# Duality and Stacking of Bosonic and Fermionic SPT Phases

Alex<EMAIL_ADDRESS>Perimeter Institute for
Theoretical Physics, 31 Caroline St N, Waterloo, ON N2L 2Y5, Canada  Minyoung
<EMAIL_ADDRESS>POSTECH Basic Science Research Institute, Pohang,
Gyeongbuk, 37673, Korea

(November 30, 2023)

###### Abstract

We study the interplay of duality and stacking of bosonic and fermionic
symmetry-protected topological phases in one spatial dimension. In general the
classifications of bosonic and fermionic phases have different group
structures under the operation of stacking, but we argue that they are often
isomorphic and give an explicit isomorphism when it exists. This occurs for
all unitary symmetry groups and many groups with antiunitary symmetries, which
we characterize. We find that this isomorphism is typically not implemented by
the Jordan-Wigner transformation, nor is it a consequence of any other duality
transformation that falls within the framework of topological holography.
Along the way to this conclusion, we recover the fermionic stacking rule in
terms of $\mathcal{G}$-pin partition functions, give a gauge-invariant
characterization of the twisted group cohomology invariant, and state a
procedure for stacking gapped phases in the formalism of symmetry topological
field theory.

###### Contents

  1. 1 Introduction
  2. 2 Bosonic and fermionic SPT phases
    1. 2.1 Group cohomology invariants
    2. 2.2 Supercohomology invariants
    3. 2.3 Bosonic partition functions
    4. 2.4 Fermionic partition functions
    5. 2.5 The Jordan-Wigner transformation
    6. 2.6 Stacking rules
  3. 3 Isomorphic classifications
    1. 3.1 The classifications as group extensions
    2. 3.2 Examples
    3. 3.3 Necessary and sufficient conditions for isomorphism
    4. 3.4 Square roots of the diagonal
    5. 3.5 Derivation of the genus rules
  4. 4 Dualities of SPT phases
    1. 4.1 Topological holography
    2. 4.2 Actions of dualities on SPT invariants
    3. 4.3 Stacking of magnetic Lagrangian algebras
    4. 4.4 An example, revisited
  5. 5 Discussion
  6. A Invertible fermionic phases
  7. B State sum evaluation of bosonic partition functions
  8. C Twisted cohomology not of order two
  9. D The Arf invariant

##  1 Introduction

The celebrated Jordan-Wigner transformation establishes a correspondence
between bosonic spin chains and chains of spinless fermions [1]. Like that of
other duality transformations, this correspondence allows one to reinterpret
features of one system in the framework of another and thus to translate
questions about one system into a language where they may be more easily
answered. In this paper, we investigate the ability of the Jordan-Wigner
transformation and boson-to-fermion dualities in general to transform a key
property of collections of bosonic and fermionic systems – namely, the their
behavior under stacking.

The systems we consider are all one dimensional, gapped, interacting models of
spins and fermions with a global symmetry $\mathcal{G}$. These systems are
classified into various topological phases, which are characterized by their
patterns of breaking and fractionalization of this symmetry. Of particular
interest to us are symmetry protected topological (SPT) phases – those which
would be topologically trivial if not for the symmetry. SPT phases of spins
(called “bosonic SPT phases”) and SPT phases of fermions are in one-to-one
correspondence under the Jordan-Wigner transformation [2, 3]. But while the
classifications of phases are the same on either side of the duality _as sets_
, there is still a sense in which they may differ. Two SPT phases may be
“stacked” to produce another SPT phase, and this operation gives the
collection of SPT phases the structure of an abelian group. The bosonic
stacking group law may differ from the fermionic one. The failure of
isomorphism of stacking rules is exemplified by systems with
$\mathcal{G}={\mathbb{Z}}_{2}^{T}\times{\mathbb{Z}}_{2}$ symmetry – a time-
reversal symmetry a unitary ${\mathbb{Z}}_{2}$ symmetry that is regarded as
fermion parity on the fermionic side of the transformation. The four bosonic
SPT phases with this symmetry form the group
$\mathcal{B}={\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ under stacking, while the
four fermionic SPT phases (which are modelled by chains of even numbers of
Majoranas) stack according to $\mathcal{F}={\mathbb{Z}}_{4}$ [2]. Looking
beyond SPT phases for a moment, stacking rules can differ even more
drastically: for example, the symmetry-breaking phase of the one dimensional
Ising model has no inverse under the stacking operation yet is transformed
into an _invertible_ topological order by the Jordan-Wigner transformation.

The difference between bosonic and fermionic SPT classification groups has its
origins in the bosonic and fermionic commutation relations, so it may be
surprising that they ever agree beyond rare accidents. On the other hand, the
Jordan-Wigner transformation which relates them is nonlocal in a way that one
might expect to compensate for the difference. Our first task is to
investigate for which symmetry groups the stacking rules agree and for which
they disagree. Our second task is to study whether the Jordan-Wigner
transformation or another boson-to-fermion duality can explain this agreement
when it occurs. We find that the stacking rules are often isomorphic: they
always agree for SPT phases with unitary symmetry (in agreement with a result
of Ref. [4]), and they may agree in the presence of antiunitary symmetry when
certain conditions are satisfied. In the cases where the stacking rules agree,
we state an explicit isomorphism between the classification groups and ask
whether it is implemented to by the Jordan-Wigner transformation or another
duality. Typically the answer to this question is “no.”

The structure of the paper is as follows. In section 2 we review of the
classifications of bosonic and fermionic SPT phases in terms of group
cohomology and supercohomology invariants for general symmetry groups, we
define topological and $\mathcal{G}$-pin partition functions that capture the
background-response of these SPT systems, we formulate the Jordan-Wigner
transformation in terms of these field theories, and we use them to rederive
the SPT stacking rules. In section 3 we derive conditions under which the
bosonic and fermionic SPT classifications are isomorphic as groups. These
conditions are stated in section 3.3. The idea is to understand the
classifications as group extensions that become equivalent when a certain
extension class vanishes and then to relate the vanishing of this class to the
vanishing of certain partition functions. In section 4, we study duality and
stacking of SPT phases from the point of view of topological holography in
order to determine that the isomorphism of classification groups typically
cannot be implemented by a physical duality transformation. This result
suggests that the stacking rules reflect an essential difference between
bosons and fermions than cannot be transformed away by duality and leaves open
the mystery of why – from a physical perspective – the rules are often
isomorphic in the abstract. To reach this conclusion, we understand the action
of dualities on the SPT invariants and state a procedure for stacking the
Lagrangian algebras that correspond to SPT phases. The appendices contain
several related results: we use topological partition functions to rederive
the stacking rule for invertible fermionic phases (including the invertible
topological order) and review related results in the literature; we use the
equivariant state sum construction to evaluate the bosonic partition function
on all orientable and nonorientable surfaces, which yields a complete set of
gauge-invariant quantities characterizing the SPT invariants; and we discuss
some interesting computations in twisted group cohomology.

The authors are grateful for the hospitality of the Asia Pacific Center for
Theoretical Physics, the Yukawa Institute for Theoretical Physics, and the
Perimeter Institute for Theoretical Physics, where parts of this work were
carried out. Research at the Perimeter Institute is supported in part by the
Government of Canada through the Department of Innovation, Science and
Economic Development and by the Province of Ontario through the Ministry of
Colleges and Universities. M.Y. is supported by Basic Science Research
Institute Fund, whose NRF grant number is 2021R1A6A1A10042944.

##  2 Bosonic and fermionic SPT phases

###  2.1 Group cohomology invariants

The group cohomology invariants of the symmetry group provide a
characterization of bosonic SPT phases and – via the Jordan-Wigner
transformation – fermionic SPT phases as well [2].

Consider bosonic phases. The symmetry class of a bosonic system is specified
by a group $\mathcal{G}$ and a homomorphism
$x:\mathcal{G}\rightarrow{\mathbb{Z}}_{2}$ encoding which elements are
represented unitarily vs antiunitarily. The classification of bosonic SPT
phases with symmetry $(\mathcal{G},x)$ is given by classes in the twisted
cohomology group [5]

$[\omega]\in H^{2}(\mathcal{G},U(1)_{x})~{}.$ (2.1)

These are group cochains $\omega:\mathcal{G}\times\mathcal{G}\rightarrow U(1)$
satisfying the twisted cocycle condition333The $U(1)$ coefficients are written
in additive notation, as $U(1)\simeq{\mathbb{R}}/{\mathbb{Z}}$.

$0=(\delta\omega)(g,h,k)=(-1)^{x(g)}\omega(h,k)+\omega(g,hk)-\omega(g,h)-\omega(gh,k)$
(2.2)

modulo twisted coboundaries (“gauge transformations”)

$(\delta\lambda)(g,h)=(-1)^{x(g)}\lambda(h)+\lambda(g)-\lambda(gh)~{}.$ (2.3)

The cohomology class $[\omega]$ may be interpreted physically as measuring the
projectivity in the representation of the symmetry on the boundary [6, 3]. In
this paper, we use a partial gauge fixing where $\omega(g,1)=\omega(1,g)=1$, a
condition which is preserved by gauge transformations with $\lambda(1)=0$.

Meanwhile, the symmetry class of a fermionic system is specified by bosonic
symmetry data $(\mathcal{G},x)$ and a distinguished unitary central involution
$p\in\mathcal{G}$ called _fermion parity_. Fermionic SPT phases with symmetry
$(\mathcal{G},p,x)$ are in one-to-one correspondence with bosonic
$(\mathcal{G},x)$-SPT phases;444We do not regard the invertible topological
order modeled by the nontrivial Majorana chain as an SPT phase, though it and
its symmetry-enrichments are discussed in appendix A. namely, they are also
associated with classes $[\omega]$. The explicit correspondence between
bosonic and fermionic SPT phases given by the Jordan-Wigner transformation
will be discussed in detail in section 2.5. To characterize certain physical
features such as their stacking rule (discussed in section 2.6), however, it
is useful to reorganize the topological data into pairs

$(\omega,\nu)\in C^{2}(\mathcal{G},U(1)_{x})\times
C^{1}(\mathcal{G},{\mathbb{Z}}_{2})$ (2.4)

subject to the constraints555The map $\tfrac{1}{2}$ embeds
${\mathbb{Z}}_{2}=\\{0,1\\}$ as the subgroup $\\{0,1/2\\}\subset U(1)$.

$\delta\omega=0~{},$ (2.5)
$\qquad\tfrac{1}{2}\nu(g)=\omega(g,p)-\omega(p,g)+x(g)\omega(p,p)$ (2.6)

modulo coboundaries. Since every closed $\omega$ satisfies the constraint for
a unique $\nu$, unchanged by coboundary shift, classes $[\omega,\nu]$
correspond one-to-one with $[\omega]$’s. The invariant $\nu$ singles out the
distinguished fermionic parity element666The invariant $\nu$ is essentially
the slant product of $\omega$ with respect to fermion parity $p$. and encodes
whether the boundary action of a symmetry is parity-even or -odd [2].

###  2.2 Supercohomology invariants

The SPT invariants can be reformulated in the language of supercohomology.
This perspective is typically employed to characterize fermionic SPT phases,
where the supercohomology invariants describe how a phase arises from a
decorated domain wall construction [7, 8, 9]. Similarly, in section 2.4 we
find supercohomology useful for studying fermionic SPT phases, as it lets us
write their explicit partition functions in all symmetry classes. We
emphasize, however, that either formulation of the invariants can be used to
characterize both bosonic and fermionic SPT phases. As we discuss in section
3.1, supercohomology has the benefit of putting the bosonic and fermionic SPT
stacking rules on an equal footing, as two group extensions of the invariants.

The choice of fermion parity $p$ realizes $\mathcal{G}$ as a central extension

${\mathbb{Z}}_{2}^{f}\rightarrow\mathcal{G}\xrightarrow[]{b}G_{b}$ (2.7)

of a group of bosonic symmetries $G_{b}\simeq\mathcal{G}/{\mathbb{Z}}_{2}^{f}$
by ${\mathbb{Z}}_{2}^{f}=\\{1,p\\}$. It is often useful to describe the
symmetry in terms of $G_{b}$ and a class $[\rho]\in
H^{2}(G_{b},{\mathbb{Z}}_{2})$ that captures the extension to $\mathcal{G}$.
Given any function $s:G_{b}\rightarrow\mathcal{G}$ with $b\circ s=1$, a
cocycle $\rho$ representing the extension class measures the failure of
$G_{b}$ multiplication in $\mathcal{G}$:

$s(g_{b})s(h_{b})=\rho(g_{b},h_{b})s(g_{b}h_{b})~{},$ (2.8)

and $\rho$ shifts by a coboundary under change of section $s$. We are often
interested in symmetries with $[\rho]=0$, in which case $\mathcal{G}$ _splits_
as $G_{b}\times{\mathbb{Z}}_{2}$. In this case, a section $s$ with $\rho=0$ is
called a splitting.

Using a section $s$, the cochain data on $\mathcal{G}$ can be pulled back to
$G_{b}$:

$\alpha=s^{*}\omega~{},\qquad\beta=s^{*}\nu~{}.$ (2.9)

The result is the following data, dubbed _supercohomology classes_ : pairs

$(\alpha,\beta)\in C^{2}(G_{b},U(1)_{x})\times C^{1}(G_{b},{\mathbb{Z}}_{2})$
(2.10)

satisfying the conditions

$\delta\alpha=\tfrac{1}{2}\beta\cup\rho~{},$ (2.11) $\delta\beta=0$ (2.12)

and modulo the relation777Ref. [10], Theorem A.4 states the relation as
$\alpha\mapsto\alpha+\delta\lambda$ for $\lambda:\mathcal{G}\rightarrow U(1)$
with $\lambda(gp)=\lambda(g)+\lambda(p)$. To get Eq. (2.13), note that
$(\delta\lambda)(s(g_{b}),s(h_{b}))=(-1)^{x(s(g_{b}))}\lambda(s(h_{b}))+\lambda(s(g_{b}))-\lambda(s(g_{b}h_{b}))-\lambda(p)\rho(g_{b},h_{b})$
and take $\lambda_{1}(g_{b})=\lambda(s(g_{b}))$ and $\lambda_{0}=\lambda(p)$.

$\alpha\mapsto\alpha+\delta\lambda_{1}+\tfrac{1}{2}\lambda_{0}\cup\rho$ (2.13)

for $\lambda_{1}:G_{b}\rightarrow U(1)$ and
$\lambda_{0}\in{\mathbb{Z}}_{2}$.888In order words, fermionic SPT phases
correspond to cohomology classes of the complex of pairs with differential
$\mathcal{D}(x,y)=(\delta x+\tfrac{1}{2}\rho\cup y,\delta y)$, a.k.a.
_supercohomology classes_. Note that $\delta\lambda_{0}=0$, so $\beta$ is not
shifted by a coboundary. The map (2.9) from group cohomology invariants to
supercohomology invariants is bijective, with inverse given on the cochain
level999Every class $[\omega]$ contains a cochain $\omega$ in the partial
gauge (2.14). The remaining freedom in $\omega$ is the relation (2.13) on
$\alpha$. by

$\omega=b^{*}\alpha+\tfrac{1}{2}b^{*}\beta\cup t~{},\qquad\nu=b^{*}\beta~{},$
(2.14)

where $t$ is the function $t:\mathcal{G}\rightarrow{\mathbb{Z}}_{2}^{f}$ such
that $t\circ s=0$ [10]. Therefore, supercohomology classes $[\alpha,\beta]$
provide an complete characterization of SPT phases. The set of SPT phases may
be expressed as

${\mathbb{A}}\times{\mathbb{B}}~{},$ (2.15)

where ${\mathbb{A}}$ is the quotient of $H^{2}(G_{b},U(1)_{x})$ by the coarser
equivalence relation (2.13), i.e. classes $[\alpha]$ modulo
$[\tfrac{1}{2}\rho]$,101010Technically, due to the condition (2.11), classes
of $\alpha$ belong not to the group ${\mathbb{A}}$ but rather to a torsor over
it. If one chooses a “zero” $[\alpha_{0}]$, then each equivalence class
$[\alpha]$ may be identified with the element $[\alpha-\alpha_{0}]$ of
${\mathbb{A}}$. and ${\mathbb{B}}$ is the subgroup of
$H^{1}(G_{b},{\mathbb{Z}}_{2})$ consisting of _valid_ $\beta$ invariants –
that is, cocycles $\beta$ for which there is a solution $\alpha$ to the
condition (2.11), i.e. for which $[\tfrac{1}{2}\beta\cup\rho]=0$.111111The
groups ${\mathbb{A}}$ and ${\mathbb{B}}$ are sometimes called
$H^{2}_{\text{rigid}}$ and $BH^{1}$, and the constraint defining
${\mathbb{B}}$ is the twisted Gu-Wen-Freed equation for one spatial dimension.
${\mathbb{A}}$ and ${\mathbb{B}}$ are the cokernel and kernel of the spectral
sequence differential $[\tfrac{1}{2}\cup\rho]$ [11]. When $\mathcal{G}$ splits
and $s$ is a splitting, the condition (2.11) and relation (2.13) reduce to the
usual ones in cohomology, and so the set (2.15) becomes121212A third invariant
$\gamma\in C^{0}(G_{b},{\mathbb{Z}}_{2})={\mathbb{Z}}_{2}$ is necessary to
classify all invertible fermionic phases with split symmetry group, including
symmetry-enrichments of the nontrivial Majorana chain [2, 12, 13]. But for SPT
phases, $\gamma$ does not appear. The full classification of invertible
fermionic phases is discussed in appendix A.

$H^{2}(G_{b},U(1)_{x})\times H^{1}(G_{b},{\mathbb{Z}}_{2})~{}.$ (2.16)

The difference between two sections takes values in ${\mathbb{Z}}_{2}^{f}$ and
so amounts to a $1$-cochain $\mu:G_{b}\rightarrow{\mathbb{Z}}_{2}^{f}$. Under
a change of section, $\rho$ shifts by $\delta\mu$ and $\alpha$ by
$\tfrac{1}{2}\beta\cup\mu$ while $\beta$ remains unchanged.131313The choice of
section is not physical, yet the classification of phases is also not reduced
modulo this shift in $\alpha$; rather, it is only the interpretation of the
abstract class $[\alpha,\beta]$ as a physical SPT phase (i.e. as a class
$[\omega]$) that depends on the section. The invariant $\alpha$ represents a
$G_{b}$-symmetric SPT order, while $\beta$ has to do with the decoration of
$G_{b}$ domain walls by complex fermions [7, 8, 9]; the invariant $\beta$ also
appears as the fermion parities of twisted sectors [12, 14].

###  2.3 Bosonic partition functions

Now we define topological partition functions parameterized by the group
cohomology and supercohomology invariants. These theories represent the
responses of SPT phases to background gauge fields for the symmetry and will
prove useful later on in section 2.6 when we discuss the stacking rule on SPT
phases and in section 3.5 when we study the stacking rule isomorphism. The
bosonic theory associated to $\omega$ has partition function

$\mathcal{Z}_{\omega}^{b}[X,\mathcal{A}]=\exp(2\pi
i\int_{X}\mathcal{A}^{*}\omega)$ (2.17)

on a closed two-dimensional spacetime $X$ with $\mathcal{G}$-valued background
gauge field $\mathcal{A}$, by which we mean a map $\mathcal{A}:X\rightarrow
B\mathcal{G}$ satisfying141414The second constraint enforces that time-
reversing holonomies of $\mathcal{A}$ appear around orientation-reversing
cycles of $X$.

