{k}{2}\,f_{2}\wedge f_{2}\biggl{]}$ (49)

and gauge transformations $\delta g_{1}=d\rho_{0}-k\lambda_{1}$, $\delta
f_{2}=d\lambda_{1}$. On a manifold with boundary the gauge variation produces
a boundary term. It is not uniquely determined because we have the freedom to
add a boundary term proportional to $g_{1}\wedge f_{2}$, but independently of
this choice the boundary gauge variation cannot be cancelled unless we couple
the 4d theory to a 3d theory of edge modes.

Consider first the case $k=0$. The gauge variation of (49) produces the
boundary term

$\delta\mathcal{P}=-\frac{1}{2\pi}\\!\int_{\partial X_{4}}d\rho_{0}\wedge
f_{2}\,.$ (50)

In order to cancel it, we place on the boundary the following 3d TQFT of edge
modes with a 1-form symmetry coupled to $f_{2}$, whose boundary value we
regard as a background field:

$\mathcal{Z}=\frac{1}{2\pi}\\!\int_{\partial X_{4}}\Bigl{[}b_{1}\wedge
dA_{1}-b_{1}\wedge f_{2}\Bigr{]}\,.$ (51)

The 1-form symmetry acts on the lines $e^{in\\!\int\\!A_{1}}$ and shifts
$\delta A_{1}=\lambda_{1}$. The gauge variation (50) is canceled by imposing
the gluing condition $g_{1}|_{\partial}=-b_{1}$, which is the boundary EOM
from the variation of the total action with respect to $f_{2}$, and is
compatible with the gauge transformation $\delta b_{1}=-d\rho_{0}$. Turning
off the background $f_{2}$, we recognize (51) as the Symmetry TFT for a 2d
theory with $U(1)$ 0-form symmetry.

There is another boundary theory we can use. It is more cleanly presented if
we first add to (49) the boundary term $g_{1}\wedge f_{2}$, which is
equivalent to recasting the Anomaly Polynomial TFT as
$\mathcal{P}^{\prime}=\frac{1}{2\pi}\\!\int_{X_{4}}\\!dg_{1}\wedge f_{2}$. The
gauge variation produces the boundary term

$\delta\mathcal{P}^{\prime}=\frac{1}{2\pi}\\!\int_{\partial X_{4}}g_{1}\wedge
d\lambda_{1}\,.$ (52)

Now we regard the boundary value of $g_{1}$ as a background field for the
0-form symmetry of a boundary 3d TQFT:

$\mathcal{Z}^{\prime}=\frac{1}{2\pi}\\!\int_{\partial
X_{4}}\Bigl{[}h_{2}\wedge d\Theta_{0}-h_{2}\wedge g_{1}\Bigr{]}\,.$ (53)

The 0-form symmetry shifts $\delta\Theta_{0}=\rho_{0}$, and the variation (52)
is canceled by imposing the gluing condition $f_{2}|_{\partial}=h_{2}$, which
follows for the variation of the total action with respect to $g_{1}$ and is
compatible with the gauge transformation $\delta h_{2}=d\lambda_{1}$. We
recognize (53) as the Symmetry TFT for a $U(1)^{(-1)}$ symmetry [43], related
in 2d by dynamical gauging to the $U(1)^{(0)}$ symmetry. It corresponds to a
shift of the theta angle $\frac{\theta}{2\pi}\\!\int_{X_{2}}\\!\mathcal{F}$
for the dynamical $U(1)$ gauge field. We observe indeed that turning off
background fields, (53) is obtained from (51) by using the map (48) that
implements dynamical gauging in the Symmetry TFT.

In the case with anomaly $k\neq 0$, a gauge variation produces the boundary
term

$\delta\mathcal{P}=\frac{1}{2\pi}\\!\int_{\partial
X_{4}}\biggl{[}-\bigl{(}d\rho_{0}-k\lambda_{1}\bigr{)}\wedge
f_{2}+\frac{k}{2}\,\lambda_{1}\wedge d\lambda_{1}\biggr{]}.$ (54)

It is canceled by the following modification of (51):

$\mathcal{Z}=\frac{1}{2\pi}\\!\int_{\partial}\biggl{[}b_{1}\wedge
dA_{1}+\frac{k}{2}A_{1}\wedge dA_{1}-\bigl{(}b_{1}+kA_{1}\bigr{)}\wedge
f_{2}\biggr{]}.$ (55)

