# Topological holography for fermions

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

###### Abstract

Topological holography is a conjectured correspondence between the symmetry
charges and defects of a $D$-dimensional system with the anyons in a
$(D+1)$-dimensional topological order: the symmetry topological field theory
(SymTFT). Topological holography is conjectured to capture the topological
aspects of symmetry in gapped and gapless systems, with different phases
corresponding to different gapped boundaries (anyon condensations) of the
SymTFT. This correspondence was previously considered primarily for bosonic
systems, excluding many phases of condensed matter systems involving fermionic
electrons. In this work, we extend the SymTFT framework to establish a
topological holography correspondence for fermionic systems. We demonstrate
that this fermionic SymTFT framework captures the known properties of $1+1D$
fermion gapped phases and critical points, including the classification, edge-
modes, and stacking rules of fermionic symmetry-protected topological phases
(SPTs), and computation of partition functions of fermionic conformal field
theories (CFTs). Beyond merely reproducing known properties, we show that the
SymTFT approach can additionally serve as a practical tool for discovering new
physics, and use this framework to construct a new example of a fermionic
intrinsically gapless SPT phase characterized by an emergent fermionic
anomaly.

###### Contents

  1. I Introduction
  2. II Background: Bosonic SymTFT
    1. II.1 Reference boundary
    2. II.2 Physical boundaries and anyon condensation
    3. II.3 Condensable algebras and gapless SPTs
    4. II.4 Edge modes from SymTFT
  3. III SymTFT for fermions: Structure and Examples
    1. III.1 Construction of the fermionic SymTFT
      1. III.1.1 Local fermion excitations
      2. III.1.2 Induced spin structure
      3. III.1.3 Symmetry defects
    2. III.2 Example: Fermionic gapped phases with $G^{F}=\mathbb{Z}_{2}^{F}$
      1. III.2.1 Spin structure and partition function of gapped phases
    3. III.3 Example: Fermionic SPTs with $G^{F}=\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$
    4. III.4 Example of a critical point: The Majorana CFT
      1. III.4.1 Review of (bosonic) Ising CFT
      2. III.4.2 Majorana CFT from SymTFT
  4. IV Intrinsically fermionic and gapless SPTs from SymTFT
    1. IV.1 Anomalous fermionic symmetries in SymTFT
    2. IV.2 Fermionic igSPT with $\mathbb{Z}_{8}$ symmetry
    3. IV.3 Field theory description
  5. V SymTFT for fermions: Systematics and Technical Details
    1. V.1 General symmetries
      1. V.1.1 Condensation on the reference boundary
    2. V.2 Spin and twisted spin structures of reference boundary
      1. V.2.1 Review of spin structure
      2. V.2.2 Spin structure in the $\mathbb{Z}_{2}^{F}$-SymTFT
      3. V.2.3 Review of $G^{F}$-spin structure
      4. V.2.4 $G^{F}$-spin structure in $G^{F}$-SymTFT
  6. VI Fermionic SPT phases from SymTFT
    1. VI.1 Partition function of fermionic SPTs from SymTFT
    2. VI.2 Stacking rules of fermionic SPTs from SymTFT
      1. VI.2.1 Review: Bosonic SPT stacking rules from SymTFT
      2. VI.2.2 Example: $G=\mathbb{Z}_{2}^{2}$
      3. VI.2.3 Fermionic SPT stacking rules in SymTFT
      4. VI.2.4 Example: stacking of Kitaev chains
      5. VI.2.5 Example: $G^{F}=\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$ and $G^{B}=\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
  7. VII Bosonization in SymTFT
    1. VII.1 Bosonization of an anomaly-free symmetry
    2. VII.2 Bosonization of anomalous symmetry
      1. VII.2.1 The $\nu=2$ anomaly of $\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$
      2. VII.2.2 The $\nu=1$ anomaly of $\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$
  8. VIII Summary and outlook
  9. A Quantum double and its condensable algebra
  10. B Fermionic boundary of toric code and spin structures
    1. B.1 Setup
    2. B.2 The $f$-boundary of toric code
    3. B.3 Lattice version of $\mathbb{Z}_{2}^{F}$-SymTFT

##  I Introduction

Symmetry provides a powerful set of organizing principles for understanding
phases of matter. Beginning from the traditional Landau spontaneous symmetry-
breaking paradigm, to the modern theory of generalized symmetries that provide
a unifying framework for numerous phenomena from symmetry-protected
topological orders (SPTs) with non-local (e.g. string) order parameters, to
topological orders that can be understood as spontaneously broken higher-form
symmetries, to non-perturbative dualities at critical points that are captured
by non-group like, non-invertible symmetries [1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13]. A key lesson from this progress is that one must generalize the
notion of symmetry beyond global unitary operators with group-like structure.
Somewhat unexpectedly, when all such generalized symmetries are taken into
account, the structure of a higher-dimensional topological order suddenly
appears [7, 8, 9, 10, 14, 13, 14, 11, 12, 15, 16, 17, 18]. For example, as
discussed in [7, 8, 15], a $1+1D$ Ising spin chain has both an ordinary
$\mathbb{Z}_{2}$ symmetry associated with spin-flip, and also a “dual
symmetry” associated with the conservation (modulo two) of domain walls. The
local action of both symmetries restricted to finite spatial “patches” unveils
a collection of line operators that have precisely the same structure as the
$e$ and $m$ anyons of a $2+1D$ toric code.

These observations have led to a conjectured _topological holography_ between
symmetries of a quantum system and a topological field theory that is one-
dimensional higher, which is referred to as symmetry topological field theory
(SymTFT) [12, 19, 20, 16, 17, 18] or symmetry topological order (SymTO) [15,
8, 10]111In this paper, we primarily focus on $2+1D$ SymTFT, and the SymTFT is
always described by some $2+1D$ topological order. Hence we will freely employ
the terminology of $2+1D$ topological order like anyons, anyon condensation,
etc. Accordingly, different phases of a quantum system with the given symmetry
correspond to different boundary conditions of the SymTFT. For a finite,
unitary symmetry $G$ of a bosonic system, the corresponding SymTFT is the
$G$-gauge theory in $2+1D$, which can be described by the quantum double
theory $D(G)$ in the categorical language. In this case, the electric and
magnetic anyon excitations of the SymTFT correspond to charged local
operators, and topological defects (domain walls) of the SymTFT, respectively.

The big advantage of SymTFT is that it disentangles topological aspects of the
symmetry (e.g. ‘t Hooft anomalies, conservation of topological defects, the
symmetry quantum numbers of topological defect operators), from dynamical
aspects (e.g. the scaling dimensions at a critical point222Still, there are
attempts in the literature to derive the scaling dimensions of $1+1D$ systems
from SymTFT. See e.g. [21].). Moreover, it makes certain “topological
manipulations”, including gauging, stacking with invertible phases and duality
transformation manifest by simply considering different boundary conditions of
the same SymTFT. Hence SymTFT makes the connections of these theories with
different appearances manifest.

In gapped systems, all universal aspects of symmetry are topological, and the
topological holography correspondence states that different gapped phases of
matter correspond to distinct gapped boundaries of the SymTFT, which are in
turn characterized by different patterns of anyon condensations. For gapless
systems, the SymTFT captures only topological rather than dynamical aspects
(e.g. it can constrain the possible symmetry actions on topological defect
operators, but does not fully determine the spectrum of scaling operators at a
critical point). This raises the intriguing opportunity to establish a non-
perturbative, topological (partial) characterization of gapless systems –
which are notoriously difficult to study except in low-dimensions.

The topological holographic correspondence has been systematically established
[7, 8, 10, 9, 12, 15, 17, 18, 16], in $1+1D$ bosonic systems, with finite,
internal (non-spacetime) symmetries, including both ordinary group-like
symmetries and non-invertible symmetries described by a general fusion
category. In the group-like symmetry case, it has been shown recently that the
SymTO can be derived from analyzing the algebra of patch operators associated
with the symmetry and the dual symmetry [15]. The correspondence can also be
pictorially understood via a dimensional reduction procedure called the
“sandwich construction” (see Fig. 1), in which the partition function for a
$G$-symmetric $1+1D$ quantum system is written in terms of that of a thin-
slab of $2+1D$ SymTFT sandwiched between two boundaries: 1) a “trivial”
reference boundary in which all the gauge-electric charges are condensed
(corresponding to Dirichlet boundary conditions in the field theory language),
and 2) a “physical” boundary that determines the topological properties of the
$1+1D$ quantum system. Different boundary conditions of the SymTFT correspond
to different patterns of anyon condensations at the boundary [8, 10, 22, 23].
Gapped systems correspond to patterns of anyon condensation on the physical
boundary that fully confine the SymTFT (such that all particles are either
condensed or confined in that boundary condensate). Anyons of the SymTFT
together define a categorical version of symmetry of the $1+1D$ quantum system
[7, 9, 8, 10]. These symmetries include both ordinary on-site unitary
symmetries, but can also capture generalized symmetries such as string-orders
of symmetry-protected topological phases (SPTs), and non-invertible symmetries
[2, 3, 19, 20, 24, 12, 16, 17, 18, 25, 26, 27] such as the Kramers-Wannier
duality of the $1+1D$ Ising CFT (for which the corresponding SymTFT is the
double Ising topological order).

Recent work has made progress in extending these ideas to gapless systems such
as gapless SPTs [28, 29, 30, 31], continuous symmetries and Goldstone phases
[32, 33], and has made forays into higher numbers of spatial dimensions [7,
26, 27]. These advances have, so far, been developed mainly for _bosonic_
systems, that is, systems where all local operators are bosonic, except for
some general discussions in a few pioneering works [34, 13] and very recently
in [35, 36]. This has limited the relevance of the framework to condensed
matter problems involving electronic degrees of freedom.

This work aims to fill this gap, and develop a topological holographic
correspondence and SymTFT formalism for _fermionic_ systems enriched by
symmetry. We focus on $1+1D$ fermionic systems with a (possibly
anomalous)333In this work, we do not consider systems with pure gravitational
anomaly, and systems with nonzero chiral central charge. symmetry $G^{F}$ ,
which is a finite unitary symmetry that contains the $\mathbb{Z}^{F}_{2}$
fermion parity conservation as a subgroup. The central structures of our
construction include:

  * •

For a non-anomalous fermionic symmetry $G^{F}$, the $2+1D$ SymTFT is given by
the $G^{F}$ gauge theory.

  * •

The SymTFT for a generic $1+1D$ fermionic symmetry is given by a “gauged
fermionic SPT” in $2+1D$.

  * •

The reference boundary of the SymTFT is defined by condensing a set of anyons
of the SymTFT that have the same fusion and braiding structure as local
symmetry charges of $G^{F}$. This involves “condensing” a set of fermionic
anyons, $f_{R}$, which is done by introducing local fermions, $c$, on the
reference boundary, and condensing the bosonic composite $cf_{R}$. The
reference boundary is therefore fermionic due to these local fermions $c$.

  * •

The dependence on spin structure (or $G^{F}$-spin structure in the general
case) of a $1+1D$ fermionic system is manifested in the SymTFT by the spin
structure dependence of the fermionic reference boundary.

  * •

Bosonization of the $1+1D$ system is manifested in the SymTFT simply by
changing the reference boundary from “fermionic” to “bosonic”.

We explain how this “fermion condensed” boundary condition [37, 38] can
reproduce universal properties of various known $1+1D$ phases with- and
without- additional symmetries, such as the Kitaev chain, the Majorana CFT,
and arbitrary fermionic SPTs. These fermionic phases have many distinctive
features absent in bosonic phases. For instance, the edge modes of the Kitaev
chain are Majorana defects with radical quantum dimension $\sqrt{2}$, the
fermion parity is an unbreakable symmetry, and the partition function of a
fermionic system depends on the spin structure of the manifold. We find that
all these features can find their places in a properly formulated SymTFT. We
show that one can obtain the full classification of $1+1D$ fermionic SPTs from
SymTFT. This includes not only the set of SPT phases, but also the stacking
rules of these phases. We also show that bosonization/Jordan-Wigner
transformation has a simple realization in the SymTFT: it amounts to a change
of reference boundary condition of the SymTFT. We recover many bosonization-
related results by simple SymTFT considerations.

Beyond providing a fresh perspective on familiar phases, the fermionic SymTFT
framework provides a convenient tool to theoretically discover new phases of
matter. As an example, we use the SymTFT framework to construct an example of
an intrinsically-fermionic and intrinsically-gapless SPT (igSPT) state, which
exhibits an emergent anomalous fermionic symmetry (a discrete version of the
$1+1D$ chiral anomaly) that protects a gapless Luttinger liquid bulk and
fractionally charged fermionic edge states. This fermionic igSPT is both
intrinsically-fermionic, i.e., its topological properties cannot arise in a
bosonic system, and intrinsically-gapless, i.e., the fractionally charged edge
modes are not equivalent to those of any gapped fermion SPT state. We show
that the topological aspects of this fermion igSPT can be readily deduced from
the SymTFT construction, which also directly facilitates a simple bosonized
field-theory description of the fermion igSPT and its edge states.

This work is organized as follows. In Section II we review the theory of
SymTFT for bosonic symmetries. Next we construct and analyze the SymTFT for
the simplest fermionic symmetry–the fermion parity symmetry
$\mathbb{Z}_{2}^{F}$ in Section III. We discuss the Kitaev chain and the
Majorana CFT within the framework of SymTFT. We also discuss how spin
structure dependence is realized in SymTFT. We construct and study an example
of fermionic igSPT in Section IV and reveal fascinating properties of the
igSPT via SymTFT methods. In Section V.1 we provide the general construction
for a fermionic symmetry with no anomaly, analyzing the structure of the
relevant fermionic condensation and the dependence on twisted spin structures.
We show in Section VI how partition function of $1+1D$ fermionic SPTs may be
derived from SymTFT methods. In Section VI.2 we address the issue of stacking
rules in SymTFT. We define the stacking of SymTFTs via a product structure of
condensable algebras and show with examples that it matches with the stacking
of the corresponding phases. In Section VII, we explain how the standard
procedure of bosonization/gauging fermion parity (or the Jordan-Wigner
transformation for $1+1D$ fermion chains) manifests in the SymTFT
construction, and use this to identify non-invertible symmetry transformations
(dualities) in Section VII.2.

##  II Background: Bosonic SymTFT

We begin by briefly reviewing the SymTFT description of a $1+1D$ bosonic
system $\sf{T}$ with a finite, unitary, internal (non-spacetime), group-like
symmetry $G$, for which the corresponding SymTFT is simply a $2+1D$ $G$-gauge
theory. For anomalous symmetry charaterized by cocycle $\omega\in
H^{3}[G,U(1)]$, the SymTFT is a twisted (Dijkgraaf-Witten) gauge theory
twisted by the same cocycle $\omega$, or the so-called gauged SPT in the
condensed matter language, and corresponds to the quantum double topological
order $D(G,\omega)$.444We focus mostly on group-like symmetries in this work,
except in VII.2 where we also use the SymTFT for non-invertible fusion
categorical symmetry to discuss bosonization of anomalous fermionic symmetry.
For a general fusion categorical symmetry $\mathcal{A}$, the SymTFT is given
by the Drinfeld center $\sf{Z}[\mathcal{A}]$. In the group-like symmetry case,
the symmetry fusion category is $\sf{Vec}_{G}^{\omega}$, and we recover the
quantum double SymTFT due to $\sf{Z}[\sf{Vec}_{G}^{\omega}]\simeq
D(G,\omega)$. The original $1+1D$ bosonic system $\sf{T}$ can be reconstructed
from the $2+1D$ SymTFT according to the “slab” or “sandwich” construction
illustrated in Fig. 1:

  1. 1.

The SymTFT lives on the manifold $M\times I$, where $M$ is a $2+1D$ (closed,
orientable) manifold and $I$ is the interval $[0,1]$.

  2. 2.

The upper boundary $M\times\\{0\\}$ is a fixed _reference boundary_
$\sf{B_{ref}}$ that encodes _topological aspects_ of the symmetry.

  3. 3.

The lower boundary $M\times\\{1\\}$ is a _physical boundary_ $\sf{B_{phys}}$
that can be varied to describe different possible $G$-symmetric phases.

  4. 4.

The original $1+1D$ bosonic system on $M$ is obtained by “dimensional
reduction” of the interval $[0,1]$.

###  II.1 Reference boundary

For group-like symmetry $G$, the reference boundary is one where all of the
gauge charges of the $G$ gauge theory are condensed (this corresponds to
Dirichlet boundary conditions in the field theory language). This is
substantiated by the following two observations. First, one defining property
of the reference boundary is that the non-trivial anyon lines that cannot be
absorbed on the reference boundary correspond to the symmetry defects of the
original $1+1D$ system $\sf{T}$, as illustrated in Fig. 1. Indeed, by choosing
to condense all gauge charges on $\sf{B_{phys}}$, the remaining non-trivial
anyon lines that cannot simply be absorbed into $\sf{B_{ref}}$ are the flux
lines. These flux lines exactly form the symmetry category
$\sf{Vec}_{G}^{\omega}$. Namely, the fluxes are labeled by elements in $G$
with the fusion rules from the group structure of $G$, and the $F$-symbol of
the flux lines matches with the anomaly $\omega$.

