# SymTFTs for Continuous non-Abelian Symmetries

Federico Bonetti  Department of Mathematical Sciences, Durham University
Michele Del Zotto  Department of Mathematics, Uppsala University, SE-75120
Uppsala, Sweden  Department of Physics and Astronomy, Uppsala University,
SE-75120 Uppsala, Sweden  Centre for Geometry and Physics, Uppsala University,
SE-75120 Uppsala, Sweden  Ruben Minasian  Institut de Physique Théorique,
Université Paris-Saclay, CNRS, CEA, F-9119, Gif-sur-Yvette, France

###### Abstract

Topological defects and operators give a far-reaching generalization of
symmetries of quantum fields. An auxiliary topological field theory in one
dimension higher than the QFT of interest, known as the SymTFT, provides a
natural way for capturing such operators. This gives a new perspective on
several applications of symmetries, but fails to capture continuous non-
Abelian symmetries. The main aim of this work is to fill this gap. Guided by
geometric engineering and holography, we recover various known features of
representation theory of the non-Abelian symmetry from a SymTFT viewpoint.
Central to our approach is a duality between (flat) free Yang-Mills and non-
Abelian BF theories. Our results extend naturally to models without
supersymmetry.

##  I Introduction and motivation

In recent years our understanding of symmetries of quantum field theories
(QFTs) has undergone a revolution based on the idea that symmetries are
encoded in topological subsectors of the spectrum of QFT operators [1]. While
ordinary symmetries are described by groups, symmetries characterized by
topological operators have a higher structure that generalizes the notion of
group multiplication, believed to be inscribed in a symmetry (higher) category
(see e.g. [2, 3, 4, 5, 6, 7, 8, 9] for recent reviews). A very convenient way
to characterise generalized finite symmetries of a $d$-dimensional QFT,
denoted $\mathbf{T}$, is given by the topological symmetry theory construction
(or SymTFT for short) [10, 11, 12, 13] (see also [14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29]). The construction is based on four
ingredients: (1) a $d$-dimensional field theory $\widehat{\mathbf{T}}$ defined
relative to (2) a bulk $d+1$ topological field theory $\mathbf{S}$, together
with (3) a topological gapped $d$-dimensional boundary condition for
$\mathbf{S}$, that we denote $\mathbf{B}$ such that there is (4) an
isomorphism of field theories

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(1)

Schematically, one can encode the topological operators of $\mathbf{T}$ in the
topological operators of $\mathbf{S}$ that are non-zero along $\mathbf{B}$.
Dually, $\mathbf{B}$ captures background configurations for the symmetry of
$\mathbf{T}$, which is gauged in the bulk (where only the Drinfeld center of
the symmetry is manifest).

While this construction is very effective for (generalized) finite symmetries,
it fails to reproduce verbatim the case of continuous global symmetries for
$\mathbf{T}$. The main reason is that while gauging a finite symmetry is a
topological operation that can be encoded in a topological boundary changing
interface of $\mathbf{S}$, in the case of continuous symmetries the gauging
operation is more involved. Gauging a continuous symmetry introduces new
degrees of freedom in the spectrum (the gauge mediators as well as various
topological solitons) that cannot be captured by a simple topological
manipulation on $\mathbf{T}$. Recent proposals in the literature address this
topic [30, 31] advancing a SymTFT for continuous $U(1)$ symmetries and
conjecturing a possible generalization for continuous non-Abelian symmetries
in terms of a non-Abelian version of BF theory. From this latter perspective,
however, it is unclear how the representation theory of the non-Abelian flavor
symmetry is realized as a bulk-boundary system.

In this letter we give a dual formulation of the topological symmetry theory
in terms of free non-Abelian Yang-Mills theory, that allows to address this
open question among others.

Our proposal is realized explicitly in the context of geometric engineering as
well as in holography, that serve as examples. We stress that the structure
which is responsible for capturing the symmetry is the LHS of the diagram in
Equation (1), namely the data (2) and (3) in the SymTFT construction, that are
completely independent from the choices of $\mathbf{T}$ as well as
$\widehat{\mathbf{T}}$:

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(2)

Here we exploit the extra features (such as superconformal invariance) as a
strategy for obtaining an explicit expression for $\mathbf{S}$. By the above
argument, these extras are not strictly necessary, and indeed our construction
holds for non-supersymmetric theories as well 111Similar to the Abelian case
discussed in [30, 31], our analysis applies to flat backgrounds of the
continuous symmetry. These backgrounds are captured by the boundary conditions
of our bulk TFT. We reserve a better treatment including more general
backgrounds for future work..