The current in parenthesis that multiplies the background field $f_{2}$ is
$\partial\mathcal{Z}/\partial(dA_{1})$. One uses the transformations $\delta
A_{1}=\lambda_{1}$, $\delta b_{1}=-d\rho_{0}$ as well as the gluing condition
$g_{1}|_{\partial}=-b_{1}-kA_{1}$ that follows from varying the total action
with respect to $f_{2}$. With the background field $f_{2}$ off, we recognize
(55) as the Symmetry TFT (21) for a $U(1)^{(0)}$ symmetry with chiral anomaly
$k$ in 2d. It turns out that the other boundary theory (53) for $k=0$ cannot
be modified in any way to cancel the gauge variation once $k\neq 0$, hence
becoming an inconsistent boundary condition. This is a signal of the ’t Hooft
anomaly, from the Anomaly Polynomial TFT viewpoint.

There is an analogous story for $U(1)$ symmetries in 4d, for which the Anomaly
Polynomial TFT takes the form:

$\mathcal{P}=\frac{1}{2\pi}\\!\int_{X_{6}}\biggl{[}g_{3}\wedge
df_{2}-\frac{k}{12\pi}\,f_{2}\wedge f_{2}\wedge f_{2}\biggr{]}\,,$ (56)

with transformations $\delta g_{3}=d\rho_{2}+\frac{k}{2\pi}\lambda_{1}\wedge
f_{2}+\frac{k}{4\pi}{\lambda_{1}\wedge d\lambda_{1}}$ and $\delta
f_{2}=d\lambda_{1}$. For $k=0$ there are two possible boundary theories, which
are the Symmetry TFTs for a 0-form and 1-form $U(1)$ symmetries, respectively,
related by dynamical gauging. For $k\neq 0$, instead, only the first one is
consistent and it takes the form:

$\displaystyle\mathcal{Z}$ $\displaystyle=\frac{1}{2\pi}\\!\int_{\partial
X_{6}}\biggl{[}b_{3}\wedge dA_{1}+\frac{k}{12\pi}\,A_{1}\wedge dA_{1}\wedge
dA_{1}$ (57)
$\displaystyle\qquad\qquad-\biggl{(}b_{3}+\frac{k}{4\pi}\,A_{1}\wedge
dA_{1}-\frac{k}{4\pi}A_{1}\wedge f_{2}\biggr{)}\wedge f_{2}\biggr{]}$

where the terms in parenthesis describe the coupling of the 1-form symmetry to
the background $f_{2}$ and correspond to
$\partial\mathcal{Z}/\partial(dA_{1})$. One finds the gluing condition
$g_{3}|_{\partial}=-b_{3}-\frac{k}{4\pi}A_{1}\wedge
dA_{1}+\frac{k}{2\pi}A_{1}\wedge f_{2}$ from the variation of the total action
with respect to $f_{2}$, compatible with the transformations $\delta
A_{1}=\lambda_{1}$, $\delta
b_{3}=-d\bigl{(}\rho_{2}-\frac{k}{4\pi}\lambda_{1}\wedge A_{1}\bigr{)}$.

##  V More examples

We present here two other examples that can be derived by dynamically gauging
a $U(1)$ 0-form symmetry.

###  V.1 Abelian 2-group symmetry in 4d

A 0-form and a 1-form symmetry can combine into one algebraic structure known
as a 2-group [44, 5, 45, 46]. Four-dimensional theories with a continuous
Abelian 2-group symmetry were discussed in [45]. Consider a 4d Abelian gauge
theory coupled to chiral fermions. The photon has a magnetic 1-form symmetry
$U(1)^{(1)}$ whose current is $J_{2}=*\,(\mathcal{F}/2\pi)$ written in terms
of the dynamical field strength $\mathcal{F}$. This current is topologically
conserved: $d*J_{2}=d\mathcal{F}/2\pi=0$. The chiral fermions transform under
a 0-form symmetry $U(1)^{(0)}$ with current $J_{1}$. Suppose that there is a
gauge-flavor-flavor triangle anomaly, so that
$d*J_{1}=\frac{k}{4\pi^{2}}\,(dA_{1})\wedge\mathcal{F}$ where $A_{1}$ is the
background gauge field coupled to $J_{1}$. It is easy to arrange the charges
such that there are no other anomalies, for instance:

$\begin{array}[]{rcccc}\text{Weyl
fermions}:&\psi_{1}&\psi_{2}&\psi_{3}&\psi_{4}\\\ \text{gauge charges
}q_{i}:&1&-1&1&-1\\\ \text{flavor charges }f_{i}:&2&1&-2&-1\end{array}$ (58)

One checks that $\sum q_{i}^{3}=\sum q_{i}^{2}f_{i}=\sum f_{i}^{3}=0$, while
$\sum q_{i}f_{i}^{2}\equiv 2k=6$ in this example 555The minimal mixed $U(1)$
Chern-Simons terms that are well defined are $\frac{1}{4\pi^{2}}A\,dB\,dB$ or
$\frac{1}{8\pi^{2}}(A\,dB\,dB+B\,dA\,dA)$.. The theory has therefore a
modified conservation equation:

$d\,{*}J_{1}-\frac{k}{2\pi}\,(dA_{1})\wedge*J_{2}=0\,,\qquad d\,{*}J_{2}=0\,.$
(59)

This type of symmetry is called a 2-group symmetry. If we couple this symmetry
to $U(1)$ background gauge fields $A_{1}$ and $C_{2}$ through the Lagrangian
terms

$\mathcal{L}_{4}\,\supset\,(*J_{1})\wedge A_{1}+(*J_{2})\wedge C_{2}\,,$ (60)

the modified conservation equations imply modified gauge transformations for
the backgroud fields:

$\displaystyle\delta A_{1}$ $\displaystyle=d\lambda_{0}\,,$
$\displaystyle\delta b_{3}$
$\displaystyle=d\gamma_{2}-\tfrac{k}{2\pi}\,\xi_{1}\wedge dA_{1}$
$\displaystyle\delta C_{2}$
$\displaystyle=d\eta_{1}-\tfrac{k}{2\pi}\,\lambda_{0}\,dA_{1}\,,\quad$
$\displaystyle\delta h_{2}$ $\displaystyle=d\xi_{1}\,.$ (61)

Here we also included the conjugated $\mathbb{R}$ gauge fields $b_{3}$ and
$h_{2}$ necessary to write a 5d Lagrangian. A gauge-invariant 5d Symmetry TFT
action we can write is:

$\mathcal{Z}=\frac{1}{2\pi}\\!\int\biggl{[}b_{3}\wedge dA_{1}+h_{2}\wedge
dC_{2}+\frac{k}{2\pi}h_{2}\wedge A_{1}\wedge dA_{1}\biggr{]}.$ (62)

This is the Symmetry TFT for a 2-group symmetry.

Indeed we can derive this theory from the Symmetry TFT of the free-fermion
theory by the procedure of gauging the $U(1)$. The Symmetry TFT of the free-
fermion theory is

$\mathcal{Z}=\frac{1}{2\pi}\\!\int\biggl{[}b_{3}\wedge dA_{1}+f_{3}\wedge
dG_{1}+\frac{k}{2\pi}G_{1}\wedge dA_{1}\wedge dA_{1}\biggr{]}$ (63)

where $A_{1}$ and $G_{1}$ are the $U(1)$ background fields for the flavor
symmetry and the to-be gauge symmetry, respectively. Gauging the symmetry is
implemented by the replacement $dG_{1}\mapsto h_{2}$ and $f_{3}\mapsto
dC_{2}$, as in (48), which indeed produces the action in (62).

###  V.2 $\boldsymbol{\mathbb{Q}/\mathbb{Z}}$ non-invertible symmetry in 4d

Using the building blocks introduced so far, we can derive the Symmetry TFT
describing the non-invertible chiral symmetry of QED-like theories [19,
*Cordova:2022ieu], which provides an intrinsic definition of an Abelian
symmetry with ABJ anomaly. We start from the 5d Symmetry TFT for two 0-form
symmetries $U(1)_{\text{A}}\times U(1)_{\text{V}}$ with a mixed AVV anomaly
and a pure AAA anomaly:

$\displaystyle\mathcal{Z}$
$\displaystyle=\frac{1}{2\pi}\\!\int_{X_{5}}\biggl{[}b_{3}\wedge
dA_{1}+c_{3}\wedge dV_{1}$ (64)
$\displaystyle\qquad\quad+\frac{l}{4\pi}\,A_{1}\wedge dV_{1}\wedge
dV_{1}+\frac{k}{12\pi}\,A_{1}\wedge dA_{1}\wedge dA_{1}\biggl{]}.$

We assumed that the VAA triangle anomaly vanishes, therefore $l$ must be even.
For $l=k=2$ this can be thought of as, for instance, the Symmetry TFT for a
Dirac fermion in four dimensions.

