iint iiint TXFiint TXFiiint aainstitutetext: Physics Institute, University of
Zurich, Switzerland bbinstitutetext: Department of Mathematical Sciences,
Durham University, United Kingdom ccinstitutetext: Department of Mathematics,
Purdue University, USA

# Toric 2-group anomalies via cobordism

Joe Davighi b and Nakarin Lohitsiri c
with a mathematical appendix by Arun Debray<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

2-group symmetries arise in physics when a 0-form symmetry $G^{[0]}$ and a
1-form symmetry $H^{[1]}$ intertwine, forming a generalised group-like
structure. Specialising to the case where both $G^{[0]}$ and $H^{[1]}$ are
compact, connected, abelian groups (i.e. tori), we analyse anomalies in such
‘toric 2-group symmetries’ using the cobordism classification. As a warm up
example, we use cobordism to study various ’t Hooft anomalies (and the phases
to which they are dual) in Maxwell theory defined on non-spin manifolds. For
our main example, we compute the 5th spin bordism group of $B|\mathbb{G}|$
where $\mathbb{G}$ is any 2-group whose 0-form and 1-form symmetry parts are
both $\mathrm{U}(1)$, and $|\mathbb{G}|$ is the geometric realisation of the
nerve of the 2-group $\mathbb{G}$. By leveraging a variety of algebraic
methods, we show that
$\Omega^{\mathrm{Spin}}_{5}(B|\mathbb{G}|)\cong\mathbb{Z}/m$ where $m$ is the
modulus of the Postnikov class for $\mathbb{G}$, and we reproduce the expected
physics result for anomalies in 2-group symmetries that appear in 4d QED.
Moving down two dimensions, we recap that any (anomalous) $\mathrm{U}(1)$
global symmetry in 2d can be enhanced to a toric 2-group symmetry, before
showing that its associated local anomaly reduces to at most an order 2
anomaly, when the theory is defined with a spin structure.

##  1 Introduction

In quantum field theory, anomalies are loosely defined to be quantum
obstructions to symmetries. More precisely, anomalies can themselves be
identified with (special classes of) quantum field theories in one dimension
higher than the original theory, via the idea of ‘anomaly
inflow’.111Mathematically, any anomalous theory in $d$ dimensions can be
described by a relative quantum field theory Freed:2012bs  between an extended
field theory in one dimension higher (the ‘anomaly theory’), and the trivial
extended field theory (see e.g. Freed:2014iua ). This modern viewpoint led to
an algebraic classification of anomalies via cobordism, which was made
rigorous by Freed and Hopkins Freed:2016rqq  following many important works
(including Witten:1985xe ; Dai:1994kq ; Kapustin:2014dxa ; Witten:2015aba ;
Witten:2019bou ). The cobordism classification includes all known anomalies
afflicting chiral symmetries of massless fermions in any number of dimensions,
as well as other more subtle anomalies that involve discrete spacetime
symmetries (see e.g. Tachikawa:2018njr ; Witten:2016cio ; Wang:2018qoy ).

The cobordism group that classifies anomalies, for $d$ spacetime dimensions
and symmetry type $\SS$,222A given symmetry type $\SS$, as defined precisely
in Freed:2016rqq , includes both the spacetime symmetry and an internal
symmetry group, as we as maps between them, extended to all dimensions.  is
the Anderson dual of bordism Freed:2016rqq ,

$H_{I\mathbb{Z}}^{d+2}\left(MT\SS\right):=[MT\SS,\Sigma^{d+2}I\mathbb{Z}],$
(1.1)

where $MT\SS$ denotes the Madsen–Tillman spectrum associated to symmetry type
$\SS$, $I\mathbb{Z}$ denotes the Anderson dual of the sphere spectrum (with
$\Sigma^{d+2}$ denoting the $d+2$-fold suspension), and $[X,Y]$ denotes the
homotopy classes of maps between a pair of spectra $X$ and $Y$. In fact, this
cobordism classification goes beyond what we would normally think of as
‘anomalies’ to cover all reflection positive invertible field theories (with
or without fermions) with symmetry $\SS$ in $d+1$-dimensions.

To unpack the meaning and significance of the cobordism group
$H_{I\mathbb{Z}}^{d+2}\left(MT\SS\right)$, it is helpful to recall that it
fits inside a defining short exact sequence Freed:2016rqq

$\text{Ext}^{1}_{\mathbb{Z}}\left(\pi_{d+1}(MT\SS),\mathbb{Z}\right)\hookrightarrow
H_{I\mathbb{Z}}^{d+2}\left(MT\SS\right)\twoheadrightarrow\operatorname*{Hom}\left(\pi_{d+2}(MT\SS),\mathbb{Z}\right),$
(1.2)

where $\pi_{k}(X)$ is the $k^{\text{th}}$ stable homotopy group of a spectrum
$X$. For the special case of chiral fermion anomalies in $d$ dimensions, the
invertible field theory in question is precisely the exponentiated
$\eta$-invariant Witten:2019bou  of Atiyah, Patodi and Singer Atiyah:1975jf ;
Atiyah:1976jg ; Atiyah:1976qjr  in dimension $d+1$. In that case, integrating
the gauge-invariant anomaly polynomial $\Phi_{d+2}$ provides an element of the
right factor $\operatorname*{Hom}\left(\pi_{d+2}(MT\SS),\mathbb{Z}\right)$.
Theories with $\Phi_{d+2}=0$, which are in the kernel of the right map and
thus in the image of the left map, correspond to residual ‘global anomalies’
which are thus captured by the left-factor. For theories with symmetry type
$\SS=\text{Spin}\times G$, where $G$ is the internal symmetry group, we have
that
$\text{Ext}^{1}_{\mathbb{Z}}\left(\pi_{d+1}(MT\SS),\mathbb{Z}\right)\cong\operatorname*{Hom}\left(\operatorname*{Tor}\Omega_{d+1}^{\text{Spin}}(BG),\mathbb{R}/\mathbb{Z}\right)$
classifies global anomalies, where $\Omega_{\bullet}^{\text{Spin}}$ denotes
spin bordism. A straightforward corollary is that if the group
$\operatorname*{Tor}\Omega_{d+1}^{\text{Spin}}(BG)$ vanishes there can be no
global anomalies, which has been applied to various particle physics
applications in recent years Freed:2006mx ; Garcia-Etxebarria:2017crf ;
Garcia-Etxebarria:2018ajm ; Wan:2018bns ; Seiberg:2018ntt ; Hsieh:2019iba ;
Davighi:2019rcd ; Wan:2019gqr ; Davighi:2020uab ; Lee:2020ewl ;
Davighi:2020kok ; Debray:2021vob ; Davighi:2022fer ; Davighi:2022bqf ;
Wang:2022eag ; Lee:2022spd ; Davighi:2022icj ; Debray:2023yrs .

Concurrent with this development of the cobordism classification of anomalies,
in the past decade the notion of ‘symmetry’ has been generalised beyond the
action of groups on local operators, in various exciting directions (for
recent reviews, see Ref. Cordova:2022ruw ; Bah:2022wot ). This includes the
notion of $p$-form symmetries Gaiotto:2014kfa , which act not on local
operators (the case $p=0$) but more generally on extended operators of
dimension $p$. These symmetries couple to background fields that are
$(p+1)$-form gauge fields. Such higher-form symmetries are found in a plethora
of quantum field theories. The most well-known examples include a pair of
$\text{U}(1)$ 1-form symmetries in 4d Maxwell theory, and a $\mathbb{Z}/N$
1-form symmetry in 4d $\text{SU}(N)$ Yang–Mills theory; the latter exhibits a
mixed anomaly with charge-parity symmetry when the $\text{SU}(N)$
$\theta$-angle equals $\pi$, which led to new insights regarding the vacua of
QCD Gaiotto:2017yup . Other important classes of generalised symmetry include
non-invertible and categorical symmetries, as well as subsystem symmetries.
These shall play no role in the present paper.

Soon after the notion of $p$-form symmetries was formalised in Gaiotto:2014kfa
, it was realised that higher-form symmetries of different degrees can ‘mix’,
leading to the notion of $p$-group symmetry. The most down-to-earth example of
this symmetry structure, which in general is described using higher category
theory (see e.g. Gripaios:2022yjy ), is the notion of a 2-group symmetry
whereby a 1-form symmetry $H^{[1]}$ and a 0-form symmetry $G^{[0]}$ combine
non-trivially. This occurs, for example Hsin:2020nts ; Lee:2021crt , when a
bunch of Wilson lines (which are charged under a discrete 1-form centre
symmetry) can be screened by a dynamical fermion (which is charged under a
0-form flavour symmetry). 2-group symmetries were first observed, in their
gauged form, in string theory in the context of the Green–Schwarz mechanism,
in which a $\text{U}(1)$ 1-form symmetry combines non-trivially with a
diffeomorphism Baez:2005sn ; Sati:2009ic ; Fiorenza:2010mh ; Fiorenza:2012tb .
Their appearance as global symmetries in field theory was appreciated first by
Sharpe Sharpe:2015mja , before many new examples were identified in Refs.
Cordova:2018cvg ; Benini:2018reh . Further instances of 2-group symmetry have
since been discovered and analysed in 4d Hsin:2020nts ; Lee:2021crt ;
Bhardwaj:2021wif ; Cvetic:2022imb ; Carta:2022fxc , 5d Cvetic:2022imb ;
Benini:2018reh ; Apruzzi:2021vcu ; DelZotto:2022fnw ; DelZotto:2022joo , and
6d Cordova:2020tij ; DelZotto:2020sop ; Apruzzi:2021mlh  quantum field
theories. A 2-group symmetry structure has recently been studied in the
Standard Model Cordova:2022qtz , in the limit of zero Yukawa couplings.

In this paper we study how the cobordism classification of anomalies can be
applied to 2-group symmetries. As a first step, we here specialise to 2-groups
for which both $G^{[0]}$ and $H^{[1]}$ are compact, connected, abelian groups
ergo tori $\cong\text{U}(1)^{n}$; we refer to such a structure, at times, as a
‘toric 2-group’. In this case, there are 1-form and 2-form Noether currents
$j^{(1)}_{G}$ and $j^{(2)}_{H}$ associated with the 0-form and 1-form
symmetries respectively, and the non-trivial 2-group ‘mixing’ between these
symmetries is reflected in a fusion rule for these currents, schematically

$j^{(1)}_{G}(x)j^{(1)}_{G}(0)\sim K(x)j^{(2)}_{H}(0)+\dots,$ (1.3)

for a known (singular) function $K(x)$. Toric 2-group symmetries appear, for
example, in chiral $\text{U}(1)$ gauge theories in 4d when there are mixed
‘operator-valued’ anomalies between global chiral symmetry currents and the
gauge current Cordova:2018cvg , as well as in 3d Damia:2022rxw  and in
hydrodynamics Iqbal:2020lrt . Toric 2-group symmetries thus naturally appear
in fermionic systems, for which anomalies can be more fruitfully analysed
using cobordism (compared to purely bosonic anomalies).