Figure 1: The SymTFT setup. (a). The SymTFT for a general symmetry category
$\mathcal{A}$ is the Drinfeld center $\sf{Z}[\mathcal{A}]$. The sandwich
(left) reduces to the original system $\sf{T}$ (right) when viewed as an
effective $1+1D$ system. (b). A non-trivial topological defect line near the
reference boundary becomes a symmetry defect operator $\sf{D}_{a}$ of $\sf{T}$
after the dimensional reduction. Figure 2: Local and non-local charges in
SymTFT. (a). A gauge charge line in the SymTFT can be absorbed by the
reference boundary. This corresponds to a local symmetry charge of $\sf{T}$
after dimensional reduction. The red dot represents an excitation created by
the anyon line at $\sf{B_{phys}}$. (b). A general anyon of the SymTFT is a
dyon, the associated defect line can not be fully absorbed by $\sf{B_{ref}}$,
instead it becomes a flux line on $\sf{B_{ref}}$(blue). After dimensional
reduction this becomes a charge that lives on the end of a symmetry defect,
i.e. a non-local charge or a string order parameter.

Secondly, the anyons condensed on the reference boundary correspond to local
charges of the $1+1D$ system. Concretely, a short anyon string that is
absorbed by the reference boundary corresponds to a local charged operator of
the original system in the sandwich construction, also illustrated in Fig. 2.
Therefore, anyons condensed on the reference boundary should have the same
structure (fusion rule, $F$-symbols) as that of local charges of the symmetry.
This is another defining property of the reference boundary. Indeed, the gauge
charges of the twisted gauge theory $D(G,\omega)$ have the same fusion
structure as local charges of the symmetry $G$, namely the fusion category of
representations of $G$, $\text{Rep}(G)$.

Besides fluxes and charges, there are dyons in the SymTFT. When a dyon line
reaches the reference boundary, it cannot be fully absorbed but becomes a flux
line living on the reference boundary. This configuration corresponds to a
charge of $\sf{T}$ that lives at the end of a symmetry defect. These string-
like charges are known as non-local charges of the symmetry. If the dyon is
bosonic, its anyon string also corresponds to string order parameters of the
symmetry $G$, whose vacuum expectation value differentiates different gapped
or gapless SPTs. See Fig. 2. Therefore the SymTFT provides a unified
description for all topological aspects of the symmetry, namely the anyons of
the SymTFT correspond to the symmetry defects and symmetry charges, and the
braiding of anyons encodes the algebraic relations among these symmetry
defects and charges.

From the dimension-reduction construction, the SymTFT sandwich computes the
partition function of the physical $1+1D$ system as an inner product between
the state defined by the physical boundary and the state defined by the
reference boundary. A generic state of the $G$-gauge theory can be written in
the field configuration basis as

$\displaystyle|\Psi\rangle=\sum_{A\in H^{1}[M,G]}\Psi_{A}|A\rangle,$ (1)

where $A$ is the gauge field on $M$.555For finite, potentially non-abelian
group $G$, the notation of non-abelian cohomology $H^{1}[M,G]$ (which is just
a set) simply denotes the set of all gauge bundles on $M$. A gauge bundle and
its associated gauge field $A$ on $M$ are completely determined by the
holonomy on the noncontractible cycles, up to conjugation action on the
holonomoy, i.e., a map $f\colon\pi_{1}(M)\rightarrow G$ with $f$ identified
with $f^{\prime}$ if their images are related by conjugation with an element
$g\in G$. When $G$ is an abelian group, the notation coincides with the usual
cohomology of $M$ with coefficient $G$, which is also denoted as $H^{1}[M,G]$.
The all-charge condensed reference boundary corresponds to the Dirichlet
boundary

$\displaystyle|\sf{Ref},A_{0}\rangle=\sum_{A\in
H^{1}[M,G]}\delta_{A,A_{0}}|A\rangle,$ (2)

that fixes the value of the gauge field to $A_{0}$ on the reference boundary.
$A_{0}$ also sets the expectation value of the Wilson loops of condensed gauge
charges on different non-contractible cycles. If the physical boundary is in
the state

$\displaystyle|\sf{Phys}\rangle=\sum_{A\in
H^{1}[M,G]}\mathcal{Z}_{\sf{T}}[A]|A\rangle,$ (3)

then the sandwich has partition function

$\displaystyle\langle\sf{Ref},A_{0}|\sf{Phys}\rangle=\mathcal{Z}_{\sf{T}}[A_{0}],$
(4)

which is the partition of a $1+1D$ system $\sf{T}$ coupled to a background
$G$-gauge field $A_{0}$.

The bulk $G$-gauge theory also has a standard Neumann boundary condition when
the twist $\omega$ is trivial. This is a topological boundary described by the
state

$\displaystyle|N,\widehat{A}_{0}\rangle:=\sum_{A\in
H^{1}[M,G]}e^{i\int_{M}\widehat{A}_{0}\cup A}|A\rangle,~{}\widehat{A}_{0}\in
H^{1}[M,\widehat{G}]$ (5)

where $\widehat{G}:=\hom(G,U(1))$ is the Pontryagin dual of $G$. In the
condensed matter language, this boundary is the all-flux condensed boundary,
and $\widehat{A}_{0}$ sets the expectation values of the ’t Hooft loops of the
condensed fluxes on different non-contractible cycles.

Gauging a global symmetry $G$ in $1+1D$ corresponds in the SymTFT to changing
the reference boundary from Dirichlet to Neumann. It is direct to verify that
taking the inner product with the Neumann boundary state gives the partition
function of the gauged theory $\sf{T}/G$. Recall that the reference boundary
defines the symmetry represented by the SymTFT. Formally, the Dirichlet
boundary defines the symmetry of the effective $1+1D$ system to be
$\text{Vec}_{G}$, while the Neumann boundary defines the symmetry to be
$\text{Rep}(G)$. The fact that these two symmetries are described by the same
SymTFT with different boundary conditions is the direct consequence of the
fact that there is a topological manipulation relating the two symmetries. In
this case the topological manipulation is gauging $G$. The SymTFT provides a
general framework for analyzing such topological manipulation. Namely any
topological manipulation, such as gauging subgroups and stacking with
invertible phases, does not change the SymTFT, and corresponds to only
changing the reference boundary in the sandwich construction.

This shows the advantage (and limitation) of the SymTFT – it isolates
_topological_ aspects of the symmetry from dynamical properties (which are
potentially non-topological in gapless systems). For gapped symmetric states,
all universal long-wavelength aspects of symmetries are topological and the
SymTFT fully captures the universal properties of the physical system. For
gapless states, such as conformal field theories (CFTs), the SymTFT does not
fully constrain the operator content and scaling dimensions of the physical
system, but does capture topological consistency conditions, and may enable a
characterization or classification of symmetry-enriched gapless phases of
matter [39, 40, 41, 10, 28, 29, 30, 31].

A major result of this work is extending this advantage of SymTFT to fermionic
symmetries. In Section VII, we show that bosonization, which is also a
topological manipulation but maps fermionic symmetries into bosonic
symmetries, also corresponds to a change of reference boundary of the SymTFT.

###  II.2 Physical boundaries and anyon condensation

In the SymTFT framework different $\mathcal{A}$-phases are represented by
different boundary conditions on the physical boundary, and it is known that
boundaries of $2+1D$ topological orders have intimate relation with anyon
condensation. [23, 22, 39, 40, 41, 10] Namely, for every boundary
condition666More precisely we assume the boundary is either gapped or CFT-
like.  of a $2+1D$ topological order $\mathcal{C}$, there is an associated
anyon condensation. An anyon is condensed on a boundary if it can be moved
from the bulk to this boundary and absorbed. Or equivalently if it can be
dragged out of the boundary without costing any energy. Anyons that are
condensed simultaneously need to satisfy a set of consistency conditions.
First, all condensed anyons must be bosons. Moreover, if $a$ and $b$ are both
condensed, then at least one fusion outcome of $a\times b$ is necessarily also
condensed, These conditions make the condensate have the structure of a
separable, connected, commutative algebra. See reference [22] for a formal
definition. Due to the physical origin of this structure, we will call it a
condensable algebra instead. For abelian topological orders, a condensable
algebra is equivalent to a subgroup of the fusion group, such that all
elements of this subgroup have trivial self and mutual statistics.

Anyon condensation in a bulk $2+1D$ topological order occurs as a phase
transition that results in a new phase. For Abelian topological orders, the
new topological order can be viewed as formed by the remaining deconfined
anyons, i.e. anyons that braid trivially with the condensate. In a non-Abelian
theory, an anyon of the initial phase may become partially
confined/deconfined, and it is less straightforward to identify the new
topological order resulting from condensation. In general a composite anyon of
the original phase becomes a simple anyon of the new phase. The composite
anyons needs to be invariant under fusing with the condensate, since the
condensate is the vacuum of the new phase. This gives the anyons in the new
phase the structure of a module over the condensable algebra $\mathcal{A}$.
That is to say, an excitation of the new phase (possibly confined) is a non-
simple anyon of the old phase$\mathcal{M}=\oplus_{i}c_{i}a_{i}$, and there is
a product $\nu_{M}:\mathcal{A}\otimes\mathcal{M}\to\mathcal{M}$. The product
needs to satisfy some natural consistency conditions [22]. Deconfined
excitations are the modules such that braiding with the condensate commute the
product, these are formally called local modules.

A useful formula regarding anyon condensation is the dimension formula. If we
denote the topological order obtained by condensing $\mathcal{A}$ in
$\mathcal{C}$ by $\mathcal{C}/\mathcal{A}$, then

$\displaystyle\dim(\mathcal{C}/\mathcal{A})=\dim(\mathcal{C})/\dim(\mathcal{A})$
(6)

where $\dim(\mathcal{C})=\sqrt{\sum_{a\in\mathcal{C}}d_{a}^{2}}$ is the total
quantum dimension of the topological order, and if $\mathcal{A}=\sum c_{a}a$,
then $\dim(\mathcal{A})=\sum_{a}c_{a}d_{a}$. If a condensation results in a
trivial topological order, the condensable algebra is called Lagrangian. We
see from the dimension formula that a condensable algebra is Lagrangian if and
only if its dimension equals the total quantum dimension of the topological
order: $\dim(\mathcal{A})=\dim(\mathcal{C})$.

The relation between anyon condensation and boundary condition of a
topological order is as follows. A gapped boundary has a Lagrangian
condensation while a gapless boundary necessarily has a non-Lagrangian
condensation. In the SymTFT setup the reference boundary is always gapped, and
different $\mathcal{A}$-phases are realized by choosing different boundary
conditions on the physical boundary. Therefore gapped $\mathcal{A}$-phases
correspond to Lagrangian condensations of $\sf{Z}[\mathcal{A}]$, while gapless
$\mathcal{A}$-phases corrspond to non-Lagrangian condensations of
$\sf{Z}[\mathcal{A}]$. In the gapped boundary case the boundary is fully
determined by the condensation, in the sense that all excitations on the
boundary can be calculated from the Lagrangian algebra. Physically they are
the anyons confined on the boundary. When the boundary is gapless, the
condensation on the boundary determines only the gapped excitations, and the
gapless excitations on the boundary can not be fully determined by the
condensation alone. However it is known that if the gapless boundary is
described by some CFT, the SymTFTconstrains the partition function of the CFT
via its modular transformation properties [10].

In the bosonic SymTFT, the reference boundary condenses all symmetry charges,
the corresponding algebra is called the electric algebra[24]. The electric
algebra in $D(G,\omega)$ is

$\displaystyle\mathcal{A}_{\sf{ref}}^{b}:=\oplus_{R\in\text{Rep}(G)}d_{R}e_{R}.$
(7)

Here we use $e_{R}$ to denote a gauge charge carrying the representation $R$,
and $d_{R}$ is the dimension of the representation. The subscript $b$ stands
for bosonic, stressing that the reference boundary is bosonic, as oppose to
the fermionic reference boundary we will introduce latter.
$\mathcal{A}_{\sf{ref}}^{b}$ has a compact description as the group algebra
$\mathbb{C}[G]$. Here $\mathbb{C}[G]$ is the linear space of complex functions
on $G$, and the product of the algebra is given by the product of functions
(point-wise multiplication). A natural basis for $\mathbb{C}[G]$ is the delta
functions $|g\rangle,g\in G$. $\mathbb{C}[G]$ carries a natural
$G$-representation $g\cdot|h\rangle:=|gh\rangle$, which constitutes a highly
reducible representation. When decomposed into irreducible representations
(irreps), $\mathbb{C}[G]\cong\oplus_{R}d_{R}R$. The quantum dimension of
$\mathcal{A}_{\sf{ref}}^{b}$ is
$\dim(\mathcal{A}_{\sf{ref}}^{b})=\sum_{R}d_{R}^{2}=|G|=\dim(D(G,\omega))$,
therefore $\mathcal{A}_{\sf{ref}}^{b}$ is Lagrangian.

If a gauge charge is also condensed on $\sf{B_{phys}}$, then one can form a
Wilson line of this charge that connects the two boundaries without creating
any excitation. Since vertical charge lines in the SymTFT correspond to local
symmetry charge of the represented $1+1D$ system, this means condensing gauge
charges on the reference boundary results in a non-zero order parameter for
the represented $1+1D$ system, signaling spontaneous breaking of the symmetry.
Therefore to describe symmetric $\mathcal{A}$-phases, the physical boundary
must not condense any gauge charges. Such a condensation/ condensable algebra
is called magnetic [24].

###  II.3 Condensable algebras and gapless SPTs

In the SymTFT sandwich construction different $G$-phases correspond to
different anyon condensations on the physical boundary. Gapped and gapless
SPTs [42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 49, 54, 55, 56], being
phases of matter enriched by symmetry, should have correspondence in the
SymTFT framework. Indeed, it has been shown that SymTFT gives a complete
characterization for gapped and gapless bosonic SPTs. More concretely, there
is a one-to-one correspondence between condensable algebras of $D(G)$ and
gapped or gapless SPTs protected by $G$. We briefly review this correspondence
here, the full theory can be found in [28]. Some details of quantum double
models are also reviewed in Appendix A.

A bosonic $\Gamma$-(g)SPT has a symmetry extension structure summarized by the
sequence [44, 46, 47]

$\displaystyle 1\to N\to\Gamma\to G\to 1.$ (8)

The physical meaning of $N,G$ is as follows. Excitations that carry non-zero
charges of the subgroup $N$ are gapped. Below the gap to these gapped
excitations, the symmetry is effectively the quotient $G=\Gamma/N$. $N$ is
also called the gapped symmetry, and $G$ the gapless symmetry. A gapped SPT is
the case where all excitations are gapped, i.e. $N=\Gamma$. The low energy
gapless sector of a gSPT can carry an anomaly of the effective symmetry $G$.
This is called the emergent anomaly of the gSPT. When the emergent anomaly is
non-trivial, the gSPT is called an intrinsically gapless SPT(igSPT) [3, 46],
since the resulting edge modes of this SPT cannot be realized in any gapped
SPT with the same symmetry. The topological aspects of the gSPT are determined
by two functions $(\eta,\epsilon)$, where $[\eta]\in H^{2}[N,U(1)]$ determines
the SPT-class of the gapped degrees of freedom, and $\epsilon:N\times\Gamma\to
U(1)$ determines the charges of edge $N$-actions under $\Gamma$. The two
functions are subject to a set of natural consistency conditions and
equivalence relations. The edge modes of a gSPT is fully determined by the
data $(\eta,\epsilon)$. The emergent anomaly of the low energy effective
symmetry $G$ can also be computed from $(\eta,\epsilon)$ [28].

On the SymTFT side, the condensable algebras of $D(\Gamma)$ have been
classified in [57, 58]. A condensable algebra of $D(\Gamma)$ is determined by
the data $(H,N,\eta,\epsilon)$, and is denoted by
$\mathcal{A}[H,N,\eta,\epsilon]$. Here $H$ is a subgroup of $\Gamma$, $N$ is a
normal subgroup of $H$. $\eta\in\mathcal{Z}^{2}[N,U(1)]$ is a 2-cocycle of
$N$, and $\epsilon$ is a $U(1)$-valued function: $N\times H\to U(1)$. The
functions $\eta,\epsilon$ need to satisfy a set of consistency equations. When
$H\neq G$, the condensable algebra $\mathcal{A}[H,N,\eta,\epsilon]$
necessarily contains gauge charges, therefore condensing a
$\mathcal{A}[H,N,\eta,\epsilon]$ at $\sf{B_{phys}}$ with $H$ strictly smaller
than $\Gamma$ describes an SSB phase of $\Gamma$. Thus for describing
symmetric phases we consider $H=\Gamma$. The algebra is then Lagrangian if and
only if $N=\Gamma$. In general, in the SymTFT sandwich construction,
condensing $\mathcal{A}[\Gamma,N,\eta,\epsilon]$ on the physical boundary
corresponds to a $1+1D$ phase whose gapped symmetry is $N$ and gapless
symmetry is $G=\Gamma/N$. Therefore gapped phases are represented by algebras
with $N=\Gamma$. In this case the function $\epsilon$ is fully determined by
$\eta$. Thus a magnetic and Lagrangian algebra of $D(\Gamma)$ is fully
determined by a 2-cocycle of $\Gamma$. This agrees with the classification of
$1+1D$ $\Gamma$-SPTs.777In the classification of SPTs two cocycles differ by a
coboundary describe the same phase, this is matched in SymTFT by the fact that
two condensable algebras are isomorphic if the 2-cocycles differ by a
coboundary. When $N\neq\Gamma$, the condensation is non-Lagrangian and
corresponds to a gapless SPT(gSPT) of $\Gamma$ with gapped symmetry $N$. In
this case the data $(\eta,\epsilon)$ defining a condensable algebra have the
same structure as those defining a $\Gamma$-gSPT. Namely they satisfy the same
set of consistency conditions and equivalence relations. The emergent anomaly
of the corresponding gSPT is matched in SymTFT by the post-condensation twist:
condensing $A[\Gamma,N,\eta,\epsilon]$ changes the SymTFT to a twisted gauge
theory of the quotient group: $D(G)_{\omega}$, and the twist
$\omega\in\mathcal{Z}^{3}[G,U(1)]$ is exactly the emergent anomaly of the
corresponding gSPT. The edge modes of the gapped/gapless SPTs can also be
recovered in SymTFT by the method we described before.