This note is organized as follows. In Section II we derive the duality between
the Horowitz non-Abelian BF theory [33] and free Yang-Mills theory. In Section
III we discuss the evidence for our proposal. Our main classes of examples are
SCFTs realized via geometric engineering in M-theory, discussed in §.III.1, as
well as holographic SCFTs, discussed in §.III.2. In particular, our argument
applies uniformly to all holographic SCFTs of $d\geq 4$, as expected since
$d+1=4$ is a critical dimension for gravitational theories in AdS. Our
analysis of free Yang-Mills as a SymTFT for continuous non-Abelian symmetries
is complemented by a first series of applications discussed in Section IV,
where we discuss how to recover the representation theory of non-Abelian Lie
groups from the bulk-boundary perspective 222We find it sacrilegious to form
sandwiches out of quiches and therefore we refrain from using the terminology
introduced in [13] in this note.. We present some conclusions and directions
of further study in Section V.

Note Added. We were informed by Fabio Apruzzi of an upcoming work [35] that
has some overlap with the results in our note.

##  II SymTFT for continuous non-Abelian symmetries

Motivated by geometric engineering in string theory and holography as
discussed in Section III below, we propose that the SymTFT for a QFT in
$d$-dimensions with a continuous non-Abelian symmetry of type $G$ contains a
$(d+1)$-dimensional free Yang-Mills theory, namely a Yang-Mills theory at zero
gauge coupling $g\to 0$. Our claim might seem in tension with the recent
proposal by [30, 31] that the relevant SymTFT is the non-Abelian Horowitz BF
theory. In this Section we briefly outline a duality of free Yang-Mills with
the Horowitz non-Abelian BF theory thus reconciling our proposals.

Following Witten (see e.g. [36], sec. 4.2), one can re-write the Yang-Mills
action as follows:

$\displaystyle\mathbf{S}\supset-{1\over
2g^{2}}\int_{M_{d+1}}\text{tr}\left(f_{2}\wedge\ast f_{2}\right)$ (3)
$\displaystyle=\int_{M_{d+1}}\bigg{[}\text{tr}\left(f_{2}\wedge
h_{d-1}\right)-\frac{g^{2}}{2}\text{tr}\left(h_{d-1}\wedge\ast
h_{d-1}\right)\bigg{]}\ ,$

where $h_{d-1}$ is a collection of Lagrange multiplier $(d-1)$-form fields
transforming in the adjoint of $\mathfrak{g}$.

The corresponding equation of motion is valued in the adjoint of
$\mathfrak{g}$ and gives $f_{2}=g^{2}\ast h_{d-1}$. Motivated by geometric
engineering and holography, to obtain the contribution to the symmetry theory
we need to take the limit $g^{2}\to 0$ of the action above, in particular this
implies only the flat connections contribute to the symmetry theory.

By the Ambrose-Singer theorem, all the holonomies of such gauge fields are
topological operators, hence we expect to have topological operators
corresponding to the Wilson lines of $\mathfrak{g}$ Yang-Mills.

In the $g\to 0$ limit, we obtain the action

$\int_{M_{d+1}}\text{tr}\left(f_{2}\wedge h_{d-1}\right)\ ,$ (4)

which is exactly the non-Abelian Horowitz BF theory [33]. In the original
action (3), the $(d-1)$-form $h_{d-1}$ has no gauge redundancy, but in (4)
$h_{d-1}$ acquires a gauge redundancy,

$h_{d-1}\sim h_{d-1}+D\lambda_{d}\ ,$ (5)

where $D$ is the exterior covariant derivative in the adjoint representation.
The topological action (4) is invariant under (5) by virtue of the Bianchi
identity $Df_{2}=0$.

As already pointed out in [30, 31], from the perspective of non-Abelian BF
theory one would like to define holonomies for the Wilson surfaces of
$h_{d-1}$, but there is not a good notion of such “surface ordered” integrals.
Here our duality with YM at zero coupling is key, since we know that the
latter theory has Gukov-Witten operators, which are codimension 2 in the bulk
symmetry theory, and become topological as well in the limit of zero coupling.
At zero-coupling one can use the description of the Gukov-Witten operators in
terms of the dual fluxes for the Cartan subalgebra of the gauge group modulo
Weyl [37, 38]. In particular, from this dual perspective, the GW operators in
the bulk give rise to a non-invertible subalgebra of the spectrum of the
SymTFT for non-Abelian symmetries.