For such a toric 2-group $\mathbb{G}$, we study the bordism theory of
manifolds equipped with ‘tangential $\mathbb{G}$-structure’. This can be
defined analogously to a tangential $\SS$-structure for ordinary symmetry type
$\SS$, given now the particular 2-group $\mathbb{G}$ plus an appropriate form
of spin structure. When the 2-group symmetry does not further mix with the
spacetime symmetry, this reduces to studying the spin bordism groups of the
topological space that classifies $\mathbb{G}$-2-bundles.

Thankfully much is known about the classifying space of $\mathbb{G}$-2-bundles
that will be of use to us, given the physics applications we have in mind.
Quite generally, the classifying space of a 2-group $\mathbb{G}$ can be
computed as $B|\mathbb{G}|$, where $|\mathbb{G}|$ is the geometric realization
of the nerve of the 2-group $\mathbb{G}$ viewed as a category
bartels2006higher ; Baez2008 , which is itself a topological 1-group — albeit
often an infinite-dimensional one. This means that Freed and Hopkins’
classification of invertible field theories with fixed symmetry type can be
applied directly, taking the tangential structure to be
$\text{Spin}\times|\mathbb{G}|$.

In the special case of a pure 1-form symmetry valued in an abelian group
$H^{[1]}=A$, this formula for the classifying space of the corresponding
2-group reduces to $B(BA)$, recovering the well-known result for classifying
abelian gerbes brylinski2007loop . This special case, which is of significant
physical interest, has already been used in the physics literature to study
1-form anomalies, for many examples with variously Spin, SO, and O structures
in Ref. Wan:2018bns . An extension of this to a pure 2-form symmetry and its
anomalies in 6d gauge theories was studied via cobordism in Ref. Lee:2020ewl .

More generally for any 2-group $\mathbb{G}$, the classifying space
$B|\mathbb{G}|$ always sits in a ‘defining fibration’

$B\left(BH^{[1]}\right)\to B|\mathbb{G}|\to BG^{[0]}\,,$ (1.4)

which allows one to leverage the usual spectral sequence methods to compute
e.g. the cohomology and/or the (co)bordism groups of $B|\mathbb{G}|$. For
‘non-trivial’ 2-groups, i.e. those for which the fibration (1.4) is non-
trivial, there is to our knowledge only a smattering of calculations in the
literature; in particular, the case $B(B\mathbb{Z}/2^{[1]})\to
B|\mathbb{G}|\to B\text{O}^{[0]}$ was studied by Wan and Wang Wan:2018bns
(Section 4), and $B(B\text{U}(1)^{[1]})\to B|\mathbb{G}|\to
B\text{SU}(2)^{[0]}$ was studied by Lee and Tachikawa Lee:2020ewl  (Appendix
B.6). In this paper we extend these works with a dedicated cobordism analysis
of theories with 2-group symmetry and their anomalies.

The content of this paper is as follows. After reviewing 2-groups and their
appearances in quantum field theory in §2, we then describe the topological
spaces $B|\mathbb{G}|$ that classify 2-group background fields, and show how
these spaces can be computed, in §3. By considering appropriate tangential
structures for 2-group symmetries, this then leads to a definition of the
cobordism groups relevant for describing 2-group symmetries and their
anomalies (§3.2). We then apply this cobordism theory to study a small
selection of examples in detail, focussing on toric 2-group symmetries as an
especially tractable case. The main examples that we study in this paper are
as follows. In §4, which is a warm up example, we study anomalies involving
the pair of 1-form symmetries in 4d Maxwell theory, defined on non-spin
manifolds. As well as the $\mathbb{Z}$-valued ‘local’ mixed anomaly between
the two 1-form symmetries Gaiotto:2014kfa ; Wang-Senthil:2016a ; Hsin:2019fhf
; Brennan:2022tyl , there is a trio of $\mathbb{Z}/2$-valued global anomalies
(involving gravity) that only appear on non-spin manifolds; the associated
phases have been studied in Jian:2020qab . In §5 we turn to our main example
of toric 2-group symmetries, with non-trivial Postnikov class, occurring in 4d
abelian chiral theories. We see, using cobordism, how turning on the Postnikov
class transmutes a $\mathbb{Z}$-valued local anomaly into a torsion-valued
global anomaly à la the Green–Schwarz mechanism – we precisely recover the
physics results derived by Córdova, Dumitrescu and Intriligator in Ref
Cordova:2018cvg . Finally, in §6 we drop down to two dimensions, where
anomalous 0-form symmetries can be recast as 2-group symmetries by using the
trivially conserved 1-form symmetry whose 2-form current is simply the volume
form Sharpe:2015mja . Our cobordism analysis highlights the prominent role
played by spin structures; when the $\text{U}(1)^{[0]}$ anomaly coefficient is
odd, the Green–Schwarz mechanism cannot in fact absorb the anomaly completely
but leaves a $\mathbb{Z}/2$-valued anomaly, which we identify as an anomaly in
the spin structure; we also show how this $\mathbb{Z}/2$-anomaly is
trivialised upon passing to $\text{Spin}_{c}$ bordism.333These examples
emphasize that bordism computations are useful tools for keeping track of such
subtle mod 2 effects, by automatically encoding various characteristic
classes’ normalisation that can differ in spin vs. non-spin theories – for
example, the class $c_{1}^{2}$ is even on a spin manifold but needn’t be on a
non-spin one.

##  2 Review: 2-groups and their classifying spaces

In this Section we recall the notion of a 2-group $\mathbb{G}$ (§2.1), and
briefly reprise their appearance as generalised symmetry structures in quantum
field theory (§2.2).

###  2.1 What is a 2-group?

We begin by sketching some equivalent definitions of a 2-group, for which our
main reference is Ref. Baez2008  by Baez and Stevenson. First and most concise
is probably the definition of a 2-group $\mathbb{G}$ as a higher category.
Loosely, a 2-group $\mathbb{G}$ is in this language a 2-category that contains
a single object, with all 1-morphisms and 2-morphisms being invertible.

(Slightly) less abstractly, a 2-group can also be described using ‘ordinary’
category theory, without recourse to higher categories, and this can be done
in two ways; either as a groupoid in the category of groups, or as a group in
the category of groupoids.444It will be implicitly assumed that all groups and
groupoids discussed carry a topology. Expanding on the former viewpoint, a
2-group is itself a category whose objects form a group $\tilde{G}^{[0]}$, and
whose morphisms also form a group, where the latter can be written as a semi-
direct product between $\tilde{G}^{[0]}$ and another group $\tilde{H}^{[1]}$,
viz.

$\operatorname*{Obj}(\mathbb{G})=\tilde{G}^{[0]},\qquad\operatorname*{Mor}(\mathbb{G})=\tilde{H}^{[1]}\rtimes\tilde{G}^{[0]},$
(2.1)

such that all the various maps involved in the definition of a groupoid are
continuous group homomorphisms (see §3 of Baez2008 ). We will usually be
interested in situations where both $\tilde{G}^{[0]}$ and $\tilde{H}^{[1]}$
are Lie groups.

This definition involving a pair of groups $(\tilde{G}^{[0]},\tilde{H}^{[1]})$
is closely related to yet another, somewhat more practical, definition of a
2-group as a topological crossed module. The data specifying a topological
crossed module consists of a quadruplet

$\mathbb{G}=(\tilde{G}^{[0]},\tilde{H}^{[1]},t,\alpha),$ (2.2)

where $\tilde{G}^{[0]}$ and $\tilde{H}^{[1]}$ can be identified with the
groups that appear in (2.1), and
$t:\tilde{H}^{[1]}\rightarrow\tilde{G}^{[0]}$,
$\alpha:\tilde{G}^{[0]}\to\text{Aut}(\tilde{H}^{[1]})$ are continuous
homomorphisms such that the pair of conditions

$t\left(\alpha(g)(h)\right)=gt(h)g^{-1},\qquad\alpha(t(h))(h^{\prime})=hh^{\prime}h^{-1}$
(2.3)

hold $\forall g\in\tilde{G}^{[0]},\;h,\,h^{\prime}\in\tilde{H}^{[1]}$, which
can be thought of as equivariance conditions on the maps. To define a 2-group
from this data, one takes the group of objects
$\operatorname*{Obj}(\mathbb{G})=\tilde{G}^{[0]}$ and the group of morphisms
$\operatorname*{Mor}(\mathbb{G})=\tilde{H}^{[1]}\rtimes\tilde{G}^{[0]}$, where
the semi-direct product is defined using the map $\alpha$ as
$(h,g)\cdot(h^{\prime},g^{\prime})=(h\alpha(g)(h^{\prime}),gg^{\prime})$, and
the various source, target, and composition maps from
$\operatorname*{Mor}(\mathbb{G})$, as well as identity maps from
$\operatorname*{Obj}(\mathbb{G})$, are defined using simple formulae (for
which we refer the interested reader to §3 of Baez2008 .)

We can tie together this circle of definitions by linking this final crossed
module definition of $\mathbb{G}$ to the first definition of $\mathbb{G}$ as a
2-category; in that picture, $\tilde{G}^{[0]}$ is the group of 1-morphisms,
and $\tilde{H}^{[1]}$ is the group of 2-morphisms from the identity 1-morphism
to all other 1-morphisms Kapustin:2013uxa . When $\tilde{G}^{[0]}$ and
$\tilde{H}^{[1]}$ are both Lie, and when the maps $t$ and $\alpha$ are both
smooth, $\mathbb{G}$ is a Lie 2-group.

The final description of a 2-group $\mathbb{G}$ that we just gave, as a
topological crossed module, is still of somewhat limited use in physics
because the groups $\tilde{G}^{[0]}$ and $\tilde{H}^{[1]}$ are not preserved
under equivalence of 2-groups as 2-categories. Moreover, a physical system
with a 2-group symmetry (possibly with associated background fields) is only
sensitive to the equivalence class of the 2-group at long distance, not the
2-group itself, as described by Kapustin and Thorngren Kapustin:2013uxa . It
is therefore more convenient to work with equivalence classes of 2-groups from
the beginning, which we describe next.

An equivalence class of 2-groups, with a particular 2-group
$\mathbb{G}=(\tilde{G}^{[0]},\tilde{H}^{[1]},t,\alpha)$ above as a
representative, can also be described by a quadruplet
$(G^{[0]},H^{[1]},\alpha,\beta)$ where Kapustin:2013uxa

$G^{[0]}:=\text{coker}\,t,\qquad H^{[1]}:=\ker t.$ (2.4)

The homomorphism $\alpha:G^{[0]}\to\text{Aut}(H^{[1]})$ is so named because it
descends from the homomorphism $\alpha:G^{[0]}\to\text{Aut}(H^{[1]})$. The new
component is the so-called _Postnikov class_

$\beta\in H^{3}(BG^{[0]};H^{[1]}).$ (2.5)

More precisely, two crossed modules
$\mathbb{G}=(\tilde{G}^{[0]},\tilde{H}^{[1]},t,\alpha)$ and
$\mathbb{G}^{\prime}=(\tilde{G}^{\prime[0]},\tilde{H}^{\prime[1]},t^{\prime},\alpha^{\prime})$
are said to be equivalent when $\tilde{G}^{\prime[0]}\cong\tilde{G}^{[0]}$,
$\tilde{H}^{\prime[1]}\cong\tilde{H}^{[1]}$ and there are homomorphisms (not
necessarily isomorphisms) $h:\tilde{H}^{[1]}\to\tilde{H}^{\prime[1]}$ and
$g:\tilde{G}^{[0]}\to\tilde{G}^{\prime[0]}$ that makes the diagram

${1}$${H^{[1]}}$${\tilde{H}^{[1]}}$${\tilde{G}^{[0]}}$${G^{[0]}}$${1}$${1}$${H^{\prime[1]}}$${\tilde{H}^{\prime[1]}}$${\tilde{G}^{\prime[0]}}$${G^{\prime[0]}}$${1}$$\scriptstyle{h}$$\scriptstyle{t}$$\scriptstyle{g}$$\scriptstyle{t^{\prime}}$
(2.6)

of two exact sequences commutative and compatible with the actions of
$\tilde{G}^{[0]}$ on $\tilde{H}^{[1]}$ and $\tilde{G}^{\prime[0]}$ on
$\tilde{H}^{\prime[1]}$. These equivalence classes are classified by the group
cohomology $\mathcal{H}^{3}(G^{[0]},H^{[1]})$ Brown-K-S:1982 , which is
isomorphic to the ordinary cohomology $H^{3}(BG^{[0]};H^{[1]})$.