The SymTFT provides a unified theory for gapped and gapless SPTs: gapped SPTs
correspond to Lagrangian magnetic algebras and gapless SPTs correspond to non-
Lagrangian magnetic algebras. We will explore generalization of this
correspondence to fermionic gSPTs in Section IV, and a more complete treatment
will appear in a separate work [59].

Figure 3: Edge modes in SymTFT. The physical boundary has two regions of
different condensations separated by a point defect, represented by a black
box. The all-flux condensation $\mathcal{A}^{0}$ corresponds to the trivial
symmetric phase. $\mathcal{A}_{\sf{T}}$ is the condensation that corresponds
to a non-trivial $1+1D$ phase $\sf{T}$. If there is an anyon string that can
end on the point defect as well as the reference boundary, then this anyon
line becomes a local zero mode supported on the edge of $\sf{T}$ after
dimensional reduction. Figure 4: Edge modes of the cluster chain from SymTFT.
The SymTFT bulk is $D(\mathbb{Z}_{2}\times\mathbb{Z}_{2})$-two copies of toric
codes. The trivial paramagnet phase of $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ is
represented by the canonical magnetic condensation $\mathcal{A}^{0}=\langle
m_{1},m_{2}\rangle$. The non-trivial SPT phase, i.e. the cluster chain, is
represented by the magnetic condensation $\mathcal{A}^{1}=\langle
e_{1}m_{2},e_{2}m_{1}\rangle$. The two panels show two linearly independent
edge modes in the SymTFT. Notice these two edge modes anti-commute, giving
rise to a two-fold GSD for every edge.

###  II.4 Edge modes from SymTFT

One characteristic of gapped or gapless SPT phases is their edge modes. This
is a set of zero modes acting on the edge of the system whose algebra is
protected by the symmetry. They are responsible for the GSD with open
boundaries and characterize the topological properties of the phases. To study
edge modes in SymTFT we consider the following setup. For a symmetry group $G$
with no anomaly there is always a canonical magnetic algebra that corresponds
to condensing all gauge fluxes: $\mathcal{A}^{0}:=\oplus_{C}m_{C}$, the sum is
over conjugacy classes of $G$ and $d_{m_{C}}=|C|$. Since
$\dim(\mathcal{A}^{0})=\sum_{C}|C|=|G|$ this algebra is Lagrangian. Condensing
$\mathcal{A}^{0}$ on $\sf{B_{phys}}$ produces a gapped symmetric phase which
is identified as the trivial $G$-phase, i.e. the vacuum. Now to study edge
modes of a phase $\sf{T}$, we consider putting the corresponding boundary
condition $\mathcal{A}^{\sf{T}}$ on a finite region of $\sf{B_{phys}}$, while
the rest of $\sf{B_{phys}}$ has the canonical magnetic condensation
$\mathcal{A}^{0}$, see Fig. 3. Then edge modes correspond in the SymTFT to
anyon line operators that are localized at the interface between
$\mathcal{A}^{\sf{T}}$ and $\mathcal{A}^{0}$, such that no excitations are
created. As an illustration, we show the SymTFT representation of edge modes
of the nontrivial $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$-SPT (a.k.a the cluster
chain) in Fig. 4.

##  III SymTFT for fermions: Structure and Examples

We now seek a generalization of the SymTFT framework for fermionic systems.
The key features of the SymTFT that we wish to preserve are that:

  1. 1.

It reduces to the original system by dimensional reduction/sandwich
construction.

  2. 2.

Non-trivial defect lines on the reference boundary have the same structure as
symmetry defects of the original system.

  3. 3.

Anyons condensed on the reference boundary have the same structure as local
charges of the original system.

The sandwich construction of the fermionic SymTFT proceeds similarly to that
for bosons with two key differences: first, the gauge group of the SymTFT
contains the fermion parity subgroup $\mathbb{Z}^{F}_{2}$ to account for the
conservation of fermion number parity, $(-1)^{F}$, and second, to describe a
system with fermion excitations we must explicitly introduce local (ungauged)
fermion excitations into the otherwise bosonic SymTFT sandwich.888In our
construction, these local fermion excitations will be located on the reference
boundary of a bosonic SymTFT. We mention that there are other proposals that
introduce local fermions into the bulk of the SymTFT construction [34, 13,
36]. See Section VIII for more discussion.

In this section, we illustrate the basic ideas of this approach through a
series of examples of how various familiar gapped topological and SPT phases,
and gapless critical points arise in the fermion SymTFT framework. In Section
IV below, we also construct a new fermionic intrinsically gapless SPT phase ,
characterized by a low-energy emergent symmetry with a fermionic anomaly. In
Section V, we formalize this construction, and address various technical
details related to spin structures.

###  III.1 Construction of the fermionic SymTFT

The symmetry group, $G^{F}$, of a fermion system necessarily includes fermion
parity $\mathbb{Z}_{2}^{F}$ generated by $(-1)^{F}$ where $F$ is the number of
fermions (modulo two), as a normal subgroup. $G^{F}$ can be regarded as an
extension of the bosonic symmetry, $G^{B}=G^{F}/\mathbb{Z}_{2}^{F}$, by
fermion parity $\mathbb{Z}_{2}^{F}$. Following the bosonic construction, we
choose the SymTFT as the $2+1D$ $G^{F}$ gauge theory. For example, without any
microscopic (bosonic) symmetries, $G^{B}=1$, $G^{F}=\mathbb{Z}_{2}^{F}$
remains non-trivial, enabling a description of distinct fermion gapped phases
such as the trivial and Kitaev chain topological superconductor phases which
have no symmetry-distinction. We consider for now the case where the symmetry
splits into the product of the fermion parity subgroup and the bosonic
symmetry group: $G^{F}=\mathbb{Z}_{2}^{F}\times G^{B}$. In this case the
$D(\mathbb{Z}_{2}^{F})$ sector of the SymTFT $D(G^{F})$ contains an abelian
bosonic flux $m^{F}$ and an abelian charge $e^{F}$,999We caution that, unlike
in the SPT literature where $m_{f}$ is sometimes used to denote a fermionic
statistics, here we use a superscript $m^{F}$ to denote that this is the flux
associated with the fermion parity sub-group of the gauge-group $G^{F}$. and a
fermionic dyon $f=e^{F}\times m^{F}$.

The reference boundary, $\sf{B_{ref}}$, must be chosen so that the condensed
anyons have the right structure of local symmetry charges of $G^{F}$. Since a
$\mathbb{Z}_{2}^{F}$-charge is necessarily fermionic, it looks as if this
requires condensing particles with fermion statistics, which is not directly
possible. To proceed, we introduce an auxiliary local fermion, $c$, that lives
only at $\sf{B_{ref}}$. We then create a gapped reference boundary,
$\sf{B_{ref}}$, by condensing the _bosonic_ pair $f\times c$. In addition,
following the bosonic SymTFT we also condense all the $G^{B}$ gauge charges
$e^{R},R\in\text{Rep}(G^{B})$ at $\sf{B_{ref}}$.

We note that this approach is analogous to that for obtaining fermion models
from bosonic string-net models for quantum doubles . Notice that the physical
boundary $\sf{B_{phys}}$ is bosonic. We have chosen to locate the fermionic
degrees of freedom on the reference boundary so that all topological aspects
of the symmetry, including the statistics of symmetry charges, are encoded by
the reference boundary.

####  III.1.1 Local fermion excitations

This choice of reference boundary condition automatically produces a fermion
excitation at the physical boundary, $\sf{B_{phys}}$, which can be thought of
as the charge of $\mathbb{Z}_{2}^{F}$. Namely, consider a short, vertical $f$
line segment that spans from $\sf{B_{phys}}$ to $\sf{B_{ref}}$, where its end
is decorated by a local fermion operator $c$ such that it can be absorbed $cf$
condensate at $\sf{B_{ref}}$. However, the other end at $\sf{B_{phys}}$ cannot
be absorbed, and instead leaves behind an excitation. Under dimensional
reduction, this line operator becomes a local point-like operator with Fermi
exchange statistics inherited from the $c$ decoration. Also, notice that with
the fermionic reference boundary, a single $f$-anyon can now by created by a
local operator. Thus, the fermionic reference boundary turns the emergent
fermions, $f$, into local physical fermions of the effective $1+1D$ system.

Figure 5: $\mathbb{Z}_{2}^{F}$-SymTFT setup. Top: An $f$-line can end on
$\sf{B_{ref}}$, with the endpoint dressed by a local fermion operator $c$,
represented by a blue dot. Notice there is no excitation created on
$\sf{B_{ref}}$. The excitation is created at the other end of the $f$-line,
represented by a red dot. After dimensional reduction this vertical line
becomes a local fermionic operator. Bottom: The local fermion operators $c$
attached to $f$-lines give anti-commutation relation between two vertical
$f$-lines. After dimensional reduction, this becomes the standard anti-
commutation relation between local fermionic operators.

####  III.1.2 Induced spin structure

A crucial difference between fermions and bosons, is that fermions can only be
defined on manifolds that permit a spin structure. Roughly speaking, a spin
structure gives a consistent set of $\pm 1$ phases obtained when transporting
a fermion around a closed loop in spacetime. In the following, we will mainly
consider the case where $\sf{B_{ref}}$ has the structure of a two-torus,
$T^{2}$, in which case the spin structure may be specified by periodic (P) or
anti-periodic (AP) boundary condition along the space-like and time-like
noncontractible loops. We will therefore denote the spin structure as
(P/AP,P/AP) where the first (second) argument indicates the boundary
conditions in space (time) respectively. To define the $fc$-condensed
reference boundary will require choosing a spin structure on $\sf{B_{ref}}$,
and this spin structure is exactly the spin structure of the effective $1+1D$
system represented by the SymTFT sandwich. Specifically, the spin structure of
the physical fermions is associated with the phase obtained when absorbing an
$f$ loop into the reference boundary. To see this, consider a short
$f$-segment that creates an $f$-anyon in the SymTFT bulk (see Fig. 5).
According to our discussion in III.1.1, this $f$-anyon corresponds to the
local physical fermion excitation of the effective $1+1D$ system. Now imagine
transporting this local fermion by dragging the $\sf{B_{phys}}$ end of the
$f$-segment around a spatial cycle, $\gamma$. As shown in Fig. 6, with a
single fermion exchange (resulting in a $(-1)$ phase) this configuration can
be deformed into the original fermion excitation (short, open $f$-segment),
and a closed $f$-string loop corresponding to a Wilson loop operator
$W_{f}(\gamma)$. The $f$ loop can then be absorbed into $\sf{B_{ref}}$
producing a phase $\langle W_{f}(\gamma)\rangle_{\sf{B_{ref}}}=\pm 1$. The
resulting overall phase is $(-1)\langle W_{f}(\gamma)\rangle_{\sf{B_{ref}}}$.
Therefore the boundary condition on $\gamma$ is related to the value of
$f$-loop by the following relation:

$\displaystyle P/AP\Leftrightarrow\langle
W_{f}(\gamma)\rangle_{\sf{B_{ref}}}=-1/+1.$ (9)

Throughout the remainder of this section, we will choose the convention that
there is a unit eigenphase for absorbing an $f$-loop into both the space- or
time- cycle $\gamma$ of $\sf{B_{ref}}=T^{2}$: $\langle
W_{f}(\gamma)\rangle_{\sf{B_{ref}}}=+1$. This corresponds to AP boundary
conditions for the physical fermions along the two cycles of $\sf{B_{phys}}$.
We note, however, that the boundary condition for the physical fermions can be
toggled by inserting a fermion parity flux ($m^{F}$) defect line as desired.
Below we will use this freedom to relate the partition functions of fermion
phases for different boundary conditions, through the SymTFT framework.

Figure 6: Diagrammatic argument for spin-structure dependence of the
$f$-condensed boundary. (a). An $f$-string is used to create an $f$ anyon (red
dot) in the bulk. The $f$-string ends on the fermionic reference boundary and
has a fermionic endpoint (blue dot). The created anyon $f$ should be viewed as
a local physical fermion of the effective $1+1D$ system represented by the
sandwich. (b). The $f$ anyon (red dot) moves around the horizontal direction
by extending the $f$-string and returns to its initial location. (c). We
perform a braiding for the $f$-string, which results in a -1 sign. (d) We
perform an $F$-move to re-arrange the string into two separate strings: a
vertical short string and a horizontal noncontractible loop. The $F$-move does
not produce any phases. (e) Deform the strings into straight lines. (f). The
horizontal $f$-loop is absorbed by the $f$-condensate at $\sf{B_{ref}}$,
producing a phase $\langle W_{f}(\gamma)\rangle_{\sf{B_{ref}}}$. Thus the
overall phase accumulated in this process is $-\langle
W_{f}(\gamma)\rangle_{\sf{B_{ref}}}$.

####  III.1.3 Symmetry defects

In addition to the standard symmetry defects of the bosonic part of the
symmetry, $G^{B}$, there are additional line defects associated with the
$\mathbb{Z}_{2}^{F}$ subgroup of the SymTFT: the horizontal $m^{F}$-line and
$e^{F}$-line. Since both defects anticommute with the $\mathbb{Z}_{2}^{F}$
charge (short vertical decorated $f$ strings described above), and neither can
be trivially absorbed into $\sf{B_{ref}}$ (unlike for the bosonic
$\sf{B_{ref}}$), we have the freedom of choosing either as the fermion parity
symmetry. These two choices are essentially equivalent, and the resulting
SymTFT dictionaries differ only by a relabelling of $e^{F}\leftrightarrow
m^{F}$ (which is an automorphism of the SymTFT). We fix this ambiguity by
choosing $m$-line as the fermion parity symmetry generator. Below, we will see
that this ambiguity precisely stems from the physical ambiguity between
labeling the trivial and topological superconducting phases (Kitaev chain) of
a Majorana chain with periodic boundary conditions.

We emphasize that this ambiguity is physical and not a deficiency of the
SymTFT framework. Consider a $1+1D$ lattice of Majorana fermions. The trivial
and topological superconducting phases differ only by a convention for which
pairs of Majorana modes constitute atomic “sites”, and which ones lie along
bonds of the chain. With periodic boundary conditions, we can transform from
one phase to the other through translation by one site. This translation
symmetry exactly corresponds to the exchange symmetry $e^{F}\rightarrow m^{F}$
in $D(\mathbb{Z}_{2}^{F})$.

Figure 7: Kitaev chain order parameter in SymTFT. In the SymTFT we consider an
$e$-line that emanate from $\sf{B_{phys}}$, it approaches the reference
boundary and becomes an $m$-line (blue line) near $\sf{B_{ref}}$. After
dimensional reduction this becomes a charge of $\mathbb{Z}_{2}^{F}$ (blue dot)
that lives at the end of a $\mathbb{Z}_{2}^{F}$-defect line. Since in the
SymTFT there is no excitation created anywhere, this non-local operator now
has non-zero vacuum expectation value.

###  III.2 Example: Fermionic gapped phases with $G^{F}=\mathbb{Z}_{2}^{F}$

We begin by studying gapped phases of one-dimensional systems without any
additional symmetries other than conservation of fermion parity, $(-1)^{F}$
where $F$ is the number of fermions (modulo two), which has group structure
$G^{F}=\mathbb{Z}_{2}^{F}$ (where the $F$ superscript is just a label
reminding that this $\mathbb{Z}_{2}$ factor arises from fermion parity). There
are two gapped phases of such a $1+1D$ fermion system without any symmetries:
a trivial superconductor and topological superconductor (Kitaev chain) with
unpaired Majorana edge modes. In a Majorana chain with periodic boundary
conditions, there is a fundamental ambiguity between these phases, which
differ only by a (Majorana) translation: the labeling of “trivial” and
“topological” amounts to declaring which pairs of Majorana fermions are paired
into atomic sites, and which pairs forms inter-site bonds. An unambiguous
property is that the interface between these two phases carries an unpaired
Majorana zero mode with radical quantum dimension $\sqrt{2}$.

Parallely, there are two distinct gapped boundaries of the SymTFT
corresponding respectively to condensing $m$ or $e$ at the physical boundary
$\sf{B_{phys}}$ in the sandwich construction. The interface between these two
gapped boundaries hosts a twist defect with quantum dimension $\sqrt{2}$.
Further, this twist defect is associated a short vertical line operator that
becomes a local Majorana fermion zero mode under dimensional reduction (See
Fig. 8). This line operator consists of a short vertical $f$ line that
terminates on the reference boundary decorated by a local fermion, $c$, and on
the physical boundary by splitting $f\rightarrow e\times m$, and absorbing the
$e$ and $m$ anyons in the appropriate domains.