$\delta\mathcal{A}=0~{},$ (2.18) $\mathcal{A}^{*}x=w_{1}~{}.$ (2.19)

The partition function is topological given the condition (2.2) and invariant
under the relation (2.3).

The $\mathcal{G}$ gauge field $\mathcal{A}$ can be split into $G_{b}$ gauge
field $A_{b}:X\rightarrow BG_{b}$ satisfying

$\delta A_{b}=0~{},$ (2.20) $A_{b}^{*}x=w_{1}$ (2.21)

and a map $A_{p}:X\rightarrow B{\mathbb{Z}}_{2}$ with

$\delta A_{p}=A_{b}^{*}\rho$ (2.22)

via the correspondence

$A_{b}=\mathcal{A}^{*}b~{},\qquad
A_{p}=\mathcal{A}^{*}t~{}\qquad\Longleftrightarrow\qquad\mathcal{A}=A_{b}^{*}s+A_{p}~{},$
(2.23)

as can be seen from $\delta t=b^{*}\rho$. In terms of the pair
$(A_{b},A_{p})$, the bosonic theory (2.17) becomes

$\mathcal{Z}_{\alpha,\beta}^{b}[X,A_{b},A_{p}]=\exp(2\pi
i\int_{X}\mathcal{A}^{*}(b^{*}\alpha+\tfrac{1}{2}b^{*}\beta\cup t))=\exp(2\pi
i\int_{X}A_{b}^{*}\alpha+\tfrac{1}{2}A_{b}^{*}\beta\cup A_{p})~{}.$ (2.24)

This partition function is section-independent by construction, but one can
also verify that the shift $\frac{1}{2}\beta\cup\mu$ in $\alpha$ under change
of section is cancelled exactly by the shift $A_{b}^{*}\mu$ in $A_{p}$.

###  2.4 Fermionic partition functions

Fermionic partition functions are sensitive to $\mathcal{G}$-pin structures
rather than to $\mathcal{G}$ gauge fields. A $\mathcal{G}$-pin structure is a
map $\eta^{\mathcal{G}}:X\rightarrow B\mathcal{G}$ satisfying151515The first
condition means that the violation of the triple overlap condition by
$\eta^{\mathcal{G}}$ is valued in ${\mathbb{Z}}_{2}$ and coincides with that
of a pin-minus structure. Compared to the treatment of $\mathcal{G}$-spin
structures in Ref. [12], we have absorbed the freedom $\phi$. See Ref. [11]
for another perspective on $\mathcal{G}$-pin structures, referred to as
twisted spin structures.

$\delta\eta^{\mathcal{G}}=w_{1}^{2}+w_{2}~{},$ (2.25)
$\eta^{\mathcal{G}*}x=w_{1}~{}.$ (2.26)

It can be re-expressed as a $G_{b}$ gauge field $A_{b}$ satisfying (2.20) and
(2.21) and a map $\eta:X\rightarrow B{\mathbb{Z}}_{2}$ satisfying

$\delta\eta=w_{1}^{2}+w_{2}+A_{b}^{*}\rho$ (2.27)

via the correspondence

$A_{b}=\eta^{\mathcal{G}*}b~{},\qquad\eta=\eta^{\mathcal{G}*}t\qquad\Longleftrightarrow\qquad\eta^{\mathcal{G}}=A_{b}^{*}s+\eta~{}.$
(2.28)

$\mathcal{G}$-pin structures generalize many familiar structures. A
$\mathcal{G}$-spin structure is realized when all symmetries are unitary
($x=0$), in which case the theory is defined only on orientable surfaces
($w_{1}=0$). When $\mathcal{G}$ splits and $s$ is a splitting, the map $\eta$
is a pin-minus structure (trivialization of $w_{1}^{2}+w_{2}$); in particular,
for $\mathcal{G}={\mathbb{Z}}_{2}^{T}\times{\mathbb{Z}}_{2}^{f}$, $A_{b}$ is
completely fixed by $w_{1}$, and so a $\mathcal{G}$-pin structure is just the
pin-minus structure $\eta$. A pin-plus structure (trivialization of $w_{2}$)
is realized for $\mathcal{G}={\mathbb{Z}}_{4}^{Tf}$ since $A_{b}$ is
completely fixed by $w_{1}$ and $\rho=x\cup x$ means $\delta\eta=w_{2}$.

To write the fermionic partition functions, we make use of the quadratic
enhancement of the cup pairing that is associated to every pin-minus structure
$e$ [15, 16, 17]. This is a map
$q_{e}:H^{1}(X,{\mathbb{Z}}_{2})\rightarrow{\mathbb{Z}}_{4}$ satisfying

$q_{e}(a)+q_{e}(b)-q_{e}(a+b)=2\int_{X}a\cup b~{},$ (2.29)
$q_{e+b}(a)-q_{e}(a)=2\int_{X}a\cup b$ (2.30)

for ${\mathbb{Z}}_{2}$ gauge fields $a$ and $b$. To work with
$\mathcal{G}$-pin structures, we introduce an auxiliary variable
$\tau:X\rightarrow B{\mathbb{Z}}_{2}$ with $\delta\tau=A_{b}^{*}\rho$, so that
$e=\eta+\tau$ is a pin-minus structure. Given the existence of a
$\mathcal{G}$-pin structure on $X$, a solution $\tau$ always exists because
every two-dimensional space is pin-minus [15]. The partition function should
not depend on the choice of $\tau$. In terms of the pair $(A_{b},\eta)$ and
the auxiliary variable $\tau$, the theory is

$\mathcal{Z}_{\alpha,\beta}^{f}[X,A_{b},\eta]=\exp(2\pi
i\int_{X}A_{b}^{*}\alpha+\tfrac{1}{2}A_{b}^{*}\beta\cup\tau)\exp(\frac{\pi
i}{2}q_{\eta+\tau}(A_{b}^{*}\beta))~{},$ (2.31)

Note that our expression (2.31) reduces to the $\mathcal{G}$-spin partition
function of Ref. [12] in the case of unitary symmetries, where $q/2$ defines a
quadratic refinement associated to a spin structure since $q$ is even on
orientable manifolds. To see that the theory (2.31) is well-defined, several
properties must be checked. First, the theory is independent of the auxiliary
variable $\tau$ since shifting $\tau$ by a closed $b$ changes each of the two
$\tau$-dependent terms by the same sign $A_{b}^{*}\beta\cup b$, as follows
from the relation (2.30). Second, the theory is independent of the section $s$
since $\alpha$ shifts by $\tfrac{1}{2}\beta\cup\mu$ and $\eta$ shifts by
$A_{b}^{*}\mu$ and so by the same rule (2.30) these variations cancel. Third,
the theory is topological since
$\delta(A_{b}^{*}\alpha+\tfrac{1}{2}A_{b}^{*}\beta\cup\tau)=0$ by the
conditions (2.11). Fourth, the theory is invariant under the relation (2.13)
since $b^{*}(\lambda_{0}\cup\rho)=\delta(b^{*}\lambda_{0}\cup t)$.

In the split case, the auxiliary variable may be eliminated by setting it to
$\tau=0$. Then the theory becomes

$\mathcal{Z}_{\alpha,\beta}^{f}[X,A_{b},\eta]=\exp(2\pi
i\int_{X}A_{b}^{*}\alpha)\exp(\frac{\pi i}{2}q_{\eta}(A_{b}^{*}\beta))~{}.$
(2.32)

###  2.5 The Jordan-Wigner transformation

Now we are ready to define the Jordan-Wigner transformation
${\mathcal{J}}{\mathcal{W}}$ on partition functions. It maps a bosonic
partition function to the fermionic partition function defined as follows:

${\mathcal{J}}{\mathcal{W}}(\mathcal{Z}^{b})[X,A_{b},\eta]:=\sum_{A_{p}}\frac{\text{Arf}(\eta+A_{p})}{\sqrt{|H^{1}(X,{\mathbb{Z}}_{2})|}}\mathcal{Z}^{b}[X,A_{b},A_{p}]~{}.$
(2.33)

Here we have used the Arf-Brown-Kervaire invariant [18]

$\text{Arf}(e):=\frac{1}{\sqrt{|H^{1}(X,{\mathbb{Z}}_{2})|}}\sum_{a\in
H^{1}(X,{\mathbb{Z}}_{2})}\exp(\frac{\pi i}{2}q_{e}(a))$ (2.34)

of a pin-minus structure $e$. Some useful properties of the Arf invariant are
collected in appendix D. Using the property (D.9), we can introduce an
auxiliary variable $\tau$ to rewrite the transformation as

${\mathcal{J}}{\mathcal{W}}(\mathcal{Z}^{b})[X,A_{b},\eta]=\frac{\text{Arf}(\eta+\tau)}{\sqrt{|H^{1}(X,{\mathbb{Z}}_{2})|}}\sum_{A_{p}}\mathcal{Z}^{b}[X,A_{b},A_{p}]\exp(\frac{-\pi
i}{2}q_{\eta+\tau}(A_{p}+\tau))~{}.$ (2.35)

As demonstrated in appendix D, the inverse of the Jordan-Wigner transformation
is given by

${\mathcal{J}}{\mathcal{W}}^{-1}(\mathcal{Z}^{f})[X,A_{b},A_{p}]=\sum_{\eta}\frac{\text{Arf}(\eta+A_{p})^{-1}}{\sqrt{|H^{1}(X,{\mathbb{Z}}_{2})|}}\mathcal{Z}^{f}[X,A_{b},\eta]~{}.$
(2.36)

These forms of the Jordan-Wigner transformation were first stated in Ref. [19]
for split, unitary symmetries and were generalized to include antiunitary
symmetries in Ref. [11].

The Jordan-Wigner transform of the bosonic theory associated with the
invariant $[\alpha,\beta]$ is the fermionic theory associated with the same
invariant, as can be seen by computing

$\displaystyle\begin{split}{\mathcal{J}}{\mathcal{W}}(\mathcal{Z}^{b}_{\alpha,\beta})[X,A_{b},\eta]&=\frac{\text{Arf}(\eta+\tau)}{\sqrt{|H^{1}(X,{\mathbb{Z}}_{2})|}}\sum_{A_{p}}\exp(2\pi
i\int_{X}A_{b}^{*}\alpha+\tfrac{1}{2}A_{b}^{*}\beta\cup A_{p})\exp(\frac{-\pi
i}{2}q_{\eta+\tau}(A_{p}+\tau))\\\ &=\exp(2\pi
i\int_{X}A_{b}^{*}\alpha+\tfrac{1}{2}A_{b}^{*}\beta\cup\tau)\exp(\frac{\pi
i}{2}q_{\eta+\tau}(A_{b}^{*}\beta))\\\
&\qquad\qquad\qquad\qquad\times\,\frac{\text{Arf}(\eta+\tau)}{\sqrt{|H^{1}(X,{\mathbb{Z}}_{2})|}}\sum_{A_{p}}\exp(\frac{-\pi
i}{2}q_{\eta+\tau}(A_{b}^{*}\beta+A_{p}+\tau))\\\
&=\mathcal{Z}^{f}_{\alpha,\beta}[X,A_{b},\eta]~{},\end{split}$ (2.37)

where the second line follows from the relation (2.29) and the third from the
expression (D.1). This means that the Jordan-Wigner transformation defines a
map from bosonic to fermionic SPT phases given by

${\mathcal{J}}{\mathcal{W}}:[\alpha,\beta]\mapsto[\alpha,\beta]~{}.$ (2.38)

###  2.6 Stacking rules

As we have just observed, the sets of bosonic and fermionic SPT phases are
equal, with the Jordan-Wigner transformation defining a bijective
correspondence between them. However, there is a natural group operation
defined on the two sets given by _stacking_ , and the classifications are
_not_ always isomorphic as groups. A simple example is given by SPT phases
with symmetry $\mathcal{G}={\mathbb{Z}}_{2}^{T}\times{\mathbb{Z}}_{2}^{f}$:
there are four SPT phases on each side of the correspondence; however, the
bosonic phases form the group
$\mathcal{B}={\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ under stacking while the
fermionic phases form the group $\mathcal{F}={\mathbb{Z}}_{4}$ [2]. This and
other examples are explored in section 3.2.

The _stack_ of two theories is the theory whose partition function is the
product.161616In the language of lattice models, the stack of two systems is
the one defined on the tensor product Hilbert space with tensor product time
evolution $U_{AB}=U_{A}\otimes U_{B}$. This means that the stacked Hamiltonian
is $H_{AB}=H_{A}\otimes\mathds{1}+\mathds{1}\otimes H_{B}$. From the
expression (2.17) for the bosonic partition function in the group cohomology
variable, it follows that

$\mathcal{Z}^{b}_{\omega_{1}}\mathcal{Z}^{b}_{\omega_{2}}=\mathcal{Z}^{b}_{\omega_{1}+\omega_{2}}~{},$
(2.39)

so the invariants stack according to the usual group law on
$H^{2}(\mathcal{G},U(1)_{x})$:

$[\omega_{1}]\otimes_{\mathcal{B}}[\omega_{2}]=[\omega_{1}+\omega_{2}]~{}.$
(2.40)

On the other hand, in the variables (2.4), the fermionic stacking rule is
given by

$[\omega_{1}]\otimes_{\mathcal{F}}[\omega_{2}]=[\omega_{1}+\omega_{2}+\tfrac{1}{2}\nu_{1}\nu_{2}]~{}.$
(2.41)

In terms of the supercohomology invariants, these stacking rules read

$\left[\begin{array}[]{c}\alpha_{1}\\\
\beta_{1}\end{array}\right]\otimes_{\mathcal{B}}\left[\begin{array}[]{c}\alpha_{2}\\\
\beta_{2}\end{array}\right]=\left[\begin{array}[]{c}\alpha_{1}+\alpha_{2}\\\
\beta_{1}+\beta_{2}\end{array}\right]$ (2.42)

and

$\left[\begin{array}[]{c}\alpha_{1}\\\
\beta_{1}\end{array}\right]\otimes_{\mathcal{F}}\left[\begin{array}[]{c}\alpha_{2}\\\
\beta_{2}\end{array}\right]=\left[\begin{array}[]{c}\alpha_{1}+\alpha_{2}+\tfrac{1}{2}\beta_{1}\beta_{2}\\\
\beta_{1}+\beta_{2}\end{array}\right]~{}.$ (2.43)

The supercohomology formulation of the fermionic stacking rule was first
stated in Ref. [19] in case case of a split, unitary symmetry group. It was
generalized to arbitrary unitary symmetries in Refs. [13, 12] and to the most
general setting in Ref. [13]. The general rule has been reproduced in Refs.
[10, 20, 21].

When the symmetry is unitary and split, the group of classes
$[\alpha,\beta]\in H^{2}(G_{b},U(1)_{x})\times H^{1}(G_{b},{\mathbb{Z}}_{2})$
under the fermionic stacking rule (2.43) is isomorphic to
$\tilde{\Omega}_{\text{spin}}^{2}(BG_{b})$, the reduced spin cobordism
group171717The “cobordism” group refers to the Pontryagin dual ${\rm
Hom}(\,\cdot\,,U(1))$ of the bordism group. The bordism group that appears is
reduced because we have specialized to SPT phases among all invertible
fermionic phases. of $BG_{b}$ [22, 23, 19, 24]. More generally, the group of
fermionic SPT phases given by _twisted_ cobordism [23, 11]:

$\mathcal{F}(G_{b},\rho,x)\simeq\tilde{\Omega}_{\text{spin}}^{2}(BG_{b},\rho,x)~{}.$
(2.44)

This is in contrast with bosonic SPT phases, whose group structure (2.42) is
that of twisted cohomology

$\mathcal{B}(G_{b},\rho,x)\simeq H^{2}(\mathcal{G},U(1)_{x})~{}.$ (2.45)

The bosonic and fermionic stacking rules can be recovered by multiplying
partition functions. From

$\displaystyle\begin{split}\mathcal{Z}_{\alpha_{1},\beta_{1}}^{b}[X,A_{b},A_{p}]\mathcal{Z}_{\alpha_{2},\beta_{2}}^{b}[X,A_{b},A_{p}]&=\exp(2\pi
i\int_{X}A_{b}^{*}(\alpha_{1}+\alpha_{2})+\tfrac{1}{2}A_{b}^{*}(\beta_{1}+\beta_{2})\cup
A_{p})\\\
&=\mathcal{Z}_{\alpha_{1}+\alpha_{2},\beta_{1}+\beta_{2}}^{b}[X,A_{b},A_{p}]~{},\end{split}$
(2.46)

one sees the rule (2.42). And due to the quadratic relation (2.29), one sees
the fermionic rule (2.43) from

$\displaystyle\begin{split}\mathcal{Z}_{\alpha_{1},\beta_{1}}^{f}[X,A_{b},\eta]&\mathcal{Z}_{\alpha_{2},\beta_{2}}^{f}[X,A_{b},\eta]\\\
&=\exp(2\pi
i\int_{X}A_{b}^{*}(\alpha_{1}+\alpha_{2})+\tfrac{1}{2}A_{b}^{*}(\beta_{1}+\beta_{2})\cup\tau)\exp(\frac{\pi
i}{2}(q_{\eta+\tau}(A_{b}^{*}\beta_{1})+q_{\eta+\tau}(A_{b}^{*}\beta_{2})))\\\
&=\exp(2\pi
i\int_{X}A_{b}^{*}(\alpha_{1}+\alpha_{2}+\tfrac{1}{2}\beta_{1}\cup\beta_{2})+\tfrac{1}{2}A_{b}^{*}(\beta_{1}+\beta_{2})\cup\tau)\exp(\frac{\pi
i}{2}q_{\eta+\tau}(A_{b}^{*}(\beta_{1}+\beta_{2})))\\\
&=\mathcal{Z}_{\alpha_{1}+\alpha_{2}+\tfrac{1}{2}\beta_{1}\cup\beta_{2},\beta_{1}+\beta_{2}}^{f}[X,A_{b},\eta]~{}.\end{split}$
(2.47)

In appendix A, this approach of multiplying partition functions is used to
extend the fermionic stacking rule to all invertible phases, including those
beyond SPT phases.

##  3 Isomorphic classifications

###  3.1 The classifications as group extensions

We now ask the following: when are the bosonic and fermionic SPT
classifications isomorphic as groups? We find that, despite their different
looking forms (2.42) and (2.43) in the supercohomology variables, the groups
are often isomorphic, even when the Jordan-Wigner transformation does not
provide an isomorphism.