The symmetry $U(1)_{\text{V}}$ has no pure anomaly, hence it can be gauged
dynamically by coupling it to a photon. This is implemented in the Symmetry
TFT as explained in Section IV, and the net result is to replace
$dV_{1}\mapsto f_{2}$ and $c_{3}\mapsto dG_{2}$. We obtain:

$\displaystyle\mathcal{Z}$
$\displaystyle=\frac{1}{2\pi}\\!\int_{X_{5}}\biggl{[}b_{3}\wedge
dA_{1}+f_{2}\wedge dG_{2}$ (65)
$\displaystyle\qquad\quad+\frac{l}{4\pi}\,A_{1}\wedge f_{2}\wedge
f_{2}+\frac{k}{12\pi}\,A_{1}\wedge dA_{1}\wedge dA_{1}\biggl{]}.$

We propose that this is the Symmetry TFT for the non-invertible
$\mathbb{Q}/\mathbb{Z}$ chiral symmetry in 4d.

Let us study this theory more carefully. For simplicity we set $l=2$ (the
generalization to other even values of $l$ being straightforward). While the
gauge transformations of $b_{3}$ and $G_{2}$ are standard, the presence of the
non-derivative term $A_{1}\wedge f_{2}\wedge f_{2}$ forces the gauge
transformations of $f_{2}$ and $A_{1}$ to also act on $b_{3}$ and $G_{2}$:

$\displaystyle\delta f_{2}$ $\displaystyle=d\lambda_{1}\,,\;$
$\displaystyle\delta b_{3}$ $\displaystyle=-\tfrac{2}{4\pi}\,\lambda_{1}\wedge
d\lambda_{1}-\tfrac{2}{2\pi}\,\lambda_{1}\wedge f_{2}\,,$ (66)
$\displaystyle\delta A_{1}$ $\displaystyle=d\rho_{0}\,,\;$
$\displaystyle\delta G_{2}$
$\displaystyle=-\tfrac{2}{2\pi}\,\rho_{0}\,(f_{2}+d\lambda_{1})-\tfrac{2}{2\pi}\,\lambda_{1}\wedge
A_{1}\,.$

Keeping this into account, we analyse the operator content of the theory.
First we have

$V_{\alpha}[\gamma_{2}]=e^{i\alpha\\!\int_{\gamma_{2}}\\!f_{2}}\,,\qquad
W_{n}[\gamma_{1}]=e^{in\\!\int_{\gamma_{1}}\\!A_{1}}$ (67)

which are both topological and gauge invariant because of the EOMs $dA_{1}=0$
and $df_{2}=0$. On the other hand, the integrals of $b_{3}$ and $G_{2}$ do not
lead to gauge-invariant operators because of (66).

We can try to construct non-genuine operators:

$\displaystyle\widetilde{U}_{\alpha}[\gamma_{3},D_{4}]$
$\displaystyle=\exp\biggl{(}i\alpha\\!\int_{\gamma_{3}}\\!b_{3}+i\,\frac{2\alpha}{4\pi}\\!\int_{D_{4}}f_{2}\wedge
f_{2}\biggr{)},$ (68)
$\displaystyle\widetilde{\mathcal{T}}_{n}[\gamma_{2},D_{3}]$
$\displaystyle=\exp\biggl{(}in\\!\int_{\gamma_{2}}\\!G_{2}+i\,\frac{2n}{2\pi}\int_{D_{3}}A_{1}\wedge
f_{2}\biggr{)}.$