##  III Evidence from geometric engineering and holography

In this section we discuss evidence for our claim arising from geometric
engineering examples (§.III.1), as well as holographic SCFTs (§.III.2).

###  III.1 SymTFT from Geometry

For the sake of brevity, in this letter we consider the simplest possible
geometric engineering scenario, where a $d$-dimensional SCFT is obtained by
geometric engineering M-theory on a conical Calabi-Yau singularity $X$ with
metric

$ds^{2}_{X}=dr^{2}+r^{2}ds^{2}_{L_{X}}\ ,$ (6)

where $L_{X}$ is a Sasaki-Einstein manifold, also known as the link of the
singularity. We use this class of geometries here as a main source of
examples, but we stress that our results are more general 333Mutatis mutandis
our results generalize straightfordwardly to type IIA geometric engineering
limits, and little more effort is needed in the context of IIB or Heterotic
scenarios, which we reserve to study in future work.. As a example consider
the case $X$ is an orbifold of $\mathbb{C}^{n}$, $\mathbb{C}^{n}/\Gamma$,
where $\Gamma$ is a finite subgroup of $SU(n)$ preserving the Calabi-Yau
structure. In these cases, the link is a lens space which has at worse
orbifold singularities $L_{X}=\mathbb{S}^{2n-1}/\Gamma$. Many examples of
$d$-dimensional SCFTs are obtained in this way, in particular choosing $n=3$,
one obtains the 5d orbifold SCFTs from M-theory [40, 41].

The geometric origin of the SymTFT for finite symmetries in these cases is
encoded in a “geometric engineering limit at infinity”, where the topological
operators of $\mathbf{S}$ have a geometric origin as membranes wrapped on
torsional cycles of $L_{X}$ [12, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52,
53, 54, 55] building on [56, 57, 58, 59, 60]. The case of continuous
symmetries deserve a special analysis. In many cases, continuous symmetries
are realized in geometry as non-compact curves of singularities, that
intersect the singular locus where the SCFT is located (see e.g. [61, 62, 63,
64, 65, 66, 67, 68, 69, 70, 71] for some examples in various spacetime
dimensions). More precisely, consider the case of continuous non-Abelian
0-form symmetries, that arise from non-compact loci of du-Val singularities
(split or non-split corresponding to simply-laced vs non-simply laced groups).
Since these non-compact loci extend at infinity, the resulting link $L_{X}$ is
also singular, with compact loci $C_{\mathfrak{g}_{i}}\subset L_{X}$ of du-Val
singularities. The latter correspond to gauge theories in one higher
dimension, with gauge couplings $1/g^{2}_{i}\sim\text{vol
}(C_{\mathfrak{g}_{i}})$. However, since the link is at infinity in geometry,
all of its compact cycles have infinite volume in the metric (6), hence the
corresponding higher dimensional gauge theory contributes to the SymTFT only
in the limit $g^{2}_{i}\to 0$. To capture the contribution to the symmetry
theory from the singularity link we can follow the procedure outlined in the
previous section for each of these factors, obtaining the action

$\int_{M_{d+1}}\text{tr}\left(f_{2,i}\wedge h_{d-1,i}\right)\ .$ (7)

An explicit example: 5d $T_{N}$ SCFTs. The 5d $T_{N}$ theory is obtained
geometrically engineering M-theory on the orbifold
$X_{N}=\mathbb{C}^{3}/(\mathbb{Z}_{N}\times\mathbb{Z}_{N})$ where the action
by $\mathbb{Z}_{N}\times\mathbb{Z}_{N}$ is $(z_{1},z_{2},z_{3})\to(\alpha
z_{1},\overline{\alpha}\beta z_{2},\overline{\beta}z_{3})$ and
$\alpha^{N}=\beta^{N}=1$ [61]. In this case the three curves $C_{1}\colon
z_{2}=z_{3}=0$, $C_{2}\colon z_{1}=z_{3}=0$ and $C_{3}\colon z_{1}=z_{2}=0$
correspond to three different non-compact loci of
$\mathbb{C}^{2}/\mathbb{Z}_{N}$ singularities, associated to the 0-form
symmetry $SU(N)^{3}$ of this theory. Consider the link of this singularity,
one obtains $L_{X}=\mathbb{S}^{5}/(\mathbb{Z}_{N}\times\mathbb{Z}_{N})\,,$
where the $\mathbb{S}^{5}$ has equation
$|z_{1}|^{2}+|z_{2}|^{2}+|z_{3}|^{2}=R^{2}$ and the limit $R\to\infty$ is
taken in the metric.