We resolve this ambiguity by fixing the convention that the microscopic
fermion parity symmetry is generated by the $m$-line (as described above). As
we now show, this choice selects the $m$ condensed boundary as the trivial
phase and the $e$ condensed boundary as the topological phase. For this
purpose, we note that the local action of the fermion parity symmetry $m$ on a
finite boundary interval corresponds to acting with an $m$ line that spans the
interval, and terminates on the edges of the interval. Consider first the
$m$-condensed physical boundary (with condensable algebra
$\mathcal{A}_{m}^{phys}:=1+m$). Here, the fermion parity defect line $m$ can
be freely absorbed by the physical boundary, so the local action of fermion
parity is trivial – i.e. the system is a trivial insulator. By contrast, for
the $e$-condensed physical boundary (with condensable algebra
$\mathcal{A}_{e}^{phys}:=1+e$, the fermion parity cannot be trivially
localized to a finite boundary region, _unless_ its end points are “decorated”
by pulling an $f$ string from the reference boundary (See Fig. 7). This
decorated $m$ string, with fermionic ends, corresponds to a non-local string
order parameter of the topological superconducting phase [60, 61, 62, 63].

Figure 8: Majorana edge mode in SymTFT. On $\sf{B_{phys}}$ there is an
interface between $\mathcal{A}^{e}$-boundary condition and $\mathcal{A}^{m}$
boundary condition (black box). An $f$ line can now connect $\sf{B_{ref}}$ and
this interface without costing any energy. There is a local fermion operator
attached to the endpoint of this line on $\sf{B_{ref}}$ (blue dot). After
dimensional reduction this line becomes an edge mode. This edge mode is
fermionic and squares to 1. The point defect on the physical boundary has
quantum dimension $\sqrt{2}$. These properties indicate the edge mode is a
Majorana mode.

We note that, for a bosonic system with $\mathbb{Z}_{2}$ symmetry, the
$e$-condensed physical boundary condition would correspond to a spontaneous
symmetry breaking (SSB) state. There, this arose because $e$ was also
condensed at the reference boundary. Unlike for ordinary bosonic symmetries,
$\mathbb{Z}_{2}^{F}$ cannot be spontaneously broken, since its charges (the
order parameter of the symmetry) are fermionic and can never have long-range
SSB order. In the SymTFT description, SSB arises when the symmetry defect line
cannot be absorbed into either boundary, such that the sandwich, and the
sandwich with a symmetry-defect “condiment” layer correspond to distinct,
degenerate ground states. Importantly, this cannot happen in the fermionic
SymTFT construction, since there is no way to condense the fermion end of the
$f$ string on the physical boundary $\sf{B_{phys}}$(recall the physical
boundary in our construction stays bosonic). Moreover, for both the $e$ and
$m$ condensed $\sf{B_{phys}}$ phases an $m$ string condiment layer that wraps
through the bulk of the sandwich can either by directly absorbed into the
physical boundary, or dressed by an $f$ string from the reference boundary and
then absorbed into the $e$-string boundary, indicating that the system has a
unique symmetric ground-state.

In addition to matching the structure of edge states, we can also directly use
the SymTFT to reproduce the partition function of gapped phases, as we now
illustrate.

####  III.2.1 Spin structure and partition function of gapped phases

In the continuum a fermionic system is defined on a manifold with a spin
structure (or a Pin-structure for un-orientable manifolds), which can loosely
be thought of as a “background gauge field of fermion parity” in the sense
that the spin structure defines whether a fermion obtains a minus sign when
dragged around a non-contractible cycle of the manifold. A spin structure on
the torus can specified by periodic or anti-periodic boundary condition along
two non-contractible loops (generators of $H_{1}[T^{2}]$). Therefore we will
denote a spin structure on the torus by $(P/AP,P/AP)$.

##### Review: Partition function of the Kitaev chain.

The partition function of a fermionic system depends on the spin structure,
$\eta$. For example, the partition function of the Kitaev chain is given by
$\text{Arf}(\eta)$ in the IR limit, where $\text{Arf}(\eta)=\pm 1$ is the Arf
invariant. The values of the Arf invariant on the torus are

$\displaystyle\text{Arf}(\eta)=\begin{cases}1&\eta=(P,AP),(AP,P),(AP,AP)\\\
-1&\eta=(P,P)\end{cases}$ (10)

Physically this comes from that fact that the ground state of the topological
phase of the Kitaev chain has opposite fermion parities for periodic and anti-
periodic boundary conditions, whereas the trivial phase is adiabatically
connected to a product state that is completely insensitive to boundary
conditions. The partition function of the topological phase of the Kitaev
chain can then be computed as follows. We denote the ground-states with $P/AP$
boundary conditions as $|GS_{P/AP}\rangle$ respectively, and define the
normalized partition function on a torus ($T^{2}$) with $X^{1},X^{0}=P/AP$
boundary conditions in the space and time cycles as
$\mathcal{Z}[T^{2},(X^{1},X^{0})]$. Then the ordinary zero-temperature thermal
partition function has $AP$ boundary conditions in time, from which we
identify:

$\displaystyle\mathcal{Z}[T^{2},(P,AP)]$
$\displaystyle=\mathrm{Tr}\,|GS_{P}\rangle\langle GS_{P}|=1$
$\displaystyle\mathcal{Z}[T^{2},(AP,AP)]$
$\displaystyle=\mathrm{Tr}\,|GS_{AP}\rangle\langle GS_{AP}|=1$ (11)

Partition functions with periodic boundary condition in time can be obtained
from these by inserting a $(-1)^{P_{f}}$ into the trace which acts like a
spacelike defect of fermion parity in a path integral evaluation of the trace:

$\displaystyle\mathcal{Z}[T^{2},(X,P)]=\langle
GS_{X}|(-1)^{P_{F}}|GS_{X}\rangle.$ (12)

The key defining property of the topological superconductor is that the
ground-state fermion parity of $P$ and $AP$ boundary conditions are opposite.
The fermion parity of either alone is non-universal, and can be toggled by
adding an additional occupied fermion orbital, only the relative fermion
parity for $P$ vs $AP$ boundary conditions is well defined. For definiteness,
we will fix the convention that the ground-state with periodic boundary
conditions has odd fermion parity, in which case:

$\displaystyle\mathcal{Z}[T^{2},(P,P)]$
$\displaystyle=\mathrm{Tr}\,(-1)^{P_{f}}|GS_{P}\rangle\langle GS_{P}|$
$\displaystyle=\langle GS_{P}|(-1)^{P_{F}}|GS_{P}\rangle$ $\displaystyle=-1$
$\displaystyle\mathcal{Z}[T^{2},(AP,P)]$
$\displaystyle=\mathrm{Tr}\,(-1)^{P_{f}}|GS_{AP}\rangle\langle GS_{AP}|$
$\displaystyle=\langle GS_{AP}|(-1)^{P_{F}}|GS_{AP}\rangle$ $\displaystyle=1$
(13)

##### Partition function of the Kitaev chain from SymTFT:

We will now show how this partition function can be obtained from SymTFT
methods, deferring some technical details to Section V. Partition function of
the SymTFT sandwich is given by an inner product between states defined on
$\sf{B_{ref}}$ and $\sf{B_{phys}}$. We have given the boundary states
corresponding to the Dirchelet and Neumann boundary condition in II in the
field configuration basis. Here we will find it useful to introduce another
basis for boundary state, suitable for a torus boundary geometry. This is the
so-called anyon label basis, $\\{|a\rangle\\}$, where the label $a$ goes
through all simple anyons of the bulk SymTFT. This basis is defined by the
property that time-like anyon loops act on them as
$W_{b}^{t}|a\rangle=\sum_{c}N^{c}_{ba}|c\rangle$, and space-like loops act on
them as $W^{x}_{b}|a\rangle=\frac{S_{a,b}}{S_{a,0}}|a\rangle$, where
$N^{c}_{ba}$ is the fusion matrix of the SymTFT, and $S_{a,b}$ is the
$S$-matrix of the SymTFT. The state $|a\rangle$ can be thought of as being
prepared by a path integral in the interior of the torus, with an anyon loop
of $a$ inserted in the center.

The anyon basis is particularly useful when a boundary is specified by an
anyon condensation. If the anyon condensation is given by a condensable
algebra $\mathcal{A}=\sum_{a}c_{a}a$, then it is straightforward to derive
that the corresponding boundary state can be written in the anyon basis as
$|\mathcal{A}\rangle:=\sum_{a}c_{a}|a\rangle$. Notice this state has the
property that $W^{t/x}_{a}$ both have eigenvalue 1, for any $a\in\mathcal{A}$.

For the $f$-condensed reference boundary of the $\mathbb{Z}_{2}^{F}$-SymTFT,
it is natural to expect the reference boundary to be given by the state
$|1\rangle+|f\rangle$. Notice that the state $|1\rangle+|f\rangle$ has the
property that $f$-loops in the space- and time-cycles both have eigenvalue 1.
To see this, we can shrink the reference boundary to a point, then in the
spacetime picture the reference boundary is now reduced to an insertion of an
$1+f$ anyon string in the center of a solid torus. See Fig. 9. Then an
$f$-loop in the time direction can be fused with the $1+f$ insertion:
$W^{t}_{f}(|1\rangle+|f\rangle)=|f\rangle+|1\rangle$. Thus $f$-loops in the
time direction has eigenvalue 1. An $f$-loop in the space direction on the
other hand is performing a braid with the $1+f$ insertion. Since this braiding
is trivial, the space-like $f$-loops also have eigenvalue $1$ when acting on
the reference boundary. Then according to the discussion on the relation
between $f$-loop eigenvalues and spin structures in III.1.2, we see that the
state $|1\rangle+|f\rangle$ describes the $f$-condensed boundary with
$(AP,AP)$ spin structure.

With this established, we can start to compute the partition function of the
Kitaev chain via SymTFT. The Kitaev chain is obtained by imposing an $e$
condensation ($\mathcal{A}^{e}=1+e$) at $\sf{B_{phys}}$. The partition
function of the SymTFT sandwich is given by the inner product between the
state on the inner circle and the state on the outer circle:
$\mathcal{Z}[T^{2},(AP,AP)]=\langle\mathcal{A}_{\sf{ref}}|\mathcal{A}_{\sf{phys}}\rangle$,
with $|\mathcal{A}_{\sf{ref}}\rangle=|1\rangle+|f\rangle$ and
$|\mathcal{A}_{\sf{phys}}\rangle=|1\rangle+|e\rangle$, resulting in:
$\mathcal{Z}[T^{2},(AP,AP)]=1$.

The spin structure (boundary conditions) for the physical fermion can be
adjusted by inserting fermion parity flux lines through space or time-like
cycles of the SymTFT bulk. Consider changing the spatial boundary condition by
inserting an $m$-loop along the time cycle. Again, if we shrink the reference
boundary to a point, the condensate $\mathcal{A}_{\sf{ref}}=1+f$ becomes an
insertion of a $1+f$ world-line in the solid torus (see Fig. 9).

Figure 9: 1\. The reference boundary is shrunk to a point and becomes an
insertion of the algebra $\mathcal{A}_{\sf{ref}}^{f}$. 2. The spacetime
picture, the reference boundary becomes a world line of $1+f$ inserted in the
center of the solid torus.

Fusing the time-like $m$ loop with the $1+f$ condensed $\sf{B_{ref}}$ line
changes it into an $m+e$ line, resulting in:

$\mathcal{Z}[T^{2},(P,AP)]=(\langle e|+\langle
m|)(|\mathcal{A}_{\sf{phys}}\rangle)=1.$

Similarly, inserting an $m$-loop encircling the spatial cycle produces
$(AP,P)$ boundary conditions. The $m$-loop has a $(-1)$ braiding phase with
the $f$-insertion at $\sf{B_{ref}}$, resulting in:

$\displaystyle\mathcal{Z}[T^{2},(AP,P)]$ $\displaystyle=(\langle 1|+\langle
f|)W^{x}_{m}|\mathcal{A}_{\sf{phys}}\rangle$ $\displaystyle=\langle
1|\mathcal{A}_{\sf{phys}}\rangle-\underset{0}{\underbrace{\langle
f|\mathcal{A}_{\sf{phys}}\rangle}}=1$ (14)

Here $W^{x}_{m}$ is the $m$-loop in the spatial direction.

Lastly, (P,P) boundary conditions are obtained by inserting
$\mathbb{Z}_{2}^{F}$ symmetry defects ($m$ loops) in both space and time
directions. Fusing the time-like defect with the $1+f$ insertion at
$\sf{B_{ref}}$ and taking into account the braiding between the space-like
defect and the $f$-line, yields

$\mathcal{Z}[T^{2},(P,P)]=\underset{0}{\underbrace{\langle
m|\mathcal{A}^{e}\rangle}}-\langle e|\mathcal{A}^{e}\rangle=-1.$

Combining these results, we confirm that the proposed fermionic SymTFT
procedure correctly reproduces the known partition function for the Kitaev
chain.

In comparison, to deduce the trivial gapped phase, corresponding to an
$m$-condensate ($\mathcal{A}_{\sf{phys}}=\mathcal{A}^{m}=1+m$), one can use
precisely the same calculation with $e\leftrightarrow m$. The sole change is
that:

$\mathcal{Z}^{\rm trivial}[T^{2},(P,P)]=\langle
m|\mathcal{A}^{m}\rangle-\underset{0}{\underbrace{\langle
e|\mathcal{A}^{m}\rangle}}-=+1$

so that $\mathcal{Z}^{\rm trivial}=1$ for all boundary conditions as required
for a topological trivial phase.

###  III.3 Example: Fermionic SPTs with
$G^{F}=\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$

The SymTFT framework also captures gapped symmetry-protected topological (SPT)
states. In this section, we illustrate the main principles through the
simplest example: $G^{F}=\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$.

It is known in the literature (see e.g. [64]) that different $1+1D$
topological phases for $\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$ are labeled by
a pair of $\mathbb{Z}_{2}$-valued topological invariants, $(\nu_{+},\nu_{-})$
that respectively label the number of Majorana edge states with even (+) or
odd (-) $\mathbb{Z}_{2}$ symmetry charge. The stacking rule for these phases
is
$(\nu_{+},\nu_{-})\boxtimes(\nu^{\prime}_{+},\nu^{\prime}_{-})=(\nu_{+}+\nu^{\prime}_{+},\nu_{-}+\nu^{\prime}_{-})$,
with addition taken modulo $2$. The four phases have a group structure
$\mathbb{Z}_{2}^{2}$ under stacking.

Figure 10: Edge modes of the $(\nu_{+}=1,\nu_{-}=1)$ SPT. The left half of the
sandwich represents the non-trivial SPT phase of
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$, while the right half of the sandwich
represents the trivial phase. (a). A $\mathbb{Z}_{2}$-neutral fermionic edge
mode. (b). A $\mathbb{Z}_{2}$-charged fermionic edge mode. (c). A
$\mathbb{Z}_{2}$-charged bosonic edge mode, obtained from the product of the
fermion modes shown in (a),(b).

To construct a SymTFT description, we consider a
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$ gauge theory,
$D(\mathbb{Z}_{2})\times D(\mathbb{Z}_{2}^{F})$. The anyons of this theory can
be labeled by two copies of those for a $\mathbb{Z}_{2}$ gauge theory:
$\\{1,e,m,f=e\times f\\}$.

We label anyons from $D(\mathbb{Z}_{2})$ with a subscript 1 and those from
$D(\mathbb{Z}_{2}^{F})$ with a subscript 2. According to the general recipe
outlined above, the reference boundary according to the general recipe Eq.(37)
is given by condensing the symmetry gauge-charge $e_{1}$, and a bound state
$f_{2}\times c$ of the $\mathbb{Z}_{2}^{F}$ emergent fermion. This corresponds
to the condensation algebra:
$\mathcal{A}^{f}_{\sf{ref}}=1+e_{1}+f_{1}+e_{2}+f_{1}e_{2}$.

Distinct gapped phases correspond to different Lagrangian (fully confining)
condensations on $\sf{B_{phys}}$. We list all Lagrangian condensations of
$D(\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F})=D(\mathbb{Z}_{2})\boxtimes
D(\mathbb{Z}_{2}^{F})$, and the corresponding SPT phases in Table 1.

Condensation Algebra | Description | SPT Invariant
---|---|---
$\mathcal{A}^{1}=1+m_{1}+m_{2}+m_{1}m_{2}$ | Trivial | $(\nu_{+}=0,\nu_{-}=0)$
$\mathcal{A}^{2}=1+m_{1}+e_{2}+m_{1}e_{2}$ | neutral Kitaev chain | $(\nu_{+}=1,\nu_{-}=0)$
$\mathcal{A}^{3}=1+e_{1}e_{2}+m_{1}m_{2}+f_{1}f_{2}$ |  $\mathbb{Z}_{2}$-charged Kitaev chain | $(\nu_{+}=0,\nu_{-}=1)$
$\mathcal{A}^{4}=1+e_{1}m_{2}+e_{2}m_{1}+m_{1}m_{2}$ | SPT | $(\nu_{+}=1,\nu_{-}=1)$
$\mathcal{A}^{5}=1+e_{1}+m_{2}+e_{1}m_{2}$ | SSB | $\times$
$\mathcal{A}^{6}=1+e_{1}+e_{2}+e_{1}e_{2}$ | Kitaev chain + SSB | $\times$
Table 1: Gapped $G^{F}=\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$ phases from
SymTFT.  The correspondence between condensable algebras of
$D(\mathbb{Z}_{2}^{2})$ and $1+1D$
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$-phases. SPT indicates an invertible
topological phase that becomes trivial without symmetry protection, and SSB
denotes spontaneous symmetry breaking of the $\mathbb{Z}_{2}$ symmetry.