Each of the two stacking rules $\mathcal{B}$ and $\mathcal{F}$ can be realized
as a group extension of ${\mathbb{B}}$ by ${\mathbb{A}}$:

${\mathbb{A}}\longrightarrow\mathcal{B}\longrightarrow{\mathbb{B}}~{},\qquad{\mathbb{A}}\longrightarrow\mathcal{F}\longrightarrow{\mathbb{B}}~{},$
(3.1)

where ${\mathbb{A}}$ and ${\mathbb{B}}$ were first defined in section 2.2 as
the following: ${\mathbb{A}}$ is the quotient of $H^{2}(G_{b},U(1)_{x})$ by
${\mathbb{Z}}_{2}=\langle\tfrac{1}{2}\rho\rangle$, and ${\mathbb{B}}$ is the
subgroup of $H^{1}(G_{b},{\mathbb{Z}}_{2})$ consisting of _valid_ $\beta$,
i.e. those satisfying $[\tfrac{1}{2}\beta\cup\rho]=0$ (equivalently, those
that pullback from $\nu$ (2.6) associated with $\omega$). Since the extensions
are central (as $\mathcal{B}$ and $\mathcal{F}$ are abelian by construction),
each is characterized by an extension class in
$[\Omega_{\mathcal{B},\mathcal{F}}]\in H^{2}({\mathbb{B}},{\mathbb{A}})$ that
appears as

$\left[\begin{array}[]{c}\alpha_{1}\\\
\beta_{1}\end{array}\right]\otimes_{\mathcal{B},\mathcal{F}}\left[\begin{array}[]{c}\alpha_{2}\\\
\beta_{2}\end{array}\right]=\left[\begin{array}[]{c}\alpha_{1}+\alpha_{2}+\Omega_{\mathcal{B},\mathcal{F}}(\beta_{1},\beta_{2})\\\
\beta_{1}+\beta_{2}\end{array}\right]~{}.$ (3.2)

The expressions (2.42) and (2.43) reveal that

$\Omega_{\mathcal{B}}=0~{},\qquad\Omega_{\mathcal{F}}=\tfrac{1}{2}\cup~{}.$
(3.3)

The bosonic classification group is simply the direct product
group181818Since, due to the condition (2.11), a equivalence class of an
$\alpha$ is not canonically identified with an element of ${\mathbb{A}}$ (but
rather to an element of a torsor over it), this isomorphism is also
noncanonical.

$\mathcal{B}\simeq{\mathbb{A}}\times{\mathbb{B}}~{},$ (3.4)

and we are interested in studying when
$\mathcal{F}\simeq\mathcal{B}\simeq{\mathbb{A}}\times{\mathbb{B}}$. Since the
direct product group can arise only from the trivial extension [25], this only
happens when $[\Omega_{\mathcal{F}}]=0$, i.e. when there exists a
$\Lambda:{\mathbb{B}}\rightarrow{\mathbb{A}}$ such that

$\Omega_{\mathcal{F}}=\delta\Lambda~{}.$ (3.5)

In this case, the map

${\mathcal{X}}:[\alpha,\beta]\mapsto[\alpha-\Lambda(\beta),\beta]$ (3.6)

gives an isomorphism of stacking group laws:

$\displaystyle\begin{split}{\mathcal{X}}[\alpha_{1},\beta_{1}]\otimes_{\mathcal{F}}{\mathcal{X}}[\alpha_{2},\beta_{2}]&=[\alpha_{1}-\Lambda(\beta_{1}),\beta_{1}]\otimes_{\mathcal{F}}[\alpha_{2}-\Lambda(\beta_{2}),\beta_{2}]\\\
&=[\alpha_{1}+\alpha_{2}-\Lambda(\beta_{1})-\Lambda(\beta_{2})+\Omega_{\mathcal{F}}(\beta_{1},\beta_{2}),\beta_{1}+\beta_{2}]\\\
&=[\alpha_{1}+\alpha_{2}-\Lambda(\beta_{1}+\beta_{2}),\beta_{1}+\beta_{2}]\\\
&={\mathcal{X}}[\alpha_{1}+\alpha_{2},\beta_{1}+\beta_{2}]\\\
&={\mathcal{X}}([\alpha_{1},\beta_{1}]\otimes_{\mathcal{B}}[\alpha_{2},\beta_{2}])~{}.\end{split}$
(3.7)

When the isomorphism ${\mathcal{X}}$ exists, it is in general not unique, as
$\Lambda$ may be shifted by any cocycle in $Z^{1}({\mathbb{B}},{\mathbb{A}})$.

We emphasize that the condition (3.5) is weaker than asking when
$\Omega_{\mathcal{F}}(\beta_{1},\beta_{2})$ vanishes in ${\mathbb{A}}$ (that
is, when $\tfrac{1}{2}\beta_{1}\cup\beta_{2}$ is cohomologous to
$\tfrac{1}{2}\rho$) for all $\beta_{1},\beta_{2}\in{\mathbb{B}}$. The
vanishing of $\Omega_{\mathcal{F}}$ for all inputs means that the Jordan-
Wigner transformation (2.38) is an isomorphism, whereas the condition (3.5)
only means that there is _some_ isomorphism
${\mathcal{X}}:\mathcal{B}\rightarrow\mathcal{F}$, not necessarily given by
${\mathcal{J}}{\mathcal{W}}$. We illustrate this subtlety with some simple
examples in section 3.2, and we ask about the physical meaning of these other
isomorphisms in section 4.

Let us briefly comment on the question of when ${\mathcal{J}}{\mathcal{W}}$ is
an isomorphism of stacking rules, in the simple case where the symmetry
$\mathcal{G}$ is finite abelian, unitary, and split. Here, the group $G_{b}$
is a product of cyclic groups, and ${\mathcal{J}}{\mathcal{W}}$ is an
isomorphism if and only if there is only one factor of even order in this
decomposition.191919This notion is well-defined because factors of even order
can not be combined since their orders are not coprime.. When there is only
one factor, the group ${\mathbb{A}}=H^{2}(G_{b},U(1)_{x})$ is trivial, so the
extension class vanishes and ${\mathcal{J}}{\mathcal{W}}$ is an isomorphism.
When there is more than one factor, there are
$\beta_{1},\beta_{2}\in{\mathbb{B}}=H^{1}(G_{b},{\mathbb{Z}}_{2})$ each
evaluating to $1$ on distinct even order cyclic generators $g,h$ and $0$ on
the others. Then $\alpha=\tfrac{1}{2}\beta_{1}\cup\beta_{2}$ is nontrivial
since it evaluates to $\tfrac{1}{2}\neq 0$ on the gauge invariant quantity
(torus partition function) $\alpha(g,h)-\alpha(h,g)$, so
${\mathcal{J}}{\mathcal{W}}$ is not an isomorphism. Further examples are
discussed in section 3.2.

###  3.2 Examples

Before answering the question in general, let us consider a few examples.

First is $\mathcal{G}={\mathbb{Z}}_{2}^{T}\times{\mathbb{Z}}_{2}^{f}$, where
there is no isomorphism. There are four SPT phases, each characterized by a
choice of supercohomology data – one of the two possible $\alpha$ and one of
the two possible $\beta$:

$\alpha_{0}(t,t)=0~{},\qquad\alpha_{1}(t,t)=\tfrac{1}{2}~{},$ (3.8)
$\beta_{0}(t)=0~{},\qquad\beta_{1}(t)=1~{}.$ (3.9)

These cocycles add according to

$\alpha_{1}+\alpha_{1}=\alpha_{0}~{},\qquad\beta_{1}+\beta_{1}=\beta_{0}~{},$
(3.10)

so, denoting each cocycle by its subscript, we have the bosonic stacking rule

$\left[\begin{array}[]{c}i\\\
j\end{array}\right]\otimes_{\mathcal{B}}\left[\begin{array}[]{c}i^{\prime}\\\
j^{\prime}\end{array}\right]=\left[\begin{array}[]{c}i+i^{\prime}\\\
j+j^{\prime}\end{array}\right]~{},$ (3.11)

which is the classification group
$\mathcal{B}={\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$. The only nontrivial cup
product of $\beta$ is

$\tfrac{1}{2}\beta_{1}\cup\beta_{1}=\alpha_{1}~{},$ (3.12)

so the fermionic stacking rule is

$\left[\begin{array}[]{c}i\\\
j\end{array}\right]\otimes_{\mathcal{F}}\left[\begin{array}[]{c}i^{\prime}\\\
j^{\prime}\end{array}\right]=\left[\begin{array}[]{c}i+i^{\prime}+jj^{\prime}\\\
j+j^{\prime}\end{array}\right]~{},$ (3.13)

which is the classification group $\mathcal{F}={\mathbb{Z}}_{4}$, generated by
the phase $[\alpha_{0},\beta_{1}]$. Therefore
$\mathcal{B}\not\simeq\mathcal{F}$.

A second example is
$\mathcal{G}={\mathbb{Z}}_{4}^{T}\times{\mathbb{Z}}_{2}^{f}$, where now $t$ is
of order $4$. Here, there is an isomorphism given by the Jordan-Wigner
transformation. Again there are two each of $\alpha$ and $\beta$:

$\alpha_{0}(t^{m},t^{n})=0~{},\qquad\alpha_{1}(t^{m},t^{n})=\tfrac{1}{4}[m]_{2}n~{},$
(3.14) $\beta_{0}(t^{m})=0~{},\qquad\beta_{1}(t^{m})=[m]_{2},$ (3.15)

where $[\,\cdot\,]_{2}$ denotes reduction mod $2$. The cochain $\alpha_{1}$ is
indeed a twisted cocycle since

$\displaystyle\begin{split}(\delta\alpha)_{1}(t^{l},t^{m},t^{n})&=(-1)^{l}\tfrac{1}{4}[m]_{2}n+\tfrac{1}{4}[l]_{2}(m+n)-\tfrac{1}{4}[l]_{2}m-\tfrac{1}{4}[l+m]_{2}n\\\
&=\tfrac{1}{4}n([m]_{2}-2[l]_{2}[m]_{2}+[l]_{2}-[l+m]_{2})\\\ &=0\end{split}$
(3.16)

and is nontrivial since

$\alpha_{1}(t,t^{3})-\alpha_{1}(t^{3},t)=\tfrac{1}{2}$ (3.17)

is a nonzero gauge invariant quantity. The cocycles add as in the previous
example (3.10), so the bosonic classification is again
$\mathcal{B}={\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$. But this time, the cup
products are all trivial in cohomology:

$(\tfrac{1}{2}\beta_{1}\cup\beta_{1})(t^{m},t^{n})=\tfrac{1}{2}[m]_{2}[n]_{2}=\tfrac{1}{4}((m-[m]_{2})+(-1)^{m}(n-[n]_{2})-(m+n-[m+n]_{2}))~{},$
(3.18)
$\Longrightarrow\qquad\tfrac{1}{2}\beta_{1}\cup\beta_{1}=\delta\lambda~{},\qquad\lambda(t^{m})=\tfrac{1}{4}(m-[m]_{2})~{},$
(3.19)

so the fermionic stacking rule is also
$\mathcal{F}={\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$. In this example, the
Jordan-Wigner transformation is an isomorphism
$\mathcal{B}\rightarrow\mathcal{F}$ since the bosonic and fermionic stacking
rules are identical in terms of $\alpha_{i},\beta_{i}$.

Finally, consider the symmetry group
$\mathcal{G}={\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}^{f}$,
where there is an isomorphism $\mathcal{B}\simeq\mathcal{F}$ but it is not
given by the Jordan-Wigner transformation. This group has two $\alpha$ and
four $\beta$:

$\alpha_{0}((a,b),(a^{\prime},b^{\prime}))=0~{},\qquad\alpha_{1}((a,b),(a^{\prime},b^{\prime}))=\tfrac{1}{2}a^{\prime}b~{},$
(3.20)
$\beta_{00}(a,b)=0~{},\qquad\beta_{10}(a,b)=a~{},\qquad\beta_{01}(a,b)=b~{},\qquad\beta_{11}(a,b)=[a+b]_{2}~{},$
(3.21)

written in terms of elements
$(a,b)\in{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}=G_{b}$. These cocycles add
according to

$\alpha_{i}+\alpha_{i^{\prime}}=\alpha_{i+i^{\prime}}~{},\qquad\beta_{j,k}+\beta_{j^{\prime},k^{\prime}}=\beta_{(j+j^{\prime}),(k+k^{\prime})}~{},$
(3.22)

so the bosonic stacking rule is $\mathcal{B}={\mathbb{Z}}_{2}^{3}$:

$\left[\begin{array}[]{c}i\\\ j\\\
k\end{array}\right]\otimes_{\mathcal{B}}\left[\begin{array}[]{c}i^{\prime}\\\
j^{\prime}\\\
k^{\prime}\end{array}\right]=\left[\begin{array}[]{c}i+i^{\prime}\\\
j+j^{\prime}\\\ k+k^{\prime}\end{array}\right]~{},$ (3.23)

where the second and third rows represent $\beta_{10}$ and $\beta_{01}$. The
cup products are

$(\tfrac{1}{2}\beta_{jk}\cup\beta_{j^{\prime}k^{\prime}})((a,b),(a^{\prime},b^{\prime}))=\tfrac{1}{2}(ja+kb)(j^{\prime}a^{\prime}+k^{\prime}b^{\prime})=\tfrac{1}{2}jj^{\prime}aa^{\prime}+\tfrac{1}{2}jk^{\prime}ab^{\prime}+\tfrac{1}{2}kj^{\prime}ba^{\prime}+\tfrac{1}{2}kk^{\prime}bb^{\prime}$
(3.24)
$\Longrightarrow\qquad\tfrac{1}{2}\beta_{jk}\cup\beta_{j^{\prime}k^{\prime}}\sim(jk^{\prime}+j^{\prime}k)\,\alpha_{1}$
(3.25)

since, in cohomology, $\tfrac{1}{2}aa^{\prime}$ and $\tfrac{1}{2}bb^{\prime}$
are trivial and $\tfrac{1}{2}ab^{\prime}\sim\tfrac{1}{2}a^{\prime}b$.
Therefore the fermionic stacking rule is

$\left[\begin{array}[]{c}i\\\ j\\\
k\end{array}\right]\otimes_{\mathcal{F}}\left[\begin{array}[]{c}i^{\prime}\\\
j^{\prime}\\\
k^{\prime}\end{array}\right]=\left[\begin{array}[]{c}i+i^{\prime}+jk^{\prime}+j^{\prime}k\\\
j+j^{\prime}\\\ k+k^{\prime}\end{array}\right]~{}.$ (3.26)

Because the bosonic and fermionic stacking rules are not identical, the
Jordan-Wigner transformation is not an isomorphism. However, a trivialization
of the extension class is given by

$\Lambda(\beta_{jk})=jk\,\alpha_{1}~{},\qquad\text{i.e.}\qquad\Lambda(\beta_{11})=\alpha_{1}~{},\quad\Lambda(\text{else})=0~{},$
(3.27)

which has

$(\delta\Lambda)(\beta,\beta^{\prime})=jk\,\alpha_{1}+j^{\prime}k^{\prime}\,\alpha_{1}-(j+j^{\prime})(k+k^{\prime})\,\alpha_{1}=-(jk^{\prime}+j^{\prime}k)\,\alpha_{1}~{}.$
(3.28)

Therefore the map

${\mathcal{X}}:\left[\begin{array}[]{c}i\\\ j\\\
k\end{array}\right]\mapsto\left[\begin{array}[]{c}i+jk\\\ j\\\
k\end{array}\right]$ (3.29)

from bosonic to fermionic SPT phases is an isomorphism, and the fermionic
stacking rule is $\mathcal{F}={\mathbb{Z}}_{2}^{3}$. We will return to this
example in section 4.4 when we study whether the map ${\mathcal{X}}$ can be
implemented by a duality.

###  3.3 Necessary and sufficient conditions for isomorphism

In order to state the necessary and sufficient conditions for the bosonic and
fermionic SPT classifications to be isomorphic, we first define several
conditions. We call these conditions “genus rules” in reference to the fact,
explained in section 3.5, that the genus $n$ rule is equivalent to the
vanishing of the bosonic partition function associated to the cocycles
$\omega=\tfrac{1}{2}b^{*}\beta\cup b^{*}\beta$, for all valid $\beta$, on the
nonorientable surface of genus $n$.

_Genus $n$ rule._ A fermionic symmetry group $(G_{b},\rho,x)$ is said to
satisfy the genus $n$ rule if $\sum_{i}^{n}b^{*}\beta(g_{i})=0\text{ mod
}2~{}.$ (3.30) for every $\beta:G_{b}\rightarrow{\mathbb{Z}}_{2}$ with
$[\tfrac{1}{2}\beta\cup\rho]=0$ and elements
$g_{1},\ldots,g_{n}\in\mathcal{G}$ with $x(g_{i})=1$ and
$\prod_{i}^{n}g_{i}^{2}=1$.

In the split case,202020If the group does not split, one must worry about
$G_{b}$ gauge fields $A_{b}=\\{g_{b1},\ldots,g_{bn}\\}$ that do not extend to
$\mathcal{G}$ gauge fields. These have $\int_{X}A_{b}^{*}\rho\neq 0$ (and no
solution $A_{p}$), so they do not distinguish when a class is trivial in
${\mathbb{A}}$. the genus $n$ rule may be framed in terms of $G_{b}$ elements
as the condition that

$\sum_{i}^{n}\beta(g_{bi})=0\text{ mod }2=0$ (3.31)

for every $\beta:G_{b}\rightarrow{\mathbb{Z}}_{2}$ and any elements
$g_{b1},\ldots,g_{bn}\in G_{b}$ with $x(g_{bi})=1$ and
$\prod_{i}^{n}g_{bi}^{2}=1$.

If the genus $n$ rule is satisfied for some $n>2$, then it is also satisfied
for $n-2$. To see this, suppose that the genus $n-2$ rule is violated, which
means that there exists valid $\beta$ and antiunitary $g_{1},\ldots,g_{n-2}$
satisfying $\prod_{i}^{n-2}g_{i}^{2}=1$ but with
$\sum_{i}^{n-2}b^{*}\beta(g_{i})=1$. Then, for any antiunitary $g$, taking
$g_{n-1}=g$ and $g_{n}=g^{-1}$ means that the sequence $g_{1},\ldots,g_{n}$
satisfies $\prod_{i}^{n}g_{i}^{2}=1$ but has
$\sum_{i}^{n}b^{*}\beta(g_{i})=1$, violating the genus $n$ rule.

When the symmetry group $\mathcal{G}$ is abelian, the relation also goes the
other way: satisfying the genus $n$ rule implies satisfying the genus $n+2$
rule. To see this, suppose that the genus $n+2$ rule is violated, which means
that there exists valid $\beta$ and antiunitary $g_{1},\cdots,g_{n+2}$ with
$\prod_{i}^{n+2}g_{i}^{2}=1$ and $\sum_{i}^{n+2}b^{*}\beta(g_{i})=1$. Then
define $g=g_{n}g_{n+1}g_{n+2}$, which is antiunitary, has
$\sum_{i}^{n-1}b^{*}\beta(g_{i})+b^{*}\beta(g)=1$, and, since the group is
abelian, satisfies $(\prod_{i}^{n-1}g_{i}^{2})g^{2}=1$. Therefore
$\mathcal{G}$ violates the genus $n$ rule with the sequence
$g_{1},\ldots,g_{n-1},g$.

A group is said to satisfy the _genus $\infty$ rule_ if it satisfies the genus
$n$ rule for every $n$. For abelian groups, the genus $\infty$ rule is
equivalent to the genus $1$ and $2$ rules together, by the argument above.

In terms of these rules, the following necessary and sufficient conditions for
an isomorphism $\mathcal{B}\simeq\mathcal{F}$ hold.

_General result._ For the groups of bosonic and fermionic SPT phases to be
isomorphic $\mathcal{B}(G_{b},\rho,x)\simeq\mathcal{F}(G_{b},\rho,x)~{},$
(3.32) • The genus $1$ rule is a necessary condition. • The genus $\infty$
rule is a sufficient condition.

The genus $\infty$ rule is _not_ necessary (only sufficient) in general: in
appendix C, we discuss the example of
$\mathcal{G}=({\mathbb{Z}}_{4}\rtimes{\mathbb{Z}}_{4}^{T})\times{\mathbb{Z}}_{2}^{f}$,
which has isomorphic SPT classifications despite violating the condition.