Each of the three curves of singularities in $X_{N}$, corresponds to a
singular locus on $L_{X}$,
$C_{i}\,\longleftrightarrow\,z_{i}\bar{z}_{i}=R^{2}$, which is an $S^{1}$ with
radius $R$.

The reduction of M-theory on $S^{1}\times\mathbb{C}^{2}/\mathbb{Z}_{N}$
correspond to a 6d dynamical $\mathfrak{su}_{N}$ field theory with gauge
coupling ${1\over g^{2}}\sim R$. In the spirit of the geometric engineering at
infinity origin of the SymTFT, it is tempting to conclude that each of these
fixed loci contributes one such gauge theory sector. However, one has to keep
track of the fact that the link of the singularity in the metric (6) is of
infinite size. Hence, we need to take the limit $R\to\infty$ in the analysis
above, which is a limit of zero gauge coupling. To capture the contribution to
the symmetry theory from the singularity link, then by the steps outlined
above we obtain $\sum_{i=1}^{3}\int_{M_{6}}\text{tr}\left(f_{2,i}\wedge
h_{4,i}\right)$, as expected.

###  III.2 SymTFT from Holography

In this section we confirm our main result for those CFTs in $d\geq 4$
dimensions with a continuous non-Abelian 0-form symmetry $G$ that have a
holographic dual in $AdS_{d+1}$ 444In the case $d=2,3$ the holographic
dictionary is slightly more complicated, as emphasized by Harlow and Ooguri
[78]. In particular, for $d=2$ the analysis is subtler due to logarithmic
scaling [90] and the possibility of quadratic Chern-Simons terms.. In AdS/CFT
the usual interpretation is that the holographic dual of a global symmetry is
indeed a gauge symmetry in the bulk [73]. In particular, the $AdS_{d+1}$ low-
energy action contains a Yang-Mills term

$S_{\rm YM}=-\frac{1}{2g^{2}}\int_{AdS_{d+1}}{\rm tr}(f_{2}\wedge*f_{2})\ ,$
(8)

as well as other interactions, depending on the model. To capture the
holographic dual of the topological symmetry theory we consider an asymptotic
limit of the bulk gravitational theory, focusing on the region close to the
AdS boundary where only a topological sector of the bulk theory survives
555See e.g. [91, 92, 93, 94, 43, 44, 46, 47, 50, 51, 53, 54], building on [95,
96].. We want to estimate the effect of the Yang-Mills term near the conformal
boundary of $AdS_{d+1}$. We work in a fixed pure $AdS_{d+1}$ background. The
metric is conformally flat,

$ds^{2}_{AdS_{d+1}}=\frac{L^{2}}{z^{2}}d\widetilde{s}^{2}_{d+1}\ ,\quad
d\widetilde{s}^{2}_{d+1}=dz^{2}+ds^{2}_{\mathbb{R}^{1,d-1}}\ ,$ (9)

with $L$ the $AdS_{d+1}$ radius and the conformal boundary located at $z=0$.
We can write the Yang-Mills term as

$\displaystyle S_{\rm
YM}=\int_{AdS_{d+1}}-\frac{1}{2g^{2}}\frac{L^{d-3}}{z^{d-3}}{\rm
tr}(f_{2}\wedge\widetilde{*}f_{2})$ (10)
$\displaystyle=\int_{AdS_{d+1}}\bigg{[}{\rm tr}(h_{d-1}\wedge
f_{2})-\frac{g^{2}}{2}\frac{z^{d-3}}{L^{d-3}}{\rm
tr}(h_{d-1}\wedge\widetilde{*}h_{d-1})\bigg{]}\ ,$

where $\widetilde{*}$ denotes the Hodge star of the flat metric
$d\widetilde{s}^{2}_{d+1}$, and we have introduced the auxiliary field
$h_{d-1}$, a $(d-1)$-form with values in the Lie algebra of $G$. As we
approach the conformal boundary $z\rightarrow 0$, the ${\rm
tr}(h_{d-1}\wedge\widetilde{*}h_{d-1})$ term is subleading. An alternative,
more physical viewpoint is the following. We have a gauge theory in
$AdS_{d+1}$. Approaching the conformal boundary probes its low-energy regime.
But a non-Abelian gauge theory in dimension $\geq 5$ is IR free, with the
effective $g^{2}$ coupling going to zero as $z\rightarrow 0$. If we neglect
the ${\rm tr}(h_{d-1}\wedge\widetilde{*}h_{d-1})$ in (10), we recover the non-
Abelian BF theory