The first four phases are symmetric, since the symmetry charge, $e_{1}$, is
not directly condensed on $\sf{B_{phys}}$. The last two have spontaneous
symmetry broken (SSB) $\mathbb{Z}_{2}$ symmetry, since the $e_{1}$ condensate
on both boundaries defines a local order parameter given by a short $e_{1}$
segment spanning from $\sf{B_{ref}}$ to $\sf{B_{phys}}$. For the symmetric
phases, the SPT invariants can be directly confirmed by constructing edge
states operators for the interface between the SPT and trivial phase (or by
checking their bulk string-order parameter). As an example, we show the edge
modes of the $(\nu_{+}=1,\nu_{-}=1)$ phase, represented by the
$\mathcal{A}^{4}$ condensation, in Fig. 10. Comparing this set of phases to
the known classification, we find that the SymTFT provides a complete
description. A natural question is how the SPT stacking rule is manifested in
SymTFT, this question was addressed for bosonic symmetries in [65], we will
address the fermionic case in Section VI.2.

###  III.4 Example of a critical point: The Majorana CFT

Having confirmed that the proposed fermionic SymTFT formulation reproduces
various known gapped phases, we now examine phase transitions between these
gapped phases, focusing on the critical point separating the Kitaev and
trivial gapped phases with no additional symmetries
($G^{F}=\mathbb{Z}_{2}^{F}$). It is well known that one possible phase
transition between these phases is a massless Majorana CFT with with fixed-
point Lagrangian:

$\displaystyle L=i\int
dtdx\left[\psi_{R}(\partial_{t}-\partial_{x})\psi_{R}+\psi_{L}(\partial_{t}+\partial_{x})\psi_{L}\right]$
(15)

where $\psi_{L,R}$ are left- and right- moving Majorana (real fermion) fields.
This Majorana CFT is related to the Ising CFT by bosonization (gauging fermion
parity by summing over the different boundary conditions for the fermion
operators).

In the following, we show that the (ungapped) “nothing condensed” physical
boundary where neither $e$ nor $m$ are condensed is compatible with this CFT,
and follow the method developed in [29] to compute the partition function of
the Majorana CFT from that for the bosonic Ising CFT. We note in passing,
that, the “nothing condensed” boundary does not fully determine the CFT in the
gapless phase, but only topological aspects such as how symmetries are
implemented on scaling operators. For example, there are other CFTs (e.g. the
tricritical Ising CFT, or non-minimal models) that are also compatible with
the $\mathbb{Z}_{2}^{F}$ SymTFT, but describe more exotic multi-critical
points. We do not consider these more exotic options here.

####  III.4.1 Review of (bosonic) Ising CFT

To set the stage, we briefly review some important facts about the Ising CFT
and its SymTFT description [29]. The Ising CFT belongs to the family of
rational CFTs (RCFT). An RCFT has finite number of primary operators and the
Hilbert space on a circle decomposes into a finite direct sum

$\displaystyle\mathcal{H}=\bigoplus_{a,b}M_{ab}\mathcal{H}_{a}\otimes\overline{\mathcal{H}}_{b}$
(16)

where $\mathcal{H}_{a}$ ($\overline{H}_{b}$) is an irreducible representation
of the left (right) chiral algebra and $M_{ab}$ are positive integers. The
representations of a chiral algebra form a modular tensor category
$\mathcal{C}$, therefore the labels $a,b$ can be thought of as taking values
in anyons of a 2+1D chiral topological order $\mathcal{C}$ and its time
reversal $\overline{\mathcal{C}}$. This relation between 1+1D RCFT and 2+1D
chiral topological orders can also be understood from a sandwich construction.
Consider the non-chiral topological order
$\mathcal{Z}[\mathcal{C}]=\mathcal{C}\boxtimes\overline{\mathcal{C}}$ and a
sandwich construction with physical boundary being the state

$\displaystyle|\Psi\rangle=\sum_{a,b\in\mathcal{C}}\chi_{a}\overline{\chi}_{b}|a,b\rangle.$
(17)

Here $\chi_{a},\overline{\chi}_{b}$ are the characters of the chiral,anti-
chiral algebra representations respectively. The reference boundary can be
expanded in the anyon basis as:

$\displaystyle|\mathcal{A}\rangle=\sum_{a,b\in\mathcal{C}}M_{a,b}|a,b\rangle.$
(18)

Then the sandwich has partition function

$\displaystyle\langle\mathcal{A}|\Psi\rangle=\sum_{a,b}M_{a,b}\chi_{a}\overline{\chi}_{b},$
(19)

which is the partition function of the RCFT. This sandwich construction can be
viewed as the SymTFT for the non-invertible categorical symmetry
$\mathcal{C}$.

As an example, the Ising CFT is a RCFT with three primaries $1,\sigma,\psi$
which form the Ising category. By the above construction it can live on the
boundary of the double Ising theory and corresponds to the state

$\displaystyle|\Psi_{\sf{Ising}}\rangle=|\chi_{1}|^{2}|1,1\rangle+|\chi_{\sigma}|^{2}|\sigma,\overline{\sigma}\rangle+|\chi_{\psi}|^{2}|\psi,\overline{\psi}\rangle.$
(20)

####  III.4.2 Majorana CFT from SymTFT

To make the connection between the Ising CFT and the
$\mathbb{Z}_{2}^{F}$-SymTFT, let us replace the $\sf{B_{phys}}$ boundary of
the SymTFT sandwich by a thin slab of Ising CFT. See Fig. 11 Crucially, there
exists an invertible, gapped domain wall, $\mathcal{I}$, between the double
Ising theory and the toric code, across which Toric code anyons can “tunnel”
to those of the Ising quantum double topological order according to:

$\displaystyle 1\leftrightarrow(1,\overline{1})+(\psi,\overline{\psi})$
$\displaystyle e\leftrightarrow(\sigma,\overline{\sigma})$ $\displaystyle
m\leftrightarrow(\sigma,\overline{\sigma})$ $\displaystyle
f\leftrightarrow(1,\overline{\psi})+(\psi,\overline{1}).$ (21)

without creating excitations at the boundary.

Figure 11: The “club sandwich” construction. Left: Below the domain wall
$\mathcal{I}$ is the double Ising theory with Ising CFT living on the physical
boundary. Above the domain wall $\mathcal{I}$ is the
$\mathbb{Z}^{F}_{2}$-SymTFTwith fermionic reference boundary $1+f$. When the
double Ising layer is thin, this setup is a $\mathbb{Z}^{F}_{2}$-SymTFT
sandwich describing a gapless $\mathbb{Z}^{F}_{2}$-symmetric(fermionic)
system. Right: the domain wall is moved up and fused with the reference
boundary. The entire bulk becomes the double Ising theory. The reference
boundary has the condensation $1+\psi+\overline{\psi}+\psi\overline{\psi}$.

The resulting three layer, “club sandwich” [30], with a $\mathbb{Z}_{2}^{F}$
gauge theory interfaced by $\mathcal{I}$ to a slab of Ising quantum double
$\sf{Z[Ising]}$, sandwiched between the fermion condensed reference boundary
$\sf{B_{ref}}$, and the physical boundary state (20), now describes a gapless
fermionic $1+1D$ system. As we will see, instead of the Ising CFT the SymTO
sandwich is now equivalent to the Majorana CFT. Crucially, the Majorana CFT is
a fermionic CFT whose partition function depends on the spin structure.

We now compute the partition function of the sandwich to reveal this spin
structure dependence. First, move the invertible domain wall $\mathcal{I}$
towards $\sf{B_{ref}}$ and fuse it with $\sf{B_{ref}}$. After this the bulk of
the sandwich becomes entirely the double Ising theory. According to the rules
$1\to 1+\psi\overline{\psi},f\to\psi+\overline{\psi}$, the fermionic reference
boundary condensation $1+f$ becomes
$1+\psi\overline{\psi}+\psi+\overline{\psi}$. Therefore our initial club
sandwich is topologically equivalent to a double Ising topological order
sandwiched between the $1+\psi\overline{\psi}+\psi+\overline{\psi}$ condensed
reference boundary and the Ising CFT physical boundary (20).

The partition function of the sandwich can now be computed as the inner
product (recalling that the reference state $|1\rangle+|f\rangle$ corresponds
to $(AP,AP)$ boundary conditions):

$\displaystyle\mathcal{Z}[T^{2},(AP,AP)]$ $\displaystyle=(\langle
1,\overline{1}|+\langle\psi,\overline{\psi}|+\langle\psi,\overline{1}|+\langle
1,\overline{\psi}|)(|\Psi_{\sf{Ising}}\rangle)$
$\displaystyle=|\chi_{1}|^{2}+|\chi_{\psi}|^{2}+\chi_{\psi}\chi_{1}^{*}+\chi_{1}\chi_{\psi}^{*}=|\chi_{1}+\chi_{\psi}|^{2}.$
(22)

We obtain other boundary conditions by inserting fermion parity defect lines
as we did for the Kitaev chain in III.2.1. Consider changing the spatial
boundary condition by inserting a symmetry defect in the time direction. The
partition function of the sandwich with $(P,AP)$ boundary condition for the
physical fermions is then given by $(\langle e|+\langle
m|)\mathcal{I}(|\Psi_{\sf{Ising}}\rangle)$. Applying the transformation rule
(21), this is equal to

$\displaystyle\mathcal{Z}[T^{2},(P,AP)]=(\langle\sigma,\overline{\sigma}|+\langle\sigma,\overline{\sigma}|)|\Psi_{\sf{Ising}}\rangle=2|\chi_{\sigma}|^{2}.$
(23)

To change the boundary condition along the time direction we insert a symmetry
defect along the spatial direction. The partition function with $(AP,P)$
boundary condition is then

$\displaystyle\mathcal{Z}[T^{2},(AP,P)]$ $\displaystyle=(\langle 1|+\langle
f|)W^{x}_{m}\mathcal{I}|\Psi_{\sf{Ising}}\rangle$ $\displaystyle=\langle
1|\mathcal{I}|\Psi_{\sf{Ising}}\rangle-\langle
f|\mathcal{I}|\Psi_{\sf{Ising}}\rangle$
$\displaystyle=|\chi_{1}|^{2}+|\chi_{\psi}|^{2}-\chi_{\psi}\chi_{1}^{*}-\chi_{1}\chi_{\psi}^{*}=|\chi_{1}-\chi_{\psi}|^{2}.$
(24)

Here $W^{x}_{m}$ is an $m$-loop in the spatial direction. Lastly, we obtain
the $(P,P)$ boundary condition by inserting symmetry defects in both space and
time directions. We have the partition function: $\langle
e|\mathcal{I}|\Psi_{\sf{Ising}}\rangle-\langle
m|\mathcal{I}|\Psi_{\sf{Ising}}\rangle=|\chi_{\sigma}|^{2}-|\chi_{\sigma}|^{2}=0$.

In summary, the SymTFT computation gives

$\displaystyle\mathcal{Z}[T^{2},(AP,AP)]=|\chi_{1}+\chi_{\psi}|^{2},$
$\displaystyle\mathcal{Z}[T^{2},(P,AP)]=2|\chi_{\sigma}|^{2},$
$\displaystyle\mathcal{Z}[T^{2},(AP,P)]=|\chi_{1}-\chi_{\psi}|^{2},$
$\displaystyle\mathcal{Z}[T^{2},(P,P)]=0$ (25)

This matches perfectly with the torus partition functions of the Majorana CFT
[66].

##  IV Intrinsically fermionic and gapless SPTs from SymTFT

As we reviewed in Section II, SymTFT provides a full characterization for
bosonic gapless SPTs, including their classification, edge modes, group
extension structures, and emergent anomalies.

Given such success, it is natural to ask whether there is also a
correspondence between SymTFT and fermionic gSPT, whether the SymTFT could
assist us in constructing new models of fermionic gSPTs and understand
topological aspects of them, etc. We provide evidence by studying a SymTFT
sandwich construction, which turns out to describe a fermionic gapless SPT
with symmetry $\mathbb{Z}_{8}\times\mathbb{Z}_{2}^{F}$ and an emergent $k=2$
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$ fermionic anomaly. We show that this
igSPT has the intriguing property of having a half-charged Majorana edge mode.

Previously, models of fermionic igSPTs [44] have only realized bosonic
anomalies: i.e. are essentially bosonic igSPTs realized from the bosonic
(spin- or Cooper pair) degrees of freedom in a Mott insulator of fermions.
While this may give a physical mechanism for their experimental realization,
the resulting low energy properties (edge states and gapless modes) are
indistinguishable from those of a purely bosonic system. Therefore it is an
interesting question whether there exists an “intrinsically-fermionic” igSPTs
whose topological features can not arise in a purely bosonic system, but
involve gapless fermion degrees of freedom in the IR. Below we construct
precisely such an intrinsically fermionic igSPT via the SymTFT methods
developed above. While lattice models and field theories of this state could
in principle be constructed without reference to SymTFT, we found that the
SymTFT actually serves as a useful algebraic, and graphical tool to quickly
prototype these phases. In this way, the SymTFT can also be a practical
technique for constructing new physics, as well as re-interpreting known
phases.

We consider the gapless sector of the igSPT to have the fermionic symmetry
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$ and an emergent anomaly. Anomalies of
this symmetry in $1+1D$ have an $\mathbb{Z}_{8}$-classification. The
$\nu\in\mathbb{Z}_{8}$ anomaly can be realized by $\nu$ “flavors” of Majorana
CFTs: For the case of interests to us here, anomalies of
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$ correspond to $2+1D$
$\mathbb{Z}_{2}$-symmetric topological superconductors, which can be viewed as
$\nu$ stacks of $\mathbb{Z}_{2}$-neutral $p+ip$ superconductor and
$\mathbb{Z}_{2}$-charged $p-ip$ superconductor pairs. The symmetry-preserving
gapless edge of these 2+1D SPTs are described by the field theory:

$\displaystyle L=\sum_{a=1}^{\nu}i\int dtdx$
$\displaystyle[\psi_{R,a}(\partial_{t}-\partial_{x})\psi_{R,a}$
$\displaystyle+\psi_{L,a}(\partial_{t}+\partial_{x})\psi_{L,a}]$ (26)

where $a$ is a flavor index, $\psi_{L/R,a}$ are left/right (L/R) moving
Majorana fermions, and the anomalous $\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$
symmetry acts by:

$\displaystyle\begin{pmatrix}\psi_{L,a}\\\
\psi_{R,a}\end{pmatrix}\overset{\mathbb{Z}_{2}}{\longrightarrow}\begin{pmatrix}\psi_{L,a}\\\
-\psi_{R,a}\end{pmatrix},~{}~{}~{}~{}\begin{pmatrix}\psi_{L,a}\\\
\psi_{R,a}\end{pmatrix}\overset{\mathbb{Z}_{2}^{F}}{\longrightarrow}\begin{pmatrix}-\psi_{L,a}\\\
-\psi_{R,a}\end{pmatrix}.$ (27)

With interactions, the anomalies have a $\mathbb{Z}_{8}$ group structure
(topologically $\nu\simeq\nu+8$) [67, 68, 69].

Previously we have established the SymTFT for non-anomalous $G^{F}$ symmetry.
In order to describe fermionic igSPTs via SymTFTwe need to discuss the
description of anomalous fermionic symmetry in SymTFT.

###  IV.1 Anomalous fermionic symmetries in SymTFT

The SymTFT of a system with an anomalous bosonic symmetry is generally given
by the twisted gauge theory obtained from gauging the corresponding higher-
dimensional SPT whose boundary realizes the anomalous symmetry. This can be
understood from the fact that any $1+1D$ anomalous system can be realized on
the boundary of a $2+1D$ SPT, and the SymTFT is simply obtained by gauging
this $2+1D$ SPT. Following this principle we may formulate SymTFT for an
anomalous fermionic symmetry(with no gravitational anomaly) as follows. An
anomaly of $G^{F}$ in $1+1D$ corresponds to an SPT of $G^{F}$ in $2+1D$, i.e.
a (non-chiral)topological superconductor with symmetry $G^{F}$. Then the
SymTFT is given by gauging this topological superconductor, which is a non-
chiral topological order that we will call a twisted $G^{F}$-gauge theory,
although the twist here no longer has a cohomology classification. Independent
of the twist, the gauge charges of the twisted $G^{F}$-gauge theory have the
same structure as those of an untwisted one, the super-Tannakian category
$\text{Rep}(G,P_{f})$. Namely the gauge charges are labelled by
representations of the group $G^{F}$, and a gauge charge in the representation
$R$ is bosonic/fermionic if fermion parity is represented as $+1/-1$.
Therefore we can still define the reference boundary of the SymTFT sandwich as
the one where all gauge charges are condensed(with the understanding that
fermionic charges are condensed together with a local fermion introduced on
the reference boundary). This completes our construction for anomalous
fermionic symmetry.