For completeness, we also define the notion of _2-group homomorphisms_ , in
terms of ordinary group homomorphisms between different elements of the
associated crossed modules Baez2008 . Represent two 2-groups $\mathbb{G}$ and
$\mathbb{G}^{\prime}$ by crossed modules
$(\tilde{G}^{[0]},\tilde{H}^{[1]},t,\alpha)$ and
$(\tilde{G}^{\prime[0]},\tilde{H}^{\prime[1]},t^{\prime},\alpha^{\prime})$. A
2-group homomorphism $f:\mathbb{G}\to\mathbb{G}^{\prime}$, which is a functor
such that $f:\text{Obj}(\mathbb{G})\to\text{Obj}(\mathbb{G}^{\prime})$ and
$f:\text{Mor}(\mathbb{G})\to\text{Mor}(\mathbb{G}^{\prime})$ are continuous
homomorphisms of topological groups, can be represented by a pair of maps
$h:\tilde{H}^{[1]}\to\tilde{H}^{\prime[1]}$ and
$g:\tilde{G}^{[0]}\to\tilde{G}^{\prime[0]}$ such that the diagram

${\tilde{H}^{[1]}}$${\tilde{G}^{[0]}}$${\tilde{H}^{\prime[1]}}$${\tilde{G}^{\prime[0]}}$$\scriptstyle{t}$$\scriptstyle{h}$$\scriptstyle{g}$$\scriptstyle{t^{\prime}}$
(2.7)

is commutative and that $h,g$ are compatible with $\alpha,\alpha^{\prime}$:

$h(\alpha(a)(b))=\alpha^{\prime}(g(a))h(b),$ (2.8)

for all $a\in\tilde{G}^{[0]},\,b\in\tilde{H}^{[1]}$.

###  2.2 2-group symmetries in quantum field theory

Physically, an equivalence class $\mathbb{G}=(G^{[0]},H^{[1]},\alpha,\beta)$
of 2-groups describes a symmetry structure appearing in quantum field theories
which have both a 0-form symmetry group $G^{[0]}$ and a 1-form symmetry group
$H^{[1]}$. When the two symmetries are not independent, the Postnikov class
$\beta$ of the corresponding 2-group symmetry is non-trivial, and the 0-form
and 1-form parts cannot be analysed in isolation.

While we will eventually specialise to the case of toric 2-groups in the bulk
of this paper, we first recap how 2-groups appear in field theory more
broadly. It is convenient to distinguish two broad categories of 2-groups
which in general have different physical origin:

  1. 1.

A _continuous 2-group_ $\mathbb{G}$ is one with a continuous 1-form symmetry
group $H^{[1]}$. This class of 2-groups arises in field theory when, for
example, the gauge transformation law for the 1-form symmetry background gauge
field is modified so that there is no operator-valued ’t Hooft anomaly
involving the background gauge field for the 0-form symmetry, in an analogue
of the Green–Schwarz mechanism for global symmetries Cordova:2018cvg . The
0-form symmetry can here be abelian or non-abelian, connected or disconnected,
and compact or non-compact. In this paper we focus our attention on the
special case where both the 0-form and 1-form symmetry are connected, compact,
and abelian groups, which we refer to as a ‘toric 2-group’.

  2. 2.

A _discrete 2-group_ $\mathbb{G}$ is one with a discrete 1-form symmetry group
$H^{[1]}$. This class of 2-group symmetries arises in gauge theories when the
gauge Wilson lines, which are charged objects under the 1-form symmetry
$H^{[1]}$, are not completely screened in the presence of the background gauge
field for the 0-form symmetry group $G^{[0]}$. The local operator
$\mathcal{O}$ that screens the gauge Wilson lines when the 0-form background
gauge field is turned off is also charged under the 0-form symmetry. Hence,
when the background gauge field of $G^{[0]}$ is turned on, $\mathcal{O}$
transmutes a gauge Wilson line into a flavour Wilson line Benini:2018reh ;
Bhardwaj:2021wif ; Lee:2021crt .

##  3 Cobordism with 2-group structure

In this Section we describe the background gauge fields on associated
principal $\mathbb{G}$-2-bundles which play an important role in field
theories with 2-group symmetries. We focus on describing the topological
spaces $B|\mathbb{G}|$ that classify these background fields, and show how to
calculate such classifying spaces in elementary examples, before introducing
the cobordism groups that are central to this paper in §3.2.

###  3.1 Background fields and their classifying spaces

A theory with a 2-group symmetry $\mathbb{G}$ can be coupled to a background
gauge field (which a physicist might wish to decompose into components of an
ordinary 1-form gauge field and a 2-form gauge field),555A more explicit
description of the background fields will be sketched in §5 where we discuss
the 2-group symmetries appearing in 4d abelian chiral gauge theory.  which is
a connection on a principal $\mathbb{G}$-2-bundle in the sense defined in
bartels2006higher .

Mathematically, Bartels moreover shows that such 2-bundles over a manifold $X$
are classified by the Čech cohomology group $\check{H}^{1}(X,\mathbb{G})$ with
coefficients valued in the 2-group $\mathbb{G}$. Baez and Stevenson prove
Baez2008  that there is a bijection

$\check{H}^{1}(X,\mathbb{G})\cong[X,B|\mathbb{G}|],$ (3.1)

where $[A,B]$ denotes the set of homotopy classes of maps from $A$ to $B$, and
$|\mathbb{G}|$ is the geometric realization of the nerve
$\mathcal{N}\mathbb{G}$ of the 2-group $\mathbb{G}$ when viewed as a groupoid
as in (2.1). The nerve $\mathcal{N}\mathbb{G}$ of the category $\mathbb{G}$
with $\operatorname*{Obj}(\mathbb{G})=G$ and
$\operatorname*{Mor}(\mathbb{G})=H\rtimes G$ is a set of simplices that we can
construct out of these objects and morphisms – we will see some explicit
examples shortly.

Quantum field theories with 2-group symmetry $\mathbb{G}$ are thus defined on
spacetime manifolds $X$ equipped with maps to $B|\mathbb{G}|$, just as a
theory with an ordinary symmetry $G$ is defined on spacetimes equipped with
maps to $BG$.

####  3.1.1 Elementary examples of $B|\mathbb{G}|$

Since the classifying space $B|\mathbb{G}|$ will play a central role in what
follows, we pause to better acquaint the reader with $B|\mathbb{G}|$ and how
it can be computed by looking at some simple examples. The calculations in
this subsection are purely pedagogical – readers who are familiar with (or not
interested in) such constructions might wish to skip ahead to §3.1.2.

#### Pure 0-form symmetries

In the case that there is no 1-form symmetry, and the 2-group symmetry simply
defines an ordinary 0-form symmetry $G$, we expect to recover the usual
classifying space of $G$. In this case, the 2-group corresponds to the
quadruplet $\mathbb{G}=(G,0,0,0)$, to use the topological crossed module
notation, and indeed $B|(G,0,0,0)|=BG$. To see this from the nerve
construction, we first view $\mathbb{G}$ as a category, which is very simple
in this case: $\text{Obj}(\mathbb{G})=G$ with only identity morphisms at each
element. So there are no non-degenerate $n$-simplices when $n>0$, while the
set of 0-simplices is $G$. Hence, the geometric realisation of the nerve
$|\mathbb{G}|$ is simply the group $G$ itself, and its classifying space is
$BG$, as claimed.

#### Pure 1-form symmetries

The case of a ‘pure 1-form symmetry’, in which there is no 0-form symmetry at
all, corresponds to 2-groups of the form $\mathbb{G}=(0,H,0,0)=:H[1]$, where
$H$ is the (abelian) 1-form symmetry group. The corresponding classifying
space is bartels2006higher

$B|H[1]|=B(BH),$ (3.2)

which we often denote simply $B^{2}H$, which coincides with the well-known
classifying space of an abelian gerbe brylinski2007loop . For example, when
$H=\text{U}(1)$, $B|H[1]|$ is an Eilenberg–Maclane space $K(\mathbb{Z},3)$.

##### An example: pure $\mathbb{Z}/2$ 1-form symmetry.

For an explicit example that illustrates how to actually calculate the
geometric realization of the nerve of such a $\mathbb{G}=H[1]$, we consider
the simplest case where $H=\mathbb{Z}/2$. Viewing this as a category as in
(2.1), we have that $\operatorname*{Obj}(\mathbb{Z}/2[1])$ consists of only
one element because $G$ is the trivial group, denoted simply by $\bullet$,
while the group $\operatorname*{Mor}(\mathbb{Z}/2[1])$ is just isomorphic to
$H=\mathbb{Z}/2$. Diagrammatically, this structure can be represented as

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The nerve of $\mathbb{Z}/2[1]$ is then a simplicial set
$\mathcal{N}\mathbb{Z}/2[1]$ built out of these objects and morphisms, whose
low dimensional components are given as follows.

$\displaystyle\left(\mathcal{N}\mathbb{Z}/2[1]\right)_{0}$
$\displaystyle=\left\\{\bullet\right\\}\qquad$ (0-simplex)
$\displaystyle\left(\mathcal{N}\mathbb{Z}/2[1]\right)_{1}$
$\displaystyle=\left\\{\bullet\xrightarrow{-1}\bullet\right\\}\qquad$
(1-simplex) $\displaystyle\left(\mathcal{N}\mathbb{Z}/2[1]\right)_{2}$
$\displaystyle=\left\\{\bullet\xrightarrow{-1}\bullet\xrightarrow{-1}\bullet\right\\}\qquad$
(2-simplex) etc.