They depend on the open regions $D_{4}$ and $D_{3}$, whose boundaries are
$\gamma_{3}$ and $\gamma_{2}$, respectively. They are gauge invariant and
topological because of the EOMs. On a boundary condition which is Dirichlet
for $A_{1}$ and $G_{2}$, that correspond to the QED-like theory, the boundary
values of $A_{1}$ and $G_{2}$ play the role of backgrounds fields for the
would-be axial $U(1)_{\text{A}}$ and the magnetic $U(1)^{(1)}$ symmetry,
respectively. The endlines of $\widetilde{\mathcal{T}}_{n}$ would seem to be
’t Hooft lines, charged under the operators $V_{\alpha}$, while the
$\widetilde{U}_{\alpha}$ lying on the boundary would seem to generate the
axial symmetry, whose charged operators are the endpoints of $W_{n}$. However
this conclusion is not correct. In the definition (68), $\gamma_{3}$ and
$\gamma_{2}$ are boundaries and hence homologically trivial. Since the
operators are topological, they are essentially trivial and cannot be used to
define boundary conditions. Moreover, differently from other cases considered
above, the non-genuine operators $\widetilde{U}_{\alpha}$ do not become
genuine even on the boundary, since $f_{2}$ is not set to zero there. One can
however do better.

Consider $\widetilde{\mathcal{T}}_{n}$ first. The bulk term
$\frac{2n}{2\pi}A_{1}\wedge f_{2}$ can be thought of as the inflow action for
a 2d pure $\mathbb{Z}_{2n}$ gauge theory, where $A_{1}$ and $f_{2}$ are viewed
as the background fields for the $\mathbb{Z}_{2n}$ 0-form and 1-form symmetry,
respectively. This implies that if we take $D_{3}$ to be a tube
$\gamma_{2}\times[0,1]$, we can place
$\exp\bigl{(}in\\!\int_{\gamma_{2}}\\!G_{2}\bigr{)}$ on one end of the tube,
and the 2d $\mathbb{Z}_{2n}$ gauge theory coupled to $A_{1}$ and $f_{2}$ on
the other end. Then we shrink the tube, so as to define a genuine 2d operator:

$\mathcal{T}_{n}[\gamma_{2}]=\widetilde{\mathcal{T}}_{n}[\gamma_{2}]\;\mathbb{Z}_{2n}[\gamma_{2};A_{1},f_{2}]\,.$
(69)

We can perform a similar operation with $\widetilde{U}_{\alpha}$. Since
$\alpha$ is a continuous parameter, we should be careful. Indeed a 3d TQFT on
which $\frac{2\alpha}{4\pi}\\!\int_{D_{4}}\\!f_{2}\wedge f_{2}$ can
topologically terminate only exists if $2\alpha=p/q\in\mathbb{Q}$ (we take
$\gcd(p,q)=1$) 666For $\alpha=1/2$ we can simply use the $U(1)_{1}$ trivial
spin-Chern-Simons theory. Indeed this is the only value that corresponds to an
invertible symmetry, namely $(-1)^{F}$.. This is the minimal TQFT with 1-form
symmetry $\mathbb{Z}_{q}$ and anomaly $p$ introduced in [40] and denoted by
$\mathcal{A}^{q,p}$. By coupling its 1-form symmetry to $f_{2}$ we are able to
define a genuine 3-dimensional topological operator

$U_{p/2q}[\gamma_{3}]=\widetilde{U}_{\alpha=p/2q}[\gamma_{3}]\;\mathcal{A}^{q,p}[\gamma_{3};f_{2}]\,.$
(70)

The operators at irrational $\alpha$, on the other hand, remain non-genuine.

The procedure we illustrated is the bulk analogue of [19, *Cordova:2022ieu],
with the novelty that not only the generator of the chiral symmetry, but also
the 2-dimensional operators $\mathcal{T}_{n}$ whose endlines are the ’t Hooft
lines, require dressing with a TQFT in order to be well defined. This fact is
harder to deduce directly from the boundary because the ’t Hooft lines are not
topological. One of the advantages of the Symmetry TFT description is that
even the topological aspects of non-topological charged objects can be derived
from properties of the topological operators in the bulk. Our result is an
example of this phenomenon.

The necessity of dressing even the ’t Hooft lines is not surprising, since the
non-invertibility of the $\mathbb{Q}/\mathbb{Z}$ symmetry is precisely encoded
in the action on ’t Hooft lines. When a line crosses the symmetry defect it
emerges with a 2-dimensional topological operator attached [19, 49]. It would
be interesting to derive this action from the bulk, studying in detail the
braiding and crossing of the various operators we introduced.