Recall that the $2+1D$ SPT corresponding to the level $\nu$ anomaly of
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$ is $\nu$ stacks of
$\mathbb{Z}_{2}$-neutral $p+ip$ superconductor and $\mathbb{Z}_{2}$-charged
$p-ip$ superconductor pairs. The theory obtained by gauging
$G^{F}=\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$ in the 2+1D bulk of this SPT
can be deduced from the stacking structure; this topological order will become
the SymTFT for the anomalous $\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$
symmetry. Denote the symmetry ($\mathbb{Z}_{2}$) flux by $\Phi$, and fermion-
parity ($\mathbb{Z}_{2}^{F}$) flux by $\Phi_{F}$. Whereas $\Phi_{F}$ effects
both the $p\pm ip$ layers (all fermions are charged under
$\mathbb{Z}_{2}^{F}$), the symmetry flux, $\Phi$, only affects the $p-ip$
layers. For a single $p\pm ip$ superconductor, the flux (superconducting
vortex), carries an unpaired Majorana zero mode with topological spin $e^{\pm
i\pi/4}$. Thus, for $\nu=1$, the $\Phi$ flux will be such a non-Abelian
particle, and the corresponding anomalous edge will also have a non-Abelian
character that prevents it from having a tensor-product Hilbert space
structure. We anticipate that this obstacle will prevent the realization of
the $\nu=1$ edge anomaly as an igSPT – since this anomaly has a
“gravitational” character that cannot be alleviated merely by extending the
symmetry group. Hence, we instead focus on even $\nu$.

The $\nu=4$ ($n=2$) anomaly is a self-anomaly of the bosonic,
$\mathbb{Z}_{2}$, part of the symmetry group and does not involve
$\mathbb{Z}_{2}^{F}$. Equivalently, the 2+1D $\nu=4$ SPT is topologically-
equivalent to a bosonic SPT, and the fermions are merely spectators.
Therefore, the only interesting candidate for an intrinsically-fermionic igSPT
is $\nu=2$ ($\nu=6\simeq-2$ will be similar), on which we now focus.

We refer to the 2+1D topological order obtained by gauging the symmetry of
this $\nu=2$ SPT: $D^{F}[\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F},\nu=2]$. The
simple anyons are generated by fluxes $\phi,\phi_{F}$, and their corresponding
charges, denoted by $b$ and $f$. The $\mathbb{Z}_{2}$ symmetry flux is a
$\pi$-flux in two $p+ip$ layers. Since a stack of two $p+ip$ superconductors
is equivalent to an integer quantum Hall state with unit Hall conductance,
this $\pi$ flux therefore binds $1/2$ of a symmetry-charge and acquiring
quantum spin $e^{i\pi/4}$. The $\mathbb{Z}_{2}^{F}$-flux is a $\pi$ flux in
all $p+ip$ and $p-ip$ layers. It therefore induces equal and opposite
contributions to spins from the $+$ and $-$ layers, and is a boson with
fractional $\mathbb{Z}_{2}$ charge (from the $+$ layers). The $\Phi^{2}$ is a
$2\pi$-flux in two $p+ip$ layers, therefore it binds a charged electron and is
identified with $f\times b$. $\phi_{F}^{2}$ is a boson, but it binds a charged
electron and 7 uncharged electrons, therefore it is charged under
$\mathbb{Z}_{2}$ and is identified with $b$. Thus the gauged $\nu=2$ SPT is
completely generated by $\Phi$ and $\Phi_{F}$. They both have order 4. The
braiding phase between $\Phi_{F}$ and $\Phi$ is the same as the braiding phase
between two $\Phi$s, which is square of the exchange phase:
$\theta_{\phi,\phi_{F}}=e^{i\pi/2}$. This completes the construction of the
gauged $\nu=2~{}\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$ SPT, which is the
SymTFT for the $\nu=2$ anomalous $\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$.

###  IV.2 Fermionic igSPT with $\mathbb{Z}_{8}$ symmetry

The topological properties of igSPTs stem from an emergent anomalous symmetry,
$G^{F}$. Instead of arising at the boundary of a higher-dimensional gapped
$G^{F}$ SPT, the anomaly is “cured” by gapped degrees of freedom that
transform under a larger group $\Gamma^{F}$, which is an extension of $G^{F}$.
In the SymTFT description, this requires that an untwisted $\Gamma^{F}$ gauge
theory can be reduced to a twisted $G^{F}$ gauge theory by partially-confining
(non-Lagrangian) anyon condensation, $\mathcal{A}$.

For the intrinsically-fermionic $\nu=2$
$G^{F}=\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$ anomaly above, one solution is
to extend the $\mathbb{Z}_{2}$ symmetry by $\mathbb{Z}_{4}$ to
$\Gamma^{F}=\mathbb{Z}_{8}\times\mathbb{Z}_{2}^{F}$. In other words, a
fermionic system with $\mathbb{Z}_{8}$ symmetry can exhibit an intrinisically-
fermionic igSPT with the $\nu=2$ $\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$
anomaly.

##### Lifting the anomaly

Denote the anyons of $D(\mathbb{Z}_{8}\times\mathbb{Z}_{2}^{F})$ as $a_{1,2}$
with $a\in{1,e,m}$ and composites thereof, and where $1,2$ subscripts refer to
the $\mathbb{Z}_{8},\mathbb{Z}_{2}^{F}$ gauge subgroups respectively. As
above, denote the canonical fermion charge as $f_{2}=e_{2}m_{2}$. Then, a
suitable partially-confining condensation that yields the desired twisted
$D(\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F},\nu=2)$ topological order is
obtained by condensing the anyon $a=e_{1}^{6}m_{1}^{2}e_{2}m_{2}$. The
remaining deconfined anyons are generated by $\phi=e_{1}m_{1}$, and
$\phi_{F}=e_{1}^{2}m_{2}$, and are topologically identical to those of the
anomalous SymTFT $D(\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F},\nu=2)$.

##### Sandwich construction

A sandwich construction of the igSPT is obtained by choosing a
$\Gamma^{F}=\mathbb{Z}_{8}\times\mathbb{Z}_{2}$ gauge theory for the bulk
SymTFT, the canonical reference boundary $\mathcal{A}_{\sf{ref}}^{f}=\langle
e_{1},e_{2}m_{2}c\rangle$, and the partially-confining (non-Lagrangian)
physical boundary, $\mathcal{A}_{\sf{phys}}=\langle
e_{1}^{6}m_{1}^{2}e_{2}m_{2}\rangle$. The physical boundary is necessarily
gapless (or symmetry broken). A candidate field theory that matches the
anomaly is given by two copies of the Majorana CFT (Eq. 26 with $\nu=2$), or
equivalently a massless Dirac fermion with a chiral symmetry anomaly.

The $\mathbb{Z}_{8}$ symmetry is generated by inserting a horizontal $m_{1}$
line “condiment” in the sandwich. For the igSPT sandwich, a pair of symmetry
generators, $m_{1}^{2}$ line, can be freely absorbed into the boundaries, by
splitting $m_{1}^{2}=a\times(e_{1}^{6}f_{2})$ and absorbing the $a$ into the
physical boundary and the $e_{1}^{6}f_{2}$ into the reference boundary. Thus,
the $\mathbb{Z}_{8}$ symmetry is effectively reduced to an IR $\mathbb{Z}_{2}$
symmetry. Denoting elements of
$\Gamma^{F}=\mathbb{Z}_{8}\times\mathbb{Z}_{2}^{F}$ as $(n,m)$, the group
extension structure is given by the short exact sequence:

$\displaystyle
1\to\mathbb{Z}_{4}\xrightarrow{i}\mathbb{Z}_{8}\times\mathbb{Z}_{2}^{F}\to\mathbb{Z}_{2}^{\text{IR}}\times\mathbb{Z}_{2}^{F}\to
1.$ (28)

Here the embedding map $i$ is $\mathbb{Z}_{4}\ni
1\to(2,0)\in\mathbb{Z}_{8}\times\mathbb{Z}_{2}^{F}$.

Figure 12: Edge mode of the $\mathbb{Z}_{8}\times\mathbb{Z}_{2}^{F}$ igSPT.
The edge mode is a verticle $e_{1}^{6}f_{2}$ line at the interface between
igSPT and trivial phase. Ths operator has charge $1/2$ under the
$\mathbb{Z}_{2}^{\text{IR}}$ symmetry(a horizontal $m_{1}$-line), and charge
$1$ under fermion parity(a horizontal $m_{2}$-line).

##### Edge modes

A defining property of igSPTs is the presence of symmetry-protected
topological edge states, that, despite the gapless bulk, are confined to the
edge. Moreover, these edge states cannot be realized in any gapped topological
phase with the same symmetry. In the SymTFT sandwich construction, edge modes
are revealed by consider a spatial interface between two different gapped
physical boundaries : a trivial phase described by condensing symmetry fluxes:
$\mathcal{A}^{0}=\langle m_{1},m_{2}\rangle$, and the igSPT boundary,
$\mathcal{A}=\langle a=e_{1}^{6}m_{a}^{2}f_{2}\rangle$ (Fig 12). The short
string segment of $e_{1}^{6}f_{2}$ localized near the trivial/igSPT interface
represents a zero-energy edge state. Decorating the $\sf{B_{ref}}$ end of this
segment by a local (Majorana) fermion, $\gamma=\frac{1}{2}(c+ic^{\dagger})$,
allows it to be absorbed into the fermionic condensate. At $\sf{B_{phys}}$,
the segment can split into an $a$ anyon and an $m_{1}^{2}$ excitation that can
respectively be absorbed into the igSPT ($\mathcal{A}$) and trivial
$(\mathcal{A}^{0})$ condensates. Since the ends of the segment can be
absorbed, the resulting operator does not produce an energetic excitation.
Moreover, the segment operator has a non-trivial symmetry action on the
ground-space of the system. The symmetry generators are horizontal $m_{1}$ or
$m_{2}$ lines. The zero mode operator anticommutes with the $m_{2}$ line, as
required since the mode is a fermion and is charged under the physical fermion
parity. Moreover, the zero mode operator has exchange phase $(-i)=e^{-2\times
2\pi i/8}$ with $m_{1}$, indicating that it carries ($-2$) units of the
$\mathbb{Z}_{8}$ symmetry charge (in terms of the IR $\mathbb{Z}_{2}$
symmetry, the fermionic zero mode appears to carry a “half” symmetry charge).

##### Ground-state degeneracy with open boundaries

For an igSPT of length $L$, with open boundary conditions (surrounded by
trivial phases), the igSPT edge zero-modes span a four-fold ground-space
degeneracy. Defining this ground-space degeneracy for a gapless system takes
some care. In a system of length for the multi-component Majorana CFT the bulk
gap scales $\sim 1/L$, whereas the topological ground-space degeneracy is
exponentially small $\sim e^{-L/\xi}$ where $\xi$ is a microscopic correlation
length related to the $a$-condensation energy scale. To see this, note that
the edge mode operator shown in Fig. 12 fails to commute up to an
$e^{2i\pi/4}$ phase with a horizontal $m_{1}$ line (which measures the
symmetry charge). This algebra cannot be represented on a unique ground-state,
but rather requires a minimum of four degenerate ground-states. Importantly,
unlike for gapped SPTs, this ground-space degeneracy does not factorize into a
pair of local Hilbert spaces for each edge. Rather, there is a single, non-
local dimension-four ground-space shared by the entire chain.

###  IV.3 Field theory description

Despite its purely formal appearance, the SymTFT sandwich construction
actually directly hints at a potential microscopic mechanism for realizing the
igSPT phase. The insights from SymTFT can also guide the construction of
lattice-models and field-theory descriptions, as we now illustrate.

Consider a $1+1D$ fermion chain with $\mathbb{Z}_{8}$ symmetry. The
$e_{1}^{6}m_{1}^{2}e_{2}m_{2}$ condensation can then be interpreted via the
SymTFT dictionary between anyons and generalized charges. The $m_{1}^{2}$
anyon represents a strength-two domain wall of the $\mathbb{Z}_{8}$ symmetry.
Thus the $e_{1}^{6}m_{2}^{2}$ part of the condensate corresponds to decorating
strength two domain walls with $6=-2$ symmetry charges of the
$\mathbb{Z}_{8}$. This decorated domain wall has fermionic statistics, which
can be screened by binding them to a local fermion (recall that in the fermion
SymTFT construction, the $e_{2}m_{2}$ anyon is the “shadow” of the local
fermion excitation on the physical boundary), and then condensing this
composite object.

The SymTFT picture suggests a simple bosonized field theory description of the
igSPT phase. We start from a pair of Luttinger liquids, consisting of a
$\mathbb{Z}_{8}$-charged boson field $e^{i\varphi_{1}}$, and its dual vortex
$e^{i\varphi_{2}}$, and a left and right moving fermion field
$\psi_{L,R}=\eta_{L,R}e^{i\phi_{L,R}}$ where $\eta$ are anticommuting Klein
factors. The (Euclidean/imaginary-time) action density for this theory reads:

$\displaystyle\mathcal{L}_{0}=$
$\displaystyle\frac{i}{4\pi}\sum_{I,J=1,2}\partial_{\tau}\varphi_{I}\sigma^{x}_{IJ}\partial_{x}\varphi_{J}+$
$\displaystyle+\frac{i}{4\pi}\sum_{I,J=L,R}\partial_{\tau}\phi_{I}\sigma^{z}_{I,J}\partial_{x}\phi_{J}+H_{v},$
(29)

where $H_{v}$ is the Hamiltonian containing (non-universal) terms that
describe the velocity of excitations such as
$v_{IJ}\partial_{x}\varphi_{I}\partial_{x}\varphi)_{J}$ where $v_{IJ}$ is a
symmetric, positive matrix (and similarly for the $\Phi$ fields), and
$\sigma^{x,z}$ are Pauli matrices in the $I,J$ flavor space(s). Quantizing the
theory yields commutation relations:
$e^{i\alpha\partial_{x}\varphi_{I}(x)/2\pi}\varphi_{J}(y)e^{-i\alpha\partial_{x}\varphi_{I}(x)/2\pi}=\varphi_{J}(y)+\alpha\sigma^{x}_{IJ}~{}\delta_{x,y}$,
and (and similarly for $\phi_{I}$ with
$\sigma^{x}_{IJ}\leftrightarrow\sigma^{z}_{IJ}$), where $\alpha$ is a
constant.

The $\mathbb{Z}_{8}$ symmetry generator, $g$, and $\mathbb{Z}_{2}^{F}$
generator, $(-1)^{F}$ acts on the fields as:

$\displaystyle g$ $\displaystyle:\varphi_{1}\rightarrow\varphi_{1}+2\pi/8,$
$\displaystyle(-1)^{F}$ $\displaystyle:\phi_{L/R}\rightarrow\phi_{L,R}+\pi$
(30)

(other, unlisted fields transform trivially under the symmetry in question).
Crucially this symmetry action is non-anomalous, and $\mathcal{L}_{0}$ may
emerge as the continuum description of a lattice model with on-site symmetry
action.

##### Domain wall operators

According to the commutation relations above, in the quantized theory, these
global symmetry transformations restricted to the interval $I=[x_{1},x_{2}]$
are implemented by operators
$g|_{I}=e^{i\int_{a}^{b}dx\partial_{x}\varphi_{2}/8}$ and
$(-1)^{F}|_{I}=e^{i\int_{x_{1}}^{x_{2}}(\phi_{R}-\phi_{L})/2}$.
Correspondingly, the operators $m_{1}(x)=e^{i\varphi_{2}(x)/8}$ and
$m_{2}(x)=e^{i(\phi_{R}(x)-\phi_{L}(x))/2}$, create symmetry domain walls at
position $x$.

If the symmetry were gauged, these domain wall operators would correspond to
boundary-termination of a a gauge-flux ($m_{1,2}$) line operator. From this
identification, one can immediately recognize a local operator that plays the
role of the decorated symmetry domain wall $a=m_{1}^{2}e_{1}^{-2}f_{2}$ whose
condensation in the SymTFT produces the igSPT boundary with anomalous
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$ symmetry action as:

$a(x)=e^{i\left(2\varphi_{2}(x)/8-2\varphi_{1}(x)+\phi_{R}(x)\right)}.$

This operator has trivial self-statistics, $[a(x),a(y)]=0$. The operator is
non-local due to the fractional coefficient in front of the $\varphi_{2}$
field, and cannot be directly added to the Hamiltonian, however its fourth
power can. Specifically, the $a$-anyon condensation can be implemented by
adding a Hamiltonian term:

$\displaystyle H_{\rm igSPT}=-\Delta\int
dx\cos\left[\varphi_{2}(x)-8\varphi_{1}(x)+4\phi_{R}(x)\right]$ (31)

where $\Delta$ is an energy scale (depending on the Luttinger liquid
parameters, this term may or may not be perturbatively relevant, however, it
can always be made relevant by cranking up the coefficient $\Delta$). $H_{\rm
igSPT}$ is clearly invariant under the terms in Eq. 30, and when relevant,
gaps out the linear combination of fields appearing in the cosine term, and
also fields that are conjugate to this linear combination.

##### Emergent anomaly

A basis for the resulting low energy fields that commute with $H_{\rm igSPT}$
is:

$\displaystyle\tilde{\phi}_{R}$ $\displaystyle=\phi_{R}-4\varphi_{1}$
$\displaystyle\tilde{\phi}_{L}$ $\displaystyle=\phi_{L},$ (32)

and an effective low-energy (IR) action-density for the null-space of $H_{\rm
igSPT}$ takes the form of a (bosonized) massless Dirac fermion:

$\mathcal{L}_{\rm
IR}=\frac{i}{4\pi}\sum_{IJ}\partial_{\tau}\tilde{\phi}_{I}\sigma^{z}_{IJ}\tilde{\phi}_{J}.$

The operators $\eta_{L,R}e^{\pm\tilde{\phi}_{L,R}}$ are fermionic fields, and
their arguments transform under the symmetry as:

$\displaystyle g$ $\displaystyle:\begin{pmatrix}\tilde{\phi}_{R}\\\
\tilde{\phi}_{L}\end{pmatrix}\rightarrow\begin{pmatrix}\tilde{\phi}_{R}-\pi\\\
\tilde{\phi}_{L}\end{pmatrix}$ $\displaystyle(-1)^{F}$
$\displaystyle:\begin{pmatrix}\tilde{\phi}_{R}\\\
\tilde{\phi}_{L}\end{pmatrix}\rightarrow\begin{pmatrix}\tilde{\phi}_{R}+\pi\\\
\tilde{\phi}_{L}+\pi\end{pmatrix}$ (33)

which is indeed the anomalous $\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$
symmetry action for the edge of the $\nu=2$ fermion SPT [70].