This is enough information for us to work out the CW complex cell structure
for its geometric realisation, which recall is the topological space that we
denote $|\mathbb{Z}/2[1]|$. The 0-cell is just a point. The 1-cell comes from
the union of the 0-cell and an interval, with the gluing rule given by the
1-simplex ${\bullet}$$\scriptstyle{-1}$ i.e. identifying both ends of the
interval with the 0-cell. Hence, the 1-cell has the topology of a 1-sphere
$S^{1}$. Similarly, the form of the 2-simplex, written more suggestively as

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tells us that the 2-cell is constructed by identifying the 1-cell $S^{1}$ as
the boundary of a 2-ball, with antipodal points on the boundary identified.
This gives the topology of the 2-dimensional real projective space
$\mathbb{R}P^{2}$. Building inductively in this way, one can show that
$|\mathbb{Z}/2[1]|$ is topologically $\mathbb{R}P^{\infty}$. On the other
hand, we know that this is the same as the classifying space of
$\mathbb{Z}/2$, i.e. $|\mathbb{Z}/2[1]|\cong B\mathbb{Z}/2$. Therefore, the
classifying space of the $\mathbb{Z}/2$ 1-form symmetry bundle is
$B|\mathbb{Z}/2[1]|\cong B^{2}\mathbb{Z}/2$.

This argument can be generalised to any 2-group of the form $H[1]=(0,H,0,0)$.
For any group $H$, it is proven that the geometrisation of the nerve $|H[1]|$
is the topological space $BH$ Segal1968 , and the classifying space of
$H[1]$-2-bundles is thus $B^{2}H$, coinciding with the known classification of
$H$-gerbes when $H$ is abelian.

#### 2-groups with a trivial map

Moving up in complexity, let us now consider a 2-group with both the
constituent 1-groups $\tilde{G}$ and $\tilde{H}$ being non-trivial, but with
at least one of the maps in the crossed module definition being trivial. To
wit, consider a particular 2-group
$\mathbb{G}=(\tilde{G},\tilde{H},t,\alpha)$, written using the notation of
(2.2), where the map $\alpha:\tilde{G}\rightarrow\text{Aut}(\tilde{H})$ is
trivial i.e. $\alpha(g)$ is the identity automorphism for any $g\in\tilde{G}$.

##### An example: $\tilde{G}=\mathbb{Z}/2$ and $\tilde{H}=\mathbb{Z}/2$.

Probably the simplest example of a 2-group of this kind is one where
$\tilde{G}$ and $\tilde{H}$ are both $\mathbb{Z}/2$. Since
$\text{Aut}(\tilde{H})\cong\text{Aut}(\mathbb{Z}/2)$ is trivial, $\alpha$ is
automatically trivial. If $\tilde{G}$ and $\tilde{H}$ do not interact at all,
meaning that the map $t$ is also trivial (the module is ‘uncrossed’), then the
2-group factorises as a product of a 0-form and a 1-form symmetry:

$|\mathbb{G}|\cong\mathbb{Z}/2\times B\mathbb{Z}/2,$

whose classifying space is just $B\mathbb{Z}/2\times B^{2}\mathbb{Z}/2$.

Since $t:\tilde{H}\rightarrow\tilde{G}$ must be a homomorphism, the only other
option for $t$ is the identity map $t:-1\mapsto-1$. Setting
$\mathbb{G}=(\mathbb{Z}/2,\mathbb{Z}/2,\text{id},0)$, then as a category
$\text{Obj}(\mathbb{G})\cong\mathbb{Z}/2$ and $\text{Mor}(\mathbb{G})=G\times
H\cong\mathbb{Z}/2\times\mathbb{Z}/2$, whose action can be fully captured in
the following diagram:

${-1}$${1}$$\scriptstyle{(1,-1)}$$\scriptstyle{(-1,-1)}$$\scriptstyle{(1,1)}$$\scriptstyle{(-1,1)}$

Since the morphisms shown in the diagram are all the morphisms in this
category, we can identify the morphisms $(1,1)$ and $(1,-1)$ as the identity
morphisms at the objects $1$ and $-1$, respectively.

The components of the nerve $\mathcal{NG}$ in low dimensions are given by

$\displaystyle\mathcal{NG}_{0}$
$\displaystyle=\left\\{\bullet_{-1},\bullet_{1}\right\\}$
$\displaystyle\mathcal{NG}_{1}$
$\displaystyle=\left\\{\bullet_{-1}\xrightarrow{(-1,-1)}\bullet_{1},\bullet_{1}\xrightarrow{(-1,1)}\bullet_{-1}\right\\}$
$\displaystyle\mathcal{NG}_{2}$
$\displaystyle=\left\\{\bullet_{-1}\rightarrow\bullet_{1}\rightarrow\bullet_{-1},\bullet_{1}\rightarrow\bullet_{-1}\rightarrow\bullet_{1}\right\\}$

From these data, it is easy to see that the 0-cell is just a pair of points,
the 1-cell is a circle $S^{1}$,
${\bullet_{-1}}$${\bullet_{1}}$$\scriptstyle{(-1,-1)}$$\scriptstyle{(-1,1)}$,
and the 2-cell is a 2-sphere $S^{2}$, constructed by joining
${\bullet_{-1}}$${\bullet_{1}}$ and ${\bullet_{1}}$${\bullet_{-1}}$ as two
“hemispheres”, with the 1-cell at the equator. Continuing ad infinitum we
obtain $|\mathbb{G}|\cong S^{\infty}$. But the infinite sphere is
contractible, which means it is homotopy equivalent to a point. Hence, the
classifying space for $\mathbb{G}$ must be trivial.

This result should not be a surprise because the crossed module under
consideration, namely $\mathbb{G}=(\mathbb{Z}/2,\mathbb{Z}/2,t,0)$ with $t$ an
isomorphism, is equivalent to the trivial 2-group because both
$G^{[0]}=\text{coker}\,t$ and $H^{[1]}=\ker t$ are trivial!

####  3.1.2 General $B|\mathbb{G}|$ as a fibration

Our discussion so far has been mostly pedagogical; the idea was to acquaint
the reader with some basic computations of the classsifying spaces of
2-groups. Thankfully, more powerful tools are available to us for the
computations that we will be interested in. First and foremost, for any
2-group $\mathbb{G}$ there is a fibration666The extensions encoded in (3.3)
are in fact classified by $H^{3}_{SM}(G^{[0]},H^{[1]})$ schommer2011central ,
the Segal–Mitchison group cohomology in degree 3 (a degree that usually
classifies 2-step extensions).

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Note that, because the 1-form symmetry $H^{[1]}$ is always an abelian group,
$BH^{[1]}$ is itself a (topological) group, and so $B(BH^{[1]})$ can be
defined using the ordinary definition of the classifying space of a group. We
will see in the next Subsection that this fibration is sufficient for our
purpose of anomaly classification.

###  3.2 Cobordism description of 2-group anomalies

In order to classify anomalies associated to a 2-group $\mathbb{G}$ via
cobordism, most of the structure contained in $\mathbb{G}$ as a 2-group can be
discarded. As discussed in the previous Subsection, $\mathbb{G}$-2-bundles are
classified by the topological space $B|\mathbb{G}|$, where $|\mathbb{G}|$ is
the geometric realization of the nerve (henceforth just ‘the nerve’) of
$\mathbb{G}$, regardless of other information contained in $\mathbb{G}$. Thus,
we can use the nerve $|\mathbb{G}|$, itself a topological group, to define a
tangential structure on spacetime just as we would for an ordinary 0-form
symmetry. For instance, if we are interested in a fermionic theory with
internal 2-group symmetry $\mathbb{G}$ (with no mixing with the spacetime
symmetry), we can take the symmetry type to be777If we do not require a spin
structure, we could use SO instead of Spin in defining the symmetry type $\SS$
– as we do in §4.

$\SS=\text{Spin}\times|\mathbb{G}|\;.$ (3.4)

Directly applying Freed–Hopkins’ classification Freed:2016rqq  of invertible,
reflection-positive field theories to this symmetry type, suggests that the
cobordism group

$H_{I\mathbb{Z}}^{d+2}(MT(\text{Spin}\times|\mathbb{G}|)\cong\text{Tors}\,\Omega^{\text{Spin}}_{d+1}(B|\mathbb{G}|)\times\operatorname*{Hom}\left(\Omega^{\text{Spin}}_{d+2}(B|\mathbb{G}|),\mathbb{Z}\right)\;$
(3.5)

correctly classifies anomalies in $d$-dimensional fermionic theories with the
2-group symmetry $\mathbb{G}$. Our detection of anomaly theories is therefore
distilled into computing the spin bordism groups of $B|\mathbb{G}|$, for which
we can use the same methods as for an ordinary symmetry.

In particular, our general strategy will be to build up the spin bordism
groups of $B|\mathbb{G}|$ by using the ‘defining fibration’ described above,
$B^{2}H^{[1]}\to B|\mathbb{G}|\to BG^{[0]}$, as follows. We first use the
Serre spectral sequence (SSS) Serre1951HomologieSD  to compute the cohomology
of $B|\mathbb{G}|$ from the fibration. Then, one can use this newly computed
cohomology to compute the spin bordism groups of $B|\mathbb{G}|$, for example
via the Adams spectral sequence (ASS) Adams:1958 . Alternatively, we can
convert the result to homology using the universal coefficient theorem, and
use it as an input of the Atiyah–Hirzebruch spectral sequence (AHSS)
AHSS1961:ProcSymp61

$E^{2}_{p,q}=H_{p}(B|\mathbb{G}|;\Omega^{\text{Spin}}_{q}(\text{pt}))\;.$
(3.6)

for another fibration $\text{pt}\to B|\mathbb{G}|\to B|\mathbb{G}|$. Kapustin
and Thorngren already used the SSS associated to (3.3) in Kapustin:2013uxa  to
compute the cohomology of $B|\mathbb{G}|$. Moreover, this approach has been
used to calculate bordism groups relevant to 2-group symmetries with non-
trivial Postnikov class; Wan and Wang did so using the ASS in Ref. Wan:2018bns
, while Lee and Tachikawa used the AHSS in Ref Lee:2020ewl . We usually adopt
the AHSS-based approach in this paper.

##  4 Maxwell revisited

As a (rather lengthy) warm-up example, let us discuss Maxwell theory. We
consider a 4d $\text{U}(1)$ gauge theory without matter, and variants thereof
in which we couple the action to various TQFTs without changing the underlying
symmetry structure. The action for vanilla Maxwell theory is

$S_{\text{Maxwell}}=-\frac{1}{4e^{2}}\int_{M_{4}}\text{d}a\wedge\star\text{d}a\,,$
(4.1)

where $a$ denotes the dynamical $\text{U}(1)$ gauge field and $\star$ denotes
the Hodge dual, assuming a metric $g$ on $M_{4}$.

This theory has two $\text{U}(1)$ 1-form global symmetries Gaiotto:2014kfa ,
referred to as ‘electric’ and ‘magnetic’ 1-form symmetries. Since each is
associated with a continuous group, one can write down the corresponding
conserved 2-form currents, which are

$j_{e}=\frac{2}{e^{2}}\text{d}a,\qquad j_{m}=\star\frac{\text{d}a}{2\pi}\,.$
(4.2)

The former is conserved ($\text{d}\star j_{e}=0$) by Maxwell’s equations
$\text{d}\star\text{d}a=0$, and the latter by $\text{d}^{2}=0$ (and using
$\star\star=1$).888For Maxwell theory in $d$ dimensions, $j_{e}$ is always a
2-form current and so the electric symmetry is always 1-form, while $j_{m}$ is
more generally a $(d-2)$-form, and so the magnetic symmetry is generally a
$(d-3)$-form symmetry.

