# Classification of 1+1D gapless symmetry protected phases via topological
holography

Rui Wen  Andrew C. Potter  Department of Physics and Astronomy, and Stewart
Blusson Quantum Matter Institute, University of British Columbia, Vancouver,
BC, Canada V6T 1Z1

###### Abstract

Symmetry topological field theory (SymTFT) gives a holographic correspondence
between systems with a global symmetry and a higher-dimensional topological
field theory. In this framework, classification of gapped phases of matter in
spacetime dimension 1+1D correspond to classifications of mechanisms to
confine the SymTFT by condensing anyons. In this work, we extend these results
to characterize gapless symmetry-protected topological states: symmetry-
enriched gapless phases or critical points that exhibit edge modes protected
by symmetry and topology. We establish a one-to-one correspondence between
1+1D bosonic gSPTs, and partially-confined boundaries of 2+1D SymTFTs. From
general physical considerations, we determine the set of data and consistency
conditions required to define a 1+1D gSPT, and show that this data precisely
matches that of symmetry-preserving partial confinement (or partially gapped
boundaries) of 2+1D quantum double models. We illustrate this correspondence
through a dimensional reduction (thin-slab) construction, which enables a
physically-intuitive derivation of how properties of the gSPT such as edge
modes, emergent anomalies, and stability to perturbations arise from the
SymTFT perspective.

###### Contents

  1. I Background: gapless SPTs, and SymTFT
    1. I.1 Gapless SPTs: Generalities
      1. I.1.1 Example of a 1+1D igSPT:
    2. I.2 The Symmetry topological field theory (SymTFT) concept
  2. II SymTFT for GSPTS?
    1. II.1 SymTFT for the $\mathbb{Z}_{4}$-igSPT
      1. II.1.1 String order parameter from SymTFT
      2. II.1.2 Edge modes from SymTFT
      3. II.1.3 Stability of the igSPT from SymTFT
  3. III Structure of 1+1D bosonic gSPTs
    1. III.1 Algebra of symmetry and consistency conditions
    2. III.2 Emergent anomaly of gSPT and a bulk-edge (1+1D/0+1-D) correspondence
  4. IV Condensation of quantum double and gSPT
    1. IV.1 Summary of results
    2. IV.2 Review of condensation in 2+1D topological orders
    3. IV.3 Condensation of 2+1D quantum double
  5. V Special cases of gSPTs and their SymTFT description
    1. V.1 igSPT of SPT-pump type
    2. V.2 Weak gSPTs
    3. V.3 Stacking gSPT with SPT
      1. V.3.1 An example
      2. V.3.2 General consideration
  6. VI Discussion and outlook
  7. A Consistency conditions on the data $\eta,\epsilon$
  8. B Connection to the LHS spectral sequence
  9. C The bulk-edge correspondence for 1+1D gSPTs
  10. D Equivalence between the post-condensation twist and gSPT emergent anomaly

Symmetry plays a crucial role in modern condensed matter physiscs. In the
presence of symmetry short range entangled states can be enriched and split
into inequivalent quantum phases, known as symmetry protected topological
(SPT) phases Chen  _et al._ (2011a, b, c); Gu and Wen (2009); Pollmann  _et
al._ (2010); Turner  _et al._ (2011); Fidkowski and Kitaev (2011); Else and
Nayak (2014); Chen  _et al._ (2013); Wang  _et al._ (2014); Kapustin  _et al._
(2015); Senthil (2015). SPT states have short-range entangled gapped bulks,
but exhibit anomalous edge states that are confined to the boundary by the
bulk gap. Recent works Kong  _et al._ (2020); Chatterjee and Wen (2023a, b);
Moradi  _et al._ (2022); Freed  _et al._ (2023); Fuchs  _et al._ (2002);
Apruzzi  _et al._ (2023); Kaidi  _et al._ (2022); Freed (2023); Kaidi  _et
al._ (2023); Zhang and Córdova (2023); Bhardwaj  _et al._ (2023); Lichtman
_et al._ (2021); Apruzzi (2022); Lin  _et al._ (2023); Cordova  _et al._
(2023) have proposed a topological holographic principle between gapped 1+1D
systems with symmetry group $\Gamma$, and anyon condensations of a symmetry
topological field theory (SymTFT): a 2+1D topological order corresponding to a
quantum double $D(\Gamma)$. This works conjectured that may be possible to
utilize the dual topological order to fully classify and characterize 1+1D
gapped as well as gapless states in a unified way. While the classification of
low-dimensional gapped phases is well understood by other methods, situation
for gapless phases and critical points are much less well understood, and any
sharp, non-perturbative, topological insights into gapless phases would be
valuable to theory. Moreover, SymTFT has been shown to be powerful in
analyzing symmetries of gapped systems, including for systems with
generalized, non-invertible symmetries.

Despite the importance of the bulk gap in defining conventional topological
phases, it has also been recently shown that topological properties such as
non-local order parameters, localized edge modes, and emergent anomalies can
also arise in _gapless_ systems Keselman and Berg (2015); Scaffidi  _et al._
(2017); Verresen  _et al._ (2018, 2021); Thorngren  _et al._ (2021); Li  _et
al._ (2023a, b); Wen and Potter (2023); Parker  _et al._ (2018); Mondal  _et
al._ (2023); Hidaka  _et al._ (2022); Ma  _et al._ (2022); Borla  _et al._
(2021); Yang  _et al._ (2023); Zhang and Levin (2023). For gapless phases or
critical points in which these topological features rely on symmetry
protection have been dubbed gapless SPT (gSPT) states. Multiple families of
gSPTs have been identified in the literature. Weak gSPTs arise as critical
points between SPT and spontaneous symmetry breaking (SSB) phases, and exhibit
a partial set of edge modes from the adjacent gapped SPT Verresen  _et al._
(2021); Senthil (2015); Li  _et al._ (2023a). In particular, weak gSPT edge
modes can be removed by stacking the system with a gapped SPT. Other gSPTs are
_intrinsically gapless_ (igSPTs), and have topological features that would be
forbidden in any gapped state Thorngren  _et al._ (2021); Li  _et al._
(2023b); Wen and Potter (2023), such as emergent bulk symmetry anomalies that
protect edge modes which could not arise at the boundary of a gapped SPT with
the same symmetry.

Given the success of SymTFT in characterizing gapped systems, it is natural to
ask if one could utilize the SymTFT to characterize gSPTs, to provide new
insights into the topological properties of gSPTs. To achieve this goal it is
crucial to understand how the topological features of gSPTs such as edge modes
and emergent anomalies enter the paradigm of SymTFT. In this work we address
these questions and develop a framework for describing gSPTs via SymTFT. To
formally establish this framework, we provide a theory of 1+1D bosonic gSPTs
that includes a set of data determing the algebra of symmetry operators and
the consistency conditions these data must satisfy. The data consists of two
phase factors $(\eta,\epsilon)$, where $\eta$ specifies the SPT class of the
gapped sector of the gSPT and $\epsilon$ determines the interplay between the
gapped sector and the gapless sector. We show that these data fully
characterizes the topological features of a gSPT including its edge modes and
emergent anomaly. Analyzing the structure of the emergent anomaly, we also re-
derive a bulk-edge (1+1D/0+1D) correspondence for gSPTs Thorngren  _et al._
(2021); Wen and Potter (2023): the edge modes determine the bulk emergent
anomaly and the bulk emergent anomaly determines the edge modes up to stacking
with a gapped SPT. Notably, precisely the same set of data is required to
specify a symmetric, partially-confining condensation of a quantum double.
Moreover, an anyon condensation in a quantum double typically results in a
twisted quantum double, and we find that the resulting post-condensation
quantum double’s twist matches the emergent anomaly of the corresponding gSPT.
These findings lay the foundation of incorporating gSPTs into the SymTFT
paradigm. We further directly illustrate the SymTFT description of gSPTs via a
dimensional reduction procedure in which we consider a thin slab of the SymTFT
with gapped boundaries specified by some anyon condensation. This construction
provides a physically intuitive, and graphical method to constrain the phase
diagram of the dual 1+1D system. We use this technique to analyze the
stability of gSPTs and their relation to other phases in the phase diagram.

This work is organized as follows. In Section I we review basics of gSPTs and
the SymTFT concept. Along the way, we generalize a thin-slab dimensional
reduction procedure Thorngren  _et al._ (2021) to describe interfaces between
gapped SPT and trivial symmetric states, which enables us to reproduce the SPT
edge states and their projective symmetry action via the braiding properties
of anyon line operators in the SymTFT dual. In Section II we initiate the
attempt to describe gSPTs by SymTFT. We use the $\mathbb{Z}_{4}$-igSPT as an
example and provide a SymTFT construction that is able characterize properties
of the $\mathbb{Z}_{4}$-igSPT. We then develop the general theory of
SymTFT/gSPT correspondence. We first analysis the structure of 1+1D gSPTs in
Section III and provide a classification. Then in Section IV we show that the
symmetric, partially-confining condensations in 2+1D quantum doubles have
exactly the same structure as that of 1+1D gSPTs. We utilize the established
duality to study two families of gSPTs and their SymTFT description in Section
V. Finally we conclude with a discussion of possible routes to generalize the
SymTFT/gSPT correspondence to higher dimensions and generalized higher-form or
non-invertible symmetries.

##  I Background: gapless SPTs, and SymTFT

###  I.1 Gapless SPTs: Generalities

Gapless SPTs are examples of SECs, in which one cannot locally distinguish
between a “trivial” critical point and a symmetry-enriched one from local bulk
measurements. Rather, the differences between different gapless SPTs are only
evident in non-local bulk probes, or local edge probes. One can make this
notion more specific by analogy to gapped SPTs. Different gapped $G$-SPT
ground-states can be connected by a finite-depth local unitary (FDLU), $U$,
that is overall symmetric ($[U,g]=0~{}\forall g\in G$) but which is not
symmetrically generated ($U\neq e^{-iH}$ for any local $G$-symmetric $H$).
This definition cannot be directly ported to the gapless setting as even
different instances of a gapless state with the same universal scaling
properties, e.g. two instances of a conformal field theory (CFT) perturbed by
different irrelevant operators, cannot be connected by an FDLU. However, we
can generalize the notion of symmetric FDLU-(in)equivalence by defining i)
that two ground-states are in the same universality class if they flow to the
same RG fixed point after applying an overall symmetric FDLU, ii) ground-
states with the same universality class are distinct gapless SPT classes if
they cannot be connected in this way by any symmetrically-generated FDLU. An
immediate corollary of this definition is that local scaling operators in
distinct gapless SPTs of the same type of criticality have the same symmetry
properties as conjugating a local operator with definite symmetry quantum
number with an overall-symmetric $U$ preserves its symmetry quantum number.
However, the symmetry properties of non-local scaling operators, such as a
disorder operator that inserts a domain wall, may change under such an overall
symmetric FDLU, leading to distinct classes of gapless SPTs. These notions can
be made more precise for 1+1D conformal field theories (CFTs) Verresen  _et
al._ (2021), for which the data specifying a universality class is well
understood and characterized by the spectrum and fusion rules for primary
scaling operators.

####  I.1.1 Example of a 1+1D igSPT:

A simple igSPT with full symmetry $\Gamma=\mathbb{Z}_{4}$ was constructed in
studied in Li  _et al._ (2023a); Wen and Potter (2023). We will use this
example extensively throughout this paper to illustrate the formal results. A
lattice Hamiltonian realizing the $\mathbb{Z}_{4}$-igSPT chain is given by Li
_et al._ (2023a)

$\displaystyle H_{\mathbb{Z}_{4}-\text{igSPT}}$
$\displaystyle=H_{0}+H_{\Delta}$ $\displaystyle H_{0}$
$\displaystyle=-g\sum_{i}\left(\tau^{z}_{i-1/2}\sigma^{x}_{i}\tau^{z}_{i+1/2}+\tau^{y}_{i-1/2}\sigma^{x}_{i}\tau^{y}_{i+1/2}\right)$
$\displaystyle H_{\Delta}$
$\displaystyle=\Delta\sum_{i}\sigma^{z}_{i-1}\tau^{x}_{i-1/2}\sigma^{z}_{i}$
(1)

with two species of spins per unit cell represented by Pauli operators
$\sigma^{\alpha},\tau^{\alpha}$. The system has a $\mathbb{Z}_{4}$ symmetry
generated by

$\displaystyle
U_{s}=\prod_{i}\sigma^{x}_{i}e^{\frac{i\pi}{4}(1-\tau^{x}_{i+1/2})}$ (2)

Below the energy scale, $\Delta$ (taken to be very large), $\tau^{x}_{i-1/2}$
is locked to $\sigma^{z}_{i-1}\sigma^{z}_{i}$, and the symmetry effectively
reduces to an _anomalous_ $G=\mathcal{Z}_{2}$ symmetry:

$\displaystyle
U_{s}\approx\prod_{i}\sigma^{x}_{i}e^{\frac{i\pi}{4}(1-\sigma^{z}_{i}\sigma^{z}_{i+1})}$
(3)

Viewed as a $\mathbb{Z}_{2}$ symmetry, this symmetry action is anomalous, and
has the same anomaly as the edge of a 2+1D SPT phase (the Levin-Gu phase Levin
and Gu (2012)). This emergent anomaly prevents the system from realizing a
symmetry-preserving short-range entangled state. Instead, the only options are
to spontaneously break the symmetry, or form a gapless state. The above choice
of $H_{0}$ realizes the latter option, reducing at low energies to the
critical spin-chain:

$\displaystyle
H_{0}\approx-\sum_{i}\sigma^{x}_{i}-\sigma^{z}_{i-1}\sigma^{x}_{i}\sigma^{z}_{i+1}.$
(4)

Further, in an open chain, one finds that acting twice with the low energy
symmetry operator “pumps” a $G=\mathcal{Z}_{2}$ charge of the low energy
symmetry group onto each end of the boundary:
$U_{s}^{2}=\sigma^{z}_{1}\sigma^{z}_{L}$, where $1,L$ are the first and last
sites in the chain respectively Li  _et al._ (2023a); Wen and Potter (2023).
This symmetry-charge pumping, locally anticommutes with the $U_{s}$ symmetry,
protecting a two-fold ground-state degeneracy, split only exponentially-weakly
in system size, $L$ by finite size effects (in contrast to polynomial scaling
with $1/L$ for gap to bulk excitations). Away from this idealized fixed point,
this ground-state degeneracy is robust, so long as symmetry remains intact,
and the gap to the $\tau^{z}$ particles remains open.

Formally, the structure of the emergent anomaly is captured by the short-exact
sequence of groups:

$\displaystyle 1\rightarrow
A\xrightarrow[]{j}\Gamma\xrightarrow[]{p}G\rightarrow 1$ (5)

which defines a group extension of $G$ by $A$ to $\Gamma$. Microscopically (in
the UV), the system has full symmetry $\Gamma$, but excitations carrying
charges of the normal subgroup $A\trianglelefteq\Gamma$ are gapped out by
$H_{\Delta}$. Therefore, below the energy scale of this gap, the symmetry is
effectively reduced to the quotient $G=\Gamma/A$. We call $A,G$ the gapped
symmetry and the gapless symmetry respectively. The gapless symmetry $G$ can
be anomalous and the anomaly is represented by a cocycle $\omega\in
H^{d+2}(G,U(1))$. In the above example, $A,G=\mathbb{Z}_{2}$, and
$\Gamma=\mathbb{Z}_{4}$, and $\omega(a,b,c)=(-1)^{abc}$ is the anomaly of the
edge of a 2+1D Levin-Gu phase Levin and Gu (2012). Since the full symmetry
$\Gamma$ is not anomalous, the emergent anomaly must be “lifted” by the gapped
sector. Formally, this implies that ,upon pull-back by the projection $p$, the
cocycle becomes a coboundary: $p^{*}(\omega)=d\alpha$. Moreover, the system
can be put in a $\Gamma$-symmetric gapped phase. However, the anomaly in the
IR has the consequence that one must first close the gap to the $A$-degrees of
freedom before tuning the system to a gapped symmetric state. Thus the
nontrivial topological properties of the system are protected by the $A$-gap,
despite the presence of gapless $G$-charged excitations.