##### Edge modes

The igSPT edge modes can be identified by considering an interface with
$H_{\rm igSPT}$ added for $x>0$, and a trivial mass term for $x<0$:

$\displaystyle H_{\rm interface}=-\Delta_{\rm
igSPT}\int_{0}^{L}dx\cos\left(\varphi_{2}-8\varphi_{1}+4\phi_{R}\right)-\Delta_{\rm
trivial}\int_{x\notin[0,L]}dx\left[\cos\varphi_{2}+\cos(\phi_{L}+\phi_{R})\right]$
(34)

To simplify the analysis, we take the extreme limit $\Delta\rightarrow\infty$
(or equivalently consider the RG fixed point limit where these terms are
relevant). In this limit, the field configurations are pinned locally to the
minima of the cosine. For the igSPT region the ground-space is given by the
four distinct minima of the corresponding cosine:

$\displaystyle\phi_{R}=2\varphi_{1}-\varphi_{2}/4+\frac{n\pi}{2}$ (35)

The operator $e^{2i\varphi_{1}(x)}$ creates a kink between these minima
whenever $0<x<L$, which costs a finite energy due to the gradient terms in
$H_{v}$, except precisely at the edges: $x=0$ or $x=L$. In the trivial
regions, there is a unique local ground-state given by $\phi_{L}=-\phi_{R}$
and $\varphi_{2}=0$. The operators $e^{\varphi_{1}(x)}$ and
$e^{i\phi_{L,R}(x)}$ create local Kink excitations in these ground-state
configurations, except when inserted exactly at the interface.

In the SymTFT, the zero-mode operator is implemented by an $e_{1}^{-2}f_{2}$
line segment that splits into an $a=m_{1}^{2}e_{2}^{-2}f_{2}$ anyon that
terminates on the igSPT side, and an $m_{1}^{-2}$ anyon that terminates on the
trivial side, and which is decorated by a local fermion on the physical
boundary (Fig. 12), decorated by a local fermion on the physical boundary.
Translating these operators into the continuum theory gives a zero-mode
operator:

$\displaystyle\Psi^{\dagger}(0,L)=\eta_{R}\exp\left\\{i\left[\phi_{R}(0,L)-2\varphi_{1}(0,L)\right]\right\\}$
(36)

$\Psi^{\dagger}$ commutes with the cosine terms in both $H_{\rm igSPT}$ and
$H_{\rm trivial}$, i.e. it is indeed a zero mode. Crucially, this operator is
only a zero mode when inserted at the igSPT/trivial interface ($x=0$ or $L$).
Otherwise, as described above, the term creates (massive) kinks in the local
ground-state configuration of the fields. Moreover, by inspection,
$\Psi^{\dagger}$ has odd fermion parity and adds $\mathbb{Z}_{8}$ two units of
$\mathbb{Z}_{8}$ symmetry charge
($g^{\dagger}\Psi^{\dagger}g=e^{i\pi/2}\Psi^{\dagger}$). As a technical
comment, the Klein factor $\eta_{R}$ is required to ensure that $\Psi$
anticommutes with all local fermion excitations such as
$\eta_{L}e^{i\phi_{L}(x)}$ (as physically required for any odd-fermion parity
excitation).

##  V SymTFT for fermions: Systematics and Technical Details

Having illustrated the fermionic SymTFT construction for a variety of specific
examples, we now generalize this framework to general finite, unitary
fermionic symmetries.

Conceptually, the construction proceeds much as the case without symmetry
above, with some technical modifications. The bulk SymTFT is a $G^{F}$ gauge
theory, whose symmetry charges divide into bosonic and fermionic ones. At the
fermionic reference boundary, in the sandwich constructions, one condenses all
symmetry gauge charges by binding the fermionic ones to a local fermion $c$ to
convert them into condensable bosons. This reference boundary is again
fermionic and requires choosing a $G^{F}$-spin structure. The remainder of
this section is devoted to making the qualitative description above
mathematically explicit.

###  V.1 General symmetries

Having worked out the topological holography for fermions without a general
symmetry, we now consider the SymTFT for a general fermionic symmetry $G^{F}$.
The fermion parity generates a central subgroup $\mathbb{Z}_{2}^{F}\lhd
G^{F}$, and the quotient $G^{B}=G^{F}/\mathbb{Z}_{2}^{F}$ is referred to as
the “bosonic part” of the symmetry. Consider a $1+1D$ fermionic system
$\sf{T}$ with symmetry $G^{F}$ living on $M$. To start, we assume the symmetry
is not anomalous. Following the construction of SymTFT for bosonic symmetries,
we may take the $2+1D$ SymTFT for $G^{F}$ to be the $G^{F}$-gauge theory. To
complete the construction we once again consider the sandwich construction. We
must choose an appropriate reference boundary condition so that the sandwich
is equivalent to the original system $\sf{T}$.

####  V.1.1 Condensation on the reference boundary

Symmetry “charges” of $G^{F}$ corresponded to representations of of $G^{F}$,
which are labeled by a $\mathbb{Z}_{2}^{F}$ fermion parity $|R|=0/1$ when the
representation $R$ has even/odd fermion parity respectively. Fermi-statistics
requires that braiding two symmetry charge excitations with representations
$R,R^{\prime}$ results in the exchange phase $(-1)^{|R|\cdot|R|^{\prime}}$,
and similarly charges should have quantum spin (self-statistics) $(-1)^{|R|}$.
Formally, the local charges of $G^{F}$ form the super-Tannakian category
$\text{Rep}(G^{F},P_{f})$ [71].

Recall anyons in $D(G^{F})$ are labeled by pairs $(\sigma,r)$, where $\sigma$
is a conjugacy class of $G^{F}$ and $r\in\text{Rep}(\mathcal{C}_{G}(\sigma))$
is a representation of the centralizer of $\sigma$. The simple anyons
correspond to the cases where the conjugacy class is minimal (not a union of
two smaller conjugacy classes) and the representation $r$ is irreducible. When
the conjugacy class is the trivial one $\sigma=\\{1\\}$, the anyons
$(1,R\in\text{Rep}(G))$ are called charges and we denote them by $e_{R}$. When
the representation $r$ is the trivial one, the anyons $(\sigma,1)$ are called
fluxes and we denote them by $m_{\sigma}$. Anyons that are neither charges nor
fluxes are called dyons. For details of quantum double models, see Appendix A.

Since all the gauge charges in $D(G^{F})$ are bosonic, the set of all gauge
charges $e_{R},R\in\text{Rep}(G^{F})$ does not have the correct structure of
$\text{Rep}(G^{F},P_{f})$. Then, to transmute the statistics of an odd fermion
parity gauge-charge charge into a fermion we can bind a fermion parity flux to
it to form a fermionic dyon. Thus the condensation having the structure of
$\text{Rep}(G^{F},P_{f})$ is given by the algebra

$\displaystyle\mathcal{A}_{\sf{ref}}^{f}=\left(\bigoplus_{|R|=0}d_{R}e_{R}\right)\bigoplus\left(\bigoplus_{|R|=1}d_{R}e_{R}m^{F}\right).$
(37)

where $d_{a}$ is the quantum dimension of anyon, $a$. The $e_{R}m^{F}$ anyons
are generalization of the $f$-particle of toric code. Similar to the
$\mathbb{Z}_{2}^{F}$-SymTFT case, the fermionic anyons $e_{R}m^{F}$ must be
bound to a local fermion $c$ in order to obtain a condensable object with
bosonic statistics.

Since the condensation involves fermions, the algebra
$\mathcal{A}_{\sf{ref}}^{f}$ is not a usual commutative Frobenius algebra.
Instead it has the structure of a super-commutative (i.e. with
$\mathbb{Z}_{2}^{F}$ graded commutation relations) Frobenius super-algebra
[37, 38]. Specifically, we can define even and odd sectors
$\mathcal{A}_{\sf{ref}}^{f,0},\mathcal{A}_{\sf{ref}}^{f,1}$ that contain the
bosonic and fermionic charges respectively. Let us rewrite
$\mathcal{A}_{\sf{ref}}^{f}$ to reveal its super-structure. The even sector is
isomorphic to the group algebra $\mathbb{C}[G^{F}]$ tensored with the trivial
$\mathbb{Z}_{2}^{F}$-representation:

$\displaystyle\mathcal{A}_{\sf{ref}}^{f,0}=\left(\bigoplus_{R,|R|=0}d_{R}e_{R}\right)=\mathbb{C}[G]\otimes_{\mathbb{Z}_{2}^{F}}\mathbb{C}_{+}.$
(38)

Here $\mathbb{C}_{+}$ is the trivial 1d representation of
$\mathbb{Z}_{2}^{F}$: $P_{f}\cdot z=z,\forall z\in\mathbb{C}_{+}$.
$\mathbb{C}[G]\otimes_{\mathbb{Z}_{2}^{F}}\mathbb{C}_{+}$ is spanned by
$|g\rangle\otimes 1$ modulo the relation $|P_{f}g\rangle\otimes
1\cong|g\rangle\otimes P_{f}\cdot 1=|g\rangle\otimes 1$. If we decompose the
first factor $\mathbb{C}[G]$ into direct sum of irreps of $G^{F}$, then the
sector $V^{R}\otimes_{\mathbb{Z}_{2}^{F}}\mathbb{C}_{+}$ is projected out if
$R$ is odd. Similarly the odd sector $\mathcal{A}_{\sf{ref}}^{f,1}$ is the
tensor product between $\mathbb{C}[G]$ and the non-trivial representation of
$\mathbb{Z}_{2}^{F}$:

$\displaystyle\mathcal{A}_{\sf{ref}}^{f,1}=\left(\bigoplus_{R,|R|=1}d_{R}e_{R}m^{F}\right)=\mathbb{C}[G]\otimes_{\mathbb{Z}_{2}^{F}}\mathbb{C}_{-}.$
(39)

$\mathbb{C}[G]\otimes_{\mathbb{Z}_{2}^{F}}\mathbb{C}_{-}$ is spanned by
$|g\rangle\otimes 1$ modulo the relation $|P_{f}g\rangle\otimes
1\cong|g\rangle\otimes P_{f}\cdot 1=-|g\rangle\otimes 1$. The even
representations in $\mathbb{C}[G]$ are projected out by the $\mathbb{C}_{-}$
factor. Put these two sectors together we arrive at a compact expression

$\displaystyle\mathcal{A}_{\sf{ref}}^{f}$
$\displaystyle=\mathbb{C}[G]\otimes_{\mathbb{Z}_{2}^{F}}(\mathbb{C}_{+}\oplus\mathbb{C}_{-})$
$\displaystyle=\mathbb{C}[G]\otimes_{\mathbb{Z}_{2}^{F}}\mathbb{C}^{1|1}.$
(40)

We used $\mathbb{C}^{1|1}$ to denote the super vector space with a 1d even
sector and a 1d odd sector. $\mathbb{C}^{1|1}$ carries a grading-preserving
algebra structure, generated by a single Majorana fermion operator $\gamma$ in
the odd sector. The grading of $\mathcal{A}_{\sf{ref}}^{f}$ is then inherited
from that of $\mathbb{C}^{1|1}$. The algebra structure of
$\mathcal{A}_{\sf{ref}}$ is defined by

$\displaystyle(|g\rangle\otimes_{\mathbb{Z}_{2}^{F}}\gamma^{i})\cdot(|h\rangle\otimes_{\mathbb{Z}_{2}^{F}}\gamma^{j})=$
$\displaystyle~{}~{}~{}~{}\delta_{g,h}(-1)^{j\cdot
gh^{-1}}|g\rangle\otimes_{\mathbb{Z}_{2}^{F}}\gamma^{i+j},$ (41)

where $\delta_{g,h}$ is 1 if $gh^{-1}$ is in $\mathbb{Z}_{2}^{F}$ and 0
otherwise. It is not hard to verify that this algebra is connected and
separable. In order for $\mathcal{A}_{\sf{ref}}^{f}$ to define a consistent
condensation the algebra also needs to be super-commutative, this means the
following diagram commutes,

${{\mathcal{A}_{\sf{ref}}^{f}\otimes\mathcal{A}_{\sf{ref}}^{f}}}$${{\mathcal{A}_{\sf{ref}}^{f}}}$${{\mathcal{A}_{\sf{ref}}^{f}\otimes\mathcal{A}_{\sf{ref}}^{f}}}$$\scriptstyle{\mu}$$\scriptstyle{R(-1)^{|\cdot|\times|\cdot|}}$$\scriptstyle{\mu}$

Here, the map $\mu$ taking the product Eq.(41), and
$R(-1)^{|\cdot|\times|\cdot|}$ is the “super-braiding”, defined as

$\displaystyle R(-1)^{|\cdot|\times|\cdot|}(a\otimes
b)=(-1)^{|a|\cdot|b|}R(a\otimes b).$ (42)

Indeed the diagram commutes,

$\displaystyle\mu\circ
R(-1)^{|\cdot|\times|\cdot|}((|g\rangle\otimes_{\mathbb{Z}_{2}^{F}}\gamma^{i})\otimes(|h\rangle\otimes_{\mathbb{Z}_{2}^{F}}\gamma^{j}))$
$\displaystyle=(-1)^{ij}\mu((P_{f}^{i}\cdot|h\rangle\otimes_{\mathbb{Z}_{2}^{F}}\gamma^{j})\otimes(|g\rangle\otimes_{\mathbb{Z}_{2}^{F}}\gamma^{i}))$
$\displaystyle=(-1)^{ij}(-1)^{ij}\mu((\cdot|h\rangle\otimes_{\mathbb{Z}_{2}^{F}}\gamma^{j})\otimes(|g\rangle\otimes_{\mathbb{Z}_{2}^{F}}\gamma^{i}))$
$\displaystyle=\delta_{g,h}(-1)^{hg^{-1}\cdot
i}|g\rangle\otimes_{\mathbb{Z}_{2}^{F}}\gamma^{i+j}$
$\displaystyle=\delta_{g,h}(-1)^{hg^{-1}\cdot
j}|h\rangle\otimes_{\mathbb{Z}_{2}^{F}}\gamma^{i+j}$
$\displaystyle=\mu((|g\rangle\otimes_{\mathbb{Z}_{2}^{F}}\gamma^{i})\otimes(|h\rangle\otimes_{\mathbb{Z}_{2}^{F}}\gamma^{j})).$
(43)

Since
$\dim(\mathcal{A}_{\sf{ref}}^{f})=\sum_{|R|=0}d_{R}^{2}+\sum_{|R|=1}d_{R}^{2}=|G^{F}|$,
the algebra $\mathcal{A}_{\sf{ref}}^{f}$ is Lagrangian and defines a gapped
boundary. We therefore define our SymTFT sandwich construction by choosing
$\mathcal{A}_{\sf{ref}}^{f}$-condensation as the reference boundary. The
SymTFT construction is shown in Fig. 13.

Figure 13: SymTFT for $G^{F}$. (a). A flux line near the reference boundary
becomes the symmetry defect of $\sf{T}$ after dimensional reduction. (b). A
vertical charge line carrying even representation, $R(P_{f})=1$ becomes a
local bosonic charge of $\sf{T}$ after dimensional reduction. (c). An odd
charge line, $R(P_{f})=-1$, is bound with the fermion parity flux line, and
its endpoint on $\sf{B_{ref}}$ is dressed by a local fermion operator(blue
dot). After dimensional reduction this line becomes a charged local fermion.
The statistics of these local fermions is guaranteed by the local fermion
operator on $\sf{B_{ref}}$(blue dot). The red dots stand for excitations.

We also note that the fermionic anyons $e_{R}m^{F},~{}|R|=1$ are still
deconfined on $\sf{B_{ref}}$: they have trivial braiding with all anyons
condensed on $\sf{B_{ref}}$. They are all equivalent to the local fermion $c$,
due to the condensate $e_{R}m^{F}c$. Therefore, up to the condensate the only
non-trivial deconfined excitation on the reference boundary is the local
fermion $c$. This means the reference boundary can be viewed as a trivial
fermionic system, whose topological excitations are described by the super-
fusion category $\sf{sVec}$. This structure matches with the construction in
[36].

###  V.2 Spin and twisted spin structures of reference boundary

As we have demonstrated in Section V.2, fermionic reference boundary with the
condensation $1+f$ is not unique. In order to fully specify a reference
boundary condition, a spin structure on $\sf{B_{ref}}$ must be specified in
addition to the condensation $1+f$. In Section V.2 we showed through a
diagrammatic arguement that this spin structure is tied to the expectation
value of $f$-loops on the reference boundary. For instance, when
$\sf{B_{ref}}\cong T^{2}$, the boundary state $|1\rangle+|f\rangle$, for which
the $f$-loops in the time- or space-direction both have value $1$, corresponds
to the spin structure (AP,AP). This dependence of the reference boundary on
spin structure is crucial to the theory of SymTFT for fermionic symmetries,
since a fermionic system is defined on a manifold with spin structure. More
generally, the symmetry of a fermionic system can be viewed as a group $G^{F}$
that contains fermion parity $\mathbb{Z}_{2}^{F}$ as a subgroup.
$G^{B}=G^{F}/\mathbb{Z}_{2}^{F}$ is referred to as the bosonic symmetry of the
system. Such a $G^{F}$-symmetric system is defined on a manifold with a
twisted version of the usual spin structure, called a $G^{F}$-spin structure.
A $G^{F}$-spin structure is a pair $(\eta,a_{b})$, where $\eta$ is a usual
spin structure, and $a_{b}$ is a $G^{B}$-background gauge field satisfying
certain constraint. Therefore, in order to describe $G^{F}$-symmetric
fermionic systems in the SymTFT framework, we must analyze how the fermionic
reference boundary with the condensation Eq. (37) depends on $G^{F}$-spin
structures.

