MnLargeSymbols’164 MnLargeSymbols’171 ††institutetext: Department of Physics,
University of California San Diego,
9500 Gilman Drive, La Jolla CA 92093-0319, USA

# Anomaly Enforced Gaplessness and Symmetry Fractionalization for $Spin_{G}$
Symmetries

T. Daniel Brennan<EMAIL_ADDRESS>
###### Abstract

Symmetries and their anomalies give strong constraints on renormalization
group (RG) flows of quantum field theories. Recently, the identification of a
theory’s global symmetries with its topological sector has provided additional
constraints on RG flows to symmetry preserving gapped phases due to
mathematical results in category and topological quantum field theory. In this
paper, we derive constraints on RG flows from ${\mathbb{Z}}_{2}$-valued pure-
and mixed-gravitational anomalies that can only be activated on non-spin
manifolds. We show that such anomalies cannot be matched by a unitary,
symmetry preserving gapped phase without symmetry fractionalization. In
particular, we discuss examples that commonly arise in $4d$ gauge theories
with fermions.

##  1 Introduction

Symmetry is one of the guiding principles of modern physics. It can be used to
derive selection rules for correlation functions, constrain non-perturbative
dynamics, and give insight into renormalization group flows. Recently, the
notion of symmetry groups has been expanded to the more general framework of
symmetry categories. In the setting of physical theories, this categorical
structure is realized via topological operators and their rules for braiding
and fusion. This is the notion of “generalized global symmetries” or
“categorical global symmetries.” For a review see Cordova:2022ruw ;
Freed:2022qnc ; Gaiotto:2014kfa ; Schafer-Nameki:2023jdn ; Brennan:2023mmt ;
Bhardwaj:2023kri ; Shao:2023gho  and references therein.

This modern perspective on global symmetries has given us new tools to analyze
quantum field theories. The fact that symmetries are realized by topological
operators means that symmetries of quantum field theories are classified by
topological field theories which themselves are described by certain types of
categories – which (at least in principle) can be completely classified.

In this paper we will restrict our attention to the case of higher form global
symmetries which are characterized by their group-like structure. Higher form
global symmetries can be decomposed into simple factors which are labeled by a
degree $p\in{\mathbb{Z}}_{\geq 0}$ and a group $G^{(p)}$: each simple factor
is referred to as a $p$-form global symmetry. A $p$-form global symmetry group
$G^{(p)}$ can be either discrete or continuous and acts on $p$-dimensional
operators via co-dimension-$(p+1)$ topological operators called symmetry
defect operators.

One of the results of axiomatizing symmetries in this way is that we are able
to better characterize how a symmetries can be matched along their RG flows.
The general picture is the following. When a theory has a symmetry group $G$,
the Hilbert space transforms under (possibly projective) representations of
$G$. Such a theory may flow at low energies to a gapped or gapless theory –
i.e. the low energy spectrum of the Hamiltonian may be discrete or a
continuous near zero. At long distances, the gapless phases are described by
scale invariant theories while gapped phases are described by topological
theories.

In addition, we can further distinguish theories by whether the symmetry $G$
is preserved or spontaneously broken. Such a symmetry $G$ is preserved if the
vacuum state transforms in a trivial representation whereas it is broken if
the vacuum state transforms in a non-trivial representation of $G$. The
breaking of $G$ is indicated by the existence of charged operators that have a
non-trivial vacuum expectation value. In the case where a symmetry is
spontaneously broken, the theory is in general nontrivial:

  * •

A spontaneously broken 0-form symmetry $G^{(0)}$ leads to a degenerate ground
state corresponding to the elements of the group – for $G^{(0)}$ continuous,
this degeneracy is encoded in a massless $G^{(0)}$-valued scalar Goldstone
boson.

  * •

A spontaneously broken $p>0$-form global symmetry does not necessarily have a
degenerate Hilbert space on any manifold, but rather leads to a non-trivial
Hilbert space on $T^{p}\times{\mathbb{R}}^{d-p}$ corresponding to wrapped
$p$-dimensional line operators which take non-trivial vevs.

Another possible behavior of symmetries along an RG flow is that it may
develop a non-trvial categorical structure.111In fact, the emergence of non-
trivial categorical structure along an RG flow can be used to infer the
behavior of the theory at intermediate points along the flow Brennan:2020ehu ;
Cordova:2018cvg ; Brennan:2023kpw ; Cordova:2022ieu ; Choi:2022jqy ;
Choi:2022rfe ; Choi:2022fgx . One of the simplest ways a non-trivial
categorical structure can emerge is by _symmetry fractionalization_ which for
our purposes describes the case where a 0-form symmetry mixes with a
(potentially emergent) 1-form symmetry Delmastro:2022pfo ; Brennan:2022tyl ;
Barkeshli:2014cna ; Chen:2014wse .222More generally, symmetry
fractionalization can describe when a $p$-form global symmetry can mix with a
$(p+1)$-form global symmetry, but we will not need the general case here. For
the more details see Delmastro:2022pfo ; Bartsch:2023pzl ; Bhardwaj:2023wzd ;
Bhardwaj:2022dyt .  This mixing can be realized by a choice of $\alpha\in
H^{2}(BG^{(0)},G^{(1)})$, which defines a central extension of $G^{(0)}$ by
$G^{(1)}$, together with a modification of the product law of the $0$-form
symmetry defect operators $U_{g}(\Sigma_{d-1})$:333 Note that a more general
action of $G^{(0)}$ on the line operators that are protected by
$G^{(1)}$-symmetry can be defined by picking an action $\rho:G^{(0)}\to
Aut(G^{(1)})$ together with a twisted cohomology group $\alpha_{\rho}\in
H^{2}_{\rho}(BG^{(0)},G^{(1)})$. This is described by a “split 2-group” (also
sometimes called “strict 2-group”) which will not be important for our
discussion. See Bartsch:2023pzl ; Bhardwaj:2023wzd ; Baez:2008hz  for more
details.

$\displaystyle\begin{split}U_{g_{1}}(\Sigma_{d-1})\cdot
U_{g_{2}}(\Sigma_{d-1})={\cal
U}_{\alpha(g_{1},g_{2})}(\partial\Sigma_{d-2})\,U_{g_{1}g_{2}}(\Sigma_{d-2})~{},\end{split}$
(1)

where $g_{1},g_{2}\in G^{(0)}$ and ${\cal U}_{h}(\Gamma_{d-2})$ are 1-form
symmetry defect operators for $h\in G^{(1)}$. Physically, the non-trivial
$G^{(0)}$ action can be realized via a world volume anomaly for the line
operators which can indicate that they describe the world line of some heavy
particle that transforms projectively under $G^{(0)}$.

The IR behavior of QFTs are highly constrained by ‘t Hooft anomalies of the
global symmetry group $G$. These anomalies must be matched along symmetry
preserving RG flows and can used rule out different possibilities for the IR
theory. These ideas go under the name of “Lieb-Schultz-Mattis type theorems”
or “anomaly enforced gaplessness” Lieb:1961fr ; Wang:2014lca ; Wang:2016gqj ;
Sodemann:2016mib ; Wang:2017txt ; Kobayashi:2018yuk ; Cordova:2019bsd ;
Cordova:2019jqi ; Apte:2022xtu . For discrete symmetries in continuum QFTs, it
was shown in Cordova:2019bsd ; Cordova:2019jqi , that $d$-dimensional unitary
theories with a global symmetry group $G$ that has an anomaly given by
$\omega_{d+1}(A)\in H^{d+1}(BG;U(1))$, cannot be matched in the IR by a
symmetry preserving gapped phase if the anomaly evaluates non-trivially on the
mapping class torus

$\displaystyle\begin{split}{\rm exp}\left\\{2\pi
i\int_{N_{d+1^{(\varphi)}}}\omega_{d+1}(A)\right\\}\neq 1~{},\end{split}$ (2)

where $N_{d+1}^{(\varphi)}=\big{(}S^{p}\times
S^{d-p}\big{)}\rtimes_{\varphi}S^{1}$ and $\varphi$ is the twist by some
$\varphi\in G$.

Recently, there has been interest in computing and understanding anomalies
that are only activated on non-spin manifolds Wang:2018qoy ; Brennan:2023mmt ;
BI ; Ang:2019txy . These anomalies are more subtle to compute than their
perturbative counter parts, often requiring more sophisticated index theorem
computations.

In this paper, we will consider the anomalies of theories which have
symmetries that are of the form

$\displaystyle\begin{split}G_{\rm total}=\frac{G\times
Spin(d)}{{\mathbb{Z}}_{2}}~{},\end{split}$ (3)

where $G$ is the internal global symmetry group, $Spin(d)$ is the
$d$-dimensional Euclidean Lorentz group, and the ${\mathbb{Z}}_{2}$ quotient
identifies $(-1)^{F}$ with a (potentially trivial) ${\mathbb{Z}}_{2}$ subgroup
of the 0-form subgroup of $G$. In other words, there is a 0-form symmetry
$(-1)^{F}$ or $(-1)^{F}\circ(-\mathds{1}_{G})$ which acts trivially on the
gauge invariant local operators. When the ${\mathbb{Z}}_{2}$ quotient acts
trivially on $G$, we call the theory bosonic and say that the theory has
bosonic symmetry, whereas when the ${\mathbb{Z}}_{2}$ quotient acts non-
trivially on $G$, we say the symmetry has a $Spin_{G}$-symmetry. Here we will
treat bosonic symmetries as a special case of $Spin_{G}$-symmetries with
trivial $G$-structure – i.e. $G$ is the trivial group.