It was already observed in Gaiotto:2014kfa  that there is a mixed ’t Hooft
anomaly between the two 1-form symmetries, which is a local anomaly that can
be represented by an anomaly polynomial $\text{d}B_{e}\wedge\text{d}B_{m}$,
where $B_{e}$ and $B_{m}$ are background 2-form gauge fields for the electric
and magnetic 1-form symmetries respectively. This kind of anomaly and its
consequences concerning various quantum numbers are discussed in Ref.
Hsin:2019fhf ; Brennan:2022tyl  (see also Wang-Senthil:2016a ). In Ref.
Jian:2020qab , a variety of 5d SPT phases that have Maxwell-like theories on
the 4d boundary, protected by electric and magnetic 1-form symmetries, were
derived. Here we show how these results are simply captured by the cobordism
classification.

###  4.1 Cobordism classification of 1-form anomalies

Maxwell theory has two global $\text{U}(1)$ 1-form symmetries, and no 0-form
global symmetries. The classifying space of this global symmetry structure
$\mathbb{G}$ is that of an abelian $H$-gerbe with
$H=\text{U}(1)_{e}\times\text{U}(1)_{m}$. From (3.2),

$B|\mathbb{G}|=B^{2}\text{U}(1)_{e}\times B^{2}\text{U}(1)_{m}\,.$ (4.3)

Since there are no fermions, we do not need a spin structure, and so can
define pure Maxwell theory on any smooth, orientable 4d spacetime
manifold.999We do not account for any time-reversal symmetry, that would
enable us to pass from oriented to unoriented bordism. The 5d invertible field
theories with this symmetry type are therefore classified by the generalized
cohomology group

$\operatorname*{Hom}\left(\operatorname*{Tor}\Omega_{5}^{\text{SO}}(B|\mathbb{G}|),\mathbb{R}/\mathbb{Z}\right)\hookrightarrow
H_{I\mathbb{Z}}^{6}\left(MT(\text{SO}\times|\mathbb{G}|)\right)\twoheadrightarrow\operatorname*{Hom}\left(\Omega_{6}^{\text{SO}}(B|\mathbb{G}|),\mathbb{Z}\right)\,,$
(4.4)

where this sequence splits, but not canonically. In terms of anomaly theories,
the factor on the right of this SES captures local anomalies, and the factor
on the left captures global anomalies; the abelian group that detects all
anomalies is isomorphic to the direct product of the two.

#### Local anomalies

In Appendix B.1 we compute that
$\Omega^{\text{SO}}_{6}\left((B^{2}\text{U}(1))^{2}\right)=\mathbb{Z}$, and so

$\operatorname*{Hom}\left(\Omega_{6}^{\text{SO}}(B^{2}\text{U}(1)_{e}\times
B^{2}\text{U}(1)_{m}),\mathbb{Z}\right)\cong\mathbb{Z}\,,$ (4.5)

corresponding to the integral of the degree-6 anomaly polynomial $dB_{e}\cup
dB_{m}$ on a generator of
$\Omega^{\text{SO}}_{6}\left((B^{2}\text{U}(1))^{2}\right)$. This detects the
local mixed ’t Hooft anomaly between the two 1-form symmetries discussed in
Gaiotto:2014kfa .

#### Global anomalies

In Appendix B.1 we also compute that
$\Omega^{\text{SO}}_{5}\left((B^{2}\text{U}(1))^{2}\right)=(\mathbb{Z}/2)^{3}$.
Thus, the group classifying global anomalies is

$\operatorname*{Hom}\left(\operatorname*{Tor}\Omega_{5}^{\text{SO}}(B^{2}\text{U}(1)\times
B^{2}\text{U}(1)),\mathbb{R}/\mathbb{Z}\right)\cong(\mathbb{Z}/2)^{3}\,.$
(4.6)

In terms of characteristic classes, these three factors of $\mathbb{Z}/2$
correspond to

$w_{2}\cup w_{3},\quad w_{2}\cup\tau_{e},\quad w_{2}\cup\tau_{m},$

where $w_{2,3}$ denote the second and third Stiefel–Whitney classes of the
tangent bundle, and $\tau_{e,m}$ is a unique generator of
$H^{3}(B^{2}\text{U}(1)_{e,m};\mathbb{Z}/2)$ (c.f. Appendix A.1) which can be
thought of as a mod 2 reduction of an integral cohomology class represented by
$\frac{dB_{e,m}}{2\pi}$.

###  4.2 Phases of non-spin Maxwell theory

The $w_{2}w_{3}$ invariant corresponds to a global gravitational anomaly seen
on non-spin manifolds, related to that in Wang:2018qoy . The $w_{2}\tau_{e,m}$
bordism invariants correspond loosely to ’t Hooft anomalies for each of the
1-form symmetries, that prevents them from being gauged on certain (non-spin)
gravitational backgrounds. We discuss these in more detail next.

#### The $w_{2}\cup\tau_{e}$ phase

To see the global anomaly that is captured by the
$w_{2}\cup\tau_{e}$,101010For simplicity, we consider the case where the
1-form symmetry is electric. The corresponding story for the magnetic 1-form
symmetry can be obtained by applying the electric-magnetic duality. we must go
beyond the vanilla Maxwell theory described by (4.1), and couple 4d Maxwell to
a topological term. The modified action is

$\begin{split}S&=-\frac{1}{4e^{2}}\int_{M_{4}}\text{d}a\wedge\star\text{d}a+\pi\text{i}\int_{M_{4}}w_{2}(TM_{4})\cup\rho_{2}\left[\frac{\text{d}a}{2\pi}\right]_{\mathbb{Z}}\\\
&=S_{\text{Maxwell}}+\pi\text{i}\int_{M_{4}}w_{2}(TM_{4})\cup
c_{1},\end{split}$ (4.7)

where $[\cdot]_{\mathbb{Z}}$ denotes an integral cohomology class and
$\rho_{2}$ is mod $2$ reduction, and where we shall write $c_{1}$ for both the
first Chern class of the $\text{U}(1)$ gauge bundle and its mod $2$ reduction.
The topological term couples a background magnetic 2-form gauge field to the
$\mathbb{Z}/2$ part of the magnetic 1-form symmetry, and then equates it to
the second Stiefel–Whitney class of the tangent bundle. The electric 1-form
symmetry shifting $a\mapsto a+\lambda$, where $\lambda$ is a closed 1-form,
remains intact.

This theory describes fermionic-monopole electrodynamics; the topological term
$\int w_{2}\cup c_{1}$ forces all monopoles to become fermionic
Thorngren:2014pza ; Ang:2019txy . To see this, following Wang:2018qoy ,
consider adding a magnetic monopole via an ’t Hooft line operator of charge
$1$ along $\ell\subset M_{4}$ that we excise from $M_{4}$ with the boundary
condition that $\int_{S^{2}_{\ell}}c_{1}=1$ on a small 2-sphere $S^{2}_{\ell}$
around $\ell$. The theory is now defined on the complement $M^{\prime}_{4}$ of
$\ell$ in the presence of the ’t Hooft operator, which is a manifold with
boundary $\partial M^{\prime}_{4}\cong S^{2}_{\ell}\times\ell$. For the
integral $\int_{M^{\prime}_{4}}w_{2}(TM^{\prime}_{4})\cup c_{1}$ to make sense
on a manifold with boundary, we need a trivialisation of the integrand on
$\partial M^{\prime}_{4}$.111111This is because for an $n$-manifold with
boundary $M^{\prime}$, the fundamental homology class $[M^{\prime}]$ is in
$H_{n}(M^{\prime},\partial M^{\prime})$. The integration of a cohomology class
on $M^{\prime}$ is in fact a pairing between a cohomology class and the
fundamental class $[M^{\prime}]$, so when we write $\int_{M^{\prime}}c$ for a
cohomology class $c\in H^{n}(M^{\prime})$, we need to first find a class in
$H^{n}(M^{\prime},\partial M^{\prime})$ that “corresponds” to $c$ to make
sense of the pairing. This is always possible because the long exact sequence
in cohomology  $\ldots\rightarrow H^{n-1}(\partial M^{\prime})\rightarrow
H^{n}(M^{\prime},\partial M^{\prime})\rightarrow H^{n}(M^{\prime})\rightarrow
0$ implies that any class $c\in H^{n}(M^{\prime})$ can always be lifted to a
class $C\in H^{n}(M^{\prime},\partial M^{\prime})$. The choice of this lift is
the choice of trivialisation of $c$ on $\partial M^{\prime}$. Now, since
$c_{1}$ is non-trivial on $S^{2}_{\ell}$ due to the non-zero monopole charge,
we have to trivialise the $w_{2}(TM^{\prime}_{4})$ factor. A trivialisation of
the second Stiefel–Whitney class is nothing but a spin structure. Since there
is a unique spin structure on $S^{2}$, a spin structure on
$S^{2}_{\ell}\times\ell$ is the same as a spin structure along $\ell$.
Therefore, to define the additional phase, we must choose a spin structure
along the monopole worldline: the monopole is a fermion.

In order to see the ’t Hooft anomaly afflicting $\text{U}(1)_{e}$, we now
couple the theory to a background electric 2-form gauge field $B_{e}$, and
promote the 1-form $\text{U}(1)_{e}$ global symmetry transformation above to a
‘local’ transformation by relaxing the condition that the 1-form parameter
$\lambda$ has to be closed. In fact, $\lambda$ does not strictly need to be a
1-form on $M_{4}$ – more precisely, we now consider shifting the gauge field
$a$ by any connection $\lambda$. For a connection $\lambda$ with non-zero
curvature $\text{d}\lambda$, the action (4.7) shifts under $a\mapsto
a+\lambda$, by

$\delta
S=\pi\text{i}\int_{M_{4}}w_{2}\cup\rho_{2}\left[\frac{\text{d}\lambda}{2\pi}\right]_{\mathbb{Z}},$
(4.8)

which encodes the ’t Hooft anomaly associated with certain ‘large 2-form gauge
transformations’, and on certain gravitational backgrounds.

For example, take $M_{4}$ to be $\mathbb{C}P^{2}$. As usual, we can
parametrize $\mathbb{C}P^{2}$ with three complex coordinates $z_{1}$, $z_{2}$,
and $z_{3}$, such that $\sum_{i=1}^{3}z_{i}^{\ast}z_{i}=1$ and with the
equivalence $z_{i}\sim e^{i\alpha}z_{i}$ for $\alpha\in\mathbb{R}/\mathbb{Z}$.
Define the 2-form
$\omega:=\frac{i}{2}\partial\overline{\partial}\log(z_{1}^{2}+z_{2}^{2})$,
which is just the volume form on the
$S^{2}\cong\mathbb{C}P^{1}\subset\mathbb{C}P^{2}$ submanifold defined by
$z_{3}=0$. Its cohomology class $[\omega]_{\mathbb{Z}}$ can be taken as a
generator for $H^{2}(\mathbb{C}P^{2};\mathbb{Z})$, and likewise
$a:=\rho_{2}[\omega]_{\mathbb{Z}}$ can be taken as a generator for
$H^{2}(\mathbb{C}P^{2};\mathbb{Z}/2)$. We also have that
$w_{2}(T\mathbb{C}P^{2})=a$. If we shift $a$ by a connection $\lambda$ with
$\left[\frac{\text{d}\lambda}{2\pi}\right]_{\mathbb{Z}}=[n\omega]_{\mathbb{Z}}=n[\omega]_{\mathbb{Z}}$,
the shift in the action is

$\delta S=\pi\text{i}\langle[M_{4}],a^{2}\rangle=n\pi\text{i}$ (4.9)

where $\langle\cdot,\cdot\rangle$ here denotes the pairing between mod 2
homology and cohomology, thus realising the $\mathbb{Z}/2$-valued global
anomaly for odd values of $n$.