The genus $1$ rule, which demands that every valid $\beta$ and $g$ with
$x(g)=1$ and $g^{2}=1$ satisfy $b^{*}\beta(g)=0$, has a simple meaning in the
split case. There, every $\beta$ – including $\beta=x$ – is valid, so the rule
says that there is no antiunitary symmetry that squares to one. When the group
does not split, interpretation of the genus $1$ rule becomes difficult: for
example,212121For another example, consider the nonabelian group
${\mathbb{Z}}_{4}^{Tf}\rtimes{\mathbb{Z}}_{2}$ and its antiunitary element
$g=(1,1)$. the group ${\mathbb{Z}}_{2}^{T}\times{\mathbb{Z}}_{4}^{f}$ has
antiunitary element $(1,0)$ squaring to $1$ yet satisfies the genus $1$ rule
since no $\beta$ that detects this element (in fact, no nonzero $\beta$ at
all) is valid.222222That no nontrivial $\beta$ are valid can be seen by
writing the $\omega$ for this group and noting that their associated $\nu$
vanish.

The general result will be proven in section 3.4 and section 3.5. First, let
us discuss several special cases. One case that we demonstrate in section 3.5
is the following:

_Central antiunitary symmetry._ When $G_{b}$ contains a central antiunitary
symmetry, the $\infty$ genus rule is necessary and sufficient.

Specializing to unitary symmetries, we have the following result, which
reveals that the isomorphism for symmetry
$\mathcal{G}={\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}^{f}$
considered in section 3.2 is not an accident but rather an example of the
general rule:

_Unitary symmetries._ When the symmetry $\mathcal{G}$ is unitary, the groups
of bosonic and fermionic SPT phases are isomorphic:
$\mathcal{B}(G_{b},\rho,0)\simeq\mathcal{F}(G_{b},\rho,0)~{}.$ (3.33)

This result is an immediate corollary of the general result because all of the
genus rules are vacuously satisfied when there are there are no suitable
elements $g_{i}$. The unitary result appeared previously in Ref. [4], where it
was proven in an abstract setting. In the present paper, we reproduce this
result while also stating the isomorphisms explicitly (c.f. section 3.4),
demystifying its relation to the Jordan-Wigner transformation (which we have
seen does _not_ typically give an isomorphism even when the groups are
isomorphic), studying its relation to more general dualities (c.f. section 4),
and extending it to include antiunitary symmetries. The unitary result was
also stated previously, at least the in split case, in the context of the
cobordism group [24]. The argument is essentially the one we generalize below:
checking that the half cup squares vanish.

Finally, we turn to abelian groups, where conditions directly about the group
structure are possible.

_Abelian symmetries._ When the symmetry $\mathcal{G}$ is abelian, • The genus
$1$ and $2$ rules together are necessary and sufficient. Moreover, when the
symmetry $\mathcal{G}$ is finite abelian and split, • It is necessary and
sufficient that $G_{b}$ is either unitary or has no ${\mathbb{Z}}_{2}$
factors.

If $\mathcal{G}$ is unitary, isomorphism of SPT classifications is automatic;
if abelian $\mathcal{G}$ contains an antiunitary symmetry, it is central, and
so the genus $\infty$ rule is necessary and sufficient. We observed above that
this rule is equivalent to the genus $1$ and $2$ rule together; therefore,
rules $1$ and $2$ are necessary and sufficient for abelian symmetries.

It is difficult to translate the genus $n$ rules into statements directly
about the group structure – without reference to $\beta$ – though we achieve
this for finite abelian groups that split. A basis of generators can always be
chosen so at most one of them is antiunitary:

$G_{b}=\prod_{i}{\mathbb{Z}}_{p_{i}^{n_{i}}}~{},\qquad\text{or}\qquad
G_{b}={\mathbb{Z}}_{2^{n}}^{T}\times\prod_{i}{\mathbb{Z}}_{p_{i}^{n_{i}}}~{},$
(3.34)

where the $p_{i}$ are (not necessarily distinct) primes. The first case, where
the symmetry is unitary, is solved. Consider the second case, where one of the
generators $t$ is antiunitary. The genus $1$ rules demands there is no valid
$\beta$ and antiunitary $g$ with $g^{2}=1$ and $\beta(g)=1$. The group has an
antiunitary $g$ squaring to $1$ if and only if $t^{2}=1$. Moreover, the
$\beta$ that evaluates to $1$ on $t$ and to $0$ on the other generators is
valid, since every $\beta$ is valid in the split case. Therefore, the genus
$1$ rule is equivalent to $t^{2}\neq 1$, i.e. $n>1$. Meanwhile, the genus $2$
rule demands that there is no $\beta$ and antiunitary $g_{1},g_{2}$ and such
that $g_{1}^{2}g_{2}^{2}=1$ and $\beta(g_{1})\neq\beta(g_{2})$. Suppose one of
the unitary generators $u$ is of order two, i.e. $p_{i}=2$, $n_{i}=1$. Then
$g_{1}=t$ and $g_{2}=ut^{-1}$ violate the genus $2$ rule with the $\beta$ that
evaluates to $1$ on $u$ and to $0$ on the other generators. On the other hand,
suppose that none of the unitary generators is of order two, which means that
every order two unitary element has a square root. Arbitrary antiunitary
elements satisfying $g_{1}^{2}g_{2}^{2}=1$ are written as $g_{1}=u_{1}t^{m}$
and $g_{2}=u_{1}t^{-m}$, which means $(u_{1}u_{2})^{2}=1$. Then $u_{1}u_{2}$
has a square root, so $0=\beta(u_{1}u_{2})=\beta(g_{1})+\beta(g_{2})$ for any
$\beta$. Therefore, the genus $2$ rule is equivalent to the lack of unitary
${\mathbb{Z}}_{2}$ factors. Together, the genus $1$ and $2$ rules means there
are no ${\mathbb{Z}}_{2}$ factors in $G_{b}$. The assumption that the symmetry
splits is crucial to this conclusion, as demonstrated by the example of
$\mathcal{G}={\mathbb{Z}}_{2}^{T}\times{\mathbb{Z}}_{4}^{f}$ mentioned above.
Similarly, the example of
$\mathcal{G}=({\mathbb{Z}}_{4}\rtimes{\mathbb{Z}}_{4}^{T})\times{\mathbb{Z}}_{2}^{f}$
(c.f. appendix C) shows the importance of the assumption that the group is
abelian.

We proceed by arguing for the general necessary and sufficient conditions in
the following steps:

  1. Lemma 1. The groups $\mathcal{B}$ and $\mathcal{F}$ are isomorphic if and only if the half cup product is trivial:

$[\tfrac{1}{2}\cup]=0\in H^{2}({\mathbb{B}},{\mathbb{A}})~{}.$ (3.35)

  2. Lemma 2. This happens if and only if, for all $\beta\in{\mathbb{B}}$, the half cup square has a square root $R_{\beta}$ in ${\mathbb{A}}$:

$2R_{\beta}\sim\tfrac{1}{2}\beta\cup\beta~{}.$ (3.36)

  3. Lemma 3. This happens if the genus $\infty$ rule is satisfied and only if the genus $1$ rule is satisfied.

Lemma 1 was argued in section 3.1. We turn to Lemma 2 in section 3.4 and then
to Lemma 3 in section 3.5.

###  3.4 Square roots of the diagonal

Choose a basis $\beta_{i}$ of generators of
${\mathbb{B}}={\mathbb{Z}}_{2}^{N}$. A general valid $\beta$ can then be
uniquely decomposed as

$\beta=\sum_{i}a_{i}\beta_{i}~{},$ (3.37)

where $a_{i}$ are ${\mathbb{Z}}_{2}$-coefficients. In this basis, the half cup
squares look like

$\tfrac{1}{2}\beta\cup\beta=\tfrac{1}{2}\sum_{ij}a_{i}a_{j}\beta_{i}\cup\beta_{j}\sim
2\,\tfrac{1}{2}\sum_{i<j}a_{i}a_{j}\beta_{i}\cup\beta_{j}+\tfrac{1}{2}\sum_{i}a_{i}a_{i}\beta_{i}\cup\beta_{i}=\tfrac{1}{2}\sum_{i}a_{i}\beta_{i}\cup\beta_{i}~{}.$
(3.38)

For every valid $\beta$, fix a square root $R_{\beta_{i}}$ of
$\tfrac{1}{2}\beta_{i}\cup\beta_{i}$ in ${\mathbb{A}}$. A square root of
$\tfrac{1}{2}\beta\cup\beta$ is given by

$R_{\beta}=\sum_{i}a_{i}R_{\beta_{i}}$ (3.39)

since

$2R_{\beta}=\sum_{i}a_{i}2R_{\beta_{i}}\sim\tfrac{1}{2}\sum_{i}a_{i}\beta_{i}\cup\beta_{i}\sim\tfrac{1}{2}\beta\cup\beta~{}.$
(3.40)

These choices of square roots satisfy

$R_{\beta+\beta^{\prime}}=\sum_{i}(a_{i}+a_{i}^{\prime}\text{ mod
}2)R_{\beta_{i}}$ (3.41)

and therefore

$R_{\beta}+R_{\beta^{\prime}}-R_{\beta+\beta^{\prime}}=\sum_{i}(a_{i}+a_{i}^{\prime}-(a_{i}+a_{i}^{\prime}\text{
mod
}2))R_{\beta_{i}}=\sum_{i}a_{i}a_{i}^{\prime}2R_{\beta_{i}}\sim\tfrac{1}{2}\sum_{i}a_{i}a_{i}^{\prime}\beta_{i}\cup\beta_{i}~{}.$
(3.42)

Define $\Lambda:{\mathbb{B}}\rightarrow{\mathbb{A}}$ as

$\Lambda(\beta)=\tfrac{1}{2}\sum_{i<j}a_{i}a_{j}\beta_{i}\cup\beta_{j}+R_{\beta}~{}.$
(3.43)

It has coboundary

$\displaystyle\begin{split}(\delta\Lambda)(\beta,\beta^{\prime})&=\Lambda(\beta)+\Lambda(\beta^{\prime})-\Lambda(\beta+\beta^{\prime})\\\
&=\tfrac{1}{2}\sum_{i<j}\left(a_{i}a_{j}+a_{i}^{\prime}a_{j}^{\prime}-(a_{i}+a_{i}^{\prime})(a_{j}+a_{j}^{\prime})\right)\beta_{i}\cup\beta_{j}+R_{\beta}+R_{\beta^{\prime}}-R_{\beta+\beta^{\prime}}\\\
&\sim\tfrac{1}{2}\sum_{i<j}\left(a_{i}a_{j}^{\prime}+a_{i}^{\prime}a_{j}\right)\beta_{i}\cup\beta_{j}+\tfrac{1}{2}\sum_{i}a_{i}a_{i}^{\prime}\beta_{i}\cup\beta_{i}\\\
&=\tfrac{1}{2}\sum_{i,j}a_{i}a_{j}^{\prime}\beta_{i}\cup\beta_{j}\\\
&=\Omega_{\mathcal{F}}(\beta,\beta^{\prime})~{}.\end{split}$ (3.44)

This means that, given the existence of square roots $R_{\beta}$, the
extension class $[\Omega_{\mathcal{F}}]=[\tfrac{1}{2}\cup]$ is trivial.
Conversely, suppose $\Omega_{\mathcal{F}}=\delta\Lambda$ for some $\Lambda$,
not necessarily of the form (3.43). The diagonal terms are

$[\tfrac{1}{2}\beta\cup\beta]=\Omega_{\mathcal{F}}(\beta,\beta)=(\delta\Lambda)(\beta,\beta)=2\Lambda(\beta)$
(3.45)

since $\Lambda(0)=\Omega_{F}(\beta,0)=0$, so $\Lambda(\beta)$ is a square root
of $\tfrac{1}{2}\beta\cup\beta$ in ${\mathbb{A}}$. This proves Lemma 2.

###  3.5 Derivation of the genus rules

Next is Lemma 3, which states a property of the symmetry group that guarantees
that its valid half cup squares have square roots (and therefore, by the
previous lemmas, that the SPT classification groups are isomorphic). A special
case occurs when these half cup squares _vanish_ in ${\mathbb{A}}$ – if they
vanish, they have zero as a square root. We argue in the following that the
genus $\infty$ rule is equivalent to them vanishing. While this condition is
sufficient for the existence of square roots (namely, zero), it isn’t
necessary, as a nonvanishing valid half cup square may still have a square
root (see the example of $G_{b}={\mathbb{Z}}_{4}\rtimes{\mathbb{Z}}_{4}^{T}$
in appendix C). The genus $1$ condition, on the other hand, must be satisfied
for the valid half cup squares to have square roots.

In turns out that when there is an antiunitary symmetry in the center of
$G_{b}$, all classes in ${\mathbb{A}}$ are of order two, so they only have
have square roots if they vanish. In this case, the genus $\infty$ rule is
necessary as well as sufficient. We prove the claim about classes of order two
in appendix C.

We proceed by studying when the valid half cup squares vanish in
${\mathbb{A}}$. Since $[\tfrac{1}{2}\rho]$ generates the kernel of
$b^{*}:H^{2}(G_{b},U(1)_{x})\rightarrow H^{2}(\mathcal{G},U(1)_{x})$, the
vanishing of a class in
${\mathbb{A}}=H^{2}(G_{b},U(1)_{x})/\langle\tfrac{1}{2}\rho\rangle$ is
equivalent to the vanishing of its pullback by $b$ in
$H^{2}(\mathcal{G},U(1)_{x})$. Therefore, we wish to know when

$\omega=\tfrac{1}{2}b^{*}\beta\cup b^{*}\beta$ (3.46)

is trivial as a $\mathcal{G}$ cocycle, for every valid $\beta$. To study this,
we employ the bosonic partition functions (2.17) associated with this cocycle.
Partition functions are gauge invariant quantities, which makes them useful
for studying the cohomology class of a cocycle. Moreover, partition functions
of unitary, invertible theories like ours completely determine232323The
complete list of partition functions is redundant. See e.g. relations between
tori, Klein bottles, and ${\mathbb{R}}P^{2}$ in Ref. [26]. their topological
invariants in the sense that two invariants $[\omega]$ and $[\omega^{\prime}]$
are equal if and only if their partition functions $\mathcal{Z}_{\omega}^{b}$
and $\mathcal{Z}_{\omega^{\prime}}^{b}$ agree on every space $(X,\mathcal{A})$
[27, 28]. Therefore, asking when an invariant is trivial amounts to asking
when its partition functions are all trivial.

The bosonic partition function associated with a cocycle $\omega$ is stated
abstractly in Eq. (2.17). Now we evaluate this expression explicitly for each
input $(X,\mathcal{A})$. The surface $X$ is either orientable – in which case
it is the connected sum of $n$ tori – or nonorientable – in which case it is
the connected sum of $n$ real projective planes; in either case, $n$ is
referred to as the “genus.” The genus $n$ orientable surface has fundamental
group

$\pi_{1}((T^{2})^{\\#n})=\\{\,a_{i},b_{i}~{},\,i=1,\ldots,n\,:\,\prod_{i}^{n}a_{i}b_{i}a_{i}^{-1}b_{i}^{-1}=1\,\\}$
(3.47)

generated by _orientation-preserving_ loops $a_{i},b_{i}$, whereas the genus
$n$ nonorientable surface has

$\pi_{1}(({\mathbb{R}}P^{2})^{\\#n})=\\{\,c_{i}~{},\,i=1,\ldots,n\,:\,\prod_{i}^{n}c_{i}^{2}=1\,\\}$
(3.48)

generated by _orientation-reversing_ loops $c_{i}$ [29]. The gauge field
$\mathcal{A}$ is determined by the map $\pi_{1}(X)\rightarrow\mathcal{G}$
induced on fundamental groups; that is, by specifying group elements
(“holonomies”) $\\{g_{i},h_{i}\\}$ or $\\{g_{i}\\}$ that satisfy the relation
(3.47) or (3.48) and the constraint $\mathcal{A}^{*}x=w_{1}$, which means
$x(g_{i})=x(h_{i})=0$ or $x(g_{i})=1$.

Next we provide two independent arguments that the partition functions
associated with the class (3.46) automatically vanish on orientable surfaces

$\mathcal{Z}^{b}_{\tfrac{1}{2}b^{*}\beta\cup
b^{*}\beta}[(T^{2})^{\\#n},\\{g_{i},h_{i}\\}]=1$ (3.49)

and evaluate to

$\mathcal{Z}^{b}_{\tfrac{1}{2}b^{*}\beta\cup
b^{*}\beta}[({\mathbb{R}}P^{2})^{\\#n},\\{g_{i}\\}]=(-1)^{\sum_{i}b^{*}\beta(g_{i})}~{}.$
(3.50)

on the nonorientable surface with holonomies $\\{g_{i}\\}$. The condition that
the nonorientable genus $n$ partition function vanishes for all valid $\beta$
is precisely the genus $n$ rule and the vanishing for all $n$ is the genus
$\infty$ rule. Therefore the genus $\infty$ rule is a sufficient condition for
the existence of square roots $R_{\beta}$ to the valid half cup squares and
thus to the isomorphism of bosonic and fermionic SPT classifications.

Our first argument uses the special form of the class (3.46), which lets us
write

$\mathcal{Z}^{b}_{\tfrac{1}{2}b^{*}\beta\cup
b^{*}\beta}[X,\mathcal{A}]=\exp(2\pi i\int_{X}\tfrac{1}{2}A_{b}^{*}\beta\cup
A_{b}^{*}\beta)=(-1)^{\langle
PD(A_{b}^{*}\beta),PD(A_{b}^{*}\beta)\rangle}~{},$ (3.51)

where $PD(A_{b}^{*}\beta)$ denotes the cycle Poincaré dual to
$A_{b}^{*}\beta\in H^{1}(X,{\mathbb{Z}}_{2})$ and $\langle\cdot,\cdot\rangle$
is the intersection form on cycles. On an orientable surface, no cycle has
even self-intersection [29], so the partition function is trivial. On the
other hand, the orientation-reversing loops $c_{i}$ in the presentation (3.48)
form a basis of cycles of the nonorientable surface and have intersection
$\langle c_{i},c_{j}\rangle=\delta_{ij}$ [29]; therefore, the self-
intersection of any cycle is the number of basis elements $c_{i}$ along which
the cycle has a component. The Poincaré dual cycle $PD(A_{b}^{*}\beta)$ is, by
definition, the cycle that intersects precisely the cycles $c$ on which
$A_{b}^{*}\beta(c)=1$; since the basis $c_{i}$ is orthogonal with respect to
the intersection form, the Poincaré dual therefore has components precisely
where it has $A_{b}^{*}\beta(c_{i})=1$, i.e. in the basis cycles labeled by
holonomies $g_{i}\in\mathcal{G}$ such that $b^{*}\beta(g_{i})=1$. We conclude
that the self-intersection of $PD(A_{b}^{*}\beta)$ is a sum of
$b^{*}\beta(g_{i})$, which gives the claimed result (3.50).