Theories with $Spin_{G}$-symmetry can be put on non-spin manifolds: this
requires turning on a ${\mathbb{Z}}_{2}$-discrete flux in $G$ for the case
with non-trivial $G$-structure. This allows for new classes of
${\mathbb{Z}}_{2}$-valued pure- and mixed-gravitational anomalies whose
anomalous phase couples to the Stiefel-Whitney class $w_{2}(TM)$. Another
important feature of $Spin_{G}$-symmetries is that the ${\mathbb{Z}}_{2}$
quotient in $G_{\rm total}$ implies $H^{2}(BG_{\rm
total},{\mathbb{Z}}_{2})\neq 0$ which allows for symmetry fractionalization.
The symmetry fractionalization is realized by line operators that transform
non-trivially under a choice of integral lift of $w_{2}(TM)+w_{2}(G)$ for
$Spin_{G}$-symmetries which describes the world-line of a heavy particle that
transforms non-trivially under $(-1)^{F}\circ(-\mathds{1}_{G})$. In the
special case of bosonic symmetries, the line operators transform non-trivially
under a choice of integral lift of $w_{2}(TM)$ which describes the world-line
of a heavy fermion.

In this paper, we will derive new constraints on the IR phase of a theory with
anomalies of such $Spin_{G}$-symmetries. Many anomalies of theories with and
$Spin_{G}$-symmetries can be analyzed on mapping class tori of a product of
spheres: such anomalies lead to constraints on the IR phase due to the theorem
of Cordova:2019bsd ; Cordova:2019jqi  described above. However, there also
exist anomalies which vanish on spin manifolds that do not lead to such
constraints. Here we will prove that such anomalies that are only activated on
non-spin manifolds do lead to constraints on RG flows. In particular, such
anomalies imply that the theory is either gapless, experiences symmetry
fractionalization, or that the symmetry group $G$ is spontaneously broken in
the deep IR.

Our anomaly can be stated more formally as follows. Here we will consider a
$d$-dimensional theory ${\cal T}_{d}$ for $d=5+p$ where $p\neq 2,4$ on a
space-time manifold $M$ with internal symmetry group $G=G_{0}\times
G_{1}=G_{0}\times\left(G^{(0)}\times G^{(1)}\times G^{(p)}\right)$ and
$Spin_{G}$-symmetry structure. Here $G_{0}$ is the part of $G$ that
participates in the $Spin_{G}$ structure – i.e. the $Spin_{G}$ structure can
be reduced to a $Spin_{G_{0}}$ structure – while $G_{1}=G^{(0)}\times
G^{(1)}\times G^{(p)}$ is a product of discrete higher form symmetry groups.
Here we make no assumption of whether or not $G_{0}$ is discrete or
continuous.

We turn on background gauge fields for the symmetry group $G_{1}$ by coupling
the theory to a non-trivial bundle $P\otimes TM$ with connection where $P$ is
a principal $G_{1}$-bundle and $TM$ is the tangent bundle of $M$. When $M$ is
a non-spin manifold, we must elevate the tangent bundle to a $Spin_{G_{0}}$
bundle which requires an additional $G_{0}/{\mathbb{Z}}_{2}$ bundle with fixed
non-trivial obstruction class $w_{2}(G_{0})=w_{2}(TM)$. We will denote the
background gauge fields for $G_{1}$ by $\\{A\\}$ and will assume that $G$ has
anomalies described by the $(d+1)$-dimensional anomaly polynomial
$\omega_{d+1}(A;w_{2}(TM))$ is a $U(1)$-valued characteristic class of
$P\otimes TN$ where $N$ is a $(d+1)$-manifold. We assume that
$\omega_{d+1}(A;0)=0$ – i.e. that the anomaly vanishes on spin manifolds for
any choice of $G_{1}$-bundle.

Result: _Consider a $d$-dimensional QFT with $Spin_{G}$-symmetry and anomaly
$\omega_{d+1}(A;w_{2}(TM))$ as above. Define the $(d+1)$-dimensional manifold
$N_{d+1}^{(g)}$ which is topologically the product manifold
$N_{d+1}^{(g)}=\mathbb{CP}^{2}\times S^{p+1}\times S^{1}$ where we twist the
fibers of $P\otimes TN$ by an element $g\in G$ along the $S^{1}$-direction
which we write as a mapping class torus:_

$\displaystyle\begin{split}N_{d+1}^{(g)}=(\mathbb{CP}^{2}\times
S^{p+1})\rtimes_{g}S^{1}~{}.\end{split}$ (4)

_If the anomaly of a theory on $N_{d+1}^{(g)}$ can evaluate to a non-trivial
phase:_

$\displaystyle\begin{split}{\rm exp}\left\\{2\pi
i\int_{N_{d+1}^{(g)}}\omega_{d+1}(A;w_{2}(TN))\right\\}\neq 1~{},\end{split}$
(5)

_then the theory ${\cal T}_{d}$ cannot flow to a symmetry preserving gapped
phase without symmetry fractionalization for $(-1)^{F}$ (which for
$Spin_{G}$-symmetries is identified with an element of $G$). _

This result generalizes those of Cordova:2019bsd ; Cordova:2019jqi  to include
$Spin_{G}$ global symmetries at the expense of a weaker constraint on the IR
behavior.

The outline of our paper is as follows. We begin by reviewing the results of
Cordova:2019bsd ; Cordova:2019jqi  which forms the cornerstone of our
analysis. In particular, we review how anomalies evaluated on the mapping
class torus of a product of spheres can imply that the theory cannot flow to a
symmetry preserving gapped phase. In Section 3 we prove our main result by
adapting the analysis of Reutter:2020bav  to the case of non-spin mapping
class tori.

We conclude with a set of examples. Of particular interest are the anomalies
of $4d$ theories. Here, we will discuss two types of anomalies that are
sensitive to the $(-1)^{F}$ symmetry:

$\displaystyle\begin{split}{\cal A}_{1}=\pi i\int x_{1}\cup y_{2}\cup
w_{2}(TM)\quad,\quad{\cal A}_{2}=\pi i\int w_{2}(TM)\cup z_{3}\end{split}$ (6)

where $x_{1},y_{2},z_{3}$ are degree -1,-2, and -3 ${\mathbb{Z}}_{2}$-valued
cohomology classes respectively. As we will discuss, these anomalies can be
matched by flowing to a phase where the $(-1)^{F}$ symmetry is fractionalize.
Such anomalies often arise in $4d$ gauge theories such as 2-flavor $SU(N_{c})$
adjoint QCD BI ; Donaldson:1983wm ; Donaldson:1990kn ; Witten:1988ze ;
Witten:1994cg ; Moore:1997pc . The results of our paper can be used to infer
characteristics of the IR limit of these theories.

##  2 Review of Anomaly Enforced Gaplessness on Spin Manifolds

As we have discussed, the derivation of “anomaly enforced gaplessness” for
theories that have anomalies for discrete symmetries on spin manifolds is
given in Cordova:2019bsd ; Cordova:2019jqi . The main result can be stated as
follows:

Result of Cordova:2019bsd ; Cordova:2019jqi : Consider a $d$-dimensional
theory ${\cal T}_{d}$ with a discrete, abelian symmetry group $G$ which has an
anomaly polynomial given by $\omega_{d+1}(A)\in H^{d+1}(BG,U(1))$. Let us
consider turning on background gauge fields for $G$, which we denote $\\{A\\}$
– i.e. we couple to a principal $G$ bundle $P$. Take the $(d+1)$-dimensional
manifold $N_{d+1}=S^{p+1}\times S^{d-p-1}\times S^{1}$ where we additionally
twist $P$ along the $S^{1}$ factor by an element $g\in G$:
$N_{d+1}^{(g)}=(S^{p+1}\times S^{d-p-1})\rtimes_{g}S^{1}$. If

$\displaystyle\begin{split}{\rm exp}\left\\{2\pi
i\int_{N_{d+1}^{(g)}}\omega_{d+1}(A)\right\\}\neq 1~{},\end{split}$ (7)

then there is no unitary, symmetry preserving TQFT with anomaly
$\omega_{d+1}(A)$. In particular, this means that the theory cannot flow to a
symmetry preserving gapped unitary TQFT.

This result can be proven as follows.