As usual, there is a dual description of this anomaly in terms of the 5d SPT
phase that captures it via inflow to the 4d boundary. If $M_{4}$ were
nullbordant, the phase would here be
$S_{\text{SPT}}=\pi\text{i}\int_{X}w_{2}\cup\tau_{e}$ where recall
$\tau_{e}=\rho_{2}\left[\frac{dB_{e}}{2\pi}\right]_{\mathbb{Z}}$, for a
5-manifold $X$ such that $\partial X=M_{4}$ and to which the background fields
are extended. To see this, first note that one could cancel the anomalous
shift (4.8) by adding a ‘counter-term’
$S_{\text{ct.}}=-\pi\text{i}\int_{M_{4}}\widehat{w_{2}}\wedge\frac{B_{e}}{2\pi}$
where $\widehat{w_{2}}$ is a closed 2-form constructed from $w_{2}$,121212To
construct $\widehat{w_{2}}$, first note that there is an integral lift
$W_{2}\in H^{2}(M_{4};\mathbb{Z})$ of $w_{2}\in H^{2}(M_{4};\mathbb{Z}/2)$
because $w_{3}=0$ for any orientable 4-manifold. (One sees this from the long
exact sequence in cohomology associated to the coefficient sequence
$0\to\mathbb{Z}\xrightarrow{\times
2}\mathbb{Z}\xrightarrow{\rho_{2}}\mathbb{Z}/2\to 0$, for which the Bockstein
connecting homomorphism $\beta:H^{2}(\cdot;\mathbb{Z}/2)\to
H^{3}(\cdot;\mathbb{Z}):w_{2}\mapsto\beta(w_{2})$ where
$\rho_{2}\beta(w_{2})=w_{3}$ ). Next, given the integral class $W_{2}$,
construct any complex line bundle over $M_{4}$ with $c_{1}=W_{2}$, and take
$\widehat{w_{2}}$ to be the curvature 2-form of that bundle.  recalling that
the gauge transformation for $B_{e}$ is $B_{e}\mapsto B_{e}+\text{d}\lambda$.
However, this ‘4d action’ is not properly quantised. (It is ‘half-quantised’,
precisely because there is a $\mathbb{Z}/2$ anomaly.) One must instead write
it as the 5d action $S_{\text{ct.}}=-\pi\text{i}\int_{X}w_{2}\cup\tau_{e}$.
Thus the fermionic-monopole electrodynamics has exactly the right anomaly to
be a boundary state of the SPT phase given by
$S_{\text{SPT}}=\pi\text{i}\int_{X}w_{2}\cup\tau_{e}$ (see also Ref.
Jian:2020qab ).131313This mixed anomaly can also be interpreted as a remnant
of the mixed anomaly between the electric and magnetic 1-form symmetry when
only the $\mathbb{Z}/2$ subgroup of the electric 1-form symmetry is coupled to
a background field $w_{2}$.

Figure 1:  A map of the four possible ’t Hooft anomalies that can afflict a 4d
theory with a pair of $\text{U}(1)$ 1-form global symmetries, defined on
orientable manifolds without a spin structure. Each ’t Hooft anomaly
corresponds to a factor in the cobordism group
$H_{I\mathbb{Z}}^{6}\cong\mathbb{Z}\times(\mathbb{Z}/2)^{3}$, and is exhibited
by Maxwell theory on its own (top, corresponding to the local
$\mathbb{Z}$-valued anomaly) or coupled to a particular TQFT (left, bottom,
and right, corresponding to the trio of global $\mathbb{Z}/2$-valued
anomalies). The red arrows illustrate the relations between these four
theories. For the global anomalies, these versions of 4d electromagnetism can
also be realised as the boundary theories for 5d SPT phases.

##### Bordism generator for the $w_{2}\cup\tau_{e}$ anomaly.

Of course, it is important to note that $\mathbb{C}P^{2}$ is not nullbordant
in $\Omega_{4}^{\text{SO}}$, given the signature $\sigma=1$ is a 4d bordism
invariant. Given we saw the anomaly explicitly on $\mathbb{C}P^{2}$, one
cannot in fact realise it via the counterterm
$S_{\text{ct.}}=-\pi\text{i}\int_{X}w_{2}\cup\tau_{e}$ because there is no
5-manifold $X$ bounded by $M_{4}$. But as is well-known Freed:2004yc , the
phase of the partition function $Z$ on such a non-nullbordant manifold suffers
from an ambiguity, since it can always be shifted by a choice of generalized
theta angle, i.e. a coupling to a non-trivial TQFT corresponding to an element
in $\operatorname*{Hom}(\Omega_{4}^{\text{SO}}(\cdot),\mathbb{R}/\mathbb{Z})$.
Thus, one can fix $\arg Z[M_{4},B_{e}]=\theta_{0}$ to a reference phase, and
then $\arg Z$ is uniquely defined on any other 4-manifold
$(M_{4}^{\prime},B_{e}^{\prime})$ that is bordant to $(M_{4},B_{e})$.

This phase can be calculated by constructing a 5d mapping torus $\tilde{X}$ by
taking a cylinder that interpolates between $(M_{4},B_{e})$ and
$(M_{4}^{\prime},B_{e}^{\prime})$, and gluing its ends to make a closed
5-manifold. This $\tilde{X}$ will be a representative of the generator of the
$\mathbb{Z}/2$ factor in $\Omega_{5}^{\text{SO}}((B^{2}\text{U}(1))^{2})$ that
we have claimed is dual to $w_{2}\cup\tau_{e}$.

In more detail, take $M_{4}=\mathbb{C}P^{2}$ and let $B_{e}^{0}$ denote any
reference choice of background 2-form gauge field for the electric 1-form
symmetry on $M_{4}$. Next take the product manifold
$\mathbb{C}P^{2}\times[0,2\pi]$ with a product metric. Over the interval
$I=[0,2\pi]$ one implements (for example linearly) the ‘large’ 2-form gauge
transformation $B_{e}\mapsto B_{e}^{\prime}:=B_{e}+n\omega$,
$n\in\mathbb{Z}$.141414We refer to this as a ‘large’ gauge transformation
because the gauge parameter $n\omega$ is closed but not exact, having a
‘winding number’ of $n$. Since $B_{e}$ and $B_{e}^{\prime}$ are gauge-
equivalent, the manifold $\mathbb{C}P^{2}\times[0,2\pi]$ can be glued at 0 and
$2\pi$ to make a closed 5-manifold with $\text{U}(1)$-2-bundle:

$\tilde{X}:=S^{1}\times\mathbb{C}P^{2},\qquad
B_{e}(\theta,z_{i})=B_{e}^{0}+\frac{t}{2\pi}n\omega,$ (4.10)

where $t\in I$.

To verify that this mapping torus can be taken as a generator of the bordism
group $\Omega^{\text{SO}}_{5}\left((B^{2}\text{U}(1))^{2}\right)$, it is
enough to evaluate the bordism invariant ‘anomaly theory’ $w_{2}\cup\tau_{e}$
on $[\tilde{X}]$ and find a non-trivial value for the phase. Physically,
evaluating $\langle w_{2}\cup\tau_{e},[\tilde{X}]\rangle$ computes the phase
accrued by the partition function on $(\Sigma,B_{e})$ upon undergoing the
2-form gauge transformation $B_{e}\to B_{e}^{\prime}$. Doing the computation,
we have that
$\tau_{e}=\left[\frac{\text{d}t}{2\pi}\right]\cup\left[n\omega\right]=y\cup
nx\in H^{3}(\tilde{X};\mathbb{Z}/2)$, where $x$ is the non-trivial element of
$H^{2}(\tilde{X};\mathbb{Z}/2)$ obtained by pulling back $a$ along the
projection $\tilde{X}\to\mathbb{C}P^{2}$, and $y$ is the non-trivial element
of $H^{2}(\tilde{X};\mathbb{Z}/2)$ obtained by pulling back the generator of
$H^{1}(S^{1};\mathbb{Z}/2)$ along $\tilde{X}\to S^{1}$. For $n$ odd, we have
that $\tau_{e}=y\cup x$, the unique non-trivial element of
$H^{3}(\tilde{X};\mathbb{Z}/2)$. Thus $w_{2}\cup\tau_{e}=x^{2}y\in
H^{5}(\tilde{X};\mathbb{Z}/2)$ for $n$ odd, and so the anomaly theory
evaluates to $1\text{~{}mod~{}}2$ on this mapping torus, while it is trivial
for $n$ even.

#### The $w_{2}\cup\tau_{m}$ phase

An identical account can be given for the magnetic 1-form symmetry, if one
replaces the topological coupling $\pi\text{i}\int w_{2}\cup c_{1}$ in (4.7)
by $\pi\text{i}\int w_{2}\cup c_{1}^{\text{dual}}$, where
$c_{1}^{\text{dual}}$ denotes the first Chern class of the electromagnetic
dual of the gauge field. In that case, the 4d theory suffers from a ’t Hooft
anomaly afflicting the magnetic 1-form symmetry, which obstructs
$\text{U}(1)_{m}[1]$ from being gauged on non-spin manifolds such as
$\mathbb{C}P^{2}$. The 5d anomaly theory is $\int_{M_{5}}w_{2}\cup\tau_{m}$.

Physically, this same anomaly can be understood from a different perspective.
Suppose one couples vanilla Maxwell theory (4.1) to a charge-1 fermion,
defined on all orientable 4-manifolds using a $\text{Spin}_{c}$ structure. The
electric 1-form symmetry is explicitly broken, but one can still probe
anomalies involving the magnetic 1-form symmetry and the gravitational
background. The magnetic 1-form symmetry remains intact, even though we
effectively have a ‘charge-$\frac{1}{2}$’ monopole due to the constraint
$c_{1}=\frac{w_{2}\text{~{}mod~{}}2}{2}$ that follows the definition of
$\text{Spin}_{c}$, because this monopole is not dynamical but rather acts as a
constraint on the field configurations that are summed in the path integral.
The symmetry type is now $\text{Spin}_{c}\times\mathbb{G}_{m}$, with
$B|\mathbb{G}_{m}|=B^{2}\text{U}(1)_{m}$, for which the cobordism group
$H_{I\mathbb{Z}}^{6}(MT(\text{Spin}_{c}\times\mathbb{G}_{m}))\cong\mathbb{Z}/2\times\mathbb{Z}^{2}$.151515The
two factors of $\mathbb{Z}$ arise simply because we have included the
‘$\text{U}(1)$ gauge symmetry’ in our definition of the symmetry type, because
it is now entangled with the spacetime symmetry. If one computed the reduced
$\text{Spin}_{c}$ bordism these factors would go, leaving only the
$\mathbb{Z}/2$-valued $w_{2}\cup\tau_{m}$ ’t Hooft anomaly for the magnetic
1-form symmetry. Of course, all the anomalies involving the electric 1-form
symmetry are absent, as is the $w_{2}w_{3}$ anomaly because the
$\text{Spin}_{c}$ requirement trivialises $w_{3}$, but the $w_{2}\cup\tau_{m}$
global anomaly remains.