###  I.2 The Symmetry topological field theory (SymTFT) concept

SymTFT is a method of studying a given symmetry via a higher dimensional
topological order. SymTFT has been proven powerful in classifying SPTs
protected by generalized symmetry, identifying anomalies of non-invertible
symmetries, revealing dualities between different phases, and beyond Moradi
_et al._ (2022); Freed  _et al._ (2023); Fuchs  _et al._ (2002); Apruzzi  _et
al._ (2023); Kaidi  _et al._ (2022); Freed (2023); Kaidi  _et al._ (2023);
Zhang and Córdova (2023); Bhardwaj  _et al._ (2023); Lichtman  _et al._
(2021); Apruzzi (2022); Lin  _et al._ (2023); Cordova  _et al._ (2023). We
here review the construction of SymTFT for ordinary internal symmetries
(invertible, 0-form symmetries).

For a system with a symmetry group $G$, the SymTFT is a $G$-gauge theory in
one higher dimension. This correspondence can be motivated in two different
ways: First, there is a relation between global gauge transformations, and a
boundary global symmetry, as we now review. Though the SymTFT correspondence
has only been established for discrete groups $G$, consider for a moment the
more familiar $U(1)$ gauge theory on a space $\Sigma$, with electric field
$E$, and charge density $\rho$. By Gauss’ law, $\nabla\cdot E=\rho$, a global
gauge transformation, reduces to a $U(1)$ symmetry transformation that acts
only on the boundary, $\partial\Sigma$ of the system:
$U_{\theta}=e^{i\theta\int_{\Sigma}\rho}=e^{i\theta\int\nabla\cdot
E}=e^{i\int_{\partial\Sigma}E}$. This boundary symmetry of a gauge theory is
also called asymptotic symmetry Harlow and Ooguri (2019); Sachs (1962);
Strominger (2014, 2018) and plays a crucial role in the recently proposed
“Higgs=SPT” correspondence Thorngren  _et al._ (2023); Verresen  _et al._
(2022). This relation between bulk gauge symmetry and edge global symmetry
also hold for discrete symmetries. Thus, a $G$-symmetric system can always be
embedded into the boundary of a $G$ gauge theory in one higher dimension. We
note that, if the $G$ symmetry action is anomalous, the corresponding bulk
gauge theory would be a twisted gauge theory, also known as the Dijkgraaf-
Witten theory with a nontrivial cocycle.

A second complementary perspective, is that one can recover a $G$-symmetric
system in $d+1$D from a dimensional reduction procedure by considering a thin
slab of $(d+1)+1$D (twisted) $G$ gauge theory, with appropriate open boundary
conditions in the thin direction, and periodic boundary conditions in the
other directions. We illustrate this dimensional reduction for the simplest
case of a 2+1D $\mathcal{Z}_{2}$ gauge theory (toric code topological order)
with anyon types $e,m,f=e\times m$. Here, consider the top boundary of the
slab to be a fixed reference gapped state. Gapped boundaries of topological
orders correspond to anyon condensations that fully confine the topological
order Kong (2014); Kitaev and Kong (2012). Here, we consider the reference top
boundary state as an $e$-condensed boundary. Define a global $\mathbb{Z}_{2}$
symmetry operation as nucleating a pair of $m$ particles, and dragging them
around the periodic cycle. Gapped states of this quasi-1d system correspond to
different gapped boundaries for the bottom of the slab. The $m$ condensed
boundary is symmetric under the above-defined symmetry, since it can absorb
the $m$ line that generates the symmetry. By contrast, the $e$ condensed
bottom boundary corresponds to a spontaneous symmetry broken state, which has
a dual symmetry under pulling an $e$ particle from the top-boundary condensate
and moving it to the bottom boundary condensate. This dual symmetry anti-
commutes with the original $\mathbb{Z}_{2}$ symmetry, due to the mutual
semionic statistics of the $e$ and $m$ particles, leading to a two-fold
degenerate ground-space. This structure of the global $\mathbb{Z}_{2}$
m-symmetry and dual $\mathbb{Z}_{2}$ e-symmetry, matches that of the lattice
spin-1/2 Ising model with global $\mathbb{Z}_{2}$ symmetry
$g_{m}=\prod_{i}\sigma^{x}_{i}$ respected by symmetric (paramagnetic) short
range entangled phases, and dual domain wall conservation symmetry:
$g_{e}=\prod_{i}\sigma^{z}_{i}\sigma^{z}_{i+1}$ respected in symmetry broken
phases: these two symmetries anticommute when restricted to overlapping finite
intervals, and were referred to as “categorical symmetries” or symmetry
topological order (SymTO) in Kong  _et al._ (2020); Chatterjee and Wen (2023a,
b); Inamura and Wen (2023); Chatterjee  _et al._ (2023). The Ising critical
point separating the two phases also corresponds to the phase transition
between the $e$ and $m$ condensed bottom boundary Lichtman  _et al._ (2021);
Chatterjee and Wen (2023b).

In 1+1D systems, this structure generalizes to any finite internal unitary
symmetries. In the broken phase of such a symmetry, a dual symmetry appears
that encodes the conservation of domain wall excitations. The original
symmetry and the dual symmetry together form the categorical symmetry of the
system and can be uniformly described by a 2+1D topological order. As we have
seen in the toric code example, the symmetry and the dual symmetry correspond
to anyon string operators of gauge flux and gauge charge respectively. The
statistics of anyons in the 2+1D topological order then encodes the
commutation relations between the symmetry and the dual symmetry. The thin-
slab construction is a way to make the categorical symmetry manifest: if the
1+1D system has a symmetry $G$ with anomaly $\omega\in H^{3}[G,U(1)]$, then
one can always put the system on the boundary of the twisted gauge theory
(a.k.a twisted quantum double) $D_{\omega}(G)$. The original/dual symmetry is
then realized by dragging fluxes/charges around the periodic direction. To
specify the symmetry and the dual symmetry, one fixes the top boundary to be
one that condenses all the charges. Different phases of the 1+1D system are
therefore realized by choosing different boundary conditions on the bottom
boundary.

This thin-slab construction, which can be viewed as a concrete realization of
the SymTFT idea, translates the question of classifying phases of matter with
symmetry in 1+1D into the question of classifying boundaries of 2+1D
topological orders. The latter question is well-understood: Boundaries of 2+1D
topological orders are completely determined by anyons that condense at the
boundaryKong (2014); Kong  _et al._ (2017); Kong and Zheng (2020, 2021);
Kitaev and Kong (2012). For abelian topological orders, an anyon condensation
is specified by a group of self and mutual bosons $\mathcal{A}$. After
condensing $\mathcal{A}$ the anyons that have non-trivial braiding with
$\mathcal{A}$ become confined and the topological order is reduced to one
whose excitations only include the deconfined anyons that braid trivially with
$\mathcal{A}$. The theory of anyon condensation for generic topological orders
is more involved and we defer the discussion to section IV.

Figure 1: SymTFT description of $\mathbb{Z}_{2}$ symmetry. The thin slab hosts
toric code in its bulk. On the top reference boundary we fix the $e$-condensed
boundary condition. The horizontal $m$-line becomes a global symmetry for the
effective 1+1D system which we identify as the generator of a $\mathbb{Z}_{2}$
symmetry. In the left figure we consider condensing $m$ on the physical
boundary. In this case the horizontal $m$-string can be absorbed by the
physical boundary and acts as identify. The effective 1+1D system is therefore
gapped and symmetric. In the right figure we consider condensing $e$ on the
physical boundary. In this case the vertical $e$-string acts as the order
paramater of the symmetry and has non-zero expectation value, signaling SSB.
Equivalently, the system is now invariant under the dual symmetry represented
by a horizontal $e$-string. Figure 2: SymTFT description of the cluster chain.
1.The condensation $\\{e_{1}m_{2},e_{2}m_{1}\\}$ describes the SPT state
protected by $\mathbb{Z}_{2}^{A}\times\mathbb{Z}_{2}^{B}$. The $m_{1}$-line
operator is the generator of $\mathbb{Z}_{2}^{A}$. 2. we pull out an
$e_{2}$-string from the $e_{2}$-condensation at the top boundary, which acts
as identity on the ground state. 3.We multiply the $e_{2}$-string and the
$m_{1}$ string to obtain an $H$-shape operator. 4. The middle part of the
$H$-shape operator is absorbed by the bottom $m_{1}e_{2}$-condensation,
leaving an operator that is supported at the left and right edge of the SPT.
Similar deformation can be perfomed for $\mathbb{Z}_{2}^{B}$ action. Therefore
the symmetry actions of the SPT can be localized to the edge in the ground
space. 5. The localization of $\mathbb{Z}_{2}^{A}$ action and
$\mathbb{Z}_{2}^{B}$ action to the right edge of the SPT is shown. Since the
two operators intersect at one point, the mutual $\pi$ statistics between
$e_{2}m_{1}$ and $m_{1}$ indicates that the two operators anti-commute.
Therefore the edge symmetry actions form a projective representation of
$\mathbb{Z}^{A}_{2}\times\mathbb{Z}_{2}^{B}$.

In general we can devide anyon condensations into two families: If the
condensation completely confines the theory and no anyons other than vacuum
stay deconfined, then we call the condensation fully-confining, otherwise we
call the condensation partially-confining. In the boundary/anyon condensation
dictionary of 2+1D topological orders Kong  _et al._ (2017); Kong and Zheng
(2020, 2021), gapped boundaries corresond to full-confining condensations and
gapless boundaries corresond to partially-confining condensations. It is then
clear that in the thin-slab construction, gapped/gapless phases of the 1+1D
system is obtained by choosing fully/partially confining condensations on the
bottom boundary. Moreover, if on the bottom boundary any of the gauge charges
is condensed, then the effective 1+1D system is invariant under a dual
symmetry implying the original symmetry is at least partially broken.
Therefore in order to describe a fully symmetric 1+1D system the bottom
boundary must condense no charges. Motivated by this relation between charge
condensation and symmetry breaking, we call a condensation symmetric if it
does not condense any gauge charges.

The SymTFT is known to be capable of characterizing gapped phases. We have
seen in the $\mathbb{Z}_{2}$ symmetry example that the dual toric code has
three boundary conditions corresonding to exactly three different phases of
the $\mathbb{Z}_{2}$ symmetry. In 1+1D there are also nontrivial SPT phases
that have edge modes protected by symmetry. The SymTFT is also capable of
describing edge modes of SPTs, as we discuss now. To illustrate the principle
we consider the symmetry $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ that supports a
non-trivial SPT known as the cluster chain. The SymTFT in this case is the
quantum double of $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$, which can be viewed as
two copies of toric code. We label the anyons in each copy by
$e_{i},m_{i},f_{i},i=1,2$. Since the cluster chain is a gapped symmetric
system, it should be dual to a symmetric, fully-confining condensation of
$D(\mathbb{Z}_{2}\times\mathbb{Z}_{2})$. Consider the option
$\mathcal{A}^{SPT}=\langle e_{1}m_{2},e_{2}m_{1}\rangle$ that satisfies all
the conditions. We then consider a thin-slab construction where on the top
boundary we condense all charges $e_{1},e_{2}$, and on the bottom boundary
there is an interface between $\mathcal{A}^{SPT}$ condensation and
$\mathcal{A}^{m}=\langle m_{1},m_{2}\rangle$ condensation. The set up is
illustrated in Fig. 2. As shown in Fig 2, the symmetry of the system, in this
case a horizontal $m_{1}$ or $m_{2}$ string, can be localized to the edge of
the SPT and the localized symmetry actions form a projective representation of
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$. Therefore the condensation
$\mathcal{A}^{SPT}$ corresponds to a gapped
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetric 1+1 system with non-trivial
edge modes, i.e. the cluster state.

##  II SymTFT for GSPTS?

SymTFT has been applied widely to analyze gapped phases. As we have seen, it
is able to characterize both symmetric and symmetry-breaking phases. In the
symmetric case, SymTFT has been shown known to be able to reproduce the
classification of SPTs Zhang and Córdova (2023). Given this capacity, one
might wonder if SymTFT could shed light on gapless phases. A preliminary
exploration of this idea was performed in Chatterjee and Wen (2023b), where
the authors conjectured that SymTFT is capable of classifying 1+1D gapless
phases with symmetry. To elucidate, the thin slab construction indicates that
a symmetric gapless 1+1D system corresponds to a symmetric, partially-
confining condensation on the bottom boundary. Thus, it’s plausible that all
symmetric, partially-confining condensations of a 2+1D quantum double $D(G)$
represent the classification for all possible 1+1D $G$-symmetric gapless
systems. Furthermore, a local low-energy equivalence principle (LLEP) was
introduced in Chatterjee and Wen (2023b). If the topological order obtained by
condensing $\mathcal{A}$ in the topological order $D(G)$ is represented as
$D(G)/\mathcal{A}$, this principle suggests that the local low-energy
properties of a 1+1D system dual to a condensation $\mathcal{A}$ of $D(G)$,
are equivalent to those of a 1+1D system dual to the trivial condensation $1$
of $D(G)/\mathcal{A}$. Using LLEP, we can derive the local low-energy
properties of 1+1D systems dual to a given partially-confining condensation of
the SymTFT. As an example, by condensing the double flux $m^{2}$ in
$D(\mathbb{Z}_{4})$, we obtain a toric code topological order
$D(\mathbb{Z}_{2})$ formed by $e^{2}$ and $m$. Thus, the condensation $m^{2}$
corresponds to a 1+1D system exhibiting local low-energy properties identical
to the Ising critical point.

However, the LLEP does not fully clarify the correspondence between 1+1D
phases and anyon condensation of 2+1D quantum doubles. For example, there
could be multiple condensations giving rise to equivalent post-condensation
topological order, then LLEP suggests that these condensations describe 1+1D
systems with identical local low energy properties. However there are non-
local properties of 1+1D systems that can be used to distinguish different
phases, the LLEP does not clarify how these non-local properties are described
in the anyon condensation picture. As a more concrete example, all SPT phases
associated with a given symmetry have the same local low energy properties:
they are adiabatically connected to a trivial product state and there is no
local order parameter that can distinguish one SPT from another. However
different SPT phases are inequivalent quantum phases if symmetry is to be
preserved and they can distinguished by non-local string order parameters.
Therefore the LLEP must only be part of a more complete dictionary. In the
gapped phases/fully-confining condensation case this is well-understood:
different symmetric full-confining condensations correspond to different 1+1D
SPT phases. However it is still unclear what is the complete 1+1D
correspondence for a partially-confining condensation other than what the LLEP
provides.

On other hand, gSPTs are gapless systems with symmetry, therefore one might
wonder how does gSPT fit into the proposed SymTFT description of gapless
systems? Specifically, are there symmetric partially-confining condensations
that are dual to gSPTs? And if there are, how does anyon condensation describe
the properties of a gSPT, such as the symmetry extension structure, emergent
anomaly, and edge modes? We address these questions in the rest of this paper
and establish a complete framework for describing gSPTs by SymTFT. We will see
that symmetric partially-confining condensations are exactly dual to gSPTs,
thereby solidify the SymTFT for gapless systems conjecture. To illustrate, let
us examine the $\mathbb{Z}_{4}$-igSPT as a case study.