Consider a $d$-dimensional unitary TQFT. First note that unitary TQFTs are by
definition reflection positive Freed:2016rqq . This means that for any
$d$-manifold $X_{d}=\overline{Y_{d}}\\#Y_{d}$ that $Z[X_{d}]\geq 0$ where
$\overline{Y_{d}}$ is the orientation reversal of $Y_{d}$. This can be seen
from the fact that the path integral on $Y_{d}$ defines a state
$\big{|}Y_{d}\big{\rangle}$ which is an element of the Hilbert space ${\cal
H}_{\partial Y_{d}}$ defined on $\partial Y_{d}$ which allows us to write

$\displaystyle\begin{split}Z[X_{d}]=\big{\langle}Y_{d}\big{|}Y_{d}\big{\rangle}\geq
0~{},\end{split}$ (8)

by unitarity.

In particular, any unitary TQFT has the condition that $Z_{\rm
TQFT}\big{[}S^{d}\big{]}=1$. This follows from the fact that $S^{d}$ can be
decomposed by cutting along the equator: $S^{d}=\overline{D^{d}}\\#D^{d}$
where $D^{q}$ is the $q$-dimensional disk. Since the vacuum state is prepared
by evaluating the path integral on $D^{d}$, which in a unitary TQFT is a non-
trivial state, we have the condition

$\displaystyle\begin{split}Z\big{[}S^{d}\big{]}=\big{\langle}0\big{|}0\big{\rangle}=1~{}.\end{split}$
(9)

Additionally, in a unitary TQFT, the Hilbert space on $S^{d-1}$ is completely
generated by the vacuum state Reutter:2020bav :

$\displaystyle\begin{split}{\cal
H}\big{[}S^{d-1}\big{]}\cong\mathbb{C}\big{[}\big{|}0\big{\rangle}\big{]}~{}.\end{split}$
(10)

This follows from the fact that there is a unique local operator that is
topologically invariant – the identity operator – which on $D^{d}$ generates
the vacuum state.

We can additionally cut $S^{d}$ along $S^{d-p-1}\times S^{p}$:

$\displaystyle\begin{split}S^{d}=\overline{M_{1}}\\#M_{2}:=\Big{(}\overline{D^{d-p}\times
S^{p}}\Big{)}\\#\Big{(}S^{d-p-1}\times D^{p+1}\Big{)}~{}.\end{split}$ (11)

This means that for any TQFT the states created by evaluating the path
integral on $M_{1},M_{2}$ are both non-trivial:

$\displaystyle\begin{split}Z_{\rm
TQFT}\big{[}S^{d}\big{]}=\big{\langle}0\big{|}0\big{\rangle}=Z_{\rm
TQFT}[\overline{M_{1}}\\#M_{2}]=\big{\langle}M_{1}\big{|}M_{2}\big{\rangle}=1~{}.\end{split}$
(12)

This further means that

$\displaystyle\begin{split}Z_{\rm TQFT}[S^{d-p-1}\times
S^{p+1}]=\big{\langle}M_{2}\big{|}M_{2}\big{\rangle}>0\end{split}$ (13)

and analogously for $M_{1}$.

We will also need the fact that if the internal global symmetry group $G$ is
not spontaneously broken, the partition function on $S^{d-p-1}\times S^{p+1}$
is non-zero for non-trivial background gauge fields:

$\displaystyle\begin{split}Z[S^{d-p-q}\times S^{p+1};A]\neq 0~{}.\end{split}$
(14)

For the moment, we will assume this – we will reproduce the proof from
Cordova:2019bsd  in the next section for completeness.

Now let us analyze a QFT that has an anomaly is that described by a
$(d+1)$-dimensional SPT phase $\omega_{d+1}(A)\in H^{d+1}(BG,U(1))$. The phase
that the partition function picks up under a constant background gauge
transformation defined by $g\in G$: $A\mapsto A^{(g)}=A$ is identical to the
phase produced by the anomaly polynomial on the product manifold
$N_{d+1}^{(g)}=\big{(}S^{d-p-1}\times S^{p+1}\big{)}\rtimes_{g}S^{1}$ where we
impose twisted boundary conditions for the principal $G$-bundle on
$S^{d-p-1}\times S^{p+1}$ along the $S^{1}$:

$\displaystyle\begin{split}Z_{\rm
QFT}\Big{[}S^{d-p-1}\times&S^{p+1};A^{(g)}\Big{]}={\rm
exp}\left\\{i\int_{N_{d+1}^{(g)}}\omega_{d+1}(A)\right\\}\times Z_{\rm
QFT}\Big{[}S^{d-p-1}\times S^{p+1};A\Big{]}~{},\end{split}$ (15)

where $\omega_{d+1}(A)$ is integrated over $N_{d+1}^{(g)}$. Since the twist
$g$ is a flat background transformation which acts trivially on the gauge
field for discrete abelian symmetries, we see that either

$\displaystyle\begin{split}{\rm
exp}\left\\{i\int_{N_{d+1}^{(g)}}\omega_{d+1}(A)\right\\}=1\quad{\rm or}\quad
Z_{\rm QFT}[S^{d-p-1}\times S^{p+1},A]=0~{}.\end{split}$ (16)

For a TQFT, the fact that $g$ acts trivially together with (14) implies that
the anomaly must satisfy

$\displaystyle\begin{split}{\rm
exp}\left\\{i\int_{N_{d+1}^{(g)}}\omega_{d+1}(A)\right\\}=1~{}.\end{split}$
(17)

This means that if

$\displaystyle\begin{split}{\rm
exp}\left\\{i\int_{N_{d+1}^{(g)}}\omega_{d+1}(A)\right\\}\neq
1\quad\Longrightarrow\quad Z_{\rm QFT}\Big{[}S^{d-p-1}\times
S^{p+1};A\Big{]}=0~{}.\end{split}$ (18)

Since we know that for a unitary, symmetry preserving TQFT, $Z_{\rm
TQFT}\Big{[}S^{d-p-1}\times S^{p+1};A\Big{]}\neq 0$, we must have that the IR
theory cannot be matched by such a TQFT, and hence the theory cannot flow to a
gapped phase.

###  2.1 Proof of (14)

Now we will prove $Z[S^{d-p-1}\times S^{p+1}]\neq 0$ implies that
$Z[S^{d-p-1}\times S^{p+1};A]\neq 0$ if the associated internal global
symmetry is not spontaneously broken. We will do this in steps. First, we will
prove the following Lemma.

Lemma:  _Let us consider a symmetry defect operator for a symmetry group $G$
wrapped on $S^{d-p}$: $U_{g}\Big{(}S^{d-p-1}\times\\{\rm pt\\}\Big{)}$ and let
us assume that $G$ is not spontaneously broken. Then, the symmetry defect
operator can be replaced by wrapping a corresponding symmetry defect operator
on a disk $D^{d-p-1}$ where we additionally impose a boundary condition $\phi$
on the boundary: _

$\displaystyle\begin{split}\big{\langle}U_{g}\Big{(}S^{d-p-1}\times\\{{\rm
pt}\\}\Big{)}...\big{\rangle}=\big{\langle}U_{g}^{(\phi)}\Big{(}D^{d-p-q}\times\\{{\rm
pt}\\}\Big{)}...\big{\rangle}~{}.\end{split}$ (19)

pf. First, if a $p$-form symmetry defect operator $U_{g}$ corresponds to a
preserved symmetry $G^{(p)}$, then $U_{g}$ has trivial winding with any non-
trivial operator on $S^{d-p-1}\times S^{p+1}$. This follows from the
assumption that $G$ is preserved, as this implies no charged operator has a
non-trivial expectation value. Now we see that we can cut the space-time
$S^{d-p-1}\times S^{p+1}$ along a $D^{d}$ surrounding a point on $U_{g}$:

$\displaystyle\begin{split}\big{\langle}U_{g}(S^{d-p-1}\times\\{{\rm
pt}\\})\big{\rangle}=\big{\langle}0_{g}\big{|}U_{g}\big{\rangle}\end{split}$
(20)

where

$\displaystyle\begin{split}\big{|}U_{g}\big{\rangle}:=Z[D^{d-p-1}_{U_{g}}\times
S^{p+1}]~{},\end{split}$ (21)

is the path integral on $D^{d-p-1}\times S^{p+1}$ with a defect operator
inserted on $D^{d-p-1}$ and $\big{|}0_{g}\big{\rangle}$ is the state on
$S^{d-1}$ created by evaluating the partition function on $D^{d}$ where
$U_{g}$ is wrapped on a $D^{d-p-1}$-disk.