#### The $w_{2}\cup w_{3}$ phase

The final $\mathbb{Z}/2$-valued global anomaly that is detected by (4.6)
corresponds to the 5d SPT phase $\int_{M_{5}}w_{2}\cup w_{3}$. To exhibit this
anomaly, one starts from vanilla Maxwell theory and couples $w_{2}(TM_{4})$ as
a background gauge field to both the electric and magnetic $\text{U}(1)$
1-form symmetries. This SPT phase is purely gravitational and has been
extensively analysed in the literature. Its boundary states include the theory
of all-fermion electrodynamics Kravec:2014aza ; Wang:2018qoy  and a fermionic
theory with an emergent $\text{SU}(2)$ gauge symmetry Wang:2018qoy .

###  4.3 Scalar QED and 1-form anomaly interplay

Now let us couple Maxwell theory, defined as before with an SO structure, to a
charge-2 boson. This coupling to matter breaks the electric 1-form symmetry
down to a discrete remnant,

$\text{U}(1)_{e}[1]\rightarrow\mathbb{Z}/2_{e}[1]\,,$ (4.11)

while the full $\text{U}(1)_{m}[1]$ magnetic 1-form symmetry is preserved.
This version of scalar QED therefore furnishes us with two global 1-form
symmetries, one discrete and one continuous, and no 0-form symmetries. The
classifying space of the global symmetry $\mathbb{G}^{\prime}$ is therefore

$B|\mathbb{G}^{\prime}|=B^{2}\mathbb{Z}/2_{e}\times B^{2}\text{U}(1)_{m}\,,$
(4.12)

and the 5d invertible field theories with this symmetry type are classified by
the sequence (4.4) but with $|\mathbb{G}|$ replaced by
$|\mathbb{G}^{\prime}|$. In Appendix B.2 we compute the relevant bordism
groups to stitch together this generalized cohomology group.

##### Local anomalies:

We compute from Appendix B.2 that

$\operatorname*{Hom}\left(\Omega_{6}^{\text{SO}}(B^{2}\mathbb{Z}/2_{e}\times
B^{2}\text{U}(1)_{m}),\mathbb{Z}\right)=0\,,$ (4.13)

and so there are no local anomalies; clearly, the mixed local ’t Hooft anomaly
‘$dB_{e}\cup dB_{m}$’ can no longer be realised now that one of the 1-form
symmetries is broken to a discrete group, because the associated 2-form gauge
field now has zero curvature.

##### Global anomalies:

On the other hand, we find that
$\Omega^{\text{SO}}_{5}\left(B^{2}\mathbb{Z}/2\times
B^{2}\text{U}(1)\right)=(\mathbb{Z}/2)^{4}$. The group classifying global
anomalies is now

$\operatorname*{Hom}\left(\operatorname*{Tor}\Omega_{5}^{\text{SO}}(B^{2}\mathbb{Z}/2_{e}\times
B^{2}\text{U}(1)_{m}),\mathbb{R}/\mathbb{Z}\right)\cong(\mathbb{Z}/2)^{4}\,,$
(4.14)

and we notice that there is an extra factor of $\mathbb{Z}/2$ corresponding to
an extra global anomaly. At the level of characteristic classes, we can
represent these four global anomalies in terms of products of Stiefel–Whitney
classes and cohomology classes of $H^{\bullet}(B^{2}\text{U}(1)\times
B^{2}\mathbb{Z}/2;\mathbb{Z}/2)$ that are in degree-5. There are five of them
in total:

$w_{2}\cup w_{3},\quad w_{2}\cup\tau_{m},\quad
w_{2}\cup\text{~{}Sq}^{1}u_{2},\quad
u_{2}\cup\tau_{m},\quad\text{~{}Sq}^{2}\text{~{}Sq}^{1}u_{2},$

where $u_{2}$ is the unique generator of
$H^{2}(B^{2}\mathbb{Z}/2;\mathbb{Z}/2)$ (see Appendix A.1). However, once
pulled back to an orientable 5-manifold $X$, Wu’s relation Wu1950  tells us
that
$\text{~{}Sq}^{2}\text{~{}Sq}^{1}u_{2}=w_{2}(TX)\cup\text{~{}Sq}^{1}u_{2}$, so
it is not an independent bordism invariant, and we end up with four
invariants, as expected from the bordism group computation.

### The anomaly interplay

Given an appropriate map $\pi$ between two spectra $MTH$ and $MTH^{\prime}$,
there is an induced pullback map between the corresponding cobordism theories,
$\pi^{\ast}:H_{I\mathbb{Z}}^{d+2}\left(MTH^{\prime}\right)\to
H_{I\mathbb{Z}}^{d+2}\left(MTH\right)$, that can be used to relate anomalies
between the two theories. This idea of ‘anomaly interplay’ has recently been
used, for example, to relate local anomalies in 4d $\text{U}(2)$ gauge theory
to Witten’s $\text{SU}(2)$ anomaly Davighi:2020bvi , to study anomalies in
$\mathbb{Z}/k$ symmetries in 2d Grigoletto:2021zyv ; Grigoletto:2021oho  with
applications to bootstrapping conformal field theories, and to derive
anomalies in non-abelian finite group symmetries in 4d Davighi:2022icj .
Physically, the idea of ‘pulling back anomalies’ from one symmetry to another
is not new, but goes back to Elitzur and Nair’s analysis of global anomalies
Elitzur:1984kr , following Witten WITTEN1983422 . In all these cases the
symmetry type takes the form $\\{\text{Spin}\text{~{}or~{}}\text{SO}\\}\times
G^{[0]}$, where $G^{[0]}$ is a 0-form symmetry. In that case, a map of spectra
$\pi$ is induced by any group homomorphism $\pi:G\to G^{\prime}$.

In the present case, where we have theories with 1-form global symmetries, it
is straightforward to adapt this notion of anomaly interplay (which is just
pullback in cobordism). Again, the crucial fact we use is that the nerves
associated with the 1-form symmetries are themselves just ordinary
(topological) groups, between which we can define group homomorphisms. Letting
$\pi:\mathbb{Z}/2\to\text{U}(1):(1,-1)\to(1,e^{\text{i}\pi})$ denote the
subgroup embedding, there is an associated map between symmetry types,
$\pi:\text{SO}\times|\mathbb{G}^{\prime}|\to\text{SO}\times|\mathbb{G}|$ and
an induced pullback map between the cobordism theories,

$\pi^{\ast}:H_{I\mathbb{Z}}^{6}(MT(\text{SO}\times|\mathbb{G}^{\prime}|))\longrightarrow
H_{I\mathbb{Z}}^{6}(MT(\text{SO}\times|\mathbb{G}|))\,.$ (4.15)

Note that, since $H_{I\mathbb{Z}}^{\bullet}$ is a contravariant functor, the
map between anomaly theories goes in the opposite direction to the subgroup
embedding that we started with. Moreover, there is a pullback diagram for the
whole short exact sequence characterizing $H_{I\mathbb{Z}}^{6}$, which encodes
the notion of ‘anomaly interplay’:

$\displaystyle\setcounter{MaxMatrixCols}{11}\begin{CD}0@>{}>{}>(\mathbb{Z}/2)^{4}@>{}>{}>(\mathbb{Z}/2)^{4}@>{}>{}>0@>{}>{}>0\\\
@A{}A{}A@A{\pi^{\ast}}A{}A@A{}A{}A\\\
0@>{}>{}>(\mathbb{Z}/2)^{3}@>{}>{}>(\mathbb{Z}/2)^{3}\times\mathbb{Z}@>{}>{}>\mathbb{Z}@>{}>{}>0\,.\end{CD}$
(4.16)

This anomaly interplay diagram can be used to track ’t Hooft anomalies in the
1-form global symmetries, through the ‘integrating in’ of a charge-2 boson.

Since all the generators of the cobordism groups can be represented by
characteristic classes, we can represent the pullback $\pi^{\ast}$ by its
action on the characteristic classes. The non-trivial action of $\pi^{\ast}$
is encoded in

$\displaystyle\pi^{\ast}:\,$ $\displaystyle\tau_{e}\cup\tau_{m}\mapsto
u_{2}\cup\tau_{m},$ (4.17) $\displaystyle w_{2}\cup\tau_{e}\mapsto
w_{2}\cup\text{~{}Sq}^{1}u_{2}.$ (4.18)

Trivially, $\pi^{\ast}$ maps $w_{2}\cup\tau_{m}$ and $w_{2}\cup w_{3}$ to
themselves.

The most interesting pullback relation is Eq. (4.17), which says that a
$\mathbb{Z}$-valued local anomaly pulls back to a $\mathbb{Z}/2$-valued global
anomaly. This is somewhat analogous to the interplay studied in
Davighi:2020bvi ; Davighi:2020uab  between $\text{U}(2)$ local anomalies and
$\text{SU}(2)$ global anomalies for 4d 0-form symmetries, where the maps there
corresponded to pulling back exponentiated $\eta$-invariants.

To see how the interplay works in this example involving 1-form symmetries,
which is arguably simpler than the story for chiral fermion anomalies, we
follow the general methodology set out in Ref. Davighi:2020uab . To wit, we
start with a 5-manifold $M_{5}$ representative of a class in
$\Omega_{5}^{\text{SO}}(B^{2}\mathbb{Z}/2_{e}\times B^{2}\text{U}(1)_{m})$
that is dual to $u_{2}\tau_{m}$ (i.e. on which $u_{2}\tau_{m}$ evaluates to 1
mod 2). We choose

$M_{5}=S^{2}_{e}\times S^{2}_{m}\times S^{1}_{\theta},$ (4.19)

equipped with a $\mathbb{Z}/2$ background 2-bundle that has support only on
$S^{2}_{e}$, and a $\text{U}(1)_{m}$ background 2-bundle that has support only
on $S^{2}_{m}\times S^{1}_{\theta}$. Specifically, the $\mathbb{Z}/2$ 2-form
connection has non-trivial 2-holonomy round $S_{e}^{2}$, viz
$\int_{S^{2}_{e}}u_{2}=$ 1 mod 2. The $\text{U}(1)_{m}$ 2-form connection
$B_{m}$ on $S^{2}_{m}\times S^{1}_{\theta}$ can be written
$B_{m}=B_{m}^{0}+\frac{\theta}{2\pi}k\omega_{m}$, $k\in\mathbb{Z}$, where
$\omega_{m}$ is the volume form on $S^{2}_{m}$ such that
$\int_{S^{2}_{m}}\omega_{m}=1$, $B_{m}^{0}$ is any connection on the
$S^{2}_{m}$ factor, and $\theta\in[0,2\pi)$ parametrizes $S^{1}_{\theta}$. The
important thing is the flux relation

$\int_{S^{2}_{m}\times
S^{1}_{\theta}}dB_{m}=k\int_{S^{1}_{\theta}}\frac{d\theta}{2\pi}\int_{S^{2}_{m}}\omega_{m}=k\,.$
(4.20)

Recalling $\tau_{m}=[dB_{m}/2\pi]_{\mathbb{Z}}$, integrating gives $\langle
x_{2}\cup\tau_{m},[M_{5}]\rangle=k$ mod 2. When $k$ is odd, $M_{5}$ is not
nullbordant, and the background fields cannot be simultaneously extended to
any 6-manifold that $M_{5}$ bounds.