###  II.1 SymTFT for the $\mathbb{Z}_{4}$-igSPT

Figure 3: String order parameter of the $\mathbb{Z}_{4}$-igSPT in the thin
slab construction. The slab describes an effective 1+1D system that we
identify as the $\mathbb{Z}_{4}$-igSPT. On the top boundary we condense the
charge $e$, on the bottom boundary we condense $e^{2}m^{2}$. The $H$-shaped
operator in blue represents the string order parameter of the
$\mathbb{Z}_{4}$-igSPT. It is a decorated $U_{s^{2}}$ action, with ends
charged under $U_{s}$, i.e. a horizontal $m$-string. Figure 4: Edge modes of
the $\mathbb{Z}_{4}$-igSPT in the thin slab construction.The slab contains a
segment of igSPT with two interfaces to the trivial symmetric gapped phase
(vacuum). The $\mathbb{Z}_{2}^{A}$ action can be localized to the interfaces,
similar to the localization of symmetry actions of SPTs. The localized
$A$-symmetry action intersects the $G$-symmetry action ($m$-string) at one
point. Since $e^{2}$ and $m$ has mutual $\pi$ statistics, the two operators
anti-commute, giving rise to a 2-fold GSD.

The $\mathbb{Z}_{4}$-igSPT is a gapless system with a non-anomalous symmetry
$\mathbb{Z}_{4}$111The $\mathbb{Z}_{2}^{G}$ anomaly of this igSPT is only
emergent and the full system has no $\mathbb{Z}_{4}$ anomaly.. Thus we expect
that it is dual to a symmetric, partially-confining condensation of
$D(\mathbb{Z}_{4})$. A condensation that satisfies these conditions is
$\mathcal{A}=\langle e^{2}m^{2}\rangle$. Let us analyze this condensation.
After condensing $e^{2}m^{2}$, the deconfined anyons are generated by
$e^{2},em$. $e^{2}$ and $em$ both have order 2 after condensing $e^{2}m^{2}$:
$(e^{2})^{2}=1$ trivially, and $(em)^{2}$ is the condensed anyon, which is
identified as vacuum. Thus we have in total 4 inequivalent anyons after the
condensation: $1,e^{2},em,e^{3}m$. Interestingly, both $em$ and $e^{3}m$ are
not bosons: $em$ is a semion with $\theta_{em}=i$ and $e^{3}m$ is an anti-
semion with $\theta_{e^{3}m}=-i$. We conclude that the post-condensation
topological order is the twisted quantum double $D_{\omega}(\mathbb{Z}_{2})$,
also known as the double semion theory. The twist
$\omega\in\mathcal{Z}^{3}[\mathbb{Z}_{2},U(1)]$ is the Levin-Gu anomaly.

By invoking the LLEP, we see that the $e^{2}m^{2}$ condensation describes a
1+1D $\mathbb{Z}_{4}$ symmetric gapless system whose local low energy
properties are the same as that of a system with $\mathbb{Z}_{2}$ symmetry and
the Levin-Gu anomaly. This means the dual 1+1D system has a low energy sector
with an effective anomalous $\mathbb{Z}_{2}$ symmetry. This is exactly the
structure of the $\mathbb{Z}_{4}$-igSPT, therefore it is tempting to identify
the condensation $\mathcal{A}=\\{1,e^{2}m^{2}\\}$ with the
$\mathbb{Z}_{4}$-igSPT. How does one describe properties of the
$\mathbb{Z}_{4}$-igSPT other than the emergent anomaly in the SymTFT language?
Using the thin-slab construction, we show below that all physical properties
of the $\mathbb{Z}_{4}$-igSPT can be extracted from the $e^{2}m^{2}$
condensation.

####  II.1.1 String order parameter from SymTFT

An important feature of a non-trivial gSPT is the existence of non-trivial
string order parameters. The charges carried by ends of these string order
parameters can be used to distinguish different gSPT phases. In the
$\mathbb{Z}_{4}$-igSPT example, this can be seen by looking at the following
string operator:
$\mathcal{S}_{a,b}:=\sigma_{a}^{z}\prod_{i=a}^{b-1}e^{\frac{i\pi}{2}(1-\tau^{x}_{i+1/2})}\sigma^{z}_{b}$.
In the low energy space below $\Delta$, we have
$\langle\mathcal{S}_{a,b}\rangle=\langle\sigma^{z}_{a}\sigma_{a}^{z}\sigma^{z}_{b}\sigma^{z}_{b}\rangle=1$.
Therefore this string operator acquires non-zero vacuum expectation value.
Notice the end point of this string operator is charged under the $U_{s}$
symmetry:
$U_{s}\mathcal{S}_{-\infty,b}U_{s}^{\dagger}=-\mathcal{S}_{-\infty,b}$. Now
let us use the thin-slab construction with the $e^{2}m^{2}$-condensation to
reproduce this result. Following the standard prescription, we put the
$\mathcal{A}^{e}=\\{1,e,e^{2},e^{3}\\}$ condensation on the top boundary and
the $\mathcal{A}^{igSPT}=\\{1,e^{2}m^{2}\\}$ condensation on the bottom
boundary. As we show in figure Fig. 3, a non-local $H$-shaped operator can
then be constructed in the thin-slab. In the middle of this operator is an
$m^{2}$-string, while near the end of it the $m^{2}$ string joins an $e^{2}$
string from the top boundary to form an $e^{2}m^{2}$ string which is then
absorbed by the bottom boundary. The structure of this operator mimics that of
the operator $\mathcal{S}_{a,b}$ constructed before. Specifically, the end of
this operator anti-commutes with an $m$ string that intersects it, see Fig 3.
Since this operator does not create any excitations, it acts as identity on
the physical state represented by the thin-slab. Therefore it is a non-local
string order parameter that has non-zero vacuum expectation value, with ends
charged under the $U_{s}$ symmetry. We see that by utilizing the thin-slab
construction, one can reconstruct the string order parameters of a gSPT from
its dual anyon condensation.

####  II.1.2 Edge modes from SymTFT

Another important property of non-trivial gSPTs is the edge modes that appear
when the system is put on an open chain. To observe the edge modes of the
$\mathbb{Z}_{4}$-igSPT in the thin-slab construction, consider a setup where
the igSPT thin-slab is put in adjacent with slabs that represent the vacuum.
The vacuum is represented as a all-flux-condensed boundary. This can be seen
from, for example, the fact that condensing all fluxes on the bottom boundary
gives a symmetric gapped 1+1D system with no non-trivial string order
parameter. Therefore in this setup we have on the bottom boundary a segment of
$e^{2}m^{2}$-condensation that is adjacent to $m$-condensations. As we show in
Fig. 4, the symmetry action by $\mathbb{Z}_{2}^{A}$, i.e. a horizontal
$m^{2}$-string, can be localized to the interfaces between the igSPT and the
vacuum, and the localized $\mathbb{Z}_{2}^{A}$ symmetry action anti-commutes
with the symmetry action by $\mathbb{Z}_{2}^{G}$, i.e. a horizontal
$m$-string. This is exactly the algebra of symmetries of the $\mathbb{Z}_{4}$
igSPT: The edge of the gapped symmetry $\widetilde{U}_{s^{2}}$ is charged
under the gapless symmetry $U_{s}$.

${1}$${{\langle
e^{2}\rangle}}$${{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\langle
e\rangle}}}$${{\mathbb{Z}_{4}^{\Gamma}-\text{Critical}}}$${{\begin{matrix}\mathbb{Z}^{\Gamma}_{4}~{}\text{Broken
to}~{}\mathbb{Z}^{A}_{2}\\\
\mathbb{Z}^{A}_{2}-\text{Critical}\end{matrix}}}$${{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathbb{Z}^{\Gamma}_{4}-\text{Broken}}}}$${{\langle
e^{2}m^{2}\rangle}}$${{\mathbb{Z}_{4}\text{-igSPT}}}$${{\langle
m^{2}\rangle}}$${{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\langle
e^{2},m^{2}\rangle}}}$${{\begin{matrix}\mathbb{Z}_{2}^{G}-\text{Critical}\\\
\mathbb{Z}_{2}^{A}-\text{paramagnet}\end{matrix}}}$${{\begin{matrix}{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathbb{Z}_{4}^{\Gamma}~{}\text{Broken
to}~{}\mathbb{Z}^{A}_{2}}\\\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathbb{Z}^{A}_{2}-\text{Paramagnet}}\end{matrix}}}$${{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\langle
m\rangle}}}$${{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mathbb{Z}_{4}-\text{Paramagnet}}}}$
Figure 5: Stability of the $\mathbb{Z}_{4}$-igSPT from SymTFT On the left we
draw a diagram where vertices are condensations of $D(\mathbb{Z}_{4})$, and
there is an arrow between two condensations if one is a subset of the other,
the arrow is pointing in the direction of the larger set. The condensations in
red are fully-confining. On the right we draw the same diagram but with
vertices replaced by the corresonding 1+1D phases, the phases in red are
gapped. The $\mathbb{Z}_{4}$-igSPT is only adjacent to the
$\mathbb{Z}_{4}$-Ising critical point and the partially symmetry broken phase,
therefore the gaplessness of the igSPT is protected.

####  II.1.3 Stability of the igSPT from SymTFT

Starting with a 1+1D phase dual to a condensation $\mathcal{A}$, going to an
adjacent phase amounts to adding anyons to the list $\mathcal{A}$ or deleting
some anyons from $\mathcal{A}$ Chatterjee and Wen (2023b). This relation
constrains the phase diagram of the dual 1+1D system. Let us now list all
possible condensations of the $\mathbb{Z}_{4}$ Toric code and their relations,
shown in Fig. 5. In this diagram, we connect two condensations if one is a
subset of the other. It is clear from the diagram that the
$\mathbb{Z}_{4}$-igSPT is stable: although the system has no
$\mathbb{Z}_{4}$-anomaly, in order to go to the gapped
$\mathbb{Z}_{4}$-symmetric phases($\mathbb{Z}_{4}$ paramagnet) the system has
to either go through the $\mathbb{Z}_{4}$-critical point where the gap to the
$A$-dof is closed, or open up a gap for the $G$-dof by breaking the
$\mathbb{Z}_{2}^{G}$ symmetry and then restore the symmetry by condensing
domian walls of $\mathbb{Z}_{2}^{G}$ and going through a
$\mathbb{Z}_{2}^{G}$-critical point. From the anyon condensation perspective,
the anyon $m$ can not be added to the condensation $\\{1,e^{2}m^{2}\\}$
because it braids non-trivially with $e^{2}m^{2}$, this indicates a direct
transition from $\mathbb{Z}_{4}$-igSPT to $\mathbb{Z}_{4}$-paramagnet is
impossible. This is an instance of “anomaly protected gaplessness” that
appears in all igSPTs, where the existence of an emergent anomaly gives
obstruction to directly gapping out the system while preserving the symmetry.
We see that the relation between different anyon condensation patterns
provides a SymTFT description of this mechanism.

The $\mathbb{Z}_{4}$-igSPT example demonstrates that the SymTFT is capable of
providing comprehensive characterization of 1+1D gSPTs. Therefore it is
plausible that every $\Gamma$-gSPT is dual to certain symmetric, partially-
confining condensation of the quantum double $D(\Gamma)$. However in order to
fully establish such a duality it is crucial to understand the classification
of two types of objects in two categories: on one hand the classification of
gSPTs in 1+1D, and on the other hand the classification of symmetric partially
confining condensations of quantum doubles. This will be the goal of the rest
of this work. After carefully examining the structure of 1+1D gSPTs, we
provide a classification of 1+1D gSPTs protected by internal unitary finite
symmetry. The classification itself turns out to provide us some surprising
insights into the structure of 1+1D gSPTs. For example, we find that there is
an edge/bulk (0+1D/1+1D) correspondence for 1+1D gSPTs in a strict sense to be
defined. After having a complete characterization of 1+1D gSPTs, we review the
established theory of anyon condensation and discuss the structure of
symmetric partially confining condensations of 2+1D quantum doubles as well as
its connection to 1+1D gSPTs. A 1-1 correspondence between the two will by
then become evident.

##  III Structure of 1+1D bosonic gSPTs

In order to establish formally the connection between 1+1D gSPTs and their
SymTFT dual, it is crucial to have a complete understanding of the structure
of 1+1D gSPTs. This is the task of this section.

###  III.1 Algebra of symmetry and consistency conditions

Let us consider a general 1+1D gSPT with a gapped sector and a gapless sector
having the structure of a symmetry extension

$\displaystyle 1\rightarrow
A\xrightarrow[]{j}\Gamma\xrightarrow[]{p}G\rightarrow 1.$ (6)

Here $A\trianglelefteq\Gamma$ is a normal subgroup of $\Gamma$, not
necessarily Abelian. Being a normal subgroup, we have $\gamma A\gamma^{-1}=A$
for any $\gamma\in\Gamma$, and this defines a $\Gamma$-action on $A$. We ask
what extra data besides the group extension is needed to specify a gSPT. From
our experience with SPTs, we know that the low energy physics of an SPT is
completely specified by the anomalous symmetry action on the boundary. For a
1+1D SPT, the algebra of edge symmetry action, namely the projective
representation of edge symmetry actions, is the origin of the SPT edge modes.
Motivated by this, we look at symmetry actions and their algebra in a 1+1D
gSPT, this algebra will similarly determine the edge modes of the gSPT. First
of all the UV $A$-symmetry actions act trivially in the bulk in the low energy
sector, since by definition there is no $A$-charges in the gapless sector in
the bulk. Therefore on a closed chain we expect to have $U^{\text{IR}}_{a}=1$.
In the following we will drop the IR superscript and it is understood that
operators are restricted to the gapless low energy subspace. However on an
open chain the $A$-action can still act non-trivially on the edge of the
chain, this is known as symmetry fractionalization principle for symmetries
acting on gapped degrees of freedom Pollmann  _et al._ (2010); Turner  _et
al._ (2011); Verresen  _et al._ (2017). Therefore an $A$-symmetry action
$U_{a}$ reduces to an edge action $U_{a}=U_{a}^{L}\otimes U_{a}^{R}$ below the
gap to $A$-dof. Similar to the edge symmetry action of SPTs, the localized
edge actions $\widetilde{U}_{a}:=U_{a}^{L}$ only need to form a projective
representation of the gapped symmetry group $A$:
$\widetilde{U}_{a}\widetilde{U}_{b}=\eta(a,b)\widetilde{U}_{ab}$, where
$\eta\in\mathcal{Z}^{2}[A,U(1)]$ is a 2-cocycle of $A$. We also need to
consider the interplay between the localized edge action $\widetilde{U}_{a}$
and the non-localized $G$-action. Consider a general $\gamma\in\Gamma$ action,
and the conjugation $U_{\gamma}\widetilde{U}_{a}U_{\gamma}^{-1}$, the group
law states that $U_{\gamma}U_{a}U_{\gamma}^{-1}=U_{\gamma a\gamma^{-1}}$,
however using the decomposition $U_{a}=U^{L}_{a}\otimes U^{R}_{a}$, we have

$\displaystyle U_{\gamma}U_{a}^{L}U_{\gamma}^{-1}\otimes
U_{\gamma}U^{R}_{a}U_{\gamma}^{-1}=U^{L}_{\gamma a\gamma^{-1}}\otimes
U^{R}_{\gamma a\gamma^{-1}}.$ (7)

Notice that $U_{\gamma}U_{a}^{L,R}U_{\gamma}^{-1}$ is an operator supported on
the left/right edge: away from the edge $U^{L,R}_{a}$ acts as identity and
$U_{\gamma}U_{a}^{L,R}U_{\gamma}^{-1}=U_{\gamma}U_{\gamma}^{-1}=1$. Therefore
both sides of (7) decompose into tensor products of operators acting on
separate Hilbert spaces, thus each factor must be equal up to a phase:

$\displaystyle
U_{\gamma}\widetilde{U}_{a}U_{\gamma}^{-1}=\epsilon(a,\gamma)\widetilde{U}_{\gamma
a\gamma^{-1}}$ (8)

The phase $\epsilon(a,\gamma)$ can be thought of as measuring the charges of
the fractionalized $\widetilde{U}_{a}$ action under $U_{\gamma}$. The data
$(\eta,\epsilon)$ naturally satisfies a set of consistency conditions. For
example, $U_{\gamma}(\widetilde{U}_{a}\widetilde{U}_{b})U_{\gamma}^{-1}$
should be equal to
$(U_{\gamma}\widetilde{U}_{a}U_{\gamma}^{-1})\cdot(U_{\gamma}\widetilde{U}_{b}U_{\gamma}^{-1})$,
which leads to an equation

$\displaystyle\eta(a,b)\epsilon(ab,\gamma)=\eta(\gamma a\gamma^{-1},\gamma
b\gamma^{-1})\epsilon(a,\gamma)\epsilon(b,\gamma).$ (9)

We perform careful analysis of consistency conditions in appendix A and
summarize them here:

Figure 6: The algebra of symmetry action of a gSPT. 1\. UV symmetry actions
$U_{a},a\in A$ reduce to edge actions in the gapless sector, and the
restriction of them to one of the endpoints make up a projective
representation of $A$ given by a 2-cocycle $\eta$. 2. The restriction of an
$A$-symmetry action to one of the endpoints fails to satisfy the commutation
relation with another symmetry action.