Our second argument uses the general expression for the partition function
associated to $\omega$ evaluated on a space $(X,\mathcal{A})$. In appendix B,
we derive these expressions from the state sum construction and compare them
to partial results in the literature. We find that the orientable partition
function is

$\displaystyle\begin{split}\mathcal{Z}^{b}_{\omega}[(T^{2})^{\\#n},\\{g_{i},h_{i}\\}]&=\exp\left(2\pi
i\sum_{i=1}^{n}\omega(x_{i},g_{i})+\omega(x_{i}g_{i},h_{i})+\omega(x_{i}g_{i}h_{i},g_{i}^{-1})\right.\\\
&\left.\qquad\qquad\qquad\qquad\qquad\phantom{\sum_{i=1}^{n}}+\omega(x_{i}g_{i}h_{i}g_{i}^{-1},h_{i}^{-1})-\omega(g_{i}^{-1},g_{i})-\omega(h_{i}^{-1},h_{i})\right)~{},\end{split}$
(3.52)

for $\\{g_{i},h_{i}\\}$ satisfying $x(g_{i})=x(h_{i})=0$ and
$\prod_{i}g_{i}h_{i}g_{i}^{-1}h_{i}^{-1}=1$, and where
$x_{i}=\prod_{j<i}g_{j}h_{j}g_{j}^{-1}h_{j}^{-1}$. And for $\\{g_{i}\\}$
satisfying $x(g_{i})=1$ and $\prod_{i}g_{i}^{2}=1$, with
$y_{i}=\prod_{j<i}g_{j}^{2}$, the nonorientable partition function is

$\mathcal{Z}^{b}_{\omega}[({\mathbb{R}}P^{2})^{\\#n},\\{g_{i}\\}]=\exp(2\pi
i\sum_{i=1}^{n}\omega(g_{i},g_{i})+\omega(y_{i},g_{i}^{2}))~{}.$ (3.53)

Gauge invariance of these expressions can be verified by plugging in
$\omega=\delta\lambda$ to obtain

$\mathcal{Z}^{b}_{\delta\lambda}[(T^{2})^{\\#n},\\{g_{i},h_{i}\\}]=\exp(2\pi
i\sum_{i=1}^{n}\lambda(x_{i})+\lambda(x_{i+1}))=1~{},$ (3.54)
$\mathcal{Z}^{b}_{\delta\lambda}[({\mathbb{R}}P^{2})^{\\#n},\\{g_{i}\\}]=\exp(2\pi
i\sum_{i=1}^{n}\lambda(y_{i})+\lambda(y_{i+1}))=1~{},$ (3.55)

using the facts that $x_{1}=1$, $x_{n}g_{n}h_{n}g_{n}^{-1}h_{n}^{-1}=1$,
$y_{1}=1$, and $y_{n}g_{n}^{2}=1$. Plugging the class (3.46) into the general
evaluations (3.52) and (3.53) and noting that
$b^{*}\beta(x_{i})=b^{*}\beta(y_{i})=0$, we see that the orientable expression
becomes trivial while the nonorientable expression matches the claimed result
(3.50).

To see the partial converse, note that the partition function on
${\mathbb{R}}P^{2}$ for an arbitrary class $[\omega]$ is of order two since

$\mathcal{Z}^{b}_{\omega}[{\mathbb{R}}P^{2}_{g}]^{2}=\exp(2\pi
i\,2\omega(g,g))=\exp(2\pi i\,(\delta\omega)(g,g,g))=1~{},$ (3.56)

due to the fundamental group relation $g^{2}=1$. For this reason, a class with
nontrivial ${\mathbb{R}}P^{2}$ partition function cannot have a square root.
We conclude that a square root $R_{\beta}$ exists only if the partition
function (3.50) vanishes at $n=1$; that is, if the genus $1$ rule is
satisfied.

##  4 Dualities of SPT phases

###  4.1 Topological holography

Duality is a tool for relating two systems via a nontrivial transformation
such that there is a correspondence between the key physical observables on
either side. Such transformations allow for solving problems about one system
in terms of its dual. For example, the Jordan-Wigner transformation [1], which
maps nonlocal strings of bosonic spin operators to local fermion operators,
has proven useful for solving spin chains such as the Ising and XY models by
translating them into fermionic Hamiltonians that can be diagonalized [30,
31]. The Jordan-Wigner transformation has also been used to study fermionic
SPT phases by mapping them into bosonic SPT phases [3, 2]. Other examples are
Kennedy-Tasaki duality [32] and its generalizations [33, 34], which interpret
the nontriviality of SPT phases as “hidden” symmetry-breaking by mapping SPT
phases to spontaneously symmetry-broken phases. Dualities can also be used to
locate critical points as points where a duality between two phases becomes a
self-duality, as was done by Kramers and Wannier [35].

While the specific duality transformations mentioned above have appeared in
separate contexts, they can be understood as instances of a general notion of
duality [36, 37, 34]. Recent years have shown progress in understanding
duality from within a unified framework known as topological holography, where
symmetric gapped phases are realized as gapped boundary conditions of higher
dimensional symmetry topological field theories (TFT). This idea has proven
especially fruitful in one dimension, where systems with finite
symmetry242424The approach applies not only to the grouplike symmetries
considered here but also to their noninvertible generalizations. and their
dualities have been classified in the language of two dimensional symmetry TFT
[38, 36, 39, 40, 41, 42]. The symmetries of one dimensional systems form a
fusion category $\mathcal{A}$ (for the nonanomalous grouplike symmetries
considered in the present paper, $\mathcal{A}=\text{Vec}_{\mathcal{G}}$) [43,
44], and the symmetry TFT is the two dimensional theory characterized by the
Drinfeld center $Z(\mathcal{A}).$ One dimensional $\mathcal{A}$-symmetric
phases (including both symmetry-protected and symmetry-breaking phases) appear
as gapped boundary conditions for $Z(\mathcal{A})$, which are mathematically
described by Lagrangian algebra objects of $Z(\mathcal{A})$, consisting of the
anyons that condense on the boundary [45, 46]. Dualities of one dimensional
systems (including operations like gauging a subgroup and stacking with SPT
phases) appear in this framework as autoequivalences of the modular tensor
category $Z(\mathcal{A})$ – that is, as permutations of the anyons that
preserve their braiding and fusion rules – which act on the Lagrangian
algebras representing the phases [47, 48]. These autoequivalences may be
physically interpreted as invertible surface operators (also called gapped
domain walls or gapped interfaces) of the two dimensional symmetry TFT [49,
38].

It is expected that the dualities formulated in the symmetry TFT language are
related to duality transformations of lattice Hamiltonians. For example, the
electric-magnetic duality of the toric code TFT, which exchanges the $e$ and
$m$ anyons, is known appear as gauging and Kramers-Wannier duality on the one
dimensional lattice. Dualities of systems with nonanomalous abelian symmetry
groups are extensively studied in Ref. [40], where dual pairs of lattice
Hamiltonians are constructed for a given autoequivalence. See Refs. [50, 51]
for a construction of dual pairs of lattice models for more general
symmetries. The Jordan-Wigner transformation has also been discussed as a
duality in the framework of symmetry TFT, though a general formalism for
boson-to-fermion dualities has yet to be fully developed [38, 50].

In section 3 it was found that the the stacking rules of bosonic and fermionic
SPT phases are often isomorphic; this happens when the symmetry is unitary as
well as when there are antiunitary symmetries as long as certain conditions
are satisfied (c.f. section 3.3). In these cases, an explicit isomorphism
${\mathcal{X}}$ (3.6) between the groups of bosonic and fermionic SPT phases
was constructed. In this section we ask whether the transformation
${\mathcal{X}}$ – like the Jordan-Wigner transformation – can be thought of as
a duality, in the sense of being implemented by an autoequivalence of the
symmetry TFT. If true, this would mean that boson-to-fermion dualities are
capable of preserving stacking rules, which are properties of entire
collections of SPT phases, similarly to how they preserve key features of
individual phases when mapping from one system to another. We show however
that ${\mathcal{X}}$ is _not_ a duality, except in the special case where the
Jordan-Wigner transformation is an isomorphism of stacking rules
($\delta\Lambda=\Omega_{\mathcal{F}}=0$). We will focus on the case where
$\mathcal{G}$ is finite abelian and unitary, but we expect the spirit of the
arguments to apply more broadly. In order to make use of the symmetry TFT
framework, which is typically formulated only for bosonic systems, we consider
the transformation

${\mathcal{J}}{\mathcal{W}}^{-1}{\mathcal{X}}:\mathcal{B}\rightarrow\mathcal{B}$
(4.1)

between bosonic SPT phases. Since ${\mathcal{J}}{\mathcal{W}}$ may be regarded
as a duality, asking whether ${\mathcal{X}}$ is a duality between bosonic and
fermionic systems is equivalent to asking whether
${\mathcal{J}}{\mathcal{W}}^{-1}{\mathcal{X}}$ is a duality. We emphasize
that, while ${\mathcal{X}}$ is by construction an isomorphism
$\mathcal{B}\rightarrow\mathcal{F}$, the composite transformation
${\mathcal{J}}{\mathcal{W}}^{-1}{\mathcal{X}}$ is typically not an
automorphism of the group of bosonic SPT phases since
${\mathcal{J}}{\mathcal{W}}$ is not typically an isomorphism.

###  4.2 Actions of dualities on SPT invariants

Anyons of abelian $Z({\rm Vec}_{\mathcal{G}})$ take the form $(a,\chi)$, where
$a\in\mathcal{G}$ represents a magnetic flux and $\chi\in{\rm
Rep}(\mathcal{G})\simeq\mathcal{G}$ an electric charge, with the topological
spin of $(a,\chi)$ given by the evaluation $\theta_{(a,\chi)}=\chi(a)$. Anyons
of the form $(1,\chi)$ are pure electric charges and form the subcategory
${\rm Rep}(\mathcal{G})\subset Z({\rm Vec}_{\mathcal{G}})$.

Gapped phases with $\mathcal{G}$ symmetry correspond to Lagrangian algebras of
$Z({\rm Vec}_{\mathcal{G}})$. These are classified by pairs $(H,\omega)$,
where $H$ is a subgroup of $\mathcal{G}$ representing the unbroken symmetry
and $\omega\in H^{2}(H,U(1))$ encodes the SPT order. The Lagrangian algebra
labeled by $(H,\omega)$ is the direct sum of anyons

$\mathcal{L}_{H,\omega}=\bigoplus_{\begin{subarray}{c}a\in
H~{},\,\,\chi\in\text{Rep}(\mathcal{G})\\\
\chi|_{H}=(i_{a}\omega)|_{H}\end{subarray}}(a,\chi)~{},$ (4.2)

where $|_{H}$ denotes restriction to $H$ and $i_{a}\omega$ denotes the
representation given by the slant product

$(i_{a}\omega)(g)=\exp(2\pi i(\omega(a,g)-\omega(g,a)))$ (4.3)

of $\omega$ with respect to $a\in H$. SPT phases correspond to the case
$H=\mathcal{G}$ and are classified by $\omega\in H^{2}(\mathcal{G},U(1))$. The
corresponding Lagrangian algebras

$\mathcal{L}_{\mathcal{G},\omega}=\bigoplus_{a\in\mathcal{G}}(a,i_{a}\omega)$
(4.4)

involve no pure electric charges $(1,\chi)$ since $i_{1}\omega=1$ on
$\mathcal{G}$; they are thus referred to as “magnetic” [52]. The Lagrangian
algebra of the trivial SPT phase involves only pure magnetic fluxes since
$i_{a}0=1$:

$\mathcal{L}_{\mathcal{G},0}=\bigoplus_{a\in\mathcal{G}}(a,1)~{}.$ (4.5)

A physical picture of the correspondence between SPT invariants and Lagrangian
algebras may be given in terms of order parameters [39, 53, 54].

Let us illustrate this story with a simple example:
$\mathcal{G}={\mathbb{Z}}_{2}^{x}\times{\mathbb{Z}}_{2}^{y}$. Let $x$ and $y$
denote the generators of the ${\mathbb{Z}}_{2}$ factors. In this basis, the
anyons generating $Z({\rm Vec}_{G})$ are

$m_{x}=((1,0),(0,0))~{},\qquad m_{y}=((0,1),(0,0))~{},\qquad
e_{x}=((0,0),(1,0))~{},\qquad e_{y}=((0,0),(0,1))~{}.$ (4.6)

The trivial SPT phase corresponds to $\omega=0$ and involves only pure
magnetic charges:

$\mathcal{L}_{\mathcal{G},0}=1\oplus m_{x}\oplus m_{y}\oplus m_{x}m_{y}.$
(4.7)

The nontrivial SPT phase corresponds to $\omega=\omega_{xy}$, the nontrivial
class in $H^{2}({\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2},U(1)).$ This class has

$i_{x}\omega_{xy}=e_{y}~{},\qquad i_{y}\omega_{xy}=e_{x}~{}.$ (4.8)

Therefore the corresponding Lagrangian algebra is

$\mathcal{L}_{\mathcal{G},\omega_{xy}}=1\oplus m_{x}e_{y}\oplus
m_{y}e_{x}\oplus m_{x}m_{y}e_{x}e_{y},$ (4.9)

reproducing the well-known expression for the
${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ SPT-phase, for example in Ref. [40].

The correspondence between SPT invariants $\omega$ and Lagrangian algebras
allows us to compute the action of autoequivalence $\mathcal{D}\in{\rm
Aut}(Z({\rm Vec}_{\mathcal{G}}))$ on SPT phases. First, observe that, since
the Lagrangian algebras of SPT phases involve no pure electric charges, any
autoequivalence $\mathcal{D}$ that maps SPT phases to SPT phases (i.e. does
not induce any symmetry-breaking) must map pure electric charges to pure
electric charges; this is, it must stabilize ${\rm
Rep}(\mathcal{G})$.252525However, if one only wishes to map _some_ SPT phase
to _some other_ SPT phase while not necessarily stabilizing the set of SPT
phases, more general autoequivalence are allowed; these will induce symmetry
breaking in _other_ SPT phases. The autoequivalences stabilizing ${\rm
Rep}(\mathcal{G})$ and their action on Lagrangian algebras of $Z({\rm
Vec}_{\mathcal{G}})$ have been studied in detail by Ref. [55].262626Ref. [55]
considers the action on Lagrangian subcategories, but Lagrangian subcategories
and and Lagrangian algebras coincide on the level of anyons when $\mathcal{G}$
is abelian – Lagrangian subcategories are fusion categories generated by some
subset of anyons of $Z({\rm Vec}_{\mathcal{G}})$, while Lagrangian algebras
correspond to direct sums of the same anyons [56]). They take the form (c.f.
Ref. [55], Cor. 6.9)

$(\zeta,a)\in H^{2}(\mathcal{G},U(1))\rtimes{\rm Out}(\mathcal{G})~{}.$ (4.10)

and act as (c.f. Ref. [55], Prop 7.11)

$(\zeta,a)\cdot
L_{\mathcal{G},\omega}=L_{\mathcal{G},a^{*}(\omega+\zeta)}~{},$ (4.11)

which means they act on the SPT phase $\omega$ by stacking with the SPT phase
$\zeta$, then acting by the map $a^{*}$ induced by the automorphism of the
group.272727The fact that the automorphism $a$ is outer is reflected in Eq.
(C.2), which says that inner automorphisms of the group act trivially on the
SPT invariants.

Now we are equipped to address the question of whether the map

${\mathcal{J}}{\mathcal{W}}^{-1}{\mathcal{X}}:[\alpha,\beta]\mapsto[\alpha-\Lambda(\beta),\beta]~{},$
(4.12)

coming from Eq. (3.6), can be implemented by an autoequivalence of $Z({\rm
Vec}_{\mathcal{G}})$. The answer is no, except in the special case where
${\mathcal{J}}{\mathcal{W}}$ is an isomorphism. Note that
${\mathcal{J}}{\mathcal{W}}^{-1}{\mathcal{X}}$ maps the trivial phase to the
trivial phase, so the stacked SPT $\zeta$ must be trivial. Then note that the
induced action $a^{*}$ on the SPT classification is an automorphism,
$a^{*}(\omega+\omega^{\prime})=a^{*}\omega+a^{*}\omega^{\prime}$, so
${\mathcal{J}}{\mathcal{W}}^{-1}{\mathcal{X}}$ is only of this form when it is
an automorphism $\mathcal{B}\rightarrow\mathcal{B}$. Since ${\mathcal{X}}$ is
an isomorphism, ${\mathcal{J}}{\mathcal{W}}$ must be an isomorphism as well,
which it typically is not. This means there is no duality transformation
implementing ${\mathcal{X}}$ in general.

###  4.3 Stacking of magnetic Lagrangian algebras

In section 4.2 we studied the interplay of stacking and duality by expressing
both in terms of the SPT invariant $\omega$. In this picture, stacking is
obvious, while the action of autoequivalences (4.11) is not. Here we consider
an alternative approach, where the action of autoequivalences is immediate but
stacking needs to be carefully defined. The idea is to express everything in
terms of anyons and Lagrangian algebras. In this section we define the
operation of stacking Lagrangian algebras and see how it plays out in the case
of $\mathcal{G}={\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$.

Given two Lagrangian algebras $\mathcal{L}_{1},\mathcal{L}_{2}$ of $Z({\rm
Vec}_{\mathcal{G}})$, thought of as gapped boundaries of the bulk symmetry
TFT, we can stack the whole bulk-boundary systems to obtain a Lagrangian
algebra $\mathcal{L}_{1}\boxtimes\mathcal{L}_{2}$ of the bulk $Z({\rm
Vec}_{\mathcal{G}})\boxtimes Z({\rm Vec}_{\mathcal{G}})$, where $\boxtimes$
denotes the Deligne product. The bulk is now a symmetry TFT for
$\mathcal{G}\times\mathcal{G}$ symmetry, which we must break into the diagonal
subgroup $\mathcal{G}$ by identifying the ${\rm Rep}(\mathcal{G})$
subcategories of the two factors. This is achieved by condensing the algebra
object

$\mathcal{E}=\bigoplus_{\alpha\in{\rm Rep}(\mathcal{G})}\alpha\boxtimes\alpha$
(4.13)

in $Z({\rm Vec}_{\mathcal{G}})\boxtimes Z({\rm Vec}_{\mathcal{G}})$ [57]. The
result of this condensation, denoted $\left(Z({\rm
Vec}_{\mathcal{G}})\boxtimes Z({\rm Vec}_{\mathcal{G}})\right)_{\mathcal{E}}$,
is equivalent to and identified with $Z({\rm Vec}_{\mathcal{G}})$ [58, 57].
The stacked Lagrangian algebra282828This procedure defines the stacking of two
general Lagrangian algebras, not necessarily magnetic. Non-magnetic Lagrangian
algebras correspond to (partially) symmetry-breaking phases, and the stacking
of these phases is indeed well-defined; they simply will not form a group
under stacking, but a commutative monoid with some noninvertible elements. is
defined to be

$\mathcal{L}_{1}\cdot\mathcal{L}_{2}:=\left(\mathcal{L}_{1}\boxtimes\mathcal{L}_{2}\right)_{\mathcal{E}}~{}.$
(4.14)

Let us illustrate this procedure concretely by returning to the example of
$\mathcal{G}={\mathbb{Z}}_{2}^{x}\times{\mathbb{Z}}_{2}^{y}$. The $16$ anyons
in $Z({\rm Vec}_{\mathcal{G}})$ are generated by $e_{x},e_{y},m_{x},m_{y}$
(4.6), and the $16^{2}$ anyons of $Z({\rm Vec}_{\mathcal{G}})\boxtimes Z({\rm
Vec}_{\mathcal{G}})$ are generated by the Deligne products of
$e_{x},e_{y},m_{x},m_{y}$. The condensable algebra (4.13) is given by

$\mathcal{E}=(1\boxtimes 1)\oplus(e_{x}\boxtimes e_{x})\oplus(e_{y}\boxtimes
e_{y})\oplus(e_{x}e_{y}\boxtimes e_{x}e_{y})~{}.$ (4.15)

The identification $\left(Z({\rm Vec}_{\mathcal{G}})\boxtimes Z({\rm
Vec}_{\mathcal{G}})\right)_{\mathcal{E}}\simeq Z({\rm Vec}_{\mathcal{G}})$ can
be made as follows. After condensing $\mathcal{E}$, we have relations
$e_{i}\boxtimes 1\sim 1\boxtimes e_{i}\sim e_{i}e_{j}\boxtimes e_{j}$, and so
forth. Let $[e_{i}]$ be the class to which $e_{i}\boxtimes 1$ belongs. This
way, we obtain four classes with trivial magnetic flux,
$[1],[e_{x}],[e_{y}],[e_{x}e_{y}]$. As for the anyons involving nontrivial
magnetic fluxes, $m_{i}\boxtimes m_{j}$ has nontrivial braiding with at least
one of the condensed anyons if $i\neq j$, and are confined. Hence deconfined
magnetic anyons take the diagonal form $m_{i}\boxtimes m_{i}$. These are then
identified via condensation as $m_{i}\boxtimes m_{i}\sim m_{i}e_{x}\boxtimes
m_{i}e_{x}$, etc. Let $[m_{i}]$ denote the equivalence class to which
$m_{i}\boxtimes m_{i}$ belongs. Equivalence classes of deconfined anyons of
$Z({\rm Vec}_{\mathcal{G}})\boxtimes Z({\rm Vec}_{\mathcal{G}})$ are generated
by the equivalence classes $[e_{i}]$ and $[m_{i}]$. We then identify $[e_{i}]$
and $[m_{i}]$ with the anyons $e_{i}$ and $m_{i}$, respectively, of $Z({\rm
Vec}_{\mathcal{G}})$.