We can show that the state $\big{|}0_{g}\big{\rangle}$ is non-trivial by
computing its norm:

$\displaystyle\begin{split}\big{\langle}0_{g}\big{|}0_{g}\big{\rangle}=\big{\langle}U_{g}(S^{d-p-1})\big{\rangle}_{S^{d}}=Z\big{[}S^{d}\big{]}=1~{},\end{split}$
(22)

since $S^{d-p-1}\subset S^{d}$ is homologous to a point. Further, since ${\cal
H}\big{[}S^{d-1}\big{]}=\mathbb{C}\big{[}\big{|}0\big{\rangle}\big{]}$,
$\big{|}0_{g}\big{\rangle}=\big{|}0\big{\rangle}$. This implies that

$\displaystyle\begin{split}\big{\langle}U_{g}(S^{d-p-1}\times\\{{\rm
pt}\\})\big{\rangle}=\big{\langle}0_{g}\big{|}U_{g}\big{\rangle}=\big{\langle}0\big{|}U_{g}\big{\rangle}=\big{\langle}U_{g}^{(\phi)}(D^{d-p-q}\times\\{pt\\})\big{\rangle}\end{split}$
(23)

and that we are simply cut a hole in the symmetry defect operator. The
argument follows similarly for correlation functions involving other operators
as long as we cut out a $d$-ball surrounding a point on $U_{g}$ that is away
from any charged operators. While we have not explicitly constructed the
boundary condition $\phi$, this proves the Lemma. For more details and the
explicit construction of $\phi$, see Cordova:2019bsd ; Cordova:2019jqi . ∎

Using this Lemma, we can prove (14). Consider the partition function
$Z[S^{d-p-1}\times S^{p+1};A]$ where the background gauge fields are non-
trivial along $S^{d-p-1}$ and $S^{p+1}$. This is equivalent to wrapping
symmetry defect operators $U_{g}$ on $S^{p+1}$ and $\tilde{U}_{g^{\prime}}$ on
$S^{d-p-1}$:

$\displaystyle\begin{split}Z[S^{d-p-1}\times
S^{p+1};A]=\big{\langle}U_{g}(S^{d-p-2}\times\\{{\rm
pt}\\})\,\widetilde{U}_{g^{\prime}}(\\{{\rm pt}\\}\times
S^{p+1})\big{\rangle}~{}.\end{split}$ (24)

Using the lemma, we can cut these symmetry defect operators

$\displaystyle\begin{split}Z[S^{d-p-1}\times
S^{p+1};A]=\big{\langle}U_{g}^{(\phi)}(D^{d-p-2}\times\\{{\rm
pt}\\})\,\widetilde{U}_{g^{\prime}}^{(\tilde{\phi})}(\\{{\rm pt}\\}\times
D^{p+1})\big{\rangle}~{},\end{split}$ (25)

and by their topological nature, contract them to an infinitesimally small
ball. We can then cut out a $D^{d}$ surrounding the contracted symmetry defect
operators and replacing it with a boundary condition $\Phi$ which corresponds
to a state $\big{|}\Phi\big{\rangle}$ in the Hilbert space on $S^{d-1}$:
${\cal H}\big{[}S^{d-1}\big{]}$:

$\displaystyle\begin{split}Z[S^{d-p-1}\times
S^{p+1};A]=\big{\langle}0\big{|}\Phi\big{\rangle}~{}.\end{split}$ (26)

However, following from our previous discussion, unitarity implies that the
state $\big{|}\Phi\big{\rangle}$ is proportional to the identity state
$\big{|}\Phi\big{\rangle}\sim\big{|}0\big{\rangle}$. We can additionally see
that this is non-zero, by computing its norm

$\displaystyle\begin{split}\big{\langle}\Phi\big{|}\Phi\big{\rangle}=\big{\langle}U_{g}(S^{d-p-2}\times\\{{\rm
pt}\\})\,\widetilde{U}_{g^{\prime}}(\\{{\rm pt}\\}\times
S^{p+1})\big{\rangle}_{S^{d}}=\big{\langle}\mathds{1}\big{\rangle}_{S^{d}}=Z\big{[}S^{d}\big{]}=1~{},\end{split}$
(27)

where here we used the fact that $S^{d-p-2}$ and $S^{p+1}$ are contractible in
$S^{d}$. Now, since $\big{|}\Phi\big{\rangle}\propto\big{|}0\big{\rangle}$, we
see that

$\displaystyle\begin{split}Z[S^{d-p-1}\times S^{p+1};A]\propto
Z[S^{d-p-1}\times S^{p+1}]\neq 0~{},\end{split}$ (28)

thus proving (14).

##  3 Anomaly Enforced Gaplessness for $Spin_{G}$ Symmetries

A salient feature of the result of Cordova:2019bsd ; Cordova:2019jqi , which
is reproduced in the previous section, is that all spheres and their products
are spin manifolds. This means that such constraints are insensitive to
${\mathbb{Z}}_{2}$-valued anomalies of $Spin_{G}$ symmetries that couple to
the spin structure. Indeed, many ${\mathbb{Z}}_{2}$-valued anomalies trivially
evaluate on the product of spheres due to the fact that they are spin
manifolds.

To be clear, by $Spin_{G}$-symmetries we mean the following. Consider a theory
that transforms faithfully under an internal symmetry group $G$. In general,
we take $G$ to be a product of higher form symmetry groups

$\displaystyle\begin{split}G=G_{0}\times G_{1}=G_{0}\times\left(G^{(0)}\times
G^{(1)}\times...\times G^{(n)}\right)~{},\end{split}$ (29)

where $G^{(p)}$ are $p$-form global symmetries. Here we assume that $G_{1}$ is
discrete and that $G_{0}$ is an additional 0-form global symmetry.

We refer to $Spin_{G}$-symmetries as ones with a faithful symmetry group444It
is also interesting to consider the case of higher group and non-invertible
global symmetries in which $(-1)^{F}$ is identified with a component of the
general symmetry structure. Here we will only focus on the case of higher
group global symmetries and leave the more general story for future
investigation.

$\displaystyle\begin{split}G_{\rm total}=\frac{G\times
Spin(d)}{{\mathbb{Z}}_{2}}~{},\end{split}$ (30)

where the ${\mathbb{Z}}_{2}$ acts as $(-1)^{F}$ in $Spin(d)$ and acts on
$G_{0}\subset G$. We call theories where the ${\mathbb{Z}}_{2}$ acts trivially
on $G$ as “bosonic” and their symmetries as bosonic symmetries. Bosonic
symmetries are a special case of $Spin_{G_{0}}$-symmetry with trivial $G_{0}$
structure: where $G_{0}$ is the trivial group.

Here the structure of the symmetry group allows us to put the theory on a non-
spin manifold. Bosonic theories can be naturally put on non-spin manifolds.
For the more general case where ${\mathbb{Z}}_{2}$ acts on both $Spin(d)$ and
$G$, the theory can only be put on a non-spin manifold by elevating the
$G$-bundle to a $Spin_{G}$-bundle as described in Wang:2018qoy . This
restricts $G$ to $G/{\mathbb{Z}}_{2}$ bundles with fixed obstruction class
$w_{2}(G)\in H^{2}(BG/{\mathbb{Z}}_{2};{\mathbb{Z}}_{2})$ such that
$w_{2}(TM)=w_{2}(G)$.

In the following analysis we will show how the previous section can be
generalized to encompass the anomalies of $Spin_{G}$-symmetries that are only
activated on non-spin manifolds. In order to generalize the previous results,
it is important to highlight what elements of the previous derivation relied
on evaluating the anomaly on the mapping class torus of a product of spheres.
As it turns out, the only (independent) statement that relied on the exact
topology of the product of spheres was to show that $Z[S^{p+1}\times
S^{d-p-1}]\neq 0$.

Thus, in order to probe the anomalies that are only activated on non-spin
manifolds, we need to find a $d$-dimensional non-spin manifold $M_{d}$ such
that

$\displaystyle\begin{split}Z[M_{d}]\neq 0~{}.\end{split}$ (31)

Given such a manifold, the other steps of proving anomaly enforced gaplessness
proceed as in the case of $S^{d-p-1}\times S^{p+1}$. Using the result from
Section 2.1, one can then show that if $G$ is not spontaneously broken, then
$Z_{TQFT}[M_{d};0]\neq 0$ implies $Z_{TQFT}[M_{d};A]\neq 0$, and then use the
mapping class argument to show that

$\displaystyle\begin{split}Z_{QFT}[M_{d};A]={\rm
exp}\left\\{i\int_{N_{d+1}^{(g)}}\omega_{d+1}(A)\right\\}Z_{QFT}[M_{d};A]\end{split}$
(32)

which again would imply that either

$\displaystyle\begin{split}{\rm
exp}\left\\{i\int_{N_{d+1}^{(g)}}\omega_{d+1}(A)\right\\}=1\quad{\rm or}\quad
Z_{QFT}[M_{d}]=0~{}.\end{split}$ (33)

Therefore, the non-trivial anomaly would imply that the theory is gapless or
that $G$ is spontaneously broken up to other possible assumptions.

It then remains to pick an appropriate non-spin manifold $M_{d}$ such that
under some conditions $Z_{TQFT}[M_{d}]\neq 0$ for unitary TQFTs.

###  3.1 $Z_{TQFT}$ on non-Spin Manifolds

A particularly nice choice of $M_{d}$ non-spin is

$\displaystyle\begin{split}M_{d}=\mathbb{CP}^{2}\times S^{p+1}~{}.\end{split}$
(34)

As we will show in this subsection, for any unitary TQFT,
$Z_{TQFT}[\mathbb{CP}^{2}\times S^{p+1}]\neq 0$ when we turn on appropriate
fluxes for the $Spin_{G}$-structure.

First we will need the following proposition from Reutter:2020bav .