A 1+1D $\Gamma$-gSPT is defined by $(A\lhd\Gamma,\eta,\epsilon)$, where $A$ is
a central subgroup of $\Gamma$, and $\eta,\epsilon$ satisfy the following
conditions:
$\displaystyle\eta\in\mathcal{Z}^{2}[A,U(1)],~{}\epsilon:A\times\Gamma\to
U(1)$
$\displaystyle\frac{\epsilon(a,\gamma)\epsilon(b,\gamma)}{\epsilon(ab,\gamma)}=\frac{\eta(a,b)}{\eta(\gamma
a\gamma^{-1},\gamma b\gamma^{-1})}$ (10)
$\displaystyle\epsilon(a,\gamma_{1}\gamma_{2})=\epsilon(\gamma_{2}a\gamma_{2}^{-1},\gamma_{1})\epsilon(a,\gamma_{2})$
(11) $\displaystyle\epsilon(a,b)=\frac{\eta(b,a)}{\eta(bab^{-1},b)},~{}\forall
a,b\in A$ (12)

Moreover, the decomposition $U_{a}=U^{L}_{a}\otimes U^{R}_{a}$ only defines
the localized $A$-action $U^{L,R}_{a}$ up to a phase. Redefining
$U^{L}_{a}\to\alpha(a)U^{L}_{a},U^{R}_{a}\to\alpha^{*}(a)U^{R}_{a}$ does not
change the overall symmetry action $U_{a}$. This will have the effect of
modifying the 2-cocycle $\eta$ by a coboundary, and will also affect the
$\epsilon$ factor by
$\epsilon(a,\gamma)\to\epsilon(a,\gamma)\frac{\alpha(\gamma
a\gamma^{-1})}{\alpha(a)}$. Since the choice of the phase factor $\alpha$ is
artifitial, two solutions to the consistency conditions related by such a
transformation should be viewed as defining the same gSPT.  Two sets of data
$(\eta,\epsilon),~{}(\eta^{\prime},\epsilon^{\prime})$ define the same gSPT,
if they are related by
$\displaystyle\eta^{\prime}(a,b)=\eta(a,b)\frac{\alpha(a)\alpha(b)}{\alpha(ab)}$
(13)
$\displaystyle\epsilon^{\prime}(a,\gamma)=\epsilon(a,\gamma)\frac{\alpha(\gamma
a\gamma^{-1})}{\alpha(a)}.$ (14) We will refer to this equivalence relation as
gauge equivalence, and a transformation of a pair $(\eta,\epsilon)$ by the
equivalence relation as a gauge transformation.The set of equivalent classes
of solutions $(\eta,\epsilon)$ to the consistency conditions is our
classification of 1+1D bosonic gSPT with unitary symmetry.

##### Finiteness of the classification.

It is not immediately clear that the classification above is finite or not,
although in many cases it can be verified by hand that the solutions are
finite modulo gauge transformation. However we will show that gSPTs are in 1-1
correspondence with symmetric partially confining condensations of 2+1D
quantum doubles, and the latter always has a finite classification for any
finite gauge group. Thus the SymTFT dual will provide a definite answer to the
finiteness question, at least for finite $\Gamma$. Thus there are only finite
many gSPTs in 1+1D for any finite symmetry group $\Gamma$. A seemingly
contradiction is that we can always stack a gSPT with a CFT that transforms
trivially under the symmetry without changing the data $(\eta,\epsilon)$. The
resolution is that we should view the classification as kinematic and does not
determine what CFT lives in the bulk. That is, it classifies possible
$\Gamma$-symmetry enriching patterns of all $G$-symmetric gapless states.
Alternatively, one can take the bulk phase as an extra data that needs to be
specified besides $(\eta,\epsilon)$. However the data $(\eta,\epsilon)$ does
put constraints on the possible gapless state. For example, the gapless sector
needs to have nontrivial $G$-symmetry, since if $G$-symmetry acts trivially on
the gapless sector, it would have been identified as a gapped symmetry. Also,
as we will show soon, the pair $(\eta,\epsilon)$ determines the emergent
anomaly of the bulk, which also needs to be matched by the gapless sector.

##### Group structure of the classification.

There is an obvious group structure of the classification. Namely since the
consistency conditions are all multiplicative in $\eta,\epsilon$, the product
of two solutions is again a solution. The solution $\eta=1,\epsilon=1$ serves
as group identity, and the inverse of an element $(\eta,\epsilon)$ is
$(\eta^{-1},\epsilon^{-1})$. It is tempting to interpret the group structure
as the physical operation of stacking two gSPTs together, however there is a
subtlty. For example, if we stack a gSPT with a CFTA bulk and another with a
CFTB bulk, then the stacked system will have CFT${}^{A}\times$CFTB bulk.
Therefore the stacking operation only defines a group structure modulo the
bulk state. This is tight to the fact that the classification is really a
classification of symmetry enriching patterns.

##### Invariants of gSPTs.

The data $\eta,\epsilon$ provides some discrete invariants for a gSPT which
will be useful in practice. Consider a given $a\in A$, let $C_{a}(\Gamma)$ be
the centralizer of $a$ in $\Gamma$, i.e. the subgroup consisting of elements
that commute with $a$. Then we have
$U_{c}\widetilde{U}_{a}U_{c}^{\dagger}=\epsilon(a,c)\widetilde{U}_{a},~{}\forall
c\in C_{a}(\Gamma)$, and the factor $\epsilon(a,c)$ is now gauge invariant: a
gauge transformation changes it to
$\epsilon^{\prime}(a,c)=\epsilon(a,c)\frac{\alpha(caa^{-1})}{\alpha(a)}=\epsilon(a,c)$.
Therefore $\epsilon$ restricted to $\\{a\\}\times C_{a}(\Gamma)$ provides an
invariant for any $a\in A$. This is nothing but the charge carried by
$\widetilde{U}_{a}$ under $C_{a}(\Gamma)$. In general $\widetilde{U}_{a}$ does
not have a well-defined $\Gamma$-charge, since a $U_{\gamma}$ conjugation maps
it to $\widetilde{U}_{\gamma a\gamma^{-1}}$ up to a phase factor. Restricting
to $C_{a}(\Gamma)$ gives a well-defined charge of $\widetilde{U}_{a}$.
However, we should stress that in general these invariants do not determine
the gSPT completely. The method of using charges of edge symmetry actions as
invariants also applies to 1+1D SPTs, and it is known that there are examples
where such charges do not fully determine the SPT (although the minimal
counter-example requires a symmetry group of order 128) Pollmann and Turner
(2012). Put the counter-example as the gapped sector will give a counter-
example for gSPTs.

###  III.2 Emergent anomaly of gSPT and a bulk-edge (1+1D/0+1-D)
correspondence

We next show that the data $(\eta,\epsilon)$ determines the emergent anomaly
of the gSPT. Since the data $(\eta,\epsilon)$ are directly related to the edge
modes of the gSPT, the result that they determine the emergent anomaly can
then be viewed as an edge-to-bulk map for gSPTs: the edge modes determine the
emergent anomaly. What’s interesting is that the converse is also (partially)
true: the emergent anomaly determines $(\eta,\epsilon)$ up to stacking with a
$\Gamma$-SPT. Therefore there is a bulk-edge correspondence222not to be
confused with the holographic duality between 1+1D gSPTs and 2+1D quantum
doubles. for gSPTs in the following sense: the edge modes determine the bulk
emergent anomaly, and the bulk emergent anomaly determines the edge modes up
to stacking with an SPT.

There are more than one way to detect the anomaly of a symmetry. For example
anomaly amounts to obstruction to consistently coupling the system to a
background gauge field, equivalently anomaly is the obstruction to
consistently inserting a defect network of the symmetry. A practical way of
computing the anomaly when the symmetry action is explicitly known is provided
by Else and Nayak (2014), where anomaly is formulated as the obstruction to
implementing the symmetry on a system with boundary. Here we review this
approach briefly.

For a symmetry action $U_{g},~{}g\in G$ acting on a 1+1D system with open
boundary condition, if we compare $U_{g}U_{h}$ and $U_{gh}$, they may fail to
be equal up to operators supported on the edge:

$\displaystyle U_{g}U_{h}=\Omega(g,h)U_{gh},$ (15)

where $\Omega(g,h)$ is a local unitary operator supported on the edge:
$\Omega(g,h)=\Omega^{L}(g,h)\otimes\Omega^{R}(g,h)$. From the definition (15)
the operators $\Omega$ must satisfy a condition from the associativity of the
$U_{g}$ operators,

$\displaystyle(U_{g}U_{h})U_{k}=U_{g}(U_{h}U_{k})$
$\displaystyle\Rightarrow\Omega(g,h)\Omega(gh,k)=U_{g}\Omega(h,k)U_{g}^{-1}\Omega(g,hk).$
(16)

However, since the operator $\Omega$ splits into tensor product of
$\Omega^{L}$ and $\Omega^{R}$, the restriction of it to one of the edges only
need to satisfy the consistency condition up to a phase:

$\displaystyle\omega_{3}(g,h,k)\Omega^{L}(g,h)\Omega^{L}(gh,k)=U_{g}\Omega^{L}(h,k)U_{g}^{-1}\Omega^{L}(g,hk).$
(17)

The phase $\omega_{3}$ can be shown to be a 3-cocycle and represents the
anomaly of the symmetry action $U_{g}$.

We now show that the data $\eta,\epsilon$ suffice to determine the emergent
anomaly of a gSPT. The group extension (6) defines a 2-cocycle of $G$ with $A$
coefficients: $e_{2}\in Z^{2}[G,A]$, with $G$ acting on $A$ by conjugation.
The role of this 2-cocycle is to specify the group law of $\Gamma$ from that
of $G$ and $A$. One can write elements of $\Gamma$ as pairs $\gamma=(a,g),a\in
A,g\in G$. The group law of $\Gamma$ is then given by $(a,g)\cdot(b,h)=(a\cdot
gbg^{-1}\cdot e_{2}(g,h),gh)$. Now with this understanding of group structure
of $\Gamma$, one can compute the composition of symmetry actions by $G$:

$\displaystyle U_{g}U_{h}=U_{(e_{2}(gh),gh)}=U_{e_{2}(g,h)}U_{gh}.$ (18)

The operator $U_{e_{2}(g,h)}$ when restricted to the gapless sector acts only
non-trivially on the edge, therefore the $G$-action(when restricted to the
gapless sector) does take the form of (15), with $\Omega(g,h)=U_{e_{2}(g,h)}$
and $\Omega^{L}(g,h)=\widetilde{U}_{e_{2}(g,h)}$. Now we can calculate the
anomaly according to equation (17):

$\displaystyle\Omega^{L}(g,h)\Omega^{L}(gh,k)=\widetilde{U}_{e_{2}(g,h)}\widetilde{U}_{e_{2}(gh,k)}$
$\displaystyle=\eta(e_{2}(g,h),e_{2}(gh,k))\widetilde{U}_{e_{2}(g,h)e_{2}(gh,k)},$
(19) $\displaystyle
U_{g}\Omega^{L}(h,k)U_{g}^{-1}\Omega^{L}(g,hk)=U_{g}\widetilde{U}_{e_{2}(h,k)}U_{g}^{-1}\widetilde{U}_{e_{2}(g,hk)}$
$\displaystyle=\epsilon(e_{2}(h,k),g)\widetilde{U}_{ge_{2}(h,k)g^{-1}}\widetilde{U}_{e_{2}(g,hk)}$
$\displaystyle=\epsilon(e_{2}(h,k),g)\eta(ge_{2}(h,k)g^{-1},e_{2}(g,hk))\widetilde{U}_{ge_{2}(h,k)g^{-1}e_{2}(g,hk)}$
(20)

The cocycle condition on $e_{2}$ is

$\displaystyle ge_{2}(h,k)g^{-1}e_{2}(g,hk)=e_{2}(g,h)e_{2}(gh,k),$ (21)

therefore the last lines of (19) and (20) only differ by a phase factor, which
is the anomaly 3-cocycle of the $G$-symmetry action:
$\displaystyle\omega_{3}(g,h,k)=\frac{\eta(ge_{2}(h,k)g^{-1},e_{2}(g,hk))}{\eta(e_{2}(g,h),e_{2}(gh,k))}\epsilon(e_{2}(h,k),g)$
(22)

##### Trivial vs. nontrivial extension.

From the expression (22) it is clear that if the group extension of a gSPT is
trivial, then $e_{2}=1$ and the emergent anomaly vanishes. Thus in order to
have an emergent anomaly the group extension must be non-trivial, i.e. the
full symmetry group $\Gamma$ can not be a direct product or semi-direct
product of the gapped symmetry $A$ and the gapless symmetry $G$.

##### gSPTs of SPT-pump type.

Any solution to the consistency conditions (10)-(12) defines a gSPT and
determines the emergent anomaly through (22). Consider a special case where
$\eta=1$ and the group extension is central. In this case elements of $A$
commute with all elements of $\Gamma$: $\gamma a\gamma^{-1}=a$, and the
consistency conditions on $\epsilon$ reduce to

$\displaystyle\epsilon(a,\gamma)\epsilon(b,\gamma)=\epsilon(ab,\gamma)$ (23)
$\displaystyle\epsilon(a,\gamma_{1}\gamma_{2})=\epsilon(a,\gamma_{1})\epsilon(a,\gamma_{2})$
(24) $\displaystyle\epsilon(a,b)=1~{}\forall a,b\in A.$ (25)

The first two conditions state that $\epsilon$ is now linear in both factors,
and the last condition states that the $\epsilon$ is well-defined on the
quotient $A\times\Gamma/A=A\times G$. Viewed from the original definition,
$U_{\gamma}\widetilde{U}_{a}U_{\gamma}^{-1}=\epsilon(a,\gamma)\widetilde{U}_{a}$,
these conditions imply that now every $a\in A$ action carries a well-defined
charge under $G$. The anomaly according to (22) is
$\omega_{3}(g,h,k)=\epsilon(e_{2}(h,k),g)=\epsilon\cup e_{2}(g,h,k)$, and the
edge modes of the gSPT can be intuitively understood as $A$-actions pumping
$G$-charges (which are 0+1-D SPTs of $G$). Such gSPTs have been studied in Wen
and Potter (2023) and the SPT-pump mechanism was also used to understand
igSPTs in higher dimensions.