Now consider the stacking of SPT phases in this example. The Lagrangian
algebras corresponding to the trivial and nontrivial
$\mathcal{G}={\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ SPT phases were stated in
Eqs. (4.7) and (4.9). We wish to compute the result of stacking the algebra
$\mathcal{L}_{\mathcal{G},\omega_{xy}}$ with itself. Consider the $16$ anyons
of
$\mathcal{L}_{\mathcal{G},\omega_{xy}}\boxtimes\mathcal{L}_{\mathcal{G},\omega_{xy}}$.
The anyon $m_{x}e_{y}\boxtimes m_{x}e_{y}$ belongs to the class $[m_{x}]$, and
thus maps to the anyon $m_{x}$ of $Z(\text{Vec}_{\mathcal{G}})$ after
condensation. Similarly, $m_{y}e_{x}\boxtimes m_{y}e_{x}$ maps to $m_{y}$.
Anyons such as $m_{x}e_{y}\boxtimes m_{y}e_{x}$ are confined. Thus we see that

$\mathcal{L}_{\mathcal{G},\omega_{xy}}\cdot\mathcal{L}_{\mathcal{G},\omega_{xy}}=\left(\mathcal{L}_{\mathcal{G},\omega_{xy}}\boxtimes\mathcal{L}_{\mathcal{G},\omega_{xy}}\right)_{\mathcal{E}}=1\oplus
m_{x}\oplus m_{y}\oplus m_{x}m_{y}=\mathcal{L}_{\mathcal{G},0}~{},$ (4.16)

the Lagrangian algebra corresponding to the trivial phase. Thus, we see that
the stacking rule of ${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ SPT phases is
recovered from the Lagrangian algebra perspective.

Now that we have made sense of stacking in the language of anyons, it is
simple to study dualities. For example, consider the autoequivalence

$\mathcal{D}:\quad m_{x}\longleftrightarrow m_{y}~{},\quad
e_{x}\longleftrightarrow e_{y}~{}.$ (4.17)

This autoequivalence is induced from an automorphism of $\mathcal{G}$ (as pure
magnetic fluxes correspond to group elements) and leaves each of
$\mathcal{L}_{\mathcal{G},0}$ and $\mathcal{L}_{\mathcal{G},\omega_{xy}}$
invariant, as can be seen from the expressions (4.7) and (4.9). Or consider
the autoequivalence

$\mathcal{D}:\quad m_{i}\longleftrightarrow m_{i}e_{j}\,,\quad j\neq i~{},$
(4.18)

which exchanges $\mathcal{L}_{\mathcal{G},0}$ and
$\mathcal{L}_{\mathcal{G},\omega_{xy}}$ and thus corresponds to stacking with
the nontrivial SPT phase. To study how an autoequivalence $\mathcal{D}\in{\rm
Aut}(Z({\rm Vec}_{\mathcal{G}}))$ interacts with the stacking of Lagrangian
algebras, one should compare two objects: first, the result of stacking then
dualizing

$\mathcal{D}(\mathcal{L}_{1}\cdot\mathcal{L}_{2})=\mathcal{D}\left((\mathcal{L}_{1}\boxtimes\mathcal{L}_{2})_{\mathcal{E}}\right)~{},$
(4.19)

and second, the result of dualizing then stacking

$\mathcal{D}(\mathcal{L}_{1})\cdot\mathcal{D}(\mathcal{L}_{2})=\left(\mathcal{D}(\mathcal{L}_{1})\boxtimes\mathcal{D}(\mathcal{L}_{2})\right)_{\mathcal{D}(\mathcal{E})}~{}.$
(4.20)

Note that the condensable algebra $\mathcal{E}$ becomes the dualized object
$\mathcal{D}(\mathcal{E})$ in this second procedure.

###  4.4 An example, revisited

Finally we return to the example of SPT phases with symmetry group
$\mathcal{G}={\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}^{f}$,
first examined in section 3.2. Recall that here the stacking rules of bosonic
and fermionic SPT phases are isomorphic despite the Jordan-Wigner
transformation not being an isomorphism. It was natural before to wonder
whether the isomorphism ${\mathcal{X}}$ could, like
${\mathcal{J}}{\mathcal{W}}$, be implemented by a duality transformation. We
have since seen in this section that the answer is no. Here we recover this
result by working explicitly with the action of dualities on the $64$ anyons
and on the $8$ magnetic Lagrangian algebras corresponding to the $8$ SPT
phases.

As discussed in section 3.2, the $8$ SPT phases are given by a choice of
$\alpha$ between $\alpha_{0}$ and $\alpha_{1}$ and a choice of $\beta$ among
$\beta_{00}$, $\beta_{10}$, $\beta_{01}$, and $\beta_{11}$. The map
${\mathcal{J}}{\mathcal{W}}^{-1}{\mathcal{X}}$ exchanges the two phases

${\mathcal{J}}{\mathcal{W}}^{-1}{\mathcal{X}}:[\alpha_{0},\beta_{11}]\longleftrightarrow[\alpha_{1},\beta_{11}]~{},$
(4.21)

while leaving the other six phases the same. We will see that no duality can
implement this transformation.

The relevant symmetry TFT is the rank $64$ abelian MTC
$Z(\rm{Vec}_{{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}})={\rm
TC}\boxtimes{\rm TC}\boxtimes{\rm TC}$, where ${\rm TC}$ denotes the Toric
Code MTC. This MTC has ${\mathbb{Z}}_{2}^{6}$ fusion rules generated by the
anyons $e_{x},e_{y},e_{p},m_{x},m_{y},m_{p}$ and has $8$ magnetic Lagrangian
algebras. The correspondence between these algebras and the $8$ SPT phases is
computed following the procedure discussed in section 4.2. The results appear
in table 1, where each Lagrangian algebra is denoted by its set of generating
anyons. These algebras also appear in Ref. [40] (c.f. Table 6).

Suppose the transformation ${\mathcal{J}}{\mathcal{W}}^{-1}{\mathcal{X}}$
(4.21) were implemented by a duality. The invariance of the trivial phase
$(\alpha_{0},\beta_{00})$ under this mapping means that the pure magnetic
fluxes must map to other pure magnetic fluxes. Then, by looking at the table,
we see that the invariance of the phase $(\alpha_{0},\beta_{10})$ means the
anyon $m_{y}$ must be invariant since it is the only pure magnetic flux for
this phase. Likewise, the invariance of the phase $(\alpha_{0},\beta_{01})$
implies the invariance of $m_{x}$. Therefore the anyon $m_{x}m_{y}$ is
invariant. However the phase $(\alpha_{0},\beta_{11})$ has the anyon
$m_{x}m_{y}=(m_{x}e_{p})(m_{y}e_{p})$ while the phase
$(\alpha_{1},\beta_{11})$ does not, so they cannot be mapped into each other
by the transformation, a contradiction. We conclude that the transformation
${\mathcal{J}}{\mathcal{W}}^{-1}{\mathcal{X}}$ – and thus also the stacking
rule isomorphism ${\mathcal{X}}:\mathcal{B}\rightarrow\mathcal{F}$ – is not
implemented by a duality.

α , β ω generating set of anyons α_0 , β_00 0 m_x m_y m_p α_0 , β_10 ω_xp m_x
e_p m_y m_p e_x α_0 , β_01 ω_yp m_x m_y e_p m_p e_y α_0 , β_11 ω_xp + ω_yp m_x
e_p m_y e_p m_p e_x e_y α_1 , β_00 ω_xy m_x e_y m_y e_x m_p α_1 , β_10 ω_xy +
ω_xp m_x e_y e_p m_y e_x m_p e_x α_1 , β_01 ω_xy +ω_yp m_x e_y m_y e_x e_p m_p
e_y α_1 , β_11 ω_xy + ω_xp + ω_yp m_x e_y e_p m_y e_x e_p m_p e_x e_y

Table 1: The $8$ SPT phases with symmetry
$\mathcal{G}={\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}^{f}$.

##  5 Discussion

We have investigated the conditions under which one dimensional bosonic and
fermionic SPT phases have isomorphic stacking rules, finding that this occurs
when the symmetry group is purely unitary and extending this result to
antiunitary symmetries, where it happens in certain circumstances. By thinking
of the bosonic and fermionic SPT classifications as group extensions that are
sometimes equivalent, we have found several necessary and sufficient
conditions for an isomorphism to hold; these results are gathered in section
3.3. A natural question is whether an isomorphism between the groups of
bosonic and fermionic phases SPT phases might exist in higher dimensions.
There, the situation is complicated by the fact that gauging the fermion
parity symmetry of a fermionic system gives rise to an anomalous higher form
symmetry ${\mathbb{Z}}_{2}$ symmetry of the resulting bosonic theory [19, 59],
rather than to the nonanomalous ordinary ${\mathbb{Z}}_{2}$ symmetry of one
dimension, so we do not expect the invertibility of phases to be preserved by
this operation. Indeed, it is known that in two dimensions the standard
bosonization procedure maps an invertible fermionic phase to a bosonic
topologically-ordered phase, and this fact has been exploited to classify
invertible fermionic phases with symmetry in terms of bosonic symmetry-
enriched topological orders [59, 60]. Indeed, the results of computing the
higher dimensional classifications of bosonic and fermionic SPT protected by
ordinary symmetries do not bear any obvious relation [8, 9], suggesting that a
correspondence – if it exists at all – is fairly subtle. The existence (under
certain conditions) of a stacking-compatible correspondence between bosonic
and fermionic SPT phases may be a special property of bosonization in one
dimension.

Where the isomorphism of SPT phases exists, we have written an explicit form
of its action on supercohomology invariants, which we contrasted with that of
the Jordan-Wigner transformation. We have then asked whether this new
transformation could be thought of as a generalized “stacking-compatible”
Jordan-Wigner transformation. In short, the answer is no, the reason being
that the isomorphism acts on the set of SPT phases in a way that a true
duality cannot. This result highlights that, even when the bosonic and
fermionic classifications are isomorphic, there are fundamental differences
between bosonic and fermionic systems than cannot be compensated for by
duality. While we treated duality in the abstract language of symmetry TFT, we
expect this answer means there is no transformation from spins to fermions on
the lattice (like the Jordan-Wigner transformation) that is compatible with
the bosonic and fermionic SPT stacking rules. It would be interesting to test
this expectation by working directly with lattice Hamiltonian models of SPT
phases. It would also be interesting to generalize this study to bosonic and
fermionic phases protected by generalized symmetries and their stacking
structures. We note that, while dualities between two given one dimensional
systems had been extensively studied from the symmetry TFT perspective in
prior work, our investigation has been concerned with the action of dualities
on the entire collection of phases and the relation of dualities to the
stacking structure on this collection. There remains the question of how to
physically interpret this nonduality – the isomorphism of stacking rules that
is not a consequence of duality, and it is curious that antiunitary symmetry
is capable of spoiling the isomorphism while unitary symmetry is not.

Along the way, we have generated several smaller results which readers may
find interesting independently of the main results. We have written explicit
$\mathcal{G}$-pin partition functions for fermionic SPT phases, in terms of
which we have rederived the stacking rule of invertible fermionic phases and
defined a nonsplit version of the Jordan-Wigner transformation; we have given
a complete gauge-invariant characterization of degree two twisted cohomology
classes in terms of bosonic partition functions on genus $n$ orientable and
nonorientable surfaces; we have studied when twisted cohomology classes are of
order two and have found examples where they are not. Many of these results
are contained in the following appendices.

##  Appendix A Invertible fermionic phases

The SPT phases discussed so far are examples of _invertible phases_ , i.e.
phases with an inverse under stacking. When the symmetry group
splits292929When the group does not split, no additional phases beyond SPT
phases are possible [2]. as $\mathcal{G}=G_{b}\times{\mathbb{Z}}_{2}^{f}$,
there are invertible phases beyond SPT phases. Rather than becoming trivial
upon forgetting the symmetry (as SPT phases do), these additional invertible
phases reduce to the nontrivial invertible topological order. This topological
order is realized on the lattice as a phase of Majorana chains [61, 2] and has
the Arf invariant as its topological partition function [23]; the full Arf
topological field theory has also been described [62, 63, 64]. In total,
invertible fermionic phases are classified by three invariants:303030When
$\gamma=1$, the invariant $\nu$ is defined differently from Eq. (2.6) and
satisfies $\nu(p)=1$ instead of $\nu(p)=0$ [2]; therefore, the pullback
$\beta=s^{*}\nu$ (2.9) satisfies $\delta\beta=\gamma\rho$. But upon choosing a
splitting, the usual $\delta\beta=0$ is recovered. the usual $\alpha$ and
$\beta$, and a new invariant

$\gamma\in H^{0}(G_{b},{\mathbb{Z}}_{2})={\mathbb{Z}}_{2}$ (A.1)

that encodes whether the phase is SPT ($\gamma=0$) or not ($\gamma=1$). The
most general invertible fermionic theory with split symmetry has partition
function

$\mathcal{Z}_{\alpha,\beta,\gamma}^{f}[X,A_{b},\eta]=\exp(2\pi
i\int_{X}A_{b}^{*}\alpha)\exp(\frac{\pi
i}{2}q_{\eta}(A_{b}^{*}\beta))\text{Arf}(\eta)^{\gamma}~{},$ (A.2)

which comes from stacking the theory (2.32) with the invertible topological
order $\text{Arf}(\eta)^{\gamma}$.

As before, the stacking rule is obtained by multiplying partition functions:

$\displaystyle\begin{split}\mathcal{Z}_{\alpha_{1},\beta_{1},\gamma_{1}}^{f}[X,A_{b},\eta]&\mathcal{Z}_{\alpha_{2},\beta_{2},\gamma_{3}}^{f}[X,A_{b},\eta]\\\
&=\exp(2\pi i\int_{X}A_{b}^{*}(\alpha_{1}+\alpha_{2}))\exp(\frac{\pi
i}{2}(q_{\eta}(A_{b}^{*}\beta_{1})+q_{\eta}(A_{b}^{*}\beta_{2})))\text{Arf}(\eta)^{\gamma_{1}+\gamma_{2}}\\\
&=\exp(2\pi
i\int_{X}A_{b}^{*}(\alpha_{1}+\alpha_{2}+\tfrac{1}{2}\beta_{1}\cup\beta_{2}))\\\
&\qquad\qquad\qquad\qquad\times\,\exp(\frac{\pi
i}{2}(q_{\eta}(A_{b}^{*}(\beta_{1}+\beta_{2}))+\gamma_{1}\gamma_{2}q_{\eta}(A_{b}^{*}x)))\text{Arf}(\eta)^{\gamma_{1}+\gamma_{2}\text{
mod }2}\\\ &=\exp(2\pi
i\int_{X}A_{b}^{*}(\alpha_{1}+\alpha_{2}+\tfrac{1}{2}\beta_{1}\cup\beta_{2}+\tfrac{1}{2}\gamma_{1}\gamma_{2}(\beta_{1}+\beta_{2})\cup
x))\\\ &\qquad\qquad\qquad\qquad\times\,\exp(\frac{\pi
i}{2}q_{\eta}(A_{b}^{*}(\beta_{1}+\beta_{2}+x)))\text{Arf}(\eta)^{\gamma_{1}+\gamma_{2}\text{
mod }2}\\\
&=\mathcal{Z}_{\alpha_{1}+\alpha_{2}+\tfrac{1}{2}\beta_{1}\cup\beta_{2}+\tfrac{1}{2}\gamma_{1}\gamma_{2}(\beta_{1}+\beta_{2})\cup
x,\beta_{1}+\beta_{2}+x,\beta_{1}+\beta_{2}}^{f}[X,A_{b},\eta]~{},\end{split}$
(A.3)

where in the second equality we used the relation (D.4) and the constraint
$A_{b}^{*}x=w_{1}$ to absorb $\text{Arf}(\eta)^{2}$ into the middle term. From
this calculation, the full stacking rule may be read off as

$\left[\begin{array}[]{c}\alpha_{1}\\\ \beta_{1}\\\
\gamma_{1}\end{array}\right]\otimes_{\mathcal{F}}\left[\begin{array}[]{c}\alpha_{2}\\\
\beta_{2}\\\
\gamma_{2}\end{array}\right]=\left[\begin{array}[]{c}\alpha_{1}+\alpha_{2}+\tfrac{1}{2}\beta_{1}\beta_{2}+\tfrac{1}{2}\gamma_{1}\gamma_{2}(\beta_{1}+\beta_{2})x\\\
\beta_{1}+\beta_{2}+\gamma_{1}\gamma_{2}x\\\
\gamma_{1}+\gamma_{2}\end{array}\right]~{}.$ (A.4)

If restricted to SPT phases ($\gamma=0$), this expression matches the SPT
stacking rule (2.43). Also note that if the symmetry is unitary ($x=0$), the
stacking rule splits as $\text{SPTs}\times{\mathbb{Z}}_{2}$.