Proposition: _For a unitary $d$-dimensional TQFT, then the partition function
$Z$ is multiplicative under connected sums: _

$\displaystyle\begin{split}Z[M\\#N]=Z[M]\,Z[N]~{}.\end{split}$ (35)

pf. Under the decomposition

$\displaystyle\begin{split}M\\#N=\big{(}M\backslash
D^{d}\big{)}\bigcup_{S^{d-1}}\big{(}N\backslash D^{d}\big{)}\end{split}$ (36)

we can write the partition function of the connected sum as

$\displaystyle\begin{split}Z\big{[}M\\#N\big{]}=Z\big{[}M\backslash
D^{d}\big{]}Z\big{[}N\backslash
D^{d}\big{]}=\big{\langle}\overline{M\backslash D^{d}}\big{|}N\backslash
D^{d}\big{\rangle}~{}.\end{split}$ (37)

Now since ${\cal
H}\big{[}S^{d-1}\big{]}\cong\mathbb{C}\left[\big{|}D^{d}\big{\rangle}\right]$,
we can insert a complete set of states:

$\displaystyle\begin{split}Z\big{[}M\\#N\big{]}&=\Big{\langle}\overline{M\backslash
D^{d}}\Big{|}\frac{\big{|}D^{d}\big{\rangle}\big{\langle}D^{d}\big{|}}{\big{\langle}D^{d}\big{|}D^{d}\big{\rangle}}\Big{|}N\backslash
D^{d}\Big{\rangle}=\frac{1}{Z\big{[}S^{d}\big{]}}\big{\langle}\overline{M\backslash
D^{d}}\big{|}D^{d}\big{\rangle}\big{\langle}D^{d}\big{|}N\backslash
D^{d}\big{\rangle}\\\
&=\frac{Z\big{[}M\big{]}Z\big{[}N\big{]}}{Z\big{[}S^{d}\big{]}}=Z\big{[}M\big{]}\,Z\big{[}N\big{]}~{}.\end{split}$
(38)

Thus proving the proposition. ∎

###  3.2 Generalized Gluck Twist and Symmetry Fractionalization

Now we would like to consider the case of a $4d$ TQFT on the manifold
$\mathbb{CP}^{2}\\#\overline{\mathbb{CP}^{2}}$. This manifold can be obtained
by taking two copies of $S^{2}\times D^{2}$ and gluing them together with a
twist called the “Gluck twist” Glu62 ; Reutter:2020bav :

$\displaystyle\begin{split}(\vec{x},t)\in S^{2}\rtimes
S^{1}\sim(R_{z}(2\pi)\cdot\vec{x},t+2\pi)~{}.\end{split}$ (39)

This means that we can identify

$\displaystyle\begin{split}Z\big{[}\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2}\big{]}=\big{\langle}S^{2}\times
D^{2}\big{|}\widehat{{\cal T}}\big{|}S^{2}\times
D^{2}\big{\rangle}~{},\end{split}$ (40)

where $\widehat{{\cal T}}$ is the “Gluck twist operator” that enacts the Gluck
twist. We then have the result that if $\widehat{{\cal T}}$ acts trivially,
then

$\displaystyle\begin{split}Z\big{[}\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2}\big{]}=\big{\langle}S^{2}\times
D^{2}\big{|}\widehat{{\cal T}}\big{|}S^{2}\times
D^{2}\big{\rangle}=\big{\langle}S^{2}\times D^{2}\big{|}S^{2}\times
D^{2}\big{\rangle}=Z\big{[}S^{2}\times S^{2}\big{]}>0~{}.\end{split}$ (41)

We then see that using the proposition, that the property
$Z\big{[}\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2}\big{]}>0$ implies

$\displaystyle\begin{split}Z\big{[}\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2}\big{]}=\frac{1}{Z\big{[}S^{4}\big{]}}Z\big{[}\mathbb{CP}^{2}\big{]}Z\big{[}\overline{\mathbb{CP}}^{2}\big{]}>0\quad\Longrightarrow\quad
Z\big{[}\mathbb{CP}^{2}\big{]}\neq 0~{}.\end{split}$ (42)

Let us be more precise about the meaning of the above statement. Consider a
$4$-dimensional unitary TQFT coupled to a $Spin_{G}$-bundle: i.e. a $4d$
unitary TQFT with $Spin_{G}$-structure. Let us consider the Hilbert space on
the boundary of $S^{2}\times D^{2}$: ${\cal H}[\partial(S^{2}\times
D^{2})]={\cal H}[S^{2}\times S^{1}]$. This Hilbert space is generated by the
set of local operators, line operators wrapping $S^{1}$, and surface operators
wrapping $S^{2}$.

In terms of symmetries, the Gluck twist can be restated as a twist by
$(-1)^{F}$ in the $S^{2}$ direction along the $S^{1}$. Because of this,
$\widehat{{\cal T}}$ will only act non-trivially on states that transform as a
spinor on the $S^{2}$ factor. The operators that are acted on non-trivially by
the twist are the local operators and the line operators wrapping the $S^{1}$.

For bosonic theories, there are no local operators that transform non-
trivially under $\widehat{{\cal T}}$. However, for the theory with
$Spin_{G}$-symmetry, the standard Gluck twist will act non-trivially. The
reason is that the structure of $\frac{G\times Spin(4)}{{\mathbb{Z}}_{2}}$ in
general identifies $(-1)^{F}$ with some generator $-\mathds{1}_{G}$ of a
${\mathbb{Z}}_{2}\subset G$. By assumption, this quotient can be reduced to an
action of ${\mathbb{Z}}_{2}\subset G_{0}$ generated by $-\mathds{1}_{G_{0}}$.
This means that there is some local operator that transforms non-trivially
under $(-1)^{F}$, although it transforms trivially under
$(-1)^{F}\circ(-\mathds{1}_{G_{0}})$. Instead, for $Spin_{G}$-symmetries, we
can construct the generalized Gluck twist operator $\widehat{{\cal T}}_{G}$
which twists by the combined action of $(-1)^{F}\circ(-\mathds{1}_{G_{0}})$.
Because the quotient acts at most on the 0-form component $G_{0}\subset G$,
there are no local operators that transform non-trivially under the
combination of $(-1)^{F}\circ(-\mathds{1}_{G_{0}})$.

Even though local operators do not transform under $\widehat{{\cal T}}_{G}$
for $Spin_{G}$-symmetries or $\widehat{{\cal T}}$ for bosonic symmetries, the
line operators can transform non-trivially under
$(-1)^{F}\circ(-\mathds{1}_{G_{0}})$ or $(-1)^{F}$ respectively. This is the
statement that the $Spin_{G_{0}}$-symmetry can fractionalize Delmastro:2022pfo
; Brennan:2022tyl ; Barkeshli:2014cna ; Chen:2014wse . As we have discussed in
the introduction, the relevant symmetry fractionalization for this setting is
that line operators (which are usually only acted on by 1-form global
symmetries) are acted on by a 0-form global symmetry which does not act on any
local operators. Because of the structure of the symmetry group, no other
operators can transform under $(-1)^{F}\circ(-\mathds{1}_{G_{0}})$ or
$(-1)^{F}$ respectively. This example of symmetry fractionalization has a
simple interpretation in terms of UV physics. If we think of the line
operators as being the world-line of some heavy particle, then the symmetry
fractionalization can be thought of as describing a heavy particle that
transforms projectively under the 0-form symmetry group.

Let us focus on the case of $Spin_{G}$-symmetry which is not bosonic. Gluing
two copies of $S^{2}\times D^{2}$ with the generalized Gluck twist will
generate $\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2}$ coupled to a fixed,
non-trivial $G_{0}/{\mathbb{Z}}_{2}$-bundle which has a discrete 2-form flux
$w_{2}(G_{0})=w_{2}(\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2})$:

$\displaystyle\begin{split}\big{\langle}D^{2}\times S^{2}\big{|}\widehat{{\cal
T}}_{G}\big{|}D^{2}\times
S^{2}\big{\rangle}=Z[\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2};w_{2}(G_{0})]~{}.\end{split}$
(43)

Now, if the generalized Gluck twist operator acts non-trivially on any line
operator $\widehat{L}$ that generates a state in ${\cal H}[S^{2}\times
S^{1}]$, then $\widehat{L}$ will also necessarily have non-vanishing
expectation value on $S^{4}$, spontaneously breaking the
${\mathbb{Z}}_{2}^{(1)}$ symmetry. To see this, let us assume there is a state
$|\psi\rangle\in{\cal H}\big{[}S^{2}\times S^{1}]$ that transforms non-
trivially under $\widehat{{\cal T}}_{G}$ which is constructed by inserting a
charged/fractionalized line operator $\widehat{L}$ in the path integral on
$D^{3}\times S^{1}:$

$\displaystyle\begin{split}\widehat{{\cal
T}}_{G}|\psi\rangle=-|\psi\rangle\quad,\quad|\psi\rangle=\widehat{L}\,|D^{3}\times
S^{1}\rangle~{}.\end{split}$ (44)

We can now compute the inner product

$\displaystyle\begin{split}\langle S^{2}\times D^{2}|\psi\rangle=\langle
S^{2}\times D^{2}|\widehat{L}|D^{3}\times
S^{1}\rangle=\langle\widehat{L}\rangle_{S^{4}}=\langle\mathds{1}\rangle_{S^{4}}=1~{},\end{split}$
(45)

where here we used the fact that $\pi_{1}(S^{4})=0$. This implies that if the
generalized Gluck operator $\widehat{{\cal T}}_{G}$ acts non-trivially on some
state on ${\cal H}[S^{2}\times S^{1}]$, then the theory will experience
symmetry fractionalization with respect to
$(-1)^{F}\circ(-\mathds{1}_{G_{0}})$. In other words, the generalized Gluck
twist acts trivially in a theory without $Spin_{G}$-symmetry
fractionalization.