##### Bulk-edge (1+1D/0+1-D) correspondence for gSPT

Interestingly the emergent anomaly determines the data $(\eta,\epsilon)$ up to
stacking with $\Gamma$-SPTs. This means up to $\Gamma$-SPTs the physics of a
$\Gamma$-gSPT is completely determined by its emergent anomaly. This has an
immediate consequence that the two definitions of igSPT are equivalent: 1. A
gSPT is intrinsic if the edge modes can not be trivialized by stacking with an
SPT. 2. A gSPT is intrinsic if it has a nontrivial emergent anomaly. We
provide a comprehensive derivation of these statements in appendix C. The idea
of the derivation can be summarized as follows. Denote the group extension as

$\displaystyle 1\rightarrow
A\xrightarrow[]{j}\Gamma\xrightarrow[]{p}G\rightarrow 1$ (26)

Since the anomaly is trivialized upon pull-back by $p$, we have on the cochain
level $p^{*}(\omega)=d(\alpha)$, where $\alpha\in\mathcal{C}^{2}[\Gamma,U(1)]$
is a 2-cochain of $\Gamma$. The composition of maps $p\circ i$ is trivial
since the sequence 26 is exact: $p\circ j=0$. This implies the composition of
pull-backs $(p\circ j)^{*}=j^{*}\circ p^{*}=0$ is also trivial. Therfore
$0=j^{*}\circ p^{*}(\omega)=j^{*}(d\alpha)=d(j^{*}\alpha)$, meaning that
$j^{*}\alpha$ is a 2-cocycle of $A$ representing a 1+1D $A$-SPT. This is to be
identified with the SPT of the gapped sector, i.e. $\eta=j^{*}\alpha$.
$\epsilon$ can be found by considering the value $\alpha((a,1),(1,g))$. Since
the trivializing cochain $\alpha$ is not unique, this procedure only
determines $\eta,\epsilon$ up to stacking with $\Gamma$-SPTs. In section V.3
we will address the question of stacking gSPTs with SPTs.

##  IV Condensation of quantum double and gSPT

The $\mathbb{Z}_{4}$-igSPT example suggests that symmetric partially-confining
condensations in the 2+1D quantum double $\mathcal{D}(\Gamma)$ correspond to
$\Gamma$-gSPTs. Now that we have a classification of 1+1D gSPTs, we are in
position to establish this correspondence in a rigorous manner. We first
briefly summarize the results.

###  IV.1 Summary of results

The SymTFT for a symmetry $\Gamma$ is the quantum double $D(\Gamma)$, also
known as the gauge theory with gauge group $\Gamma$. Anyons of $D(\Gamma)$ are
gauge charges, gauge fluxes and composites of charge and fluxes (dyons). An
anyon condensation amounts to specifying a set of condensed anyons and also a
fusion channel for fusing any two condensed anyons. For quantum doubles, an
anyon condensation can be specified as follows. One can first separate the
condensation into two steps. In the first step we only condense pure charges,
which will lead to gauge symmetry breaking from $\Gamma$ to some subgroup
$H<\Gamma$. This is dual to a global symmetry breaking in the SymTFT setup,
therefore for describing symmetric 1+1D systems one can neglect this step and
set $H=\Gamma$. In the second step we only condense fluxes and/or dyons. In
general a dyon has a flux sector and a charge sector. The contents of the
condensed dyons and their fusion channels are then specified by 3 objects: a
normal subgroup $A<\Gamma$ and two phase factors $\epsilon:A\times\Gamma\to
U(1),\eta:\Gamma\times\Gamma\to U(1)$. $A$ specifies the flux sectors of the
condensed dyons, $\epsilon$ determines the charge sector of each condensed
dyon, and $\eta$ determines the fusion phase of fusing two condensed dyons. In
general fluxes take value in $\Gamma$. Then the condensed dyons must have flux
sectors taking values in the normal subgroup $A$. For a given dyon with flux
sector $a\in A$, $\epsilon(a,\gamma)$ then measures the charge sector. It can
be thought of as the phase that is picked up by braiding a flux of value
$\gamma$ around the dyon with flux sector $a$. By collecting the phases
$\epsilon(a,\gamma)$ resulting from braiding with all the fluxes one can then
determine the charge sector of the dyon. Finally $\eta(a,b)$ is the fusion
phase of fusing two condensed dyons with flux sectors $a\in A$ and $b\in A$.
One immediately sees that the set of data needed to specify a symmetric
condensation is identical to that is needed to specify a 1+1D $\Gamma$-gSPT.
It can be shown that in order for the data $A,\epsilon,\eta$ to define a
consistent boson condensation, they must satisfy certain consistency
conditions. These turn out to be identical to the consistency conditions of
1+1D gSPTs (10)-(12). Moreover, a condensation gives rise to a new topological
order. It can be shown that the condensation specified by the data
$A,\epsilon,\eta$ is fully-confining if and only if $A=\Gamma$. And when
$A\neq\Gamma$, the post-condensation topological order is in general a twisted
quantum double of the quotient group $G:=\Gamma/A$: $D_{\omega}(G)$, and the
twist $\omega$ turns out to be given by the same formula as (22). This then
establishes a 1-1 correspondence between symmetric partially-confining
condensations of $D(\Gamma)$ and $\Gamma$-gSPTs.

In the rest of this section we review the formal theory of anyon condensation
and discuss the structure of anyon condensation in 2+1D quantum doubles.

###  IV.2 Review of condensation in 2+1D topological orders

We first review aspects of UMTC relevant to our discussion and then discuss
the description of anyon condensation in this language. A general 2+1D
topological order is described by a unitary modular tensor category(UMTC)
$\mathcal{C}$. Objects of $\mathcal{C}$ correspond to anyons of the
topological order. A UMTC is equipped with a tensor product of pairs of
objects $\otimes$, which describes the fusion of anyons, an associator
$\alpha_{X,Y,Z}:(X\otimes Y)\otimes Z\to X\otimes(Y\otimes Z)$ that relates
different orders of fusing three anyons, and a braiding $c_{X,Y}:X\otimes Y\to
Y\otimes X$ that describes the process of exchanging two anyons. The data
$\alpha_{X,Y,Z},c_{X,Y}$ satisfy certain consistency conditions, including the
pentagon equation of the associators and the hexagon equation that relates the
associator with the braiding Kitaev (2006); Bonderson (2012); Kong and Zhang
(2022).

A condensation amounts to bringing a large number of anyons in a region
together to form a new phase from $\mathcal{C}$. This physical process
corresponds to obtaining a new UMTC from $\mathcal{C}$ by specifying a
condensable algebra $\mathcal{A}$ of $\mathcal{C}$ Kong (2014). A condensable
algebra describes the contents of condensed anyons, if several simple anyons
$a_{i}$ are condensed, we can view $\mathcal{A}$ as the composite anyon
$\mathcal{A}=a_{1}+a_{2}+\cdots$. After condensing $\mathcal{A}$, we obtain a
new phase $\mathcal{D}$, with a new fusion rule $\otimes_{\mathcal{D}}$. In
the new phase $\mathcal{A}$ is identified as the vacuum,
$\mathcal{A}=1_{\mathcal{D}}$. The vacuum of the new phase must fuse trivially
with itself: $\mathcal{A}\otimes_{\mathcal{D}}\mathcal{A}=\mathcal{A}$. Since
$\mathcal{A}$ is also an anyon of the original phase, and fusion channels of
the new phase $\mathcal{D}$ come from those of the original phase
$\mathcal{C}$, this means when viewed as an anyon of $\mathcal{C}$, there must
be a nontrivial fusion channel
$\mu:\mathcal{A}\otimes_{\mathcal{C}}\mathcal{A}\to\mathcal{A}$. This makes
$\mathcal{A}$ an algebra in $\mathcal{C}$. There are extra conditions on the
algebra due to the physical requirements that the condensed anyons should be
self and mutual bosons. These conditions make $\mathcal{A}$ a so called
connected separable commutative algebra, or a condensable algebra for
simplicity. Now consider an excitation $M$ that lives in the post-condensation
phase $\mathcal{D}$, the fusion $M\otimes_{\mathcal{D}}\mathcal{A}$ is equal
to $M$ since now $\mathcal{A}$ is the vacuum:
$M\otimes_{\mathcal{D}}\mathcal{A}=M\otimes_{\mathcal{D}}1_{\mathcal{D}}=M$.
On the other hand both $M$ and $A$ are also anyons(possibly composite ones) of
the original phase, and fusion channels of $\mathcal{D}$ come from that of the
origin phase $\mathcal{C}$. Therefore when viewed as anyons of $\mathcal{C}$,
there should be a nontrivial fusion channel
$\nu_{M}:M\otimes_{\mathcal{C}}\mathcal{A}\to M$. This makes $M$ a module over
$\mathcal{A}$. For example, in the toric code the composite anyon $m+f$ is a
module over $1+e$, therefore is an excitation after condensing $1+e$.
Deconfined anyons in the new phase must braid trivially with vacuum, then when
viewed as anyons of the original phase, this means the diagram

${{M\otimes_{\mathcal{C}}\mathcal{A}}}$${M}$${{\mathcal{A}\otimes_{\mathcal{C}}M}}$${{M\otimes_{\mathcal{C}}\mathcal{A}}}$$\scriptstyle{\nu_{M}}$$\scriptstyle{c_{M,\mathcal{A}}}$$\scriptstyle{c_{\mathcal{A},M}}$$\scriptstyle{\nu_{M}}$

commutes for any deconfined anyon represented by a module $M$. This condition
makes $M$ a so called local module over $\mathcal{A}$. Thus the deconfined
excitations in the new phase are given by local modules over $\mathcal{A}$ in
the original phase. The category of local modules over $\mathcal{A}$,
$\mathcal{C}^{loc}_{\mathcal{A}}$ has a full UMTC structure induced from
$\mathcal{C}$ Kong (2014), therefore describes a valid topological order.

###  IV.3 Condensation of 2+1D quantum double

Let us now focus on a specific kind of topological order of interests to us,
the 2+1D quantum doubles. The quantum double $D(\Gamma)$ is a gauge theory
with gauge group $\Gamma$. The anyons in $D(\Gamma)$ can generally be
identified as gauge charges, gauge fluxes, and composites of charge and flux.
Gauge fluxes are in general labelled by conjugation classes of the group
$\Gamma$, and the charges are labelled by irreducible representations of the
group. A unified way of describing anyons of $D(\Gamma)$ is to identify an
anyon of $D(\Gamma)$ as a $\Gamma$-graded vector space
$V=\oplus_{\gamma\in\Gamma}V_{\gamma}$, together with a $\Gamma$-action, such
that $\gamma(V_{\gamma^{\prime}})=V_{\gamma\gamma^{\prime}\gamma^{-1}}$. In
this way a charge corresponds to the case where the graded vector space is
only supported at the identity element: $V=V_{1_{\Gamma}}$, then the anyon is
completely determined by a representation of the group $\Gamma$, recovering
the fact that charges are labelled by representations of $\Gamma$. On the
other hand a flux corresponds to the case where the vector space is supported
on at least one nontrivial element while the $\Gamma$-action is trivial. The
vector space has to be supported on conjugation classes of $\Gamma$ to be
invariant under the $\Gamma$-action, therefore a flux is labelled by a
conjugation class of $\Gamma$. A generic bond state of charge and flux then
corresponds to a graded vector space with nontrivial support as well as
nontrivial $\Gamma$-action.

Condensable algebras of $D(\Gamma)$ have been classified in Davydov (2009). A
condensation in $D(\Gamma)$ can always be done in two steps. In the first
step, only charges are condensed, reducing $D(\Gamma)$ to $D(H)$, for some
subgroup $H<\Gamma$. This step can be thought of as gauge symmetry breaking
from $\Gamma$ to $H$. The condensable algebra for this step is given by the
algebra of functions over the coset $\Gamma/H$: $\mathbb{C}(\Gamma/H)$. As an
element of $D(\Gamma)$, $\mathbb{C}(\Gamma/H)$ is a graded vector space
supported only at the identify $1_{\Gamma}$, with the $\Gamma$-action defined
by $(\gamma\cdot f)([\gamma^{\prime}]):=f([\gamma^{\prime}\gamma^{-1}])$ for
any function $f$ on the coset $\Gamma/H$. Clearly this defines a
representation of $\Gamma$ and corresponds to a condensation of charges. Local
modules over $\mathbb{C}(\Gamma/H)$ can be shown to form a UMTC equivalent to
the quantum double $D(H)$. In the second step no charges are condensed, and
the condensable algebra is determined as follows:

(Davydov (2009), Thm. 3.5.3) A condensable algebra of $D(H)$ that does not
condense any charges are determined by $\eta\in H^{2}(H,U(1))$, a normal
subgroup $A\lhd H$, and a function $\epsilon:A\times H\to U(1)$, such that the
following conditions are met:

$\displaystyle\frac{\epsilon(a,h)\epsilon(b,h)}{\epsilon(ab,h)}=\frac{\eta(a,b)}{\eta(hah^{-1},hbh^{-1})}$
(27)
$\displaystyle\epsilon(a,h_{1}h_{2})=\epsilon(h_{2}ah_{2}^{-1},h_{1})\epsilon(a,h_{2})$
(28) $\displaystyle\epsilon(a,b)=\frac{\eta(b,a)}{\eta(bab^{-1},b)},~{}\forall
a,b\in A$ (29)

The algebra $\mathcal{A}[A,\eta,\epsilon]$ determined by these data can be
described as follows. First $\mathcal{A}[A,\eta,\epsilon]$ should be an
(composite) anyon of $D(H)$, i.e. an $H$-graded vector space with an
$H$-action. As a graded vector space,
$\mathcal{A}[A,\eta,\epsilon]=\oplus_{a\in A}\mathbb{C}_{a}$ is supported on
the subgroup $A$, let the basis of $\mathbb{C}_{a}$ be $e_{a}$. The $H$-action
is defined by $h(e_{a}):=\epsilon(a,h)e_{hah^{-1}}$. The algebra structure on
$\mathcal{A}[A,\eta,\epsilon]$ is defined on the basis $e_{a}$ by $e_{a}\cdot
e_{b}:=\eta(a,b)e_{ab}$. Requiring the algebra $\mathcal{A}[A,\eta,\epsilon]$
to be well-defined and condensable then leads to the consistency condition on
$\epsilon$ and $\eta$ (27)-(29). It was also shown in Davydov (2009) that the
condensation is fully-confining if and only if $A=H$, in which case the
function $\eta$ determines $\epsilon$ through (29) and the condensation is
determined solely by a subgroup $H<\Gamma$ and a 2-cocycle of $H$, which we
recall is the same as the classification of 1+1D gapped $\Gamma$-phases, i.e.
an SSB from $\Gamma$ to $H$, and an SPT of $H$. On the other hand, if no
charge is condensed and the condensation is not fully-confining, meaning that
$H=\Gamma$, and $A$ is a proper subgroup of $G$, the data becomes exactly the
same as what determines an igSPT: a group extension $1\to A\to\Gamma\to
G:=\Gamma/A\to 1$, a 2-cocycle of $A$, and a function $\epsilon:A\times H\to
U(1)$. And the conditions (27)-(29) are exactly the same as the consistency
conditions in (10)-(12). We conclude that there is a 1-1 correspondence
between symmetric partially-confining condensations of $D(\Gamma)$ and 1+1D
$\Gamma$-gSPTs.