The stacking rule was first stated for split, unitary symmetry groups in Ref.
[19] and derived for arbitrary unitary symmetries in Refs. [13, 12], where it
takes the form (A.4) with $x=0$. The stacking rule for invertible fermionic
phases with general symmetry first appeared in Ref. [13], where it takes the
form313131In the stacking rule for $\gamma_{1}=\gamma_{2}=1$ (Eq. 132) of Ref.
[13], the factor of
$\tfrac{1}{2}\tilde{\beta}_{1}(g)(\tilde{\beta}_{2}(h)+x(h))+\tfrac{1}{4}(\tilde{\beta}_{2}(g)+x(g)+\tilde{\beta}_{2}(h)+x(h)-\tilde{\beta}_{2}(gh)-x(gh))$
can be rewritten as
$\tfrac{1}{2}\tilde{\beta}_{1}\cup(\tilde{\beta}_{2}+x)+\tfrac{1}{2}(\tilde{\beta}_{2}+x)^{2}$,
which is equal to
$\tfrac{1}{2}\tilde{\beta}_{1}\tilde{\beta}_{2}+(\tilde{\beta}_{1}+\tilde{\beta}_{2}+x)x$.
(in our notation)

$\left[\begin{array}[]{c}\tilde{\alpha}_{1}\\\ \tilde{\beta}_{1}\\\
\gamma_{1}\end{array}\right]\otimes_{\mathcal{F}}\left[\begin{array}[]{c}\tilde{\alpha}_{2}\\\
\tilde{\beta}_{2}\\\
\gamma_{2}\end{array}\right]=\left[\begin{array}[]{c}\tilde{\alpha}_{1}+\tilde{\alpha}_{2}+\tfrac{1}{2}\tilde{\beta}_{1}\tilde{\beta}_{2}+\tfrac{1}{2}\gamma_{1}\gamma_{2}(\tilde{\beta}_{1}+\tilde{\beta}_{2}+x)x\\\
\tilde{\beta}_{1}+\tilde{\beta}_{2}+\gamma_{1}\gamma_{2}x\\\
\gamma_{1}+\gamma_{2}\end{array}\right]~{}.$ (A.5)

The extra factor of $\tfrac{1}{2}\gamma_{1}\gamma_{2}x^{2}$ relative to the
rule (A.4) can be absorbed by the following redefinition:

$\tilde{\alpha}=\alpha+\tfrac{1}{2}\gamma\beta x$ (A.6)
$\tilde{\beta}=\beta+\gamma x~{},$ (A.7)

which shifts the $\alpha$ stacking rule by
$\tfrac{1}{2}\gamma_{1}\gamma_{2}x^{2}$, precisely the difference between the
expressions. Also note that, under the redefinition (A.7) of $\beta$ alone,
the $\alpha$ rule shifts by
$\tfrac{1}{2}\gamma_{1}\beta_{2}x+\tfrac{1}{2}\gamma_{2}\beta_{1}x+\tfrac{1}{2}\gamma_{1}\gamma_{2}x^{2}$,
resulting in a rule

$\left[\begin{array}[]{c}\tilde{\alpha}_{1}\\\ \beta_{1}\\\
\gamma_{1}\end{array}\right]\otimes_{\mathcal{F}}\left[\begin{array}[]{c}\tilde{\alpha}_{2}\\\
\beta_{2}\\\
\gamma_{2}\end{array}\right]=\left[\begin{array}[]{c}\tilde{\alpha}_{1}+\tilde{\alpha}_{2}+\tfrac{1}{2}\beta_{1}\beta_{2}+\tfrac{1}{2}(1-\gamma_{1})\gamma_{2}\beta_{1}x+\tfrac{1}{2}(1-\gamma_{2})\gamma_{1}\beta_{2}x\\\
\beta_{1}+\beta_{2}+\gamma_{1}\gamma_{2}x\\\
\gamma_{1}+\gamma_{2}\end{array}\right]~{}.$ (A.8)

This form of the rule was first derived in Ref. [10] and reproduced in Refs.
[20, 21]. It is in these variables that the ${\mathbb{Z}}_{8}$ stacking rule
for time-reversal symmetric Majorana chains was first expressed [2]. A further
redefinition (A.6) in $\alpha$ shifts the $\alpha$ rule by
$\tfrac{1}{2}\gamma_{1}\beta_{1}x+\tfrac{1}{2}\gamma_{2}\beta_{2}x-\tfrac{1}{2}(\gamma_{1}+\gamma_{2})(\beta_{1}+\beta_{2}+\gamma_{1}\gamma_{2}x)x=\tfrac{1}{2}\gamma_{1}\beta_{2}x+\tfrac{1}{2}\gamma_{2}\beta_{1}x$,
bringing it into the form (A.4) derived by stacking partition functions.

This differences between the formulations (A.4), (A.5), and (A.8) of the
stacking rule have the following interpretations. In a system with $\gamma=1$,
the symmetry $G_{b}\times{\mathbb{Z}}_{2}$ acts on an algebra of local
operators ${\rm End}(U_{b})\otimes C\ell(1)$ in a manner specified by the
invariants $[\alpha,\beta]$. The invariant $\beta$ measures whether a symmetry
commutes or anticommutes with the generator $\Gamma$ of ${\mathbb{C}}\ell(1)$
squaring to $+1$ [2, 10], whereas $\tilde{\beta}$ measures commutation with
$i\Gamma$ [13]; for this reason, $\beta$ and $\tilde{\beta}$ differ by $x$.
The difference between $\alpha$ and $\tilde{\alpha}$ has to do with a choice
in boundary condition. Following Ref. [65] (see arXiv v2), a boundary
condition amounts to a choice $\delta\in H^{1}(G_{b},{\mathbb{Z}}_{2})$ of
compatible symmetry action on the module $U_{b}\otimes{\mathbb{C}}^{1|1}$ over
the algebra:

$g_{b}\mapsto Q(g_{b})\otimes X^{\delta(g_{b})}Z^{\beta(g_{b})}~{},$ (A.9)

where $Q$ is a projective representation of $G_{b}$ of class
$[\tilde{\alpha}]$. In total, the projectivity class of this action is

$\tilde{\alpha}+\tfrac{1}{2}\beta\cup\delta~{}.$ (A.10)

The choice $\delta=0$ is often made implicitly and realizes $\tilde{\alpha}$
as the projectivity in the boundary symmetry action. On the other hand, the
choice $\delta=x$ corresponds to $\alpha$. The formulas (A.4) and (A.8) are
the stacking rule written in variables where the $2$-cocycle is the one
measuring the boundary projectivity – either $\alpha$ or $\tilde{\alpha}$. For
example, in the case of symmetry
$\mathcal{G}={\mathbb{Z}}_{2}^{T}\times{\mathbb{Z}}_{2}^{f}$, the three
invariants $[\alpha],[\beta],[\gamma]\in{\mathbb{Z}}_{2}$ form the
classification group ${\mathbb{Z}}_{8}$ according to the binary expansion
[66], whereas $[\tilde{\alpha}],[\beta],[\gamma]$ form the ${\mathbb{Z}}_{8}$
a different way [2, 10]. This difference appears in whether, for phase
$3\in{\mathbb{Z}}_{8}$, the boundary action of time-reversal symmetry squares
to $+1$ (binary) or $-1$ (usual), which is a choice of boundary condition
independent of the topological invariants.

Finally, we remark that passing an odd ($\gamma=1$) fermionic phase through
the inverse transformation yields

$\displaystyle\begin{split}{\mathcal{J}}{\mathcal{W}}^{-1}&(\mathcal{Z}^{f}_{\alpha,\beta,1})[X,A_{b},A_{p}]\\\
&=\sum_{\eta}\exp(2\pi i\int_{X}A_{b}^{*}\alpha)\exp(\frac{\pi
i}{2}q_{\eta}(A_{b}^{*}\beta))\text{Arf}(\eta)\exp(\frac{\pi
i}{2}q_{\eta+\tau}(A_{p}))\frac{\text{Arf}(\eta)^{-1}}{\sqrt{|H^{1}(X,{\mathbb{Z}}_{2})|}}\\\
&=\frac{1}{\sqrt{|H^{1}(X,{\mathbb{Z}}_{2})|}}\exp(2\pi
i\int_{X}A_{b}^{*}\alpha+\tfrac{1}{2}A_{b}^{*}\beta\cup
A_{p})\sum_{\eta}\exp(\frac{\pi i}{2}q_{\eta}(A_{b}^{*}\beta+A_{p}))\\\
&=\sqrt{|H^{1}(X,{\mathbb{Z}}_{2})|}\,\delta(A_{b}^{*}\beta+A_{p})\,\mathcal{Z}^{b}_{\alpha,\beta}[X,A_{b},A_{p}]~{},\end{split}$
(A.11)

where the rule (2.29) was used in the second line and the relation (D.7) in
the third. The $\delta$-function constraint on the ${\mathbb{Z}}_{2}^{f}$
background gauge field $A_{p}$ indicates that the ${\mathbb{Z}}_{2}^{f}$
symmetry is broken, with the pattern of symmetry breaking depending on the
topological invariant $\beta$.

##  Appendix B State sum evaluation of bosonic partition functions

To compute the evaluations (3.52) and (3.53) of partition functions
$\mathcal{Z}_{\omega}^{b}[X,\mathcal{A}]$ stated in section 3.5, we use the
state sum construction [67, 68, 26, 69]. The idea is to define the partition
function on a discretization of the space $(X,\mathcal{A})$ and then check
that the result is independent of the choice of discretization. A
discretization consists of a triangulation of $X$, a branching structure on
the edges of the triangulation (i.e. an assignment of edge directions such
that there are no closed loops), and group labels $g\in\mathcal{G}$ on the
edges such that the product of group labels (with labels inverted when the
edge directions point against the path direction) around each cycle in the
edges of $X$ equals the holonomy of $\mathcal{A}$ along that cycle. The
partition function is defined on such a discretization as the product of
contributions

$\mathcal{Z}^{b}_{\omega}(\Delta_{g,h}^{\pm})=\exp(\pm 2\pi i\,\omega(g,h))$
(B.1)

for each triangle whose two legs pointing in the same direction are labeled by
$g$ and $h$, as depicted in fig. 1. On an orientable surface, one fixes a
global orientation and inverts the contribution if the local orientation
determined by the branching structure disagrees with the global one ($-$
instead of $+$). The independence of this partition function on the
discretization is guaranteed by the cocycle condition on $\omega$.

Figure 1: (left) A triangle that makes a contribution (B.1) to the state sum.
(center) A discretization of the orientable surface of genus $n=2$. (right) A
discretization of the nonorientable surface of genus $n=4$.

First consider the orientable surface of genus $n$ labeled by holonomies
$\\{g_{i},h_{i}\\}$ satisfying $x(g_{i})=x(h_{i})=0$ and
$\prod_{i}g_{i}h_{i}g_{i}^{-1}h_{i}^{-1}=1$. A discretization of this space by
$4n-2$ triangles is depicted in fig. 1. On this discretization, the state sum
evaluates to

$\mathcal{Z}^{b}_{\omega}[(T^{2})^{\\#n},\\{g_{i},h_{i}\\}]=\exp(2\pi
i\sum_{i=1}^{n}\omega(x_{i},g_{i})+\omega(x_{i}g_{i},h_{i})-\omega(x_{i}g_{i}h_{i}g_{i}^{-1},g_{i})-\omega(x_{i}g_{i}h_{i}g_{i}^{-1}h_{i}^{-1},h_{i}))~{},$
(B.2)

where again $x_{i}=\prod_{j<i}g_{j}h_{j}g_{j}^{-1}h_{j}^{-1}$. Since
$x_{1}=x_{n}g_{n}h_{n}g_{n}^{-1}h_{n}^{-1}=1$, the expression contains only
$4n-2$ terms. Using the cocycle conditions

$-\omega(x_{i}g_{i}h_{i}g_{i}^{-1},g_{i})=\omega(x_{i}g_{i}h_{i},g_{i}^{-1})-\omega(g_{i}^{-1},g_{i})~{},$
(B.3)
$-\omega(x_{i}g_{i}h_{i}g_{i}^{-1}h_{i}^{-1},h_{i})=\omega(x_{i}g_{i}h_{i}g_{i}^{-1},h_{i}^{-1})-\omega(h_{i}^{-1},h_{i})~{},$
(B.4)

we obtain the form (3.52) stated in section 3.5. The check of gauge invariance
performed in section 3.5 suggests that this answer is correct. But as one
extra reality check, one can compute the torus ($n=1$, $ghg^{-1}=h$)

$\mathcal{Z}^{b}_{\omega}[T^{2}_{g,h}]=\exp(2\pi
i\,(\omega(g,h)-\omega(h,g)))$ (B.5)

and see that it matches the known result [26, 69].

Next consider the nonorientable surface of genus $n$ labeled by holonomies
$\\{g_{i}\\}$ satisfying $x(g_{i})=1$ and $\prod_{i}g_{i}^{2}=1$. A
discretization of this space is depicted in fig. 1. We obtain the expression

$\mathcal{Z}^{b}_{\omega}[({\mathbb{R}}P^{2})^{\\#n},\\{g_{i}\\}]=\exp(2\pi
i\sum_{i=1}^{n}\omega(y_{i},g_{i})+\omega(y_{i}g_{i},g_{i}))~{},$ (B.6)

where again $y_{i}=\prod_{j<i}g_{j}^{2}$. This form is related to the
expression (3.53) stated in section 3.5 by the cocycle condition

$\omega(y_{i},g_{i})+\omega(y_{i}g_{i},g_{i})=\omega(g_{i},g_{i})+\omega(y_{i},g_{i}^{2})~{}.$
(B.7)

Computing ${\mathbb{R}}P^{2}$ ($n=1$, $g^{2}=1$)

$\mathcal{Z}^{b}_{\omega}[{\mathbb{R}}P^{2}_{g}]=\exp(2\pi
i\,\omega(g,g))~{},$ (B.8)

the known result is recovered [26]. Now test it on the Klein bottle ($n=2$,
$g^{2}h=h^{-1}$):

$\mathcal{Z}^{b}_{\omega}[K_{g,h}]=\exp(2\pi
i\,(\omega(g,g)+\omega(g^{2},h)+\omega(h^{-1},h)))~{}.$ (B.9)

To compare to Ref. [26], substitute $h\mapsto g^{-1}k^{-1}$ in the expression.
The new $k$ has $x(k)=0$ and $gk^{-1}=kg$, and the expression becomes

$\mathcal{Z}^{b}_{\omega}[K_{g,g^{-1}k^{-1}}]=\exp(2\pi
i\,(\omega(g,g)+\omega(g^{2},g^{-1}k^{-1})+\omega(kg,g^{-1}k^{-1})))~{}.$
(B.10)

Use the twisted cocycle conditions

$\omega(g,g)+\omega(g^{2},g^{-1}k^{-1})=\omega(g,k^{-1})-\omega(g,g^{-1}k^{-1})~{},$
(B.11)
$\omega(kg,g^{-1}k^{-1})=\omega(k,k^{-1})-\omega(k,g)+\omega(g,g^{-1}k^{-1})$
(B.12)

to obtain

$\mathcal{Z}^{b}_{\omega}[K_{g,g^{-1}k^{-1}}]=\exp(2\pi
i\,(\omega(g,k^{-1})+\omega(k,k^{-1})-\omega(k,g)))~{},$ (B.13)

which matches the expression of Ref. [26].

##  Appendix C Twisted cohomology not of order two

In this section, we argue that the presence of an antiunitary symmetry in the
center of $\mathcal{G}$ implies that the twisted cohomology classes of
$\mathcal{G}$ are all of order two:

$2[\omega]=0~{},\qquad[\omega]\in H^{2}(\mathcal{G},U(1)_{x})~{}.$ (C.1)

We also provide examples of classes _not_ of order two that arise when no
antiunitary symmetry lies in the center, and we discuss how they spoil the
converse to Lemma 3. These examples are surprising, as the groups with
antiunitary symmetry that are common in the physics literature have classes
only of order two (see for example the table in Ref. [5]) and it had been
suggested that this may be a general rule [22, 23, 70].

Conjugating the arguments of a cocycle is a gauge transformation up to a sign,
as we can see by applying the cocycle conditions on the triplets
$(kgk^{-1},khk^{-1},k)$, $(kgk^{-1},k,h)$, and $(k,g,h)$ to obtain

$\displaystyle\begin{split}\omega(kgk^{-1},khk^{-1})&=(-1)^{x(g)}\omega(khk^{-1},k)-\omega(kghk^{-1},k)+\omega(kgk^{-1},kh)\\\
&=(-1)^{x(g)}\omega(khk^{-1},k)-\omega(kghk^{-1},k)+\omega(kgk^{-1},k)-(-1)^{x(g)}\omega(k,h)+\omega(kg,h)\\\
&=(-1)^{x(g)}\omega(khk^{-1},k)-\omega(kghk^{-1},k)+\omega(kgk^{-1},k)-(-1)^{x(g)}\omega(k,h)\\\
&\qquad\qquad\qquad+\,\omega(k,gh)-\omega(k,g)+(-1)^{x(k)}\omega(g,h)\\\
&=(-1)^{x(k)}\omega(g,h)+(\delta\lambda)(g,h)~{}.\end{split}$ (C.2)

where $\lambda(g)=\omega(kgk^{-1},k)-\omega(k,g)$. If $k$ is antiunitary
($x(k)=1$) and commutes with $g$ and $h$, this reads

$2\omega(g,h)=(\delta\lambda)(g,h)~{}.$ (C.3)

Therefore, when this holds for all $g,h$ (i.e. when $k$ is central), we have
$2[\omega]=0$, as claimed.

Let us comment on a physical interpretation of this fact. The partition
function of a unitary, invertible theory becomes complex conjugated (its
$\omega$ negated) by reversing the orientation of spacetime:

$\mathcal{Z}[M,\mathcal{A}]=\mathcal{Z}[M^{\text{rev}},\mathcal{A}^{\text{rev}}]^{*}~{}.$
(C.4)

This property appears on the cochain level as the fact that swapping and
inverting the arguments of a cocycle (which represents reversing the spacetime
orientation) is a gauge transformation up to a sign:

$\displaystyle\begin{split}\omega(g,h)&=(-1)^{x(g)}\omega(h,h^{-1})-\omega(gh,h^{-1})\\\
&=(-1)^{x(g)}\omega(h,h^{-1})+\omega(g,g^{-1})-\omega(gh,h^{-1}g^{-1})+(-1)^{x(gh)}\omega(h^{-1},g^{-1})\\\
&=(\delta\lambda)(g,h)-(-1)^{x(gh)}\omega(h^{-1},g^{-1})~{},\end{split}$ (C.5)

where $\lambda(g)=\omega(g^{-1},g)$.323232If $\omega(g,g^{-1})=0$ for all $g$,
this transformation does nothing and the cocycle is said to be in “cyclic
symmetric gauge” [71]. When $g$ and $h$ are unitary, orientation reversal
amounts to negation of $\omega$. Each symmetry $k$ acts on partition functions
by conjugating the gauge field and – if $k$ is antiunitary – reversing the
orientation:

$u\cdot\mathcal{Z}[M,\mathcal{A}]=\mathcal{Z}[M,u\mathcal{A}u^{-1}]~{},\qquad
t\cdot\mathcal{Z}[M,\mathcal{A}]=\mathcal{Z}[M^{\text{rev}},t\mathcal{A}^{\text{rev}}t^{-1}]~{}.$
(C.6)

Therefore, invariance under an antiunitary symmetry $t$ means

$\mathcal{Z}[M,\mathcal{A}]=t\cdot\mathcal{Z}[M,\mathcal{A}]=\mathcal{Z}[M^{\text{rev}},t\mathcal{A}^{\text{rev}}t^{-1}]=\mathcal{Z}[M,t\mathcal{A}t^{-1}]^{*}~{}.$
(C.7)

When $t$ is central, $t\mathcal{A}t^{-1}=\mathcal{A}$ and so the partition
function is of order two.

Now we give examples where $[\omega]$ is not of order two, despite
$\mathcal{G}$ having a (noncentral) antiunitary symmetry. As noted previously,
the partition function of ${\mathbb{R}}P^{2}$ is always of order two; however,
the higher order of the class shows up in other partition functions – both
orientable and nonorientable – that do not square to one.