Therefore, in a $4d$ unitary TQFT without $Spin_{G}$-symmetry
fractionalization we have

$\displaystyle\begin{split}Z_{{\rm
Spin}_{G}}\big{[}\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2};w_{2}(G_{0})\big{]}=Z_{{\rm
Spin}_{G}}\big{[}S^{2}\times S^{2}\big{]}\neq 0~{},\end{split}$ (46)

which implies that

$\displaystyle\begin{split}Z_{{\rm
Spin}_{G}}\big{[}\mathbb{CP}^{2};w_{2}(G_{0})\big{]}\neq 0~{}.\end{split}$
(47)

In the case of bosonic theories, the above results generalize straight-
forwardly and the analogous statement is that in a bosonic $4d$ TQFT without
$(-1)^{F}$-symmetry fractionalization

$\displaystyle\begin{split}Z_{\rm bos}[\mathbb{CP}^{2};0]\neq
0~{}.\end{split}$ (48)

###  3.3 Anomaly Enforced Gaplessness for $Spin_{G}$ Symmetries in General
Dimensions

Now we are prepared to prove our main result. As in Section 2, the partition
function of a $d$-dimensional unitary TQFT has a non-trivial partition
function on $S^{d}$: $Z\big{[}S^{d}\big{]}=1$. Let us consider the case where
$d\geq 4$.

Let us now consider the case of $Spin_{G}$-symmetries. Again, by cutting the
sphere, along a $S^{3}\times S^{p+1}$ we can deduce that $Z\big{[}S^{4}\times
S^{p+1}\big{]}\neq 0$. We can again cut along $S^{2}\times S^{1}\times
S^{p+1}$:

$\displaystyle\begin{split}S^{4}\times S^{p+1}=(\overline{D^{3}\times
S^{1}\times S^{p+1}})\\#(S^{2}\times D^{2}\times S^{p+1})\end{split}$ (49)

to see that

$\displaystyle\begin{split}Z_{{\rm Spin}_{G}}\big{[}S^{4}\times
S^{p+1}\big{]}=\big{\langle}D^{3}\times S^{1}\times S^{p+1}\big{|}S^{2}\times
D^{2}\times S^{p+1}\big{\rangle}\neq 0\end{split}$ (50)

which further implies that $Z_{{\rm Spin}_{G}}\big{[}S^{2}\times S^{2}\times
S^{p+1}\big{]}\neq 0$.

From our discussion of the generalized Gluck twist, we now can see that

$\displaystyle\begin{split}Z_{{\rm
Spin}_{G}}\big{[}(\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2})\times
S^{p+1};w_{2}(G_{0})\big{]}=\big{\langle}D^{2}\times S^{2}\times
S^{p+1}\big{|}\widehat{{\cal T}}_{G}\big{|}D^{2}\times S^{2}\times
S^{p+1}\big{\rangle}\end{split}$ (51)

where $\widehat{{\cal T}}_{G}=(-1)^{F}\circ(-\mathds{1}_{G_{0}})$. In the case
where this acts trivially on the vacuum – i.e. there is no $Spin_{G}$-symmetry
fractionalization – we then have the relation:

$\displaystyle\begin{split}Z_{{\rm
Spin}_{G}}\big{[}(\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2})\times
S^{p+1};w_{2}(G_{0})\big{]}=Z_{{\rm Spin}_{G}}\big{[}S^{2}\times S^{2}\times
S^{p+1}\big{]}\neq 0~{}.\end{split}$ (52)

We can then use the Proposition (35) to deduce

$\displaystyle\begin{split}Z_{{\rm
Spin}_{G}}\big{[}(\mathbb{CP}^{2}\\#\overline{\mathbb{CP}}^{2})\times
S^{p+1};w_{2}(G_{0})\big{]}&=Z_{{\rm Spin}_{G}}\big{[}\mathbb{CP}^{2}\times
S^{p+1};w_{2}(G_{0})\big{]}\,Z_{{\rm
Spin}_{G}}\big{[}\overline{\mathbb{CP}}^{2}\times
S^{p+1};w_{2}(G_{0})\big{]}\\\ &\neq 0\end{split}$ (53)

which then implies

$\displaystyle\begin{split}Z_{{\rm Spin}_{G}}\big{[}\mathbb{CP}^{2}\times
S^{p+1};w_{2}(G_{0})\big{]}\neq 0~{}.\end{split}$ (54)

Now, if we further assume that the $G_{1}$ symmetry is not spontaneously
broken, then we can use the Lemma from Section 2.1 to show that the partition
function on $\mathbb{CP}^{2}\times S^{p+1}$ is non-zero when we turn on
(additional) background gauge fields $\\{A\\}$ for $G_{1}$:

$\displaystyle\begin{split}Z_{{\rm Spin}_{G}}\big{[}\mathbb{CP}^{2}\times
S^{p+1};w_{2}(G_{0}),A\big{]}\neq 0~{}.\end{split}$ (55)

The case of bosonic symmetries follows directly from the above result by
replacing the generalized Gluck twist with the standard Gluck twist:
$\widehat{{\cal T}}_{G}\mapsto\widehat{{\cal T}}$. As discussed above, this
has effect of not turning on the discrete $w_{2}(G_{0})$ flux which leads to
the result:

$\displaystyle\begin{split}Z_{\rm bos}\big{[}\mathbb{CP}^{2}\times
S^{p+1};A\big{]}\neq 0~{}.\end{split}$ (56)

Now we can analyze a QFT that has an anomaly described by the
$(d+1)$-dimensional cohomology class $\omega_{d+1}(A)$ where here we use $A$
to denote the background fields as well as dependence on the Stiefel-Whitney
classes of $\mathbb{CP}^{2}\times S^{p+1}$ and corresponding
$Spin_{G}$-structure specified by $w_{2}(G_{0})$. Again, the phase that the
partition function picks up under a constant background gauge transformation
$g\in G$ is identical to the evaluation of $\omega_{d+1}(A)$ on the product
manifold $N_{d+1}^{(g)}=(\mathbb{CP}^{2}\times S^{p+1})\rtimes_{g}S^{1}$ where
we twist the $Spin_{G}$ bundle along the $S^{1}$ direction by $g$:

$\displaystyle\begin{split}Z_{QFT}\big{[}\mathbb{CP}^{2}\times
S^{p+1};A^{(g)}\big{]}={\rm
exp}\left\\{i\oint_{N_{d+1}^{(g)}}\omega_{d+1}(A)\right\\}\times
Z_{QFT}\big{[}\mathbb{CP}^{2}\times S^{p+1};A\big{]}~{}.\end{split}$ (57)

Again, since $A^{(g)}=A$, we can see that either

$\displaystyle\begin{split}{\rm
exp}\left\\{i\oint_{N_{d+1}^{(g)}}\omega_{d+1}(A)\right\\}=1\quad{\rm or}\quad
Z_{QFT}\big{[}\mathbb{CP}^{2}\times S^{p+1};A\big{]}=0~{}.\end{split}$ (58)

If this anomaly is to be matched by a symmetry preserving, unitary TQFT
without $Spin_{G}$ symmetry fractionalization, then (55) implies that

$\displaystyle\begin{split}{\rm
exp}\left\\{i\oint_{N_{d+1}^{(g)}}\omega_{d+1}(A)\right\\}=1~{}.\end{split}$
(59)

This means that if

$\displaystyle\begin{split}{\rm
exp}\left\\{i\oint_{N_{d+1}^{(g)}}\omega_{d+1}(A)\right\\}\neq 1\end{split}$
(60)

then $Z_{QFT}\big{[}\mathbb{CP}^{2}\times S^{p+1};A\big{]}=0$ which cannot be
matched by a unitary, symmetry preserving TQFT unless it spontaneously breaks
$G$ or exhibits $Spin_{G}$ symmetry fractionalization. This implies that the
theory cannot flow in the IR to a symmetry preserving gapped phase (i.e.
symmetry preserving unitary TQFT). This proves our main result.