$S_{\rm BF}=\int_{AdS_{d+1}}{\rm tr}(h_{d-1}\wedge f_{2})\ .$ (11)

An explicit example: the 6d (1,0) E-string theory. The E-string theory of rank
$N$ is the 6d (1,0) SCFT of a stack of $N$ M5 branes probing one M9 brane in
M-theory. The M9 brane has an $E_{8}$ gauge symmetry and the E-string inherits
an $E_{8}$ global symmetry. The holographic dual of the E-string theory is
given by $AdS_{7}\times\mathbb{S}^{4}/\mathbb{Z}_{2}$ where $\mathbb{Z}_{2}$
acts by
$(x_{1},x_{2},x_{3},x_{4},x_{5})\mapsto(x_{1},x_{2},x_{3},x_{4},-x_{5})$ [75].
There is a fixed $\mathbb{S}^{3}$ at $x_{5}=0$. On such a boundary, by the
Hořava-Witten construction, there is an M9 brane that by inflow supports a 10d
$E_{8}$ gauge theory. The latter is reduced on $\mathbb{S}^{3}\times AdS_{7}$.
By the reasoning in this paper, the resulting $E_{8}$ gauge fields in
$AdS_{7}$ have a SymTFT action that contains the term (11) corresponding to
the $E_{8}$ global symmetry of the 6d (1,0) rank $N$ E-string SCFT (at large
$N$).

Remarks:

  1. 1.

The charged operators in the CFTd that correspond to the end-points for the
topological Wilson lines and Gukov-Witten operators of the $g^{2}=0$ gauge
theory map to extended operators that end on the boundary and extend in the
bulk. Away from the boundary the gauge coupling is not suppressed anymore and
the extended gauge-theory operators are dressed by non-topological terms. For
consistency with the completeness hypothesis in quantum gravity we expect all
these operators end on charged objects in the bulk [76, 77]. These endings in
the bulk are the holographic dual to the extended charged operators of the
CFT. The boundary conditions at infinity are the holographic counterpart of
the boundary conditions $\mathbf{B}$.

  2. 2.

A more refined approach to symmetries indicates that splittable global
symmetries of the boundary CFT (which are the kind we are considering here due
to the presence of Noether currents) are dual to a long-range gauge symmetries
in the bulk [78]. This is consistent with our proposal: in the zero-coupling
limit, the gauge mediators in the bulk are essentially photon fields valued in
the Cartan of the gauge group (modulo Weyl), which are definitely long-range
fields and indeed are the only ones that survive in the asymptotic region
close to the AdS boundary.

  3. 3.

From the holographic perpsective Abelian symmetries can be treated similarly,
hence our analysis also confirms that the SymTFT for Abelian $U(1)$ symmetries
is indeed the one proposed in [30, 31].

##  IV Continuous non-Abelian topological symmetries

From the geometric engineering analysis above as well as from the holographic
approach it is natural to interpret the SymTFT for continuous gauge symmetries
of a $d$-dimensional QFT as a gauge theory in $d+1$ dimensions with gauge
group $G$ at $g_{\rm YM}=0$ 666We are especially indebted to Kantaro Ohmori
for a discussion that helped clarifying and shaping some key parts of this
section, in particular, how the non-Abelian symmetry is realized along the
boundary.. The $(d+1)$-dimensional theory has the following topological
extended operators:

  * •

Wilson line operators $W_{1}^{(\mathbf{R})}$ labeled by an irreducible
representation $\mathbf{R}$ of $G$;

  * •

$(d-1)$-dimensional Gukov-Witten operators $T_{d-1}^{([g])}$ labeled by a
conjugacy class $[g]$ of elements in $G$.

In the limit $g_{\rm YM}\rightarrow 0$, all Gukov-Witten operators of $G$
Yang-Mills theory become topological [38]. We have argued above that Yang-
Mills theory in the limit $g_{\rm YM}\rightarrow 0$ is captured by the non-
Abelian BF theory (4). Then, $a_{1}$ is flat and the Wilson lines become
topological as well.