A first example is the group

$\mathcal{G}={\mathbb{Z}}_{3}\times D_{6}^{T}~{},$ (C.8)

where $x$ is $1$ on reflections in the dihedral group
$D_{6}^{T}={\mathbb{Z}}_{3}\rtimes{\mathbb{Z}}_{2}^{T}$. The group law is

$(a,b,c)\cdot(a^{\prime},b^{\prime},c^{\prime})=(a+a^{\prime},b+(-1)^{c}b^{\prime},c+c^{\prime})~{}.$
(C.9)

A projective representation of the group is given by

$Q(a,b,c)=X^{a}Z^{b}{\mathcal{K}}^{c}~{},$ (C.10)

where $X$ and $Z$ denote the “shift matrix” and “clock matrix” [72]

$X=\begin{pmatrix}0&0&1\\\ 1&0&0\\\ 0&1&0\end{pmatrix}~{},\qquad
Z=\begin{pmatrix}1&0&0\\\ 0&\omega_{3}&0\\\
0&0&\omega_{3}^{2}\end{pmatrix}~{},$ (C.11)

where $\omega_{3}$ is a third root of unity, and ${\mathcal{K}}$ denotes
complex conjugation in this basis. These operations satisfy

$X^{3}=Z^{3}={\mathcal{K}}^{2}=1~{},\qquad
ZX=\omega_{3}XZ~{},\qquad{\mathcal{K}}X=X{\mathcal{K}}~{},\qquad{\mathcal{K}}Z=Z^{-1}{\mathcal{K}}~{}.$
(C.12)

Therefore

$\displaystyle\begin{split}Q(a,b,c)Q(a^{\prime},b^{\prime},c^{\prime})&=X^{a}Z^{b}{\mathcal{K}}^{c}X^{a^{\prime}}Z^{b^{\prime}}{\mathcal{K}}^{c^{\prime}}\\\
&=X^{a}Z^{b}X^{a^{\prime}}Z^{(-1)^{c}b^{\prime}}{\mathcal{K}}^{c+c^{\prime}}\\\
&=\omega_{3}^{a^{\prime}b}X^{a+a^{\prime}}Z^{b+(-1)^{c}b^{\prime}}{\mathcal{K}}^{c+c^{\prime}}\\\
&=\omega_{3}^{a^{\prime}b}Q((a,b,c)\cdot(a^{\prime},b^{\prime},c^{\prime}))~{}.\end{split}$
(C.13)

The values

$\omega((a,b,c),(a^{\prime},b^{\prime},c^{\prime}))=\omega_{3}^{a^{\prime}b}$
(C.14)

are guaranteed to satisfy the twisted cocycle condition by the associativity
of multiplication in the projective representation, and they are of order
three, not two. This cocycle defines torus partition functions that can be
nontrivial third roots of unity, as the formula (B.5) shows:

$\mathcal{Z}^{b}_{\omega}[T^{2},\\{(a,b,0),(a^{\prime},b^{\prime},0)\\}]=\exp(2\pi
i\,(\omega((a,b,0),(a^{\prime},b^{\prime},0))-\omega((a^{\prime},b^{\prime},0),(a,b,0))))=\omega_{3}^{a^{\prime}b-ab^{\prime}}~{},$
(C.15)

noting that arbitrary unitary symmetries commute and so respect the torus
fundamental group. Klein bottle partition functions (B.9) are defined for any
$a+a^{\prime}\in\\{0,2\\}$ since $1=g^{2}h^{2}=(2a+2a^{\prime},0,0)$:

$\displaystyle\begin{split}&\mathcal{Z}^{b}_{\omega}[K,\\{(a,b,1),(a^{\prime},b^{\prime},1)\\}]\\\
&\qquad\qquad\qquad=\exp(2\pi
i\,(\omega((a,b,1),(a,b,1))+\omega((2a,0,0),(a^{\prime},b^{\prime},1))+\omega((-a^{\prime},b^{\prime},1),(a^{\prime},b^{\prime},1))))\\\
&\qquad\qquad\qquad=\omega_{3}^{ab+a^{\prime}b^{\prime}}~{},\end{split}$
(C.16)

using $(a,b,c)^{-1}=(-a,(-1)^{c+1}b,c)$. We see they can also be nontrivial
third roots of unity. This example generalizes straightforwardly to the
symmetry $\mathcal{G}={\mathbb{Z}}_{n}\times D_{2n}^{T}$, which has a twisted
cohomology class of order $n$ that gives a torus and Klein bottle partition
functions in $n^{\text{th}}$ roots of unity.

A second example is the group

$\mathcal{G}={\mathbb{Z}}_{4}\rtimes{\mathbb{Z}}_{4}^{T}~{},$ (C.17)

where $x:{\mathbb{Z}}_{4}^{T}\mapsto{\mathbb{Z}}_{2}^{T}$ is the nontrivial
map. The group law is

$(a,b)\cdot(a^{\prime},b^{\prime})=(a+(-1)^{b}a^{\prime},b+b^{\prime})~{}.$
(C.18)

A projective representation of the group is given by

$Q(a,b)=Z^{a}(X{\mathcal{K}})^{b}~{},$ (C.19)

where $X$ and $Z$ are the obvious rank $4$ generalizations of the shift and
clock matrices above. Then observe

$\displaystyle\begin{split}Q(a,b)Q(a^{\prime},b^{\prime})&=Z^{a}X^{b}{\mathcal{K}}^{b}Z^{a^{\prime}}X^{b^{\prime}}{\mathcal{K}}^{b^{\prime}}\\\
&=Z^{a}X^{b}Z^{(-1)^{b}a^{\prime}}X^{b^{\prime}}{\mathcal{K}}^{b+b^{\prime}}\\\
&=\omega_{4}^{(-1)^{b}a^{\prime}b}Z^{a+(-1)^{b}a^{\prime}}X^{b+b^{\prime}}{\mathcal{K}}^{b+b^{\prime}}\\\
&=\omega_{4}^{(-1)^{b}a^{\prime}b}Q((a,b)\cdot(a^{\prime},b^{\prime}))\end{split}$
(C.20)

The cocycle

$\omega((a,b),(a^{\prime},b^{\prime}))=\omega_{4}^{(-1)^{b}a^{\prime}b}$
(C.21)

has torus

$\mathcal{Z}^{b}_{\omega}[T^{2},\\{(a,b),(a^{\prime},b^{\prime})\\}]=\exp(2\pi
i\,(\omega((a,b),(a^{\prime},b^{\prime}))-\omega((a^{\prime},b^{\prime}),(a,b))))=\omega_{4}^{a^{\prime}b-ab^{\prime}}~{},$
(C.22)

defined for $b,b^{\prime}=0,2$, and Klein bottle

$\displaystyle\begin{split}\mathcal{Z}^{b}_{\omega}[K,\\{(a,b),(a^{\prime},b^{\prime})\\}]&=\exp(2\pi
i\,(\omega((a,b),(a,b))+\omega((0,2b),(a^{\prime},b^{\prime}))+\omega((a^{\prime},-b^{\prime}),(a^{\prime},b^{\prime}))))\\\
&=\omega_{4}^{-ab+2a^{\prime}b+a^{\prime}b^{\prime}}\\\
&=\omega_{4}^{-ab-a^{\prime}b^{\prime}}~{},\end{split}$ (C.23)

defined for $b,b^{\prime}=1,3$, where we used $(a,b)^{-1}=((-1)^{b+1}a,-b)$,
then $2(b+b^{\prime})=0$ in the last line. As in the other example, the Klein
bottle can be any fourth root of unity; but this time, the torus takes values
in $\pm 1$ due to $b,b^{\prime}=0,2$. In fact, for any orientable surface, the
partition function for $\omega$ is the same as that for the pullback of
$\omega$ to $\ker x$. In this example, $\ker
x={\mathbb{Z}}_{4}\times{\mathbb{Z}}_{2}$ has cohomology group
${\mathbb{Z}}_{2}$, which means that all of its orientable partition functions
will be of order two. The nonorientable partition functions are,

$\mathcal{Z}^{b}_{\omega}[({\mathbb{R}}P^{2})^{\\#n},\\{(a_{i},b_{i})\\}]=\exp(2\pi
i\sum_{i=1}^{n}(\omega((a_{i},b_{i}),(a_{i},b_{i}))+\omega((0,2(i-1)),(0,2b_{i}))))=\omega_{4}^{-\sum_{i=1}^{n}a_{i}b_{i}}~{},$
(C.24)

for $b_{i}=1,3$. Now we claim that this $\omega$ is a square root

$2[\omega]=[\tfrac{1}{2}\nu\cup\nu]$ (C.25)

of the half cup square333333A projective representation with this cocycle is
given by $Q(a,b)=Z_{2}^{a}X_{2}^{a}{\mathcal{K}}^{b}$ with $X_{2}$ and $Z_{2}$
the Pauli matrices. of

$\nu(a,b)=a\text{ mod }2~{}.$ (C.26)

This is equivalent to the partition functions associated with $2\omega$ and
$\tfrac{1}{2}\nu\cup\nu$ agreeing on every space. Since $\omega$ is order two
on orientable surfaces, its square vanishes on them, consistent with the
result for $\tfrac{1}{2}\nu\cup\nu$ (3.49). On nonorientable surfaces, the
square of the expression (C.24) is

$\mathcal{Z}^{b}_{\omega}[({\mathbb{R}}P^{2})^{\\#n},\\{(a_{i},b_{i})\\}]^{2}=(-1)^{-\sum_{i=1}^{n}a_{i}b_{i}}=(-1)^{\sum_{i=1}^{n}a_{i}}=(-1)^{\sum_{i=1}^{n}\beta(a_{i},b_{i})}~{},$
(C.27)

also consistent with the result for $\tfrac{1}{2}\nu\cup\nu$ (3.50). A
counterexample to the full converse of Lemma 3 is therefore obtained from the
fermionic symmetry group

$\mathcal{G}=({\mathbb{Z}}_{4}\rtimes{\mathbb{Z}}_{4}^{T})\times{\mathbb{Z}}_{2}^{f}$
(C.28)

by taking $\beta$ to be the $\nu$ (C.26) defined above and the square root
$R_{\beta}$ to be the $\omega$ (C.21). The half cup square of this $\beta$ has
a square root, despite failing the genus $n$ rule for $n$ even, and – since
the other choices of $\beta$ for this group also have square roots (namely,
vanishing ones)343434The half cup square of $\beta(a,b)=b\text{ mod }2$ is
trivial and unrelated to the nontrivial degree two cohomology class of
${\mathbb{Z}}_{4}^{T}$. – the SPT classifications are isomorphic.

##  Appendix D The Arf invariant

In this appendix, we review some basic properties of the Arf invariant.

The inverse is given by

$\text{Arf}(e)^{-1}=\frac{1}{\sqrt{|H^{1}(X,{\mathbb{Z}}_{2})|}}\sum_{a}\exp(\frac{-\pi
i}{2}q_{e}(a))~{},$ (D.1)

as we can verify by computing

$\displaystyle\begin{split}\text{Arf}(e)\text{Arf}(e)^{-1}&=\frac{1}{|H^{1}(X,{\mathbb{Z}}_{2})|}\sum_{a,a^{\prime}}\exp(\frac{\pi
i}{2}(q_{e}(a)-q_{e}(a^{\prime})))\\\
&=\frac{1}{|H^{1}(X,{\mathbb{Z}}_{2})|}\sum_{a^{\prime\prime}=a+a^{\prime}}\exp(\frac{\pi
i}{2}q_{e}(a^{\prime\prime}))\sum_{a}\exp(\pi i\int_{X}a\cup
a^{\prime\prime})\\\ &=\sum_{a^{\prime\prime}}\exp(\frac{\pi
i}{2}q_{e}(a^{\prime\prime}))\delta(a^{\prime\prime})\\\ &=1~{},\end{split}$
(D.2)

where we have used the relation

$\sum_{a}\exp(\pi i\int_{X}a\cup
b)=|H^{1}(X,{\mathbb{Z}}_{2})|\,\delta(b)~{}.$ (D.3)

The square is given by

$\displaystyle\begin{split}\text{Arf}(e)^{2}&=\frac{1}{|H^{1}(X,{\mathbb{Z}}_{2})|}\sum_{a,a^{\prime}}\exp(\frac{\pi
i}{2}(q_{e}(a)+q_{e}(a^{\prime})))\\\
&=\frac{1}{|H^{1}(X,{\mathbb{Z}}_{2})|}\sum_{a^{\prime\prime}=a+a^{\prime}}\exp(\frac{\pi
i}{2}q_{e}(a^{\prime\prime}))\sum_{a}\exp(\pi
i\int_{X}a\cup(a+a^{\prime\prime}))\\\ &=\sum_{a^{\prime\prime}}\exp(\frac{\pi
i}{2}q_{e}(a^{\prime\prime}))\delta(a^{\prime\prime}+w_{1})\\\
&=\exp(\frac{\pi i}{2}q_{e}(w_{1}))~{},\end{split}$ (D.4)

where we have used the relation

$\sum_{a}\exp(\pi
i\int_{X}a\cup(a+b))=|H^{1}(X,{\mathbb{Z}}_{2})|\,\delta(b+w_{1})~{},$ (D.5)

which in turn comes from the Wu relation $w_{1}\cup a=a\cup a$ [73]. It
follows that the Arf invariant is an eighth root of unitary and that its
fourth power is independent of the pin structure since

$\text{Arf}(e)^{4}=\exp(\frac{\pi i}{2}(q_{e}(w_{1})+q_{e}(w_{1})))=\exp(\pi
i\int_{X}w_{1}^{2})\in\pm 1~{}.$ (D.6)

Summing over pin structures yields

$\displaystyle\begin{split}\sum_{e}\exp(\frac{\pi
i}{2}q_{e}(a))&=\sum_{b}\exp(\frac{\pi i}{2}q_{e_{0}+b}(a))\\\
&=\exp(\frac{\pi i}{2}q_{e_{0}}(a))\sum_{b}\exp(\pi i\int_{X}a\cup b)\\\
&=\exp(\frac{\pi i}{2}q_{e_{0}}(a))|H^{1}(X,{\mathbb{Z}}_{2})|\,\delta(a)\\\
&=|H^{1}(X,{\mathbb{Z}}_{2})|\,\delta(a)~{},\end{split}$ (D.7)

using rule (2.30). Therefore

$\frac{1}{\sqrt{|H^{1}(X,{\mathbb{Z}}_{2})|}}\sum_{e}\text{Arf}(e)=\frac{1}{|H^{1}(X,{\mathbb{Z}}_{2})|}\sum_{a,e}\exp(\frac{\pi
i}{2}q_{e}(a))=1~{}.$ (D.8)

Finally, we have the relation

$\displaystyle\begin{split}\exp(\frac{\pi
i}{2}q_{e}(a))\text{Arf}(e+a)&=\frac{1}{\sqrt{|H^{1}(X,{\mathbb{Z}}_{2})|}}\sum_{b}\exp(\frac{\pi
i}{2}(q_{e}(a)+q_{e+a}(b)))\\\
&=\frac{1}{\sqrt{|H^{1}(X,{\mathbb{Z}}_{2})|}}\sum_{b}\exp(\frac{\pi
i}{2}(q_{e}(a)+q_{e}(b)))\exp(\pi i\int_{X}a\cup b)\\\
&=\frac{1}{\sqrt{|H^{1}(X,{\mathbb{Z}}_{2})|}}\sum_{b}\exp(\frac{\pi
i}{2}q_{e}(a+b))\\\ &=\text{Arf}(e)~{}.\end{split}$ (D.9)

We now return to the Jordan-Wigner transformation (2.33). The expression
(2.36) is indeed the inverse:

$\displaystyle\begin{split}{\mathcal{J}}{\mathcal{W}}^{-1}{\mathcal{J}}{\mathcal{W}}(\mathcal{Z}^{b})[X,A_{b},A_{p}^{\prime}]&=\sum_{A_{p},\eta}\frac{\text{Arf}(\eta+A_{p})\text{Arf}(\eta+A_{p}^{\prime})^{-1}}{|H^{1}(X,{\mathbb{Z}}_{2})|}\mathcal{Z}^{b}[X,A_{b},A_{p}]\\\
&=\frac{1}{|H^{1}(X,{\mathbb{Z}}_{2})|}\sum_{A_{p},\eta}\exp(\frac{\pi
i}{2}q_{\eta+A_{p}}(A_{p}+A_{p}^{\prime}))\mathcal{Z}^{b}[X,A_{b},A_{p}]\\\
&=\sum_{A_{p}}\delta(A_{p}+A_{p}^{\prime})\,\mathcal{Z}^{b}[X,A_{b},A_{p}]\\\
&=\mathcal{Z}^{b}[X,A_{b},A_{p}^{\prime}]~{},\end{split}$ (D.10)

by the rules (D.9) and (D.7). Alternatively, write the inverse as

${\mathcal{J}}{\mathcal{W}}^{-1}(\mathcal{Z}^{f})[X,A_{b},A_{p}]=\sum_{\eta}\mathcal{Z}^{f}[X,A_{b},\eta]\exp(\frac{\pi
i}{2}q_{\eta+\tau}(A_{p}+\tau))\frac{\text{Arf}(\eta+\tau)^{-1}}{\sqrt{|H^{1}(X,{\mathbb{Z}}_{2})|}}~{}.$
(D.11)

and check

$\displaystyle\begin{split}{\mathcal{J}}{\mathcal{W}}^{-1}&(\mathcal{Z}^{f}_{\alpha,\beta})[X,A_{b},A_{p}]\\\
&=\sum_{\eta}\exp(2\pi
i\int_{X}A_{b}^{*}\alpha+\tfrac{1}{2}A_{b}^{*}\beta\cup\tau)\exp(\frac{\pi
i}{2}q_{\eta+\tau}(A_{b}^{*}\beta))\exp(\frac{\pi
i}{2}q_{\eta+\tau}(A_{p}+\tau))\frac{\text{Arf}(\eta+\tau)^{-1}}{\sqrt{|H^{1}(X,{\mathbb{Z}}_{2})|}}\\\
&=\exp(2\pi i\int_{X}A_{b}^{*}\alpha+\tfrac{1}{2}A_{b}^{*}\beta\cup
A_{p})\sum_{\eta}\exp(\frac{\pi
i}{2}q_{\eta+\tau}(A_{b}^{*}\beta+A_{p}+\tau))\frac{\text{Arf}(\eta+\tau)^{-1}}{\sqrt{|H^{1}(X,{\mathbb{Z}}_{2})|}}\\\
&=\exp(2\pi i\int_{X}A_{b}^{*}\alpha+\tfrac{1}{2}A_{b}^{*}\beta\cup
A_{p})\sum_{\eta}\frac{\text{Arf}(\eta+A_{b}^{*}\beta+A_{p})^{-1}}{\sqrt{|H^{1}(X,{\mathbb{Z}}_{2})|}}\\\
&=\mathcal{Z}^{b}_{\alpha,\beta}[X,A_{b},A_{p}]~{}.\end{split}$ (D.12)

Here we used the freedom in the auxiliary variables to set the $\tau$ of
$\mathcal{Z}^{f}_{\alpha,\beta}$ equal to that of
${\mathcal{J}}{\mathcal{W}}^{-1}$, used the relation (D.9) in the third line,
and used the relation (D.8) in the last.

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