Moreover, the post-condensation topological order is in general a twisted
quantum double given by the following:

(Davydov and Simmons (2016), Thm. 2.17) The category of local modules over
$\mathcal{A}(A,\eta,\epsilon)$ is equivalent to $D(G:=\Gamma/A)_{\omega}$ as a
UMTC, with the twist $\omega$ given by:

$\displaystyle\omega_{3}(g,h,k)=\epsilon(e_{2}(g,h),k^{-1})$
$\displaystyle\times\eta(e_{2}(g,hk),e_{2}(h,k))\eta(e_{2}(g,hk)e_{2}(h,k),k^{-1}e_{2}(g,h)^{-1}k)$
(30)

Although appears to be different from (22), it can be shown that (30) is
equivalent to the emergent anomaly of gSPT (22) as cocycles. We show the
equivalence between the two in appendix D. In summary, there is a 1d-2d
correspondence between gSPTs and symmetric partially-confining condensations
of quantum doubles:  A gSPT defined by the data $(A\lhd\Gamma,\eta,\epsilon)$
is described in SymTFT by the condensable algebra
$\mathcal{A}(A\lhd\Gamma,\eta,\epsilon)$ of $D(\Gamma)$. If the post-
condensation topological order is $D(\Gamma)^{loc}_{\mathcal{A}}\cong
D_{\omega}(G)$, then the emergent anomaly of the gSPT is equal to the twist
$\omega\in H^{3}[G,U(1)]$. It fact it is readily seen that the symmetry of a
gSPT has the structure of a $\Gamma$-graded vector space with $\Gamma$-action.
The localized edge actions $\widetilde{U}_{a}$ span a graded vector space
supported on $A$, with a $\Gamma$-action given by the conjugation
$U_{\gamma}\widetilde{U}_{a}U_{\gamma}^{\dagger}=\epsilon(a,\gamma)\widetilde{U}_{\gamma
a\gamma^{-1}}$. Therefore the symmetry algebra of a $\Gamma$-gSPT naturally
defines an object of $D(\Gamma)$. The product of operators:
$\widetilde{U}_{a}\widetilde{U}_{b}=\eta(a,b)\widetilde{U}_{ab}$ endows the
object with an algebra structure, and the consistency conditions(10)-(12) make
sure the algebra is condensable.

##  V Special cases of gSPTs and their SymTFT description

Having established the formal relation between anyon condensation and gSPT,
let us now look at several typical classes of gSPTs and how they are described
by the SymTFT as applications of this formalism. There are two families of
gSPTs that have been studied in systematic manners. One is the gSPTs of SPT-
pump type, another is the week gSPTs that can thought of as transition from
SPT to spontaneous symmetry breaking. We discuss how they are incorporated
into our general classification in terms of the data $(\eta,\epsilon)$ as well
as their SymTFT description.

###  V.1 igSPT of SPT-pump type

A class of igSPTs whose edge modes can be explained by the mechanism of SPT-
pump was studied in Wen and Potter (2023). In terms of the data
$\eta,\epsilon$, these gSPTs are solutions with $\eta=0$, and whose emergent
anomaly is given by $\omega_{3}=\epsilon\cup e_{2}$. In this section we show
how gSPTs of SPT-pump type are described by the SymTFT. For simplicity let us
consider the case where $\Gamma$ is Abelian. The quantum double $D(\Gamma)$
that is dual to the igSPT is an Abelian gauge theory whose excitations are
charges, fluxes and their bond states. In general fluxes are labelled by
elements of the gauge group $\Gamma$, while charges are labelled by
irreducible representations of the group, or in the abelian case the dual
group $\widehat{\Gamma}:=H^{1}[\Gamma,U(1)]$. Therefore we denote the fluxes
as $m_{\gamma},\gamma\in\Gamma$ and the charges as
$e_{\hat{\gamma}},\hat{\gamma}\in\widehat{\Gamma}$. Both charges and fluxes
are self-bosons, and the mutual statistics between a charge and a flux is
given by the pairing between $\Gamma$ and $\widehat{\Gamma}$:
$S_{e_{\hat{\gamma}},m_{\gamma}}=\hat{\gamma}(\gamma)\in U(1)$. From our
understanding of igSPTs of SPT-pump type, we will be able to reverse engineer
the corresponding anyon condensation. If the igSPT has a group extension of
the form

Figure 7: Edge modes of igSPTs of SPT-pump type in the thin slab construction.
On the top reference boundary we fix the charge-condensed boundary condition.
On the physical boundary we consider a condensation determined by the SPT-pump
factor $\epsilon$ where fluxes of $A$ are binded with charges $\epsilon(a,-)$
and then condensed. The resulting slab has the expected SPT-pump edge modes:
an $a\in A$ action is localized to the edge of the igSPT, and its charge under
a $g\in G$ action is $\epsilon(a,g)$.

$\displaystyle 1\to A\to\Gamma\to G\to 1,$ (31)

then we expect an action by $a\in A$ to pump a charge $\epsilon(a,-)$. In the
$\mathbb{Z}_{4}$-igSPT example, the $m^{2}$ anyon is binded with $e^{2}$
before condensation, which is essentially what is pumped by the
$\mathbb{Z}_{2}^{A}$-action. Therefore in order for $a$-action to pump a
charge $\epsilon(a,-)\in\widehat{G}$ 333The phase $\epsilon(a,\gamma)$ in
general is defined on $A\times\Gamma$, but when $\eta=0$, the consistency
condition (12) states that $\epsilon|_{A\times A}=1$. $\epsilon$ is now also
linear in both variables by (10)-(11), therefore $\epsilon$ is now an element
of $H^{1}[A\times G,U(1)]=H^{1}[A,\widehat{G}]$, we expect the bond states
$m_{a}e_{\epsilon(a)}$ to be condensed, where we defined
$\epsilon(a):=\epsilon(a,-)$ which is an element of $\widehat{G}$,
corresponding to a gauge charge. It is readily seen that anyons
$m_{a}e_{\epsilon(a)}$ can be condensed simultaneously: the quantum spin is
$\theta(m_{a}e_{\epsilon(a)})=S_{m_{a},e_{\epsilon(a)}}=\epsilon(a,a)=1$ since
$\epsilon|_{A\times A}=1$, the mutual statistics between
$m_{a}e_{\epsilon(a)}$ and $m_{b}e_{\epsilon(b)}$ is
$S_{m_{a}e_{\epsilon(a)},m_{b}e_{\epsilon(b)}}=\epsilon(a,b)\epsilon(b,a)=1$.
By the same diagramatic reasoning that leads to edge modes of the
$\mathbb{Z}_{4}$-igSPT, we see that the condensation
$\mathcal{A}=\\{m_{a}e_{\epsilon(a)},a\in A\\}$ makes the $A$-actions
localized to the edge while the $G$-actions are not localizable. The charges
binded with $m_{a}$ give the expected SPT-pump structure. Therefore this
condensation should be dual to the $\Gamma$-igSPT of SPT-pump type. Since we
also know that these igSPTs have emergent anomalies given by
$\omega_{3}=\epsilon\cup e_{2}$, we arrive at the following conclusion:  Let
$A<\Gamma$ be a subgroup, $G:=\Gamma/A$, $e_{2}$ is the extension class
associated with $1\to A\to\Gamma\to G\to 1$. For any $\epsilon\in
H^{1}[A,\widehat{G}]$, condensing $\mathcal{A}=\\{m_{a}e_{\epsilon(a)},a\in
A\\}$ in the $D(\Gamma)$ quantum double results in the topological order
$D_{\omega}(G=\Gamma/A)$, with $\omega=\epsilon\cup e_{2}$. We see that via
the SymTFT/igSPT duality and by invoking our knowledge about igSPTs in 1+1D,
we have inferred information regarding condensations in 2+1D quantum doubles.
See Fig. 7 for an illustration of the thin slab construction of igSPTs of SPT-
pump type.

Similar to the $\mathbb{Z}_{4}$-igSPT, igSPTs of SPT-pump types are all stable
due to their emergent anomalies. From the anyon condensation perspective, this
means it is impossible to further condense anyons to arrive at a fully-
confining condensation that does not condense charges. This can be seen from
that fact that the post-condensation topological order is a twisted quantum
double, and it is well known that in a twisted quantum double there is no
fully-confining condensation that does not condense charges Zhang and Córdova
(2023). Therefore in order to arrive at a gapped symmetric phase one has to
either condense charges, corresponding to breaking the symmetry, or
“uncondense” some anyons in $\mathcal{A}$, which corresponds to closing the
gap to $A$ degrees of freedom. This anyon condensation picture matches with
our expectation from the 1+1D analysis: as the gap to $A$-dof is
perturbatively stable, closing the gap to $A$ degrees of freedom will require
a phase transition out of the igSPT phase. On other hand the anomaly of the
gapless sector forbids opening a gap directly without breaking the symmetry.

###  V.2 Weak gSPTs

There is also a family of gSPTs named weak gSPTs that are transitions between
SPT phases and symmetry breaking phases, these have been studied in Verresen
_et al._ (2021); Li  _et al._ (2023a). Here we discuss some generalities of
such gSPTs and show how they fall into our classification in terms of the pair
$(\eta,\epsilon)$ as well as their SymTFT description. The common mechanism
for such gSPTs is that the full symmetry has the form $G^{A}\times G^{B}$, and
on one side of the transition is an SPT of decorated domain wall(DDW) type,
where domain walls of $G^{A}$ are decorated with charges of $G^{B}$ and vice
versa. On the other side of the transition the symmetry is broken down to
$G^{A}$, and system is a $G^{B}$-paramagnet. The transition between the two is
thus in the same universality class as a usual $G^{B}$-SSB transition, which
is further enriched by $G^{A}\times G^{B}$ to form a gSPT. The $G^{A}$
symmetry is intact in both the SPT and the $G^{B}$-SSB phases, therefore acts
as a gapped symmetry at the transition point. Thus $G^{A}$-action still
localizes to edge actions at the transition, and the restriction to one edge
carries charges of $G^{B}$ from the DDW structure of the SPT phase. This leads
to edge modes and makes the transition a nontrivial gSPT. On the other hand
the $G^{B}$-action, which acts on gapless dof, no longer fractionalizes and
the edge modes of the gSPT are reduced from that of the nearby $G^{A}\times
G^{B}$-SPT. As the group extension is trivial, there is no emergent anomaly.
The gSPT can be gapped out directly while preserving symmetry-it is adjacent
to a gapped SPT by construction. Thus such gSPTs are not stable, and the edge
modes of them are nothing but residue of SPT edge modes, hence the name weak
gSPTs. Nevertheless, the symmetry enriching pattern and nontrivial edge modes
make it distinct from a usual $G^{B}$-SSB transition with a trivial
$G^{A}$-action. In particular, the charge carried by edge $G^{A}$-action
provides a discrete invariant that can not be altered without changing the
bulk universality class. To go to a trivial $G^{B}$-SSB transition the system
has to either open up a gap or go through a multi-critical point with a
different bulk universality class.

Examples of weak gSPTs include the Haldane chain to SSB transition, which has
an Ising CFT bulk and a 2-fold GSD on an opne chain, and similarly
$Z_{3}\times Z_{3}$-SPT to SSB transitions, which have bulk three-state Potts
CFT with central charge c = 4/5 and 3-fold GSD on an open chain Verresen  _et
al._ (2021). Numerical results on the phase diagrams of these models have been
obtained in Verresen  _et al._ (2021), in either case it was confirmed that
the weak gSPT is separated from a trivial one by a multi-critical point. We
will later provide a rather simple derivation of the phase diagram using
SymTFT.

Let us now see how the weak gSPTs fall into our classification in terms of
$\eta,\epsilon$. A weak gSPT has a symmetry group $G^{A}\times G^{B}$, and on
one side of the transition is in an SPT of DDW type. In the cohomology
classification this means the SPT is described by a cocycle $\beta_{2}\in
H^{1}[G^{A},H^{1}[G^{B},U(1)]]$, which is a subgroup of the full cohomology
group $H^{2}[G^{A}\times G^{B},U(1)]$ by the Künneth formula. This means
$\beta_{2}=\beta_{2}(g^{A},h^{B})$ is linear in both factors and describes the
charges carried by edge $G^{A}$ and $G^{B}$ actions via

$\displaystyle
U_{g^{B}}\widetilde{U}_{g^{A}}U_{g^{B}}^{\dagger}=\beta_{2}(g^{A},g^{B})\widetilde{U}_{g^{A}}$
(32) $\displaystyle
U_{g^{A}}\widetilde{U}_{g^{B}}U_{g^{A}}^{\dagger}=\beta_{2}(g^{A},g^{B})^{-1}\widetilde{U}_{g^{B}}$
(33)

At the transition point the subgroup $G^{B}$ becomes gapless and the second
equation above is lost. However we still have the first equation from which we
see that it corresponds to a gSPT with
$\eta_{2}=1,\epsilon(a=g^{A},\gamma=(h^{A},h^{B}))=\beta_{2}(g^{A},h^{B})$.
444We note that a weak gSPT can be defined more broadly as any SPT to
(partial)SSB transition. Let $\beta_{2}\in H^{2}[\Gamma,U(1)]$ represent a
$\Gamma$-SPT. If one drives the SPT to the $\Gamma$ broken down to
$A\unlhd\Gamma$ transition, then the transition is a gSPT with extension $1\to
A\xrightarrow{i}\Gamma\to G\to 1$ and
$\eta=i^{*}(\beta_{2}),\epsilon(a,\gamma)=\frac{\beta_{2}(\gamma,a)}{\beta_{2}(\gamma
a\gamma^{-1},\gamma)}$.

We next discuss the SymTFT description of weak gSPTs. We take
$\Gamma=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ as an example, and generalization
to generic weak SPTs is straightforward. Since the full symmetry is
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$, we search for symmetric partially-
confining condensations that do not condense charges in
$D(\mathbb{Z}_{2}\times\mathbb{Z}_{2})$, i.e. two copies of toric code. One
choice is the $\mathcal{A}=\\{e_{1}m_{2}\\}$ condensation. The post-
condensation topological order is generated by $e_{2}m_{1}$ and $e_{1}$, both
of which are bosons and these is a mutual $\pi$ statistics between them. Thus
the post-condensation topological order is equivalent to the toric code,
$D(\mathbb{Z}_{2}^{2})/\mathcal{A}\cong D(\mathbb{Z}_{2})$, and the
condensation describes a 1+1D gSPT with a trivial group extension
$1\to\mathbb{Z}^{A}_{2}\to\mathbb{Z}_{2}^{A}\times\mathbb{Z}_{2}^{B}\to\mathbb{Z}_{2}^{B}\to
1$. Using the thin slab construction, it is readily seen that
$\mathbb{Z}_{2}^{A}$-action is localized to the edge and the edge action is
charged under the $\mathbb{Z}_{2}^{B}$-action. In terms of the pair
$(\eta,\epsilon)$, this means we have
$\eta=1,\epsilon(a,\vec{b})=(-1)^{ab^{1}}$. Since the extension is trivial,
there is no emergent anomaly. Nevertheless, the nontrivial $\epsilon$ leads to
a 2-fold GSD on an open chain. Starting with the condensation $e_{1}m_{2}$,
further condensing $e_{2}m_{1}$ is allowed and will result in the condensation
$\langle e_{1}m_{2},e_{2}m_{1}\rangle$, which is dual to the cluster chain. On
the other hand further condensing $e_{1}$ or $m_{2}$ will result in the
condensation $\langle e_{1},m_{2}\rangle$, which is dual to a
$\mathbb{Z}_{2}^{A}$-SSB/$\mathbb{Z}_{2}^{B}$-paramagnet phase. Therefore the
condensation $\langle e_{1}m_{2}\rangle$ describes the transition from the
cluster chain to the $\mathbb{Z}_{2}^{A}$-SSB/$\mathbb{Z}_{2}^{B}$-paramagnet
phase. The trivial gSPT with the same bulk universality class is described by
the condensation $\langle m_{2}\rangle$, which corresponds to the
$\mathbb{Z}_{2}^{A}\times\mathbb{Z}_{2}^{B}$-paramagnet to
$\mathbb{Z}_{2}^{A}$-SSB/$\mathbb{Z}_{2}^{B}$-paramagnet transition. From the
SymTF it is clear that the two gSPTs are not connected if symmetry is
preserved. Both gSPTs are in the Ising-CFT universality class, but in order to
go from the $e_{1}m_{2}$ condensation to the $m_{2}$ condensation one has to
either condense charges, corresonding to breaking the symmetry, or uncondense
$e_{1}m_{2}$, which will lead to the $\\{1\\}$ condensation that is dual to a
$\mathbb{Z}_{2}^{A}\times\mathbb{Z}_{2}^{B}$-multi-critical point. The same
phase diagram has been obtained in Verresen  _et al._ (2021) via DMRG.