Recall that Gukov-Witten operators in $G$ Yang-Mills theory are defined as
follows: inserting $T_{d-1}^{([g])}$ on $M_{d-1}$ corresponds to performing
the path integral on $G$ gauge field $a_{1}$ configurations with a prescribed
holonomy along the $\mathbb{S}^{1}$ linking $M_{d-1}$. The holonomy is only
specified up to conjugation. This is necessary for bulk gauge invariance,
because the holonomy ${\rm hol}_{\gamma}(a_{1})$ of $a_{1}$ along a closed
path $\gamma$ based at a point $P$ transforms as ${\rm
hol}_{\gamma}(a_{1})\mapsto h(P)^{-1}{\rm hol}_{\gamma}(a_{1})h(P)$ under the
gauge transformation $a_{1}\mapsto h^{-1}(A+d)h$ 777It would be interesting to
investigate the connection between the Gukov-Witten operators we discuss in
this note and the non-invertible averages introduced by Córdova, Ohmori and
Rudelius in [37]..

We can adopt the same stategy to define Gukov-Witten operators in the non-
Abelian BF theory (4). If we adopt this viewpoint, we avoid difficulties
related to defining “surface ordered” integrals of $h_{d-1}$ [30].

Having described the bulk theory $\mathbf{S}$ as the $g_{\rm YM}\rightarrow 0$
limit of $G$ Yang-Mills theory, or equivalently the non-Abelian BF theory (4),
let us now turn to the symmetry boundary $\mathbf{B}$. We define it by
choosing Dirichlet boundary conditions for the $G$-gauge field $a_{1}$.
Standard arguments then imply that $\mathbf{B}$ carries a global $G$ symmetry.
In particular, we have topological $(d-1)$-dimensional operators
$S^{(g)}_{d-1}$ living on $\mathbf{B}$ and labeled by a group element $g\in
G$.

Figure 1: A Gukov-Witten operator $T_{d-1}^{([g])}$ in the bulk induces a
prescribed holonomy along a small circle linking it. The holonomy is $g$ up to
conjugation. The topological operator $S_{d-1}^{(g)}$ is “anchored” to
$\mathbf{B}$. It induces a prescribed value $g$ for the quantity
$\mathrm{Pexp}\int_{P}^{Q}a_{1}$ where $P$, $Q$ are the endpoints of a small
semicircle linking $S_{d-1}^{(g)}$.

Crucially, in the bulk theory the Gauss’s law constraint holds and, because of
gauge invariance, we can only write down topological operators labeled by
conjugacy classes in the bulk gauge group, that are gauge invariant. On the
contrary, gauge transformation asymptote to the identity at the boundary, and
for this reason a global 0-form $G$ gauge symmetry is realized 888This is
reminiscent of the construction of the Drinfeld Center of a symmetry
category..

We can describe the topological operators $S^{(g)}_{d-1}$ on $\mathbf{B}$ as
operators of $(d+1)$-dimensional $G$ Yang-Mills theory defined on a half-
space. The support $M_{d-1}$ of $S^{(g)}_{d-1}$ is linked by a semicircle.
Inserting $S^{(g)}_{d-1}$ means performing the path integral over $a_{1}$ on
the half-space, with Dirichlet boundary conditions on the boundary
$\mathbf{B}$, and imposing that the path-ordered exponential of the integral
of $a_{1}$ along the semicircle equals $g$. See Figure 1. Crucially, this
integral of $a_{1}$ on the semicircle is gauge invariant. This is because we
can only perform gauge transformations in the half-space that are trivial at
the boundary $\mathbf{B}$, and the endpoints of the semicircle lie on
$\mathbf{B}$.

We emphasize that the topological operators $S^{(g)}_{d-1}$ are “anchored” to
$\mathbf{B}$, and cannot be brought into the bulk of $\mathbf{S}$ (this would
violate bulk gauge invariance, per the discussion above). We can, however,
start with a Gukov-Witten operator $T_{d-1}^{([g])}$ in the bulk, and project
it parallel onto the boundary $\mathbf{B}$. The resulting topological operator
on $\mathbf{B}$ is not simple, but rather a sum of operators
$S^{(g^{\prime})}_{d-1}$ with $g^{\prime}\in[g]$ 999This would be an averaged
operator over the Haar measure of the group, reminiscent of the discussion in
[37].. Only these averaged operators can move from the boundary into the bulk
and viceversa.