In this section we discuss this dependence in detail. We take the approach
that the reference boundary can be viewed as a boundary state, expanded in
field configuration basis as

$\displaystyle|\Psi_{\sf{ref}}\rangle=\sum_{A\in
H^{1}[M,G^{F}]}\Psi_{\sf{ref}}(A)|A\rangle.$ (44)

We will see that there are multiple allowed boundary states with the anyon
condensation $\mathcal{A}_{\sf{ref}}^{f}$, each labelled by a $G^{F}$-spin
structure $(\eta,a_{b})$. We denote these reference states by
$|\mathcal{F}(\eta,a_{b})\rangle$, with $\mathcal{F}$ standing for
“fermionic”. For a given physical boundary state $|\Psi_{phs}\rangle$, the
partition function of the sandwich is given by the inner product

$\displaystyle\mathcal{Z}_{\Psi}(\eta,a_{b}):=\langle\mathcal{F}(\eta,a_{b})|\Psi_{\sf{phys}}\rangle$
(45)

which is the partition function of a $1+1D$ system in the presence of a
$G^{F}$-spin structure $(\eta,a_{b})$.

We will begin by reviewing definition and properties of spin structure. We
then discuss the reference boundary state of the $\mathbb{Z}_{2}^{F}$-SymTFT
for arbitrary boundary manifold and spin structure. Finally we generalize to
$G^{F}$-SymTFT, and derive the boundary states
$|\mathcal{F}(\eta,a_{b})\rangle$ for arbitrary $G^{F}$-spin structure
$(\eta,a_{b})$.

####  V.2.1 Review of spin structure

Recall that the spin structure on a torus amounts to choosing periodic(P) or
anti-periodic(AP) boundary conditions along a basis for the noncontractible
loops of the torus. This gives us all four spin structures on the torus.
101010 Mathematically, AP/P boundary condition corresponds to bounding/non-
bounding spin structure of the noncontractible loop, and is also known as
NS(Neveu-Schwarz)/R(Ramond) sector in the context of string theory. On a
general two dimensional (spacetime) manifold $M$, the spin structure can be
similarly defined as a choice of P or AP boundary condition for every loop on
the manifold. We can then define an indicator $q(\gamma)$, where $\gamma$ is a
loop on $M$, and $q(\gamma)=1,-1$ when the boundary condition on $\gamma$ is
AP,P respectively. The value of this indicator function should be invariant
under deformation of the loop, and can be extended to any
$\mathbb{Z}_{2}$-coefficient formal sum of loops by deforming the formal sum
into a single loop. Therefore $q$ is a function on the homology group
$H_{1}[M,\mathbb{Z}_{2}]$. A fermion loop around the composite cycle
$\gamma_{1}+\gamma_{2}$ can be deformed into separate loops encircling
$\gamma_{1}$ and $\gamma_{2}$, where the deformation involves a number of
fermion exchanges equal to $\gamma_{1}\cdot\gamma_{2}$: Here
$\gamma_{1}\cdot\gamma_{2}$ is the number of intersection points of
$\gamma_{1}$ and $\gamma_{2}$, which is homologically invariant $\mod 2$. This
requires that $q$ satisfy the consistency condition:

$\displaystyle
q([\gamma_{1}]+[\gamma_{2}])=q([\gamma_{1}])q([\gamma_{2}])(-1)^{\gamma_{1}\cdot\gamma_{2}}.$
(46)

Functions satisfying this identity are called quadratic forms on
$H_{1}[M,\mathbb{Z}_{2}]$.

Alternatively, the spin structure can be equivalently defined as a
$\mathbb{Z}_{2}$-valued $1$-cochain, $\eta$ that trivializes the 2nd Stiefel-
Whitney class: $d\eta=w_{2}$. There is a one-to-one correspondence between
quadratic forms, $q_{\eta}$ and the associated cochain, $\eta$. Any two spin
structures, $\eta,\eta^{\prime}$ satisfy $d(\eta-\eta^{\prime})=0$, i.e. any
two spin structures are related by the addition of a flat,
$\mathbb{Z}_{2}$-valued gauge field. Formally, the set of spin structures is
referred to as a torsor of $H^{1}[M,\mathbb{Z}_{2}]$.

The Arf-invariant, which appears in the partition function of the Kitaev
chain, is defined by summing over the loops in the quadratic form:

$\displaystyle\text{Arf}(\eta):=\frac{1}{\sqrt{H_{1}[M,\mathbb{Z}_{2}]}}\sum_{\gamma\in
H_{1}[M,\mathbb{Z}_{2}]}q_{\eta}(\gamma),$ (47)

which takes values in $\\{1,-1\\}$ and only depends on the spin structure of
the manifold.

####  V.2.2 Spin structure in the $\mathbb{Z}_{2}^{F}$-SymTFT

We now generalize our discussion of spin structure dependence in section
III.1.2 for the $\mathbb{Z}_{2}^{F}$-SymTFT to general boundary manifolds. We
can simply apply the argument of moving an $f$-anyon in Fig. 6 to any loop
$\gamma$ on the boundary $\sf{B_{ref}}$. Following the same procedure as in
Fig. 6, we can see that the boundary condition along $\gamma$ is still related
to the expectation value of an $f$-loop on $\gamma$ via

$\displaystyle P/AP\Leftrightarrow\langle
W_{f}(\gamma)\rangle_{\sf{B_{ref}}}=-1/+1.$ (48)

Comparing with the definition of the quadratic form $q_{\eta}$, we see that
the expectation value of the $f$-loops on $\sf{B_{ref}}$ is simply $q_{\eta}$:

$\displaystyle\langle W_{f}(\gamma)\rangle_{\sf{B_{ref}}}=q_{\eta}(\gamma).$
(49)

This relation tells us that the collection of values of $f$-loops on the
reference boundary is in one-to-one correspondence with the spin structure of
the reference boundary. Below we will utilize this relation to determine the
boundary state with given a spin structure.111111We note that this continuum
argument can also be explicitly verified in microscopic toric-code-like
lattice models with $cf$-condensed boundaries. We provide this lattice
construction in Appendix B.

We now compute the state of the reference boundary with a given spin structure
$\eta$. We write an $f$-condensed boundary state in the field configuration
basis as

$\displaystyle|\mathcal{F}\rangle=\sum_{a}\mathcal{F}(a)|a\rangle,$ (50)

here $a\in H^{1}[\sf{B_{ref}},\mathbb{Z}_{2}]$ is the value of
$\mathbb{Z}_{2}$-gauge field on the boundary, $\mathcal{F}$ stands for
“fermionic” reference boundary state. Our goal is now to determine the
coefficients $\mathcal{F}(a)$ such that the state $|\mathcal{F}\rangle$
corresponds to an $f$-condensed boundary with spin structure $\eta$. An
$f$-condensed boundary should be able to absorb an $f$-loop on it. We can
write an $f$-loop on $\sf{B_{ref}}$ as a product of an $e$-loop and an
$m$-loop,

$\displaystyle W_{f}[\gamma]:=W_{e}[\gamma]W_{m}[\gamma],\gamma\in
H_{1}[\sf{B_{ref}},\mathbb{Z}_{2}],$ (51)

where we choose an arbitrary framing for $\gamma$ so that $W_{f}[\gamma]$ is
unambiguously defined. The $e$-loop $W_{e}[\gamma]$ is the Wilson loop of the
$\mathbb{Z}_{2}$-gauge field $a$:

$\displaystyle
W_{e}[\gamma]=(-1)^{\oint_{\gamma}a}=(-1)^{\int_{\sf{B_{ref}}}a\cup\tilde{\gamma}}$
(52)

where we used the Poincare dual of $\gamma$, denoted by $\tilde{\gamma}\in
H^{1}[\sf{B_{ref}},\mathbb{Z}_{2}]$, to write the Wilson loop as an
integration over the entire boundary $\sf{B_{ref}}$. On the other hand the
$m$-loop is the Wilson loop of the dual gauge field $\hat{a}$,

$\displaystyle
W_{m}[\gamma]=(-1)^{\oint_{\gamma}\hat{a}}=(-1)^{\int_{\sf{B_{ref}}}\hat{a}\cup\tilde{\gamma}}$
(53)

The $e/m$-loops act on the basis $|a_{0}\rangle$ as follows,

$\displaystyle W_{e}[\gamma]|a_{0}\rangle$
$\displaystyle=(-1)^{\int_{\sf{B_{ref}}}a_{0}\cup\tilde{\gamma}}|a_{0}\rangle,$
(54) $\displaystyle W_{m}[\gamma]|a_{0}\rangle$
$\displaystyle=(-1)^{\int_{\sf{B_{ref}}}\hat{a}\cup\tilde{\gamma}}|a_{0}\rangle=|a_{0}+\tilde{\gamma}\rangle.$
(55)

Therefore the $f$-loop acts on the boundary state as

$\displaystyle W_{f}[\gamma]|\mathcal{F}\rangle$
$\displaystyle=\sum_{a}(-1)^{\int_{M}a\cup\tilde{\gamma}}\mathcal{F}(a)|a+\tilde{\gamma}\rangle$
$\displaystyle=\sum_{a}(-1)^{\int_{M}a\cup\tilde{\gamma}}\mathcal{F}(a+\tilde{\gamma})|a\rangle.$
(56)

Figure 14: Schematic picture of the SymTFT for $\mathbb{Z}_{2}^{F}$. (a). The
SymTFT is the $\mathbb{Z}_{2}$ quantum double topological order(toric code).
The reference boundary depends on a spin structure $\eta\in
C^{1}[\sf{B_{ref}},\mathbb{Z}_{2}]~{}d\eta=w_{2}$, the corresponding state is
given by Eq.(66). The sandwich with $|\mathcal{F}(\eta)\rangle$ reference
boundary describes a $1+1D$ fermionic system with spin structure $\eta$. (b)
For a boundary spin structure $\eta$, an $f$-loop $W_{f}(\gamma)$ on the
reference boundary has eigenvalue given by the quadratic form of the spin
structure, $q_{\eta}(\gamma)$.

Requiring $|\mathcal{F}\rangle$ to be an eigenstate of $W_{f}[\gamma]$ leads
to the condition (we will write $\tilde{\gamma}$ as $\gamma$ from now on)

$\displaystyle\sum_{a}(-1)^{\int_{M}a\cup\gamma}\mathcal{F}(a+\gamma)|a\rangle=f(\gamma)\sum_{a}\mathcal{F}(a)|a\rangle,$
(57)

for some eigenvalues $f(\gamma)$. By comparing coefficients on both sides we
have

$\displaystyle(-1)^{\int_{M}a\cup\gamma}\mathcal{F}(a+\gamma)=f(\gamma)\mathcal{F}(a),$
(58)

for all $a,\gamma\in H^{1}[\sf{B_{ref}},\mathbb{Z}_{2}]$. Setting $a=0$ we see
$\mathcal{F}(\gamma)=f(\gamma)\mathcal{F}(0)$. Therefore $\mathcal{F}$
satisfies the condition

$\displaystyle\frac{\mathcal{F}(a+\gamma)}{\mathcal{F}(0)}=(-1)^{\int_{M}a\cup\gamma}\frac{\mathcal{F}(\gamma)}{\mathcal{F}(0)}\frac{\mathcal{F}(a)}{\mathcal{F}(0)},$
(59)

which states that $\mathcal{F}(a)/\mathcal{F}(0)$ is a quadratic form on
$H^{1}[\sf{B_{ref}},\mathbb{Z}_{2}]$. We may then write
$\mathcal{F}(a)=\mathcal{F}(0)q_{\eta}(a)$ for some spin structure $\eta$. Up
to normalization an $f$-condensed boundary is then determined by a spin
structure on $\sf{B_{ref}}$, and given by the state

$\displaystyle|\mathcal{F}(\eta)\rangle\propto\sum_{a\in
H^{1}[\sf{B_{ref}},\mathbb{Z}_{2}]}q_{\eta}(a)|a\rangle.$ (60)

The state $|\mathcal{F}(\eta)\rangle$ is an eigenstate of $f$-loops on
$\sf{B_{ref}}$ with eigenvalue $q_{\eta}(\gamma)$. Therefore according to the
relation Eq.(49), the state $|\mathcal{F}(\eta)\rangle$ is exactly the state
corresponding to the spin structure $\eta$ on $\sf{B_{ref}}$.

However, there is still a phase ambiguity for every state
$|\mathcal{F}(\eta)\rangle$. These phases for different $\eta$s are not
independent of each other. To see this, notice that in the SymTFT fermion
parity is represented by an $m$-line on $\sf{B_{ref}}$, therefore insertion of
an $m$-line should change the spin structure according to
$W_{m}[\gamma]|\mathcal{F}(\eta)\rangle=|\mathcal{F}(\eta+\gamma)\rangle$.
Therefore once a state $|\mathcal{F}(\eta)\rangle$ with a given spin structure
$\eta$ is chosen, the states with other spin structures are uniquely
determined by the relation
$W_{m}[\gamma]|\mathcal{F}(\eta)\rangle=|\mathcal{F}(\eta+\gamma)\rangle$.
This will fix the phase ambiguity of the states $|\mathcal{F}(\eta)\rangle$ up
to an overall common phase, which is inconsequential. To see this, assume the
state $|\mathcal{F}(\eta)\rangle$ has an arbitrary phase factor
$\mathcal{T}(\eta)$:

$\displaystyle|\mathcal{F}(\eta)\rangle=\mathcal{T}(\eta)\sum_{a}q_{\eta}(a)|a\rangle.$
(61)

Then the action of an $m$-loop on the state is

$\displaystyle W_{m}[\gamma]|\mathcal{F}(\eta)\rangle$
$\displaystyle=\mathcal{T}(\eta)\sum_{a}q_{\eta}(a)|a+\gamma\rangle$
$\displaystyle=\mathcal{T}(\eta)\sum_{a}q_{\eta}(a+\gamma)|a\rangle.$ (62)

The state $|\mathcal{F}(\eta+\gamma)\rangle$ on the other hand is given by

$\displaystyle|\mathcal{F}(\eta+\gamma)\rangle=\mathcal{T}(\eta+\gamma)\sum_{a}q_{\eta+\gamma}(a)|a\rangle.$
(63)

By comparing coefficients we have the relation

$\displaystyle\mathcal{T}(\eta)q_{\eta}(a+\gamma)=\mathcal{T}(\eta+\gamma)q_{\eta+\gamma}(a).$
(64)

Summing over $a\in H^{1}[M,\mathbb{Z}_{2}]$ on both sides, we have

$\displaystyle\mathcal{T}(\eta)\text{Arf}(\eta)=\mathcal{T}(\eta+\gamma)\text{Arf}(\eta+\gamma).$
(65)

We see that the solution for $\mathcal{T}$ is
$\mathcal{T}(\eta)=c\cdot\text{Arf}(\eta)$, where $c\in\mathbb{C}^{\times}$ is
arbitrary. Take $c=1$, we have the phase-fixed states

$\displaystyle|\mathcal{F}(\eta)\rangle$
$\displaystyle=\text{Arf}(\eta)\sum_{a\in
H^{1}[M,\mathbb{Z}_{2}]}q_{\eta}(a)|a\rangle$ $\displaystyle=\sum_{a\in
H^{1}[M,\mathbb{Z}_{2}]}\text{Arf}(\eta+a)|a\rangle,$ (66)

where we used relation $\text{Arf}(\eta+a)=\text{Arf}(\eta)q_{\eta}(a)$.

Eq.(66) completes our construction for the $\mathbb{Z}_{2}^{F}$-SymTFT. The
schematic of the SymTFT construction is shown in Fig. 14.

As an application of this result, we can obtain the partition function of the
Kitaev chain on any orientable manifold as follows: the physical boundary is
the $e$-condensed Dirichlet boundary $|a_{0}=0\rangle$, therefore the
partition function with spin structure $\eta$ is

$\displaystyle\mathcal{Z}_{\eta}^{\text{Kitaev}}=\langle\mathcal{F}(\eta)|a_{0}=0\rangle=\text{Arf}(\eta).$
(67)

####  V.2.3 Review of $G^{F}$-spin structure

A general fermionic symmetry is described by the short exact sequence

$\displaystyle
1\to\mathbb{Z}_{2}^{F}\xrightarrow{i}G^{F}\xrightarrow{\pi}G^{B}\to 1.$ (68)

For the convenience of later discussion, let us also specify set-theoretic
section maps $s:G^{B}\to G^{F},t:G^{F}\to\mathbb{Z}_{2}^{F}$ that satisfy
$\pi\circ s=\text{id}_{G^{B}},~{}t\circ i=\text{id}_{\mathbb{Z}_{2}^{F}}$. The
extension class of this sequence is $\rho:=\delta s\in