###  V.3 Stacking gSPT with SPT

One can always stack a gSPT with an SPT to obtain a new gSPT. We discuss some
generalities of this kind of construction in this section.

####  V.3.1 An example

Consider the topological order
$D(\mathbb{Z}_{4}^{A}\times\mathbb{Z}_{4}^{B})$, which is two copies of
$\mathbb{Z}_{4}$-Toric codes. Denote the charges and fluxes in each copy as
$e_{1,2}$ and $m_{1,2}$. Consider the condensation $\mathcal{A}=\langle
m_{1}^{2}e_{1}^{2}e_{2}^{2},m_{2}e_{1}\rangle$. The post-condensation
topological order consists of the deconfined anyons
$m_{1}e_{1}e_{2}^{3},m_{1}e_{1}^{3}e_{2}^{3}$ and $m_{2}^{2}$, which are
semion, anti-semion and boson respectively. We conclude that the condensation
leaves a twisted quantum double $D_{\omega}(\mathbb{Z}_{2})$. Let us consider
the slab construction with the condensation $\mathcal{A}$ on the physical
boundary. Following the same arguement as in Fig. 2, one can see that the
$\mathbb{Z}_{2}^{a}$ subgroup of $\mathbb{Z}_{4}^{a}$ and the full
$\mathbb{Z}_{4}^{b}$ subgroup can be localized to the edge. This corresponds
to a gapped symmetry $A=\mathbb{Z}_{2}^{a}\times\mathbb{Z}_{4}^{b}$, giving a
group extension is
$1\to\mathbb{Z}_{2}^{a}\times\mathbb{Z}_{4}^{b}\to\mathbb{Z}_{4}^{a}\times\mathbb{Z}_{4}^{b}\to\mathbb{Z}_{2}\to
1$. However now the edge $\mathbb{Z}_{2}$ action and edge $\mathbb{Z}_{4}$
action of $A$ anti-commute. This property is similar to the symmetry action of
a non-trivial $\mathbb{Z}_{2}\times\mathbb{Z}_{4}$-SPT. We conclude that this
condensation corresponds to a gSPT with a non-trivial $\eta\in
H^{2}[\mathbb{Z}_{2}\times\mathbb{Z}_{4},U(1)]\cong\mathbb{Z}_{2}$. Also, the
action by both $\mathbb{Z}_{2}^{a}$ and $\mathbb{Z}_{4}^{b}$ of $A$ on the
edge anti-commute with the $G$-action, i.e. a horizontal $m_{1}$-string.
Therefore this condensation/gSPT also has a non-trivial $\epsilon$ factor.

The condensation and the thin slab construction provide a complete description
of the dual gSPT: the group extension, the edge modes and the emergent
anomaly. As a consistency check, one can verify that
$\eta(\vec{a},\vec{b})=(-1)^{a^{1}b^{2}},~{}\epsilon(\vec{a},\vec{\gamma})=(-1)^{a^{1}\gamma^{1}+a^{1}\gamma^{2}}i^{a^{2}\gamma^{1}}$
is a solution to the consistency conditions on $(\eta,\epsilon)$, and by
inspection this solution gives a gSPT with the same structure as that is given
by the condensation $\mathcal{A}=\langle
m_{1}^{2}e_{1}^{2}e_{2}^{2},m_{2}e_{1}\rangle$.

Notably this igSPT can be viewed as obtained by stacking the
$\mathbb{Z}_{4}$-igSPT with a $\mathbb{Z}_{4}\times\mathbb{Z}_{4}$-SPT. In
other words the $A$-SPT given by $\eta$ can be cancelled by stacking it with a
$\Gamma$-SPT. In this case $\Gamma=\mathbb{Z}_{4}\times\mathbb{Z}_{4}$-SPTs
have a $\mathbb{Z}_{4}$ classification. Take the generating phase, the pull-
back of this $\Gamma$-SPT to the subgroup
$A=\mathbb{Z}_{2}\times\mathbb{Z}_{4}$ is exactly the non-trivial
$\mathbb{Z}_{2}\times\mathbb{Z}_{4}$-SPT, therefore cancels the $A$-SPT of
this igSPT upon stacking. The SPT+igSPT will still have the symmetry extension
$1\to\mathbb{Z}_{2}^{a}\times\mathbb{Z}_{4}^{b}\to\mathbb{Z}_{4}^{a}\times\mathbb{Z}_{4}^{b}\to\mathbb{Z}_{2}\to
1$, but now the second factor $\mathbb{Z}_{4}^{b}$ of $\Gamma$ plays no role
and the system is equivalent to the $\mathbb{Z}_{4}^{a}$-igSPT with an extra
trivial $\mathbb{Z}_{4}^{b}$-action.

####  V.3.2 General consideration

Let us now consider the effect on the data $(\eta,\epsilon)$ of stacking with
an SPT specified by $\omega_{2}\in H^{2}[\Gamma,U(1)]$. For simplicity we
assume all groups are abelian here, generalization to non-abelian groups is
straightforward. The stacked system has symmetry
$\Gamma^{SPT}\times\Gamma^{gSPT}$, but we are only interested in the diagonal
symmetry $\Gamma$ and can assume the symmetry is broken down to this diagonal
subgroup by perturbations. Now the symmetry extension is unchanged by the
stacking, because still only the $A$-subgroup acts trivially on the gapless
sector, and the quotient $G=\Gamma/A$ acts nontrivially on the gapless sector
coming from the gSPT. The $A$-action is now modified by the $\Gamma$-SPT,
namely $\eta$ is now modified by $j^{*}(\omega_{2})$, where $j$ is the
inclusion $A\xrightarrow{j}\Gamma$ and $j^{*}$ is the pull-back of
$\omega_{2}$ to $A$. Recall the factor $\epsilon$ is the measurement of the
charge of the edge $A$-actions under $\Gamma$. An SPT also defines such a
factor: the edge of an $a\in A$ action carries charge
$\frac{\omega_{2}(\gamma,a)}{\omega_{2}(a,\gamma)}$ under $\gamma$. Therefore
the factor $\epsilon$ is modified by $i_{a}\omega(\gamma)$, where we used the
slant product
$i_{a}\omega(\gamma):=\frac{\omega_{2}(\gamma,a)}{\omega_{2}(a,\gamma)}$.
Notice the set of data $(j^{*}(\omega_{2}),i_{a}\omega_{2}(\gamma))$ satisfies
the consistency conditions (10)-(12). The product of two solutions is again a
solution, therefore the data $(\eta
j^{*}(\omega_{2}),\epsilon(a,\gamma)i_{a}\omega_{2}(\gamma))$ is a valid
solution and corresponds to the gSPT obtained by stacking the
$(\eta,\epsilon)$-gSPT with the $\omega_{2}$-SPT. It can also be shown by
direct computation that two gSPTs differ by stacking with an SPT have the same
emergent anomaly.

In the $\mathbb{Z}_{4}^{a}\times\mathbb{Z}_{4}^{b}$-gSPT example studied
earlier, one can verify that choosing the $\omega_{2}$ corresponding to the
generating element of $H^{2}[\mathbb{Z}_{4}\times\mathbb{Z}_{4},U(1)]$ will
lead to a pull-back $j^{*}(\omega_{2})$ that is equal to the generating
element of $H^{2}[\mathbb{Z}_{2}\times\mathbb{Z}_{4},U(1)]$, therefore
trivializing the $\eta$ factor of the
$\mathbb{Z}_{4}^{a}\times\mathbb{Z}_{4}^{b}$-gSPT upon stacking, leaving an
igSPT of SPT-pump type.

##  VI Discussion and outlook

In this work established a one-to-one correspondence between 1+1D bosonic
gapless SPTs (gSPTs) and symmetric, partially-confining anyon condensations in
2+1D quantum doubles. These results suggest a number of directions for future
exploration. Fermionic igSPTs have been studied in Thorngren  _et al._ (2021),
including an example that is similar to the bosonic $\mathbb{Z}_{4}$-igSPT but
with fermion parity playing the role of the gapped symmetry group. Can one
construct a SymTFT dual to such fermionic igSPTs? This would first require
understanding how to incorporate fermion parity, which differs in important
ways from ordinary zero-form symmetries, into the SymTFT description.

Moreover, it is natural to consider gSPTs protected by generalized symmetries.
In 1+1D non-invertible symmetries are described by fusion categories.
Anomalies of non-invertible symmetries have been studied in Kaidi  _et al._
(2023); Zhang and Córdova (2023); Thorngren and Wang (2019); Bhardwaj  _et
al._ (2023). The SymTFT for 1+1D non-invertible symmetry is also well
understood, if the symmetry forms a fusion category $\mathcal{A}$, then the
SymTFT is given by the center $\mathcal{Z}[\mathcal{A}]$, which can also be
viewed as the Levin-Wen string-net model Levin and Wen (2005) with input
$\mathcal{A}$. Following the SymTFT dictionary, a condensation of
$\mathcal{Z}[\mathcal{A}]$ that i) does not confine the topological order
completely ii) does not condense generalized charges Zhang and Córdova (2023)
would be dual to a gSPT protected by $\mathcal{A}$. The post-condensation
topological order is in general another Levin-Wen model
$\mathcal{Z}[\mathcal{B}]$. If $\mathcal{A}$ is itself gappable while
$\mathcal{B}$ is not, then we can say that the gSPT has an emergent anomaly.
It would be interesting to extend the algebraic analysis of symmetries we
performed for gSPTs in this work to non-invertible symmetries and compare with
the known classification of anyon condensations of Levin-Wen models Kitaev and
Kong (2012).

In higher dimensions it is also natural to consider gSPTs protected by higher-
form symmetries or mixture of ordinary- and higher-form- symmetries. SymTFT
for higher form symmetries have also been studied Kong  _et al._ (2020);
Chatterjee and Wen (2023a) and can be viewed as (twisted) higher-form gauge
theories. In fact, it is known that gauging a subgroup $A\lhd\Gamma$ of a
0-form symmetry results in a symmetry $\Gamma/A\times\widehat{A}^{D-2}$, where
$\widehat{A}^{D-2}$ is a $D-2$-form symmetry generated by Wilson loops of the
$A$-gauge field. There is a mixed anomaly between $\Gamma/A$ and
$\widehat{A}^{D-2}$ Tachikawa (2020). One can replace the hard gauging where
the Hilbert space only consists of gauge invariant states, by the so called
soft gauging Borla  _et al._ (2021); Thorngren  _et al._ (2023); Verresen  _et
al._ (2022) where Gauss’s law is imposed only energetically. In this way, the
mixed anomaly becomes emergent, with a symmetry extension structure $1\to
A\to\Gamma\times\widehat{A}^{D-2}\to\Gamma/A\times\widehat{A}^{D-2}\to 1$.
Therefore soft gauging a subgroup gives a gSPT protected by a mixture of zero
and higher-form symmetry in spacetime dimension greater than two. A similar
construction was discussed in Su (2023). It would be interesting to see if a
correspondence between gSPTs and SymTFT holds in the presence of higher form
symmetries, and to extend the classification to gSPTs protected by such
symmetries.

Acknowledgements –  We thank Weicheng Ye, Joseph Sullivan, Fiona Burnell, and
Julio Parra-Martinez for insightful discussions. This work is supported by the
Natural Sciences and Engineering Research Council of Canada (NSERC), and by an
Alfred P. Sloan Foundation Fellowship (A.C.P.). We thank the Aspen Center for
Physics where part of this work was completed

Note –  During the completion of this manuscript, we became aware of Ref.
Huang and Cheng (2023), which examines SymTFT duals gapless states and
critical points and has some overlap with our results.

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##  Appendix A Consistency conditions on the data $\eta,\epsilon$

First of all we can compute
$U_{\gamma}\widetilde{U}_{a}\widetilde{U}_{b}U_{\gamma}^{-1}$ in two ways,

$\displaystyle
U_{\gamma}\widetilde{U}_{a}\widetilde{U}_{b}U_{\gamma}^{-1}=\eta(a,b)U_{\gamma}\widetilde{U}_{ab}U_{\gamma}^{-1}=\eta(a,b)\epsilon(ab,\gamma)\widetilde{U}_{\gamma
ab\gamma^{-1}},$ (34)

we also have

$\displaystyle U_{\gamma}\widetilde{U}_{a}\widetilde{U}_{b}U_{\gamma}^{-1}$
$\displaystyle=U_{\gamma}\widetilde{U}_{a}U_{\gamma}^{-1}U_{\gamma}\widetilde{U}_{b}U_{\gamma}^{-1}$
(35) $\displaystyle=\epsilon(a,\gamma)\epsilon(b,\gamma)\widetilde{U}_{\gamma
a\gamma^{-1}}\widetilde{U}_{\gamma b\gamma^{-1}}$
$\displaystyle=\epsilon(a,\gamma)\epsilon(b,\gamma)\eta(\gamma
a\gamma^{-1},\gamma b\gamma^{-1})\widetilde{U}_{ab},$ (36)

we conclude

$\displaystyle\eta(a,b)\epsilon(ab,\gamma)=\eta(\gamma a\gamma^{-1},\gamma
b\gamma^{-1})\epsilon(a,\gamma)\epsilon(b,\gamma)$ (37)

Secondly we can compute

$\displaystyle
U_{\gamma_{1}}U_{\gamma_{2}}\widetilde{U}_{a}U_{\gamma_{2}}^{-1}U_{\gamma_{2}}^{-1}$
$\displaystyle=\epsilon(a,\gamma_{2})U_{\gamma_{1}}\widetilde{U}_{\gamma_{2}a\gamma_{2}^{-1}}U_{\gamma_{1}}^{-1}$
(38)
$\displaystyle=\epsilon(a,\gamma_{2})\epsilon(\gamma_{2}a\gamma_{2}^{-1},\gamma_{1})\widetilde{U}_{\gamma_{1}\gamma_{2}a\gamma_{2}^{-1}\gamma_{1}^{-1}}.$
(39)

On the other hand

$\displaystyle
U_{\gamma_{1}}U_{\gamma_{2}}\widetilde{U}_{a}U_{\gamma_{2}}^{-1}U_{\gamma_{2}}^{-1}$
$\displaystyle=U_{\gamma_{1}\gamma_{2}}\widetilde{U}_{a}U_{\gamma_{1}\gamma_{2}}=\epsilon(a,\gamma_{1}\gamma_{2})\widetilde{U}_{\gamma_{1}\gamma_{2}a\gamma_{2}^{-1}\gamma_{1}^{-1}}.$
(40)

We conclude that $\epsilon$ satisfies:

$\displaystyle\epsilon(a,\gamma_{1}\gamma_{2})=\epsilon(\gamma_{2}a\gamma_{2}^{-1},\gamma_{1})\epsilon(a,\gamma_{2}).$
(41)

Lastly, there are two ways of computing
$U_{\gamma}\widetilde{U}_{a}U_{\gamma}^{-1}$ when $\gamma=b$ is an element of
$A$:

$\displaystyle U_{b}\widetilde{U}_{a}U_{b}^{-1}$
$\displaystyle=\widetilde{U}_{b}\widetilde{U}_{a}\widetilde{U}_{b}^{-1}=\eta(b,a)\widetilde{U}_{ba}\widetilde{U}_{b}^{-1}$
$\displaystyle=\eta(b,a)\eta(ba,b^{-1})\widetilde{U}_{bab^{-1}}$
$\displaystyle=\frac{\eta(b,a)}{\eta(bab^{-1},b)}\widetilde{U}_{bab^{-1}},$
(42)

where we used $\eta(ba,b^{-1})=1/\eta(bab^{-1},a)$ which comes from cocycle
condition on $\eta$ (we assume cocycles are normalized so that
$\eta(1,a)=\eta(a,a^{-1})=1$). Since we also have

$\displaystyle
U_{b}\widetilde{U}_{a}U_{b}^{-1}=\epsilon(a,b)\widetilde{U}_{bab^{-1}},$ (43)

we conclude that $\epsilon(a,b)=\frac{\eta(b,a)}{\eta(bab^{-1},b)},~{}\forall
a,b\in A$.