Finally let us mention that $U_{p/2q}[\gamma_{3}]$ can be further dressed with
an other genuine 3-dimensional operator corresponding to a Chern-Simons term
for $A_{1}$, as we did in Section III.3:

$U_{(p/2q,m)}[\gamma_{3}]=U_{p/2q}[\gamma_{3}]\times
e^{i\frac{m}{4\pi}\\!\int_{\gamma_{3}}A_{1}\wedge dA_{1}}\,.$ (71)

Studying the triple linking of these operators allows one to characterize the
anomaly of the non-invertible chiral symmetry, for which the term $A_{1}\wedge
dA_{1}\wedge dA_{1}$ in (65) is responsible.

##  VI Discussion

Let us mention two open questions. First, is it possible to generalize the
construction to non-Abelian 0-form symmetries? A natural candidate for the
Symmetry TFT is the non-Abelian BF theory of [50]:

$\mathcal{Z}=\frac{1}{2\pi}\int_{X_{d+1}}\\!\operatorname{Tr}_{\mathfrak{g}}\bigl{(}b_{d-1}\wedge
F_{2}\bigr{)}\,.$ (72)

Here the fundamental fields are a standard gauge field $A$ for the Lie group
$G$, and a collection $b_{d-1}$ of as many $\mathbb{R}$ $(d-1)$-form gauge
fields as $\dim(\mathfrak{g})$ (where $\mathfrak{g}$ is the Lie algebra of
$G$) that transform in the adjoint representation of $G$. Then $F_{2}=dA-
iA\wedge A$ is the non-Abelian field strength of $A$, and
$\operatorname{Tr}_{\mathfrak{g}}$ is the Killing form on $\mathfrak{g}$ 777In
fact, as spelled out in [50], one can use a $b_{d-1}$ that takes values in the
dual to $\mathfrak{g}$, so that a trace is not used and $\mathfrak{g}$ does
not need to be semisimple.. Thus the theory has two sets of gauge
transformations:

$A\to UAU^{-1}-i\,dU\,U^{-1}\,,\qquad b_{d-1}\to Ub_{d-1}U^{-1}$ (73)

where $U$ takes values in $G$, and

$b_{d-1}\to b_{d-1}+D\lambda_{d-2}$ (74)

where $D$ is the covariant derivative constructed with $A$, while $\lambda$ is
a globally-defined section of the adjoint bundle defined by $A$. The EOMs are
$F_{2}=0$ and $Db_{d-1}=0$, therefore the theory has topological Wilson line
operators

$W_{\mathfrak{R}}[\gamma_{1}]=\operatorname{Tr}_{\mathfrak{R}}\,\operatorname{Pexp}\bigl{(}\mbox{$i\int_{\gamma_{1}}A$}\bigr{)}$
(75)

where $\mathfrak{R}$ are representations of $G$. We expect that there exists a
boundary condition in which these lines can end on the boundary, and describe
local operators transforming in representations $\mathfrak{R}$ of the symmetry
$G$. On the other hand, the non-Abelian symmetry defects of the boundary
theory are charged under themselves, therefore we do not expect them to exist
as topological bulk operators, but rather to only appear in the presence of a
boundary. We believe this theory deserves more study.

Second, the study of TQFTs with an infinite (possibly uncountable) number of
objects opens the possibility for new mathematical studies, as well for new
physical phenomena. For instance, as we observed in Section III.1, even a 3d
TQFT with Lagrangian algebras can admit condensations which are not maximal,
and cannot be made such. This implies that new theories without gapped
boundaries can be derived from simple theories with gapped boundaries. It
would be interesting to explore the consequences of this simple observation
for the physical QFTs at the boundary.

###### Acknowledgements.

We are grateful to Diego Delmastro, Giovanni Galati, Andrea Grigoletto,
Pierluigi Niro, Giovanni Rizi, and Marco Venuti for useful discussions. We are
partially supported by the ERC-COG grant NP-QFT No. 864583 “Non-perturbative
dynamics of quantum fields: from new deconfined phases of matter to quantum
black holes”, by the MUR-FARE grant EmGrav No. R20E8NR3HX “The Emergence of
Quantum Gravity from Strong Coupling Dynamics”, by the MUR-PRIN grant No.
2022NY2MXY, as well as by the INFN “Iniziativa Specifica ST&FI”.

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