Next, let us discuss charged local operators and how $G$ acts on them.
Generalized charges have been studied extensively in recent works [83, 84, 85,
86, 23]. In the remainder of this section we follow closely the discussion in
[83, 23]. For simplicity, we restrict to genuine local operators transforming
in a representation $\mathbf{R}$. In the SymTFT picture these operators are
engineered by stretching a Wilson line $W_{1}^{(\mathbf{R})}$ along the
interval direction, connecting a point $\mathcal{M}$ on $\widehat{\mathbf{T}}$
to a point $\mathcal{E}$ on $\mathbf{B}$, see Figure 2. This configuration is
allowed by our choice of Dirichlet boundary conditions for $a_{1}$ on
$\mathbf{B}$.

Figure 2: Sandwich construction for a genuine local operator
$\mathcal{O}^{(\mathbf{R})}$ of the theory $\mathbf{T}$ in the representation
$\mathbf{R}$ in terms of a Wilson line $W_{1}^{(\mathbf{R})}$ stretched
between a topological endpoint $\mathcal{E}$ on $\mathbf{B}$ and a non-
necessarily topological endpoint $\mathcal{M}$ on $\widehat{\mathbf{T}}$.

As stressed above, the topological operators $S_{d-1}^{(g)}$ labeled by
elements of $G$ live on $\mathbf{B}$. Thus, the $G$-action is encoded on the
action of $S_{d-1}^{(g)}$ on the endpoint $\mathcal{E}$ of
$W_{1}^{(\mathbf{R})}$ on $\mathbf{B}$. This is to be contrasted, for
instance, with the case of an Abelian global symmetry, for which the
$G$-action can also be seen via linking in the bulk.

To describe more precisely the endpoint $\mathcal{E}$, we proceed as follows.
We can consider the Wilson line $W_{1}^{(\mathbf{R})}$ and project it parallel
onto the symmetry boundary $\mathbf{B}$. The result is

$nS_{1}^{(\rm id)}:=\overbrace{S_{1}^{(\rm id)}\oplus\cdots\oplus S_{1}^{(\rm
id)}}^{n\text{ times}}\ ,$ (12)

where $S_{1}^{(\rm id)}$ is the identity topological line on $\mathbf{B}$ and
$n$ is the dimension of the representation $\mathbf{R}$, see figure 3.
(Indeed, due to Dirichlet boundary conditions, ${\rm
Tr}_{\mathbf{R}}\mathrm{Pexp}\int a_{1}$ collapses to ${\rm
Tr}_{\mathbf{R}}\mathbb{I}=\dim\mathbf{R}=n$.) Then, we can identify the
possible endpoints $\mathcal{E}$ as the possible topological junctions $S_{0}$
between the lines $nS_{1}^{(\rm id)}$ and $S_{1}^{(\rm id)}$ in $\mathbf{B}$,
see Figure 4. The vector space of such topological junctions is isomorphic to
$\mathbb{C}^{n}$. This supports the identification of the possible endpoints
$\mathcal{E}$ as the components of an operator transforming in the
representation $\mathbf{R}$.

Figure 3: When the Wilson line $W_{1}^{(\mathbf{R})}$ is projected parallel to
the boundary $\mathbf{B}$ it yields $nS_{1}^{(\mathrm{id})}$, denoted with a
double dashed line. Projecting an L-shaped configuration for
$W_{1}^{(\mathbf{R})}$ gives the canonical endpoint on $\mathbf{B}$ connecting
$W_{1}^{(\mathbf{R})}$ and $nS_{1}^{(\mathrm{id})}$. When
$W_{1}^{(\mathbf{R})}$ is projected on top of an operator $S_{d-1}^{(g)}$, we
get the point-like topological junction $J_{0}^{(g)}$.  Figure 4: The endpoint
$\mathcal{E}$ of $W_{1}^{(\mathbf{R})}$ is a junction with the trivial line
$S_{1}^{\rm(id)}$ on $\mathbf{B}$. By projecting a segment of
$W_{1}^{(\mathbf{R})}$ below $\mathcal{E}$, we resolve $\mathcal{E}$ as a
topological junction $S_{0}$ between the lines $nS_{1}^{(\rm id)}$ and
$S_{1}^{(\rm id)}$.