##  4 Examples in $4d$

We will now describe some examples of anomalies in $4d$ theories that obstruct
symmetry preserving gapped phases. Here we will only present examples for
which the constraint arises from the theory on a non-spin manifold. Two such
anomalies are given by:

$\displaystyle\begin{split}{\cal A}_{1}=\pi i\int x_{1}\cup y_{2}\cup
w_{2}(TM)\quad,\quad{\cal A}_{2}=\pi i\int w_{2}(TM)\cup z_{3}~{}.\end{split}$
(61)

We will refer to these anomalies as “Type 1” and “Type 2”
${\mathbb{Z}}_{2}$-anomalies according to their subscript.

###  4.1 Type 1 Anomaly

Here we will study the Type 1 ${\mathbb{Z}}_{2}$-anomaly. This anomaly
describes a mixed anomaly between a 0-form ${\mathbb{Z}}_{2}^{(0)}$-symmetry
with background gauge field $x_{1}$, 1-form ${\mathbb{Z}}_{2}^{(1)}$-symmetry
with background gauge field $y_{2}$, and gravity.

This anomaly can be activated on the $5d$ mapping class torus
$N_{5}=\mathbb{CP}^{2}\rtimes_{-\mathds{1}}S^{1}$ where we activate
$y_{2},w_{2}(TM)$ as the generator of
$H^{2}(\mathbb{CP}^{2};{\mathbb{Z}}_{2})$ and twist the 0-form symmetry by
$-\mathds{1}$ along the $S^{1}$: i.e. pick $x_{1}$ so that
$H^{1}(S^{1};{\mathbb{Z}}_{2})={\mathbb{Z}}_{2}[x_{1}]$. With this choice of
background gauge fields we can evaluate

$\displaystyle\begin{split}{\rm exp}\left\\{\pi i\int_{N_{5}}x_{1}\cup
y_{2}\cup w_{2}(TM)\right\\}={\rm exp}\left\\{\pi
i\int_{\mathbb{CP}^{2}}w_{2}(TM)\cup w_{2}(TM)\right\\}=e^{\pi
i}=-1~{}.\end{split}$ (62)

This anomaly can be matched by a $4d$ ${\mathbb{Z}}_{2}$-gauge theory with
action

$\displaystyle\begin{split}S=\pi i\int\big{(}b_{2}\cup\beta(a_{1})+b_{2}\cup
w_{2}(TM)+x_{1}\cup y_{2}\cup a_{1}\big{)}~{},\end{split}$ (63)

where $a_{1},b_{2}$ are dynamical ${\mathbb{Z}}_{2}$-valued 1- and 2-form
fields. However, this TQFT spontaneously breaks $(-1)^{F}$ symmetry. This can
be seen explicitly by considering the effect of the middle term

$\displaystyle\begin{split}S=...+\pi i\int b_{2}\cup w_{2}(TM)~{}.\end{split}$
(64)

This theory has a single non-trivial line operator and a single non-trivial
surface operator. The above term has the effect of making the line operator a
fermion which means that $(-1)^{F}$ has fractionalized.555Note that another
way this can be matched by a ${\mathbb{Z}}_{2}$-gauge theory in the IR is by
$\displaystyle\begin{split}S^{\prime}=\pi
i\int\big{(}b_{2}\cup\beta(a_{1})+b_{2}\cup y_{2}+x_{1}\cup w_{2}(TM)\cup
a_{1}\big{)}~{},\end{split}$ (65) where again $a_{1},b_{2}$ are dynamical
${\mathbb{Z}}_{2}$-gauge fields. However, now the middle term implies that the
$a_{1}$ Wilson line operator is now charged under the 1-form
${\mathbb{Z}}_{2}^{(1)}$ symmetry which spontaneously breaks the symmetry.

These anomalies arise naturally in the setting of $4d$ QCD-like theories with
a 1-form center symmetry and $Spin_{G}$ structure. In this setting, $U(1)$
0-form symmetries which act on fermions as a “flavor symmetry” are generically
broken to a discrete subgroup $G^{(0)}$ due to ABJ anomalies.666Here we would
like to point out that in the more general setting, these symmetries can
sometimes be expanded to non-invertible symmetries as discussed in
Cordova:2022ieu ; Choi:2022jqy ; Choi:2022rfe ; Choi:2022fgx . When there is
an additional 1-form center symmetry $G^{(1)}$, the ABJ anomaly often leads to
a mixed anomaly between the remaining 0-form symmetry and the 1-form center
symmetry:

$\displaystyle\begin{split}{\cal A}\sim 2\pi i\frac{p}{q}\int x_{1}\cup{\cal
P}(y_{2})\end{split}$ (66)

where here $x_{1}$ is the $G^{(0)}$ background gauge field, ${\cal P}$ is the
Pontryagin square and $y_{2}$ is the $G^{(1)}$ background gauge field. Here
the fraction $\frac{p}{q}$ depends on the global form of the gauge group and
the representations of the fermions.

When $\frac{p}{q}=\frac{1}{2}~{}{\rm mod}_{\mathbb{Z}}$, this anomaly reduces
to a ${\mathbb{Z}}_{2}$ anomaly that can only be activated on a non-spin
manifold. This is because of the series of mod2 relations

$\displaystyle\begin{split}{\cal P}(y_{2})=y_{2}\cup y_{2}~{}{\rm
mod}_{2}=y_{2}\cup w_{2}(TM)~{}{\rm mod}_{2}\end{split}$ (67)

where we used the Wu formula, so that

$\displaystyle\begin{split}{\cal A}=\pi i\int x_{1}\cup y_{2}\cup
w_{2}(TM)\end{split}$ (68)

which is of Type 1.

###  4.2 Type 2 Anomaly

Now we will study the Type 2 ${\mathbb{Z}}_{2}$-anomaly. This anomaly can be
thought of as a mixed anomaly between gravity and a 1-form
${\mathbb{Z}}_{2}^{(1)}$ global symmetry where $z_{3}=\beta(y_{2})$ is the
Bockstein of the ${\mathbb{Z}}_{2}^{(1)}$ background gauge field.

This anomaly is also activated on the $5d$ mapping class torus
$N_{5}=\mathbb{CP}^{2}\rtimes_{-\mathds{1}}S^{1}$ where we twist the
$Spin_{G}$ bundle by $-\mathds{1}_{G}$ so that $z_{3}$ decomposes as

$\displaystyle\begin{split}z_{3}=y_{2}\cup w_{1}\quad,\quad y_{2}\in
H^{2}(\mathbb{CP}^{2};{\mathbb{Z}}_{2})\quad,\quad w_{1}\in
H^{1}(S^{1};{\mathbb{Z}}_{2})~{}.\end{split}$ (69)

See Wang:2014lca ; BI  for an explicit construction of mapping class tori that
activate such an anomaly.

Again, this anomaly can be matched in the IR via a ${\mathbb{Z}}_{2}$-gauge
theory with actoin

$\displaystyle\begin{split}S=\pi i\int\left(b_{2}\cup\beta(a_{1})+b_{2}\cup
w_{2}(TM)+z_{3}\cup a_{1}\right)~{}.\end{split}$ (70)

However, here we again see that the TQFT exhibits $(-1)^{F}$ symmetry
fractionalization.

These types of anomalies also appear in many interesting examples. One of the
simplest is $4d$ $SU(2)$ gauge theory with a Weyl fermion in the 4-dimensional
representation which has an anomaly where $z_{3}=w_{3}(TM)$ Wang:2018qoy .
This theory admits a symmetry preserving deformation that triggers an RG flow
to “all fermion electrodynamics” which is a manifestly gapless theory that
matches the anomaly.

Historically, the first instance of this anomaly was in $4d$ $SU(2)$ ${\cal
N}=2$ supersymmetric Yang-Mills theory where $z_{3}=\beta(y_{2})$ where
$y_{2}$ is the background gauge field for the ${\mathbb{Z}}_{2}^{(1)}$ center
symmetry Donaldson:1983wm ; Donaldson:1990kn ; Witten:1988ze ; Witten:1994cg ;
Moore:1997pc . As shown by Seiberg-Witten Seiberg:1994rs ; Seiberg:1994aj ,
this theory flows to a gapless supersymmetric $U(1)$ gauge theory. This also
has a SUSY breaking, yet symmetry preserving deformation that flows to $4d$
$SU(2)$ non-supersymmetric Yang-Mills theory with two adjoint fermions which
is also believed to flow to a gapless $\mathbb{CP}^{1}$ non-linear sigma model
Shifman:2013yca ; Cordova:2018acb . Note that in these examples, the theory
has a $Spin_{SU(2)}$ structure and $G_{1}={\mathbb{Z}}_{2}^{(1)}$.

Theories with Type 2 anomalies studied recently by Brennan:2022tyl ; BI  in
which the authors have constructed robust methods using RG flows to abelian
phases and index theorems to probe for these anomalies. With the results
presented here, we can further see that any theory which has such an anomaly
must either flow to a spontaneously broken gapped phase, symmetry
fractionalized gapped phase, or to a gapless theory in the deep IR.

## Acknowledgements

The author would like to thank Sungwoo Hong, Ken Intriligator, Po-Shen Hsin,
Zhengdi Sun, Kantaro Ohmori, and Clay Córdova for helpful discussions and
related collaborations. TDB is supported by Simons Foundation award 568420
(Simons Investigator) and award 888994 (The Simons Collaboration on Global
Categorical Symmetries).