##  Appendix B Connection to the LHS spectral sequence

Here we show that the consistency conditions (10)-(12) as well as the emergent
anomaly (22) have the structure of the LHS spectral sequence. We focus on
computation aspects of the LHS sequence and work with cochain-level
expressions. We also assume the group $A$ is abelian in this appendix.

Consider a 3-cocycle of $G$: $\omega_{3}\in Z^{3}[G,U(1)]$ that is trivialized
via a group extension

$\displaystyle 1\rightarrow
A\xrightarrow[]{j}\Gamma\xrightarrow[]{p}G\rightarrow 1.$ (44)

This means that there is some 2-cochain of $\Gamma$:
$\alpha\in\mathcal{C}^{2}[\Gamma,U(1)]$, such that
$d\alpha((a_{1},g_{1}),(a_{2},g_{2}),(a_{3},g_{3}))=\omega(g_{1},g_{2},g_{3})$.
Here we have written elements of $\Gamma$ as pairs $(a,g),a\in A,G\in G$. We
also denote the action of $G$ on $A$ by ${}^{g}a:=gag^{-1}$.The group
extension (44) specifies an extension class $e_{2}\in\mathcal{Z}^{2}[G,A]$. By
modifying with coboundaries one can always decompose the 2-cochain $\alpha$
into the following form Wang  _et al._ (2021):

$\displaystyle\alpha((a,g),(b,h))$
$\displaystyle=F^{0,2}(^{\bar{g}\bar{h}}a,^{\bar{h}}b)F^{0,2}(^{\bar{g}\bar{h}}e_{2}(g,h),^{\overline{gh}}a^{\bar{h}}b)F^{1,1}(^{\bar{g}}a,h)F^{2,0}(g,h),$
(45)

where $F^{0,2}\in\mathcal{C}^{2}[A,U(1)]$,
$F^{1,1}\in\mathcal{C}^{1}[A,\mathcal{C}^{1}[G,U(1)]]$,
$F^{2,0}\in\mathcal{C}^{2}[G,U(1)]$. Requiring that
$d\alpha((a,g),(b,h),(c,k))$ only depends on $g,h,k$ results in the following
conditions:

$\displaystyle\delta_{0}F^{0,2}=1,$ (46)
$\displaystyle\delta_{1}F^{0,2}\delta_{0}F^{1,1}=1,$ (47)
$\displaystyle\delta_{2}F^{0,2}\delta_{1}F^{1,1}\delta_{0}F^{0,2}=1.$ (48)

Here the differentials $\delta_{i}$ are defined as follows. $\delta_{0}$ is
the cochain differential with respect to $A$: $\delta_{0}=d_{A}$, $\delta_{1}$
is the cochain differential with respect to $G$: $\delta_{1}=d_{G}$, and

$\displaystyle\delta_{2}F_{0,2}(a,g,h):=\frac{F^{0,2}(a,e_{2}(g,h))}{F^{0,2}(e_{2}(g,h),a)}.$
(49)

Once these conditions are met, the trivializing relation
$\omega(g,h,k)=d\alpha((a,g),(b,h),(c,k))$ becomes

$\displaystyle\omega(g,h,k)=\delta_{3}F^{0,2}\delta_{2}F^{1,1}\delta_{1}F^{2,0}$
(50)

where $\delta_{3},\delta_{2}$ are defined by

$\displaystyle\delta_{3}F^{0,2}(g,h,k):=\frac{F^{0,2}(e_{2}(g,hk),^{g}e_{2}(h,k))}{F^{0,2}(e_{2}(gh,k),e_{2}(g,h))}$
(51) $\displaystyle\delta_{2}F^{1,1}(g,h,k):=F^{1,1}(e_{2}(g,h),k)$ (52)

Since $\delta_{1}=d_{G}$, we see that on the cocycle level we can neglect
$F^{2,0}$ and $\omega\cong\delta_{3}F^{0,2}\delta_{2}F^{1,1}$ as $G$-cocycles.
Now we can identify $F^{0,2}(a,b)=\eta(b,a)$ and $F^{1,2}=\epsilon(a,g)$. The
condition that $\eta$ is a 2-cocycle of $A$ is the same as
$\delta_{0}F^{0,2}=1$. The consistency condition (10) is

$\displaystyle\frac{\epsilon(a,g)\epsilon(b,g)}{\epsilon(ab,g)}=\frac{\eta(a,b)}{\eta(^{g}a,^{g}b)}$
(53)

which is the same as $d_{A}\epsilon d_{G}\eta=1$, or
$\delta_{0}F^{0,2}\delta_{1}F^{1,1}=1$. Lastly using the condition (11),

$\displaystyle\epsilon(^{h}a,g)\epsilon(a,h)=\epsilon(a,ghe_{2}(g,h))=\epsilon(a,gh)\frac{\eta(a,e_{2}(g,h))}{\eta(e_{2}(g,h),a)}$
(54)

which is the same as $\delta_{1}F^{1,1}\delta_{2}F^{0,2}=1$. The emergent
anomaly (22) can be readily seen to be the same as
$\delta_{3}F^{0,2}\delta_{2}F^{1,1}$. Since now
$d_{\Gamma}\alpha=\delta_{3}F^{0,2}\delta_{2}F^{1,1}$, this means the anomaly
(22) is the differential of the 2-cochain $\alpha$, therefore is trivialized
by the group extension.

##  Appendix C The bulk-edge correspondence for 1+1D gSPTs

A 1+1D gSPT is defined by a group extension (44) and a pair $(\eta,\epsilon)$.
The emergent anomaly is determined by $(\eta,\epsilon)$ via (22). This can be
viewed as an edge to bulk map: the edge modes determines the bulk emergent
anomaly. To show the converse the question is equivalent to finding
$\eta,\epsilon$ for a given emergent anomaly. To start with, one finds a
trivializing cochain: $d\alpha=\omega,\alpha\in\mathcal{C}^{2}[\Gamma,U(1)]$.
From the decomposition (45) we see that $\alpha((a,1),(b,1))=F^{0,2}(a,b)$.
Formally this means the pull back of $\alpha$ by $j$ is a 2-cocycle of $A$:
$dj^{*}(\alpha)=j^{*}d\alpha=j^{*}p^{*}\omega=(p\circ j)^{*}\omega=1$, where
we used $(p\circ j)^{*}=1$ because the sequence (44) is exact. Similarly,
other terms in the decomposition $F^{1,1},F^{0,2}$ can be extracted from
$\alpha$ by the Lyndon’s algorithm Lyndon (1948). For example

$\displaystyle\epsilon(a,g)=F^{1,1}(a,g)=i_{a}\alpha(g):=\frac{\alpha((a,1),(1,g))}{\alpha((1,g),(a,1))}.$
(55)

Therefore $\epsilon,\eta$ can be reconstructed once the trivializing cochain
$\alpha$ is found. However, the choice of the trivializing cochain is not
unique: if $d\alpha=d\alpha^{\prime}=\omega$, then
$d(\alpha\alpha^{\prime-1})=1$. Thus any two $\Gamma$-cochains
$\alpha,\alpha^{\prime}$ that differ by a $\Gamma$-cocycle give the same
emergent anomaly $\omega$. Therefore the data $\eta,\epsilon$ can only be
determined from $\omega$ up to a 2-cocycle of $\Gamma$. Consider another
trivializing cochain $\gamma^{\prime}=\gamma\delta$, where
$\delta\in\mathcal{Z}^{2}[\Gamma,U(1)]$. Then the data
$\eta^{\prime},\epsilon^{\prime}$ associated with $\gamma^{\prime}$ is

$\displaystyle\eta^{\prime}=j^{*}(\alpha^{\prime})=j^{*}\alpha\cdot
j^{*}\delta=\eta\cdot j^{*}\delta,$ (56)
$\displaystyle\epsilon^{\prime}(a,g)=i_{a}\alpha^{\prime}(g)=i_{a}\alpha(g)\cdot
i_{a}\delta(g)=\epsilon(a,g)i_{a}\delta(g)$ (57)

Recall from section V.3 that stacking a gSPT defined by $(\eta,\epsilon)$ with
an SPT defined by $\delta\in\mathcal{Z}^{2}[\Gamma,U(1)]$ results in a new
gSPT with $\eta^{\prime}=\eta\cdot
j^{*}\delta,\epsilon^{\prime}(a,g)=\epsilon(a,g)\cdot i_{a}\delta(g)$.
Comparing with (56)-(57), we see that the new data
$\eta^{\prime},\epsilon^{\prime}$ associated with the new trivializing cochain
$\alpha^{\prime}$ differs from the original data $\eta,\epsilon$ associated
with $\alpha$ by stacking with a $\Gamma$-SPT given by $\delta$. Therefore we
conclude that for a given emergent anomaly $\omega$, the gSPT is only
determined by $\omega$ up to stacking with $\Gamma$-SPTs. The fact that
stacking with SPTs does not change the emergent anomaly may not very
surprising, given that SPTs do not have bulk anomalies, what is surprising is
that all gSPTs with the same emergent anomaly are obtained in this way.
Therefore the gSPT bulk-edge correspondence can stated as follows: The edge
modes of a gSPT determine the bulk emergent anomaly, and the bulk emergent
anomaly determines the edge modes of the gSPT up to stacking with SPTs.

Now with the help of the bulk-edge correspondence we can prove that the two
definitions of igSPT are equivalent: 1. A gSPT is intrinsic if the edge modes
can not be trivialized by stacking with an SPT. 2. A gSPT is intrinsic if it
has a nontrivial emergent anomaly. 1$\Rightarrow$ 2: if the emergent anomaly
is trivial: $\omega=1$, then $\eta_{0}=\epsilon_{0}=1$ is clearly a solution
to the consistency conditions. By the bulk-edge correspondence any other
solution with the same $\omega=1$ must differ from $\eta_{0}=\epsilon_{0}=1$
by stacking with a $\Gamma$-SPT. Therefore if the gSPT has non-trivial edge
modes, $\eta\neq 1$ and/or $\epsilon\neq 1$, they can be eliminated by
stacking with a $\Gamma$-SPT. $2\Rightarrow 1$: If the edge modes of a gSPT
can be trivialized by stacking with a $\Gamma$-SPT, then it means the data
$\eta,\epsilon$ differs from the tivial one $\eta_{0}=1,\epsilon_{0}=1$ by
stacking with a $\Gamma$-SPT. Since stacking with $\Gamma$-SPTs does not
change the emergent anomaly, the data $\eta,\epsilon$ gives the same emergent
anomaly as that is given by the trivial one, $\omega=1$. $\Box$

##  Appendix D Equivalence between the post-condensation twist and gSPT
emergent anomaly

In the computation of emergent anomaly, we defined
$U_{g}U_{h}=\Omega(g,h)U_{gh}$. Alternatively one can define
$U_{g}U_{h}=U_{gh}\Omega^{\prime}(g,h)$. Similarly
$\Omega^{\prime}(g,h)=\Omega^{\prime L}(g,h)\otimes\Omega^{\prime R}(g,h)$ The
associativity condition on $\Omega^{\prime}$ is

$\displaystyle\Omega^{\prime}(gh,k)U^{-1}_{k}\Omega^{\prime}(g,h)U_{k}=\Omega^{\prime}(g,hk)\Omega^{\prime}(h,k)$
(58)

Restrict to $\Omega^{\prime L}$, the associativity condition is then only
satisfied up to a phase,

$\displaystyle\omega^{\prime}(g,h,k)\Omega^{\prime
L}(gh,k)U^{-1}_{k}\Omega^{\prime L}(g,h)U_{k}=\Omega^{\prime
L}(g,hk)\Omega^{\prime L}(h,k)$ (59)

The $\Omega^{\prime}(g,h)$ is related to $\Omega(g,h)$ via
$\Omega^{\prime}(g,h)=U^{\dagger}_{gh}\Omega(g,h)U_{gh}$. Using this relation
one can show that $\omega^{\prime}=\omega$:

$\displaystyle\Omega^{\prime L}(gh,k)U^{-1}_{k}\Omega^{\prime L}(g,h)U_{k}$
$\displaystyle=U^{-1}_{ghk}\Omega^{L}(gh,k)U_{ghk}U_{k}^{-1}U^{-1}_{gh}\Omega^{L}(g,h)U_{g}hU_{k}$
$\displaystyle=U^{-1}_{ghk}\Omega^{L}(gh,k)\Omega^{L}(g,h)U_{{ghk}}$ (60)

and

$\displaystyle\Omega^{\prime L}(g,hk)\Omega^{\prime
L}(h,k)=U^{-1}_{ghk}\Omega^{L}(g,hk)U_{ghk}U^{-1}_{hk}\Omega^{L}(h,k)U_{hk}$
$\displaystyle=U^{-1}_{ghk}(\Omega^{L}(g,hk)U_{g}\Omega^{L}(h,k)U_{g}^{-1})U_{ghk}.$
(61)

Comparing with (17), one sees that $\omega^{\prime}=\omega$. Next notice that
$\Omega^{\prime
L}(g,h)=U^{\dagger}_{gh}\Omega(g,h)^{L}U_{gh}=U^{\dagger}_{gh}\widetilde{U}_{e_{2}(g,h)}U_{gh}=\epsilon(e_{2}(g,h),gh)\widetilde{U}_{e_{2}(g,h)}$.
Therefore replacing $\Omega^{\prime}L(g,h)$ by $\widetilde{U}_{e_{2}(g,h)}$ in
(59) will only affect the cocycle $\omega$ by a coboundary (since every
$\Omega^{\prime L}(g,h)$ factor only gets modified by a phase factor
$\epsilon(e_{2}(g,h),gh)$). Therefore we can take $\Omega^{\prime
L}(g,h)=\widetilde{U}_{e_{2}(g,h)}$ in (59). Write (59) as

$\displaystyle\omega(g,h,k)\widetilde{U}_{e_{2}(gh,k)}=\widetilde{U}_{e_{2}(g,hk)}\widetilde{U}_{e_{2}(h,k)}U^{-1}_{k}(\widetilde{U}_{e_{2}(g,h)})^{-1}U_{k}.$
(62)

Use $\widetilde{U}_{a}\widetilde{U}_{b}=\eta(a,b)\widetilde{U}_{ab}$, together
with $U_{g}\widetilde{U}_{a}U_{g}^{-1}=\epsilon(a,g)\widetilde{U}_{gag^{-1}}$,
we arrive at the expression

$\displaystyle\omega(g,h,k)=\epsilon(e_{2}(g,h),k^{-1})$
$\displaystyle\times\eta(e_{2}(g,hk),e_{2}(h,k))\eta(e_{2}(g,hk)e_{2}(h,k),k^{-1}e_{2}(g,h)^{-1}k)$
(63)

which is the twist in (30). Therefore the twist (30) and the anomaly (22) are
equivalent as cocycles of $G$.