Let us now consider an insertion of a topological operator $S_{d-1}^{(g)}$
inside $\mathbf{B}$. When we project the bulk Wilson line
$W_{1}^{(\mathbf{R})}$ parallel onto $\mathbf{B}$, we obtain a topological
point-like junction $J_{0}^{(g)}$ between $S_{d-1}^{(g)}$ and the line
$nS_{1}^{(\rm id)}$, see Figure 3. The vector space of such topological
junctions is isomorphic to linear maps from
$\mathbb{C}^{n}\rightarrow\mathbb{C}^{n}$, or $n\times n$ complex matrices. As
shown in [23], these matrices do indeed form a representation of $G$. Finally,
the action of $S_{d-1}^{(g)}$ on the endpoint $\mathcal{E}$ corresponds to
fusing $J_{0}^{(g)}$ and $S_{0}$, to obtain a new $S_{0}^{\prime}$ junction
between the lines $nS_{1}^{(\rm id)}$ and $S_{1}^{(\rm id)}$. Here $S_{0}$ and
$S_{0}^{\prime}$ are identified with vectors in $\mathbb{C}^{n}$,
$J_{0}^{(g)}$ with an $n\times n$ matrix, and the fusion implements the matrix
multiplication $S_{0}^{\prime}=J_{0}^{(g)}S_{0}$. We have recovered the action
of $g\in G$ on the components of the operator transforming in the
representation $\mathbf{R}$.

A remark on gauging. In SymTFT for finite symmetries the gauging operation is
obtained via a boundary-changing topological interface, exchanging Dirichlet
with Neumann boundary conditions. There is a sharp difference in the case of
the SymTFT for continuous non-Abelian symmetries. In the absence of ’t Hooft
anomalies, the gauging operation consist of gauging the global symmetry
$G^{(0)}$ along the boundary $\mathbf{B}$. This is the analogue of a Neumann
boundary condition for a finite symmetry. Indeed, from this perspective, a ’t
Hooft anomaly for $G^{(0)}$ is encoded in additional couplings in the bulk
SymTFT which obstruct Neumann boundary conditions for the $G$ gauge field. The
gauging process is non-topological and acts on the bulk-boundary system as a
projector, that averages to zero all gauge non-invariant configurations. The
resulting theory along the isomorphism in (1) is $\mathbf{T}$ coupled to a
$d$-dimensional $G$ Yang-Mills sector by gauging its global $G$ symmetry.

##  V Conclusions and outlook

In this letter we have given a derivation in geometric engineering as well as
in holography of the SymTFT for continuous non-Abelian zero-form symmetries.
We find the latter is a $d+1$ dimensional Yang-Mills theory at zero coupling,
whose flat subsector is dual to the non-Abelian BF theory with the same gauge
group. For this reason our results can be seen as a derivation of the proposal
in [31, 30]. However, the most convenient duality frame to study this system
is provided by the gauge theory side, that can be used to address several key
features of the SymTFT. A natural generalization of our findings indicates
that for field theories in fewer dimensions, the relevant SymTFT can include
topological limits of models that are not just gauge theories. This is the
case of 4d SCFTs geometrically engineered in M-theory via conical
singularities with $G(2)$-holonomy [70], see also [87].

Other natural questions that we leave open are in the context of the interplay
of our SymTFT with the usual features of ordinary global symmetries, such as
anomalies or spontaneous breaking. We plan to return on these topics in future
work, thanks to another interesting duality, after ’t Hooft [88]. Namely one
can view the $g_{\rm YM}=0$ limit as a $U(1)^{r}\rtimes\text{Weyl}(G)$ gauge
theory, where $r$ and $\text{Weyl}(G)$ are the rank and Weyl group of $G$.
This gives a very explicit way of characterizing the various operators at hand
[37, 38]. In particular, one of the new features of topological defects as
symmetries is that they have a higher structure (see e.g. [89] for a recent
discussion). It would be interesting to explore the higher structure of non-
Abelian continuous symmetries and its interplay with anomalies from the
perspective advocated in this work.

As a final more speculative remark, our findings suggests a pathway to
describe topological defects for spacetime symmetries as opposed to internal
ones in terms of a gravitational $d+1$ bulk theory at zero coupling.

$|$

## Acknowledgments

We would like to thank Fabio Apruzzi, Francesco Benini, and Christian Copetti
for interesting conversations during the Nordita Program Categorical Aspects
of Symmetries, which inspired this work. We also thank Fabio Apruzzi for
interesting discussions at the 2023 Simons Collaboration on Global Categorical
Symmetries Annual Meeting in November related to the projects [35, 55]. We
especially thank Kantaro Ohmori for sharing with us his vision and insights.
The work of MDZ has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation program (grant
agreement No. 851931). MDZ also acknowledges support from the Simons
Foundation Grant #888984 (Simons Collaboration on Global Categorical
Symmetries). The work of FB is supported by the Simons Collaboration Grant on
Global Categorical Symmetries. The work of RM is partially supported by ERC
grants 772408-Stringlandscape and 787320-QBH Structure.

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