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deviations are $>$100$\times$ larger than the basic MBM effects. So any
mechanism that is creating the scatter must be symmetric (positive and
negative), must have the typical size of the scattering effect be
$>$100$\times$ larger than the theory effects, and must average close to zero
in most nova systems.

How accurate or consistent must this time-average be? A minimum criterion is
that the nova-to-nova variation in the time-average must be sufficiently small
so as to produce a fairly sharp-edged Period Gap. Figure 14 of Knigge et al.
(2011) shows that a 2$\times$ variation in the strength of the magnetic
braking component will lead to an unacceptably poor sharpness of the upper
edge of the Period Gap. (To be specific, the edge varies from 2.8 to 3.6 hours
as their scaling parameter varies from 0.25 to 1.0, so any time-averaging that
has a residual variation of 2$\times$ or more will not produce the observed
sharpness of the Period Gap edge.) For CVs just above the Period Gap, the MBM
gives a $\dot{P}$ of $-$5.1$\times$10-13. To keep the edge of the Period Gap
sufficiently narrow, the $\dot{P}$ must be in the range $-$2.5$\times$10-13 to
$-$10$\times$10-13. The width of this range is 7.5$\times$10-13, and any
averaging must return a value with an accuracy better then this limit.

A severe problem arises for the time-averaging idea because the novae do not
have enough time for averaging to get the required small scatter. For my
observations over $\Delta Y$, I see no changes in $\dot{P}$ over intervals up
to 100 years, so novae appear to have episodes of relatively constant period
changes lasting $>$100 years. Presumably, each nova will have many episodes
over the 107 year averaging interval, with these combining to produce a mean
close to zero. The number of episodes is $<$105. Averaging over this many
episodes will reduce the scatter of the mean by a factor of $\sqrt{10^{5}}$.
For a $\dot{P}$ scatter of 10-9 for individual episodes, the
$\langle\dot{P}\rangle$ will have a scatter of 10-11.5. The full range (i.e.,
two-sided) is 6.3$\times$10-12. This is to be compared to the maximum full
range of 7.5$\times$10-13 required to keep narrow the edge of the Period Gap.
Importantly, the best accuracy of averaging $\dot{P}$ over ten-million years
can only narrow the range down to 8.3$\times$ larger than required. Time-
averaging cannot reduce the observed scatter of $\dot{P}$ by enough to explain
the Period Gap.

This severe problem (how CVs average their period changes so as to produce a
narrow-edged Period Gap) is broader than simply time-averaging some unknown
mechanism, labelled as $\dot{P}_{\rm X}$, to have a spread of less than
2$\times$. In general, to keep a narrow-edge, the scatter of
$\langle\dot{P}\rangle$=$\langle\dot{P}_{\rm
total}\rangle$+$\langle\dot{P}_{\rm X}\rangle$ must be kept small both as an
average over time for individual systems plus as an average over all the
individual CVs. For an unknown mechanism, the standard theory cannot predict
the scatter of $\langle\dot{P}_{\rm X}\rangle$. For the $\langle\dot{P}_{\rm
total}\rangle$ contribution, the individual CVs should have only modest
variability over time, so averaging over ten-million years should return small
scatter. However, for standard theory, $\langle\dot{P}_{\rm total}\rangle$
will vary substantially from CV-to-CV, making for a substantial scatter in
$\langle\dot{P}\rangle$. The MBM solves this problem by insisting that for a
given $P$ all CVs have the same stellar masses, the same accretion rates, the
same $\Delta P$, and the same recurrence time-scales101010The failure of these
predictions is so extreme as to constitute a refutation of the MBM scheme of a
single unique evolutionary track for CVs. That is, for any given $P$, the
observed properties vary over a large range, in defiance of the strong
requirements of the MBM model. For CVs just above the Period Gap, $M_{\rm WD}$
is observed to vary from 0.4 to 1.3 M⊙, while $M_{\rm WD}$ is observed to vary
over the range 0.20 to 0.73 M⊙ (Ritter & Kolb 2003; Pala et al. 2022). This
greatly violates the conditions and predictions of the MBM model. There is no
possibility to average out these masses over some hypothetical cycle. Further,
the measured $\dot{M}$ for CVs of the same $P$ varies with a range of around
1000$\times$. For example, just above the Period Gap, the accretion rate has a
range from 10-10.5 to 10-7.5 M⊙ yr-1 (Patterson 1984; Pala et al. 2022). With
this, we know that the basic MBM scheme of a single track for evolution is not
working in practice, instead failing its predictions by up to three orders-of-
magnitude.. The standard theory for $\dot{P}$ recognizes that CV properties
can change greatly from CV-to-CV, and calculates the theoretical $\dot{P}_{\rm
total}$ for each system. The scatter in $\dot{P}_{\rm total}$ will necessarily
be large due to the known CV-to-CV variations in just $M_{\rm WD}$ and $M_{\rm
comp}$. In particular, for a given range of $P$, say just above the Period
Gap, the observed $M_{\rm WD}$ varies roughly uniformly between 0.4–1.3 M⊙,
while $M_{\rm comp}$ varies between 0.20–0.73 M⊙ (Ritter & Kolb 2003; Pala et
al. 2022). For CVs just above the Period Gap, those with well-measured $P$,
$M_{\rm WD}$, and $M_{\rm comp}$ from Ritter & Kolb, I calculate $\dot{P}_{\rm
total}$ assuming that all the other CV properties (including $\dot{M}$,
$M_{\rm ejecta}$, and $\tau_{\rm rec}$) are held constant for all the systems.
With these calculated $\dot{P}_{\rm total}$ values, the total range is
4$\times$. That is, just the known variations in the stellar masses makes for
a variation of 2$\times$ around the $\langle\dot{P}\rangle$ value. This by
itself is enough to smear the edge of the Period Gap to wider than is
observed. And this smearing will not be changed by averaging over time, since
the masses do not change greatly over each averaging interval. The additional
scatter of $\dot{P}_{\rm total}$ caused by the large observed variations in
$\dot{M}$ (and hence variations in $\dot{P}_{\rm ml}$) are greatly larger
still. For CVs just above the Period Gap, the observed $\dot{M}$ varies from
under 10-10.5 to 10-7.5 M⊙ yr-1 (Patterson 1984; Pala et al. 2022), so the
$\dot{P}_{\rm mt}$ varies by up to a factor of 1000$\times$, and
$\dot{P}_{total}$ varies from $-$1$\times$10-12 to $+$2$\times$10-10 for real
CVs with the same $P$. So just ordinary observed CV-to-CV variations of
properties makes for a large scatter of the predicted $\dot{P}_{\rm total}$,
with this being more than large enough to make for the predicted edge of the
Period Gap to be significantly wider than is observed. In all, the averaging
of the standard theory $\dot{P}_{total}$ cannot reduce the scatter enough so
that $\langle\dot{P}\rangle$ can follow any single evolutionary track so as to
produce a narrow edge for the Period Gap. This can only be viewed as a
critical failure of the standard theory, where the key prediction of a Period
Gap is not actually realized from the actual population of CVs.

###  11.6 A new mechanism is required to explain the large $\dot{P}$ scatter

The extreme scatter about the theory predictions forces us to realize that
some unmodelled mechanism is at work for most novae. That is, some additional
mechanism (past the first three in Section 11.1) must be introducing this
scatter. The large deviations from the MBM predictions must arise from some
physical mechanism, variations on the known physical mechanisms cannot explain
the deviations, so the physical mechanism for the deviations is not one of the
modelled mechanisms. The forced conclusion is that there must be some now-
unidentified physical mechanism that is not included in MBM or standard
theory. This strong conclusion has far-reaching implications.

A critical implication is that the MBM and standard theory is incomplete, with
the missing mechanism dominating by orders of magnitude over the modelled
mechanisms. A realist would acknowledge that the MBM calculations are often
off by large factors, but expect that the incomplete standard theory has
captured the basic situation correctly. Realists in our community should be
actively working on the theory of new physical mechanisms, and new
evolutionary models should be constructed with the empirical reality of the
observed period changes. The primary question in CV studies is their
evolution, and this is driven by the period changes, now revealed to be
missing the dominant effects. Thus, the measured $\Delta P$ and $\dot{P}$
should be setting the top of the agenda for future work on CVs.

##  12 ACKNOWLEDGEMENTS

I thank Arto Oksanen, Gordon Myers, Shawn Dvorak, Franz-Josef Hambsch, David
Cejudo Fernandez, Tonny Vanmunster, Bill Goff, James Boardman, Geoffrey Stone,
Libert Monard, Peter Nelson, Richard Sabo, Joseph Ulowetz, Walt Cooney, Gary
Walker, and Lew Cook for their photometry that produced many minimum timings.

I thank Tom Maccarone, Christian Knigge, Saul Rappaport, and Juhan Frank for
valuable, long, and detailed discussions.

##  13 DATA AVAILABILITY

All data are publicly available, either in the cited literature, the cited
databases, or in the tables.

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