Now, to show that this global anomaly is the pullback under $\pi^{\ast}$ of
the local anomaly $\tau_{e}\cup\tau_{m}$, we simply use the subgroup embedding
$\pi:\mathbb{Z}/2\to\text{U}(1)$ to embed the $\mathbb{Z}/2$ 2-connection in
an $\text{U}(1)$ 2-connection $B_{e}$. One can take

$B_{e}=\frac{n}{2}\omega_{e},\qquad n\in
2\mathbb{Z}+1,\qquad\int_{S^{2}_{e}}\omega_{e}=1.$ (4.21)

The 5-manifold (4.19) equipped with these structures $B_{e}$ and $B_{m}$ can
be regarded as the pushforward in bordism of the $M_{5}$ that we started with,
$\pi_{\ast}M_{5}$. This is nullbordant in
$\Omega_{5}^{\text{SO}}(B^{2}\text{U}(1)_{e}\times B^{2}\text{U}(1)_{m})$; it
can be realised as the boundary of a six-manifold $M_{6}=D^{3}_{e}\times
S^{2}_{m}\times S^{1}_{\theta}$, where $D^{3}_{e}$ is one half of a 3-sphere
$S^{3}$ that is bounded by $S^{2}_{e}$, to which the electric 2-form
connection (4.21) can now be extended with
$\int_{S^{3}}\frac{dB_{e}}{2\pi}=2n$.161616We emphasize that $B_{e}$ is no
longer a flat connection when extended into the $S^{3}$ bulk, even though it
restricts to a flat connection on the boundary of $D^{3}_{e}$. Using Stokes’
theorem, we thus have

$\int_{S^{2}_{e}\times S^{2}_{m}\times
S^{1}_{\theta}}\frac{B_{e}}{2\pi}\wedge\frac{dB_{m}}{2\pi}=\int_{D^{3}_{e}}\frac{dB_{e}}{2\pi}\int_{S^{2}_{m}\times
S^{1}_{\theta}}\frac{dB_{m}}{2\pi}=nk\,,$ (4.22)

obtaining the same phase from $\tau_{e}\tau_{m}$ as we did from
$u_{2}\tau_{m}$.

On the other hand, we only need to see that the characteristic class
$\tau_{e}$ reduces to the class $\text{~{}Sq}^{1}u_{2}$ when we restrict to a
flat 2-bundle with mod 2 2-holonomy, in order to show that $w_{2}\cup\tau_{e}$
pulls back to $w_{2}\cup\text{~{}Sq}^{1}u_{2}$. To see this most clearly, it
is best to use the language of cochains instead of differential forms. By
identifying $\text{U}(1)$ with $\mathbb{R}/\mathbb{Z}$, we represent the
$\text{U}(1)$ 2-form gauge field $B_{e}$ by a real 2-cochain $b_{e}$. If the
gauge field is flat, $\delta b_{e}$ must be trivial as a cochain valued in
$\mathbb{R}/\mathbb{Z}$. In other words, $\delta b_{e}$ is an integral
3-cochain, whose cohomology class can be identified with $\tau_{e}$. To use
this flat $\text{U}(1)$ 2-cochain to describe a $\mathbb{Z}/2$ 2-form gauge
field, we further impose that it must have mod 2 2-holonomy, i.e. $b_{e}$ must
be half-integral-valued, not just any real cochain. Then
$\tilde{b}_{e}:=2b_{e}$ is an integral cochain whose mod 2 reduction defines a
cohomology class in $H^{2}(M_{5};\mathbb{Z}/2)$ that coincides with $u_{2}$.
As $\text{~{}Sq}^{1}u_{2}=\beta_{2}(u_{2})$, where $\beta_{2}$ is the
Bockstein homomorphism in the long exact sequence for cohomology

$\ldots\rightarrow H^{n}(M_{5};\mathbb{Z}/2)\rightarrow
H^{n}(M_{5};\mathbb{Z}/4)\rightarrow
H^{n}(M_{5};\mathbb{Z}/2)\xrightarrow{\beta_{2}}H^{n+1}(M_{5};\mathbb{Z}/2)\rightarrow\ldots$

induced by the short exact sequence
$0\rightarrow\mathbb{Z}/2\rightarrow\mathbb{Z}/4\rightarrow\mathbb{Z}/2\rightarrow
0$, it can be represented by the mod 2 reduction of
$\frac{1}{2}\delta\tilde{b}_{e}$. But this is exactly the same as $\delta
b_{e}$ which represents $\tau_{e}$. Therefore, the mod 2 reduction of
$\tau_{e}$ is $\text{~{}Sq}^{1}u_{2}$ when we embed the $\mathbb{Z}/2$ 1-form
symmetry inside the $\text{U}(1)$ 1-form symmetry.

There will be no further examples of anomaly interplay in the present paper;
in particular, we do not consider any examples with non-trivially fibred
2-groups (although see footnote 28 for some comments along these lines).

##  5 QED anomalies revisited

The Maxwell examples in the previous Section exhibited only 1-form global
symmetries. In this Section, we move on to theories with both 0-form and
1-form global symmetries that are fused together in a non-trivial 2-group
structure.

Quantum electrodynamics (QED) in 4d with certain fermion content furnishes us
with such a theory. Here both the 0-form and 1-form symmetry groups are
$\text{U}(1)$, and the 2-group structure is non-trivial when there is a mixed
’t Hooft anomaly between the global and gauged 0-form $\text{U}(1)$ currents,
as was discovered in Cordova:2018cvg . There is a further possible ’t Hooft
anomaly in the 2-group structure that comes from the usual cubic $\text{U}(1)$
anomaly for the 0-form symmetry, but, rather than being a $\mathbb{Z}$-valued
local anomaly, the 2-group structure transmutes this cubic anomaly to a
discrete, $\mathbb{Z}/m$-valued global anomaly Cordova:2018cvg , with $m$
given by the modulus of the integral Postnikov class of the 2-group. After
reviewing the physics arguments for these statements, we derive them from the
cobordism perspective. Our spectral sequence calculations reproduce the order
of the finite $\mathbb{Z}/m$ anomaly.

###  5.1 From the physics perspective

Consider a system of Weyl fermions coupled to a $\text{U}(1)_{a}$ gauge group
with dynamical gauge field $a$. Assuming that a fermion with unit charge is
present, the electric 1-form symmetry is broken completely. This leaves only
the magnetic $\text{U}(1)^{[1]}$ 1-form symmetry, with 2-form current
$j^{m}=\star\frac{f}{2\pi}$ as in (4.2), where $f=\text{d}a$. Given enough
Weyl fermions, one can find a global $\text{U}(1)^{[0]}$ 0-form symmetry that
does not suffer from the ABJ anomaly. It is possible, however, that upon
coupling a background gauge field $A$ to this global 0-form symmetry, there is
an operator-valued mixed anomaly that is captured by the anomaly polynomial
term

$\Phi_{6}\supset\frac{\kappa}{16\pi^{3}}f\wedge F\wedge F\,,$ (5.1)

where $F=dA$ is the field strength for the background gauge field $A$. It
describes, via the usual descent procedure, the shift in the effective action
under the background gauge transformation $A\mapsto A+\text{d}\lambda^{(0)}$,
with $\lambda^{(0)}$ a $2\pi$-periodic scalar, by

$\delta S=\frac{\text{i}\kappa}{2}\int_{M_{4}}\lambda^{(0)}\frac{f\wedge
F}{4\pi^{2}}\,.$ (5.2)

It was realised in Ref. Cordova:2018cvg  that we should not interpret this
term as an anomaly, but rather as a non-trivial 2-group structure.

To see why this is the case, we first couple a 2-form background gauge field
$B^{(2)}$, which satisfies the usual normalisation condition

$\int_{M_{3}}\frac{\text{d}B^{(2)}}{2\pi}\in\mathbb{Z}$ (5.3)

on closed 3-manifolds, to the magnetic 1-form symmetry via the coupling

$S_{\text{coupling}}=\frac{\text{i}}{2\pi}\int_{M_{4}}f\wedge B^{(2)}.$ (5.4)

Then the potential anomaly (5.2) can be cancelled by modifying the background
gauge transformation for $B^{(2)}$ from an ordinary 1-form gauge
transformation $B^{(2)}\mapsto B^{(2)}+\text{d}\lambda^{(1)}$, that is
independent of the 0-form gauge transformation of $A$, to a 2-group gauge
transformation

$A\mapsto A+\text{d}\lambda^{(0)},\qquad B^{(2)}\mapsto
B^{(2)}+\text{d}\lambda^{(1)}+\frac{\hat{\kappa}}{2\pi}\lambda^{(0)}F,$ (5.5)

provided that we identify $\hat{\kappa}$ with $-\kappa/2$.171717This is well-
defined because it can be shown that $\kappa$ is always even. Here the gauge
transformation parameter $\lambda^{(1)}$ is a properly normalised
$\text{U}(1)$ gauge field.

This modified transformation mixes the magnetic $\text{U}(1)^{[1]}$ 1-form
symmetry with the $\text{U}(1)^{[0]}$ 0-form symmetry, and encodes the 2-group
structure at the level of the background fields. In our quadruplet notation,
the 2-group is

$\mathbb{G}=(\text{U}(1),\text{U}(1),0,\hat{\kappa}),$ (5.6)

with the Postnikov class $\hat{\kappa}\in
H^{3}(B\text{U}(1);\text{U}(1))\cong\mathbb{Z}$, where we use the fact that
$H^{3}(B\text{U}(1);\text{U}(1))$ with continuous $\text{U}(1)$ coefficients
is isomorphic to $H^{4}(B\text{U}(1);\mathbb{Z})\cong\mathbb{Z}$ via the
universal coefficient theorem. The nerve of such a 2-group is the extension

$|\mathbb{G}|=\text{U}(1)^{[0]}\times_{\hat{\kappa}}\text{U}(1)^{[1]}.$ (5.7)

The non-trivial 2-group structure can also be seen without turning on the
background gauge fields explicitly. If we write $j$ for the 1-form current
associated with $\text{U}(1)^{[0]}$, then the Ward identity takes the non-
trivial form

$\partial^{\mu}j_{\mu}(x)j_{\nu}(y)=\frac{\hat{\kappa}}{2\pi}\partial^{\lambda}\delta^{(4)}(x-y)j^{m}_{\nu\lambda}(y)\,,$
(5.8)

where $j^{m}_{\nu\lambda}$ are the components of the magnetic 2-form current
$j^{m}$. This Ward identity shows how the 0-form and 1-form currents are
fused, realising Eq. (1.3).

The 2-group symmetry $\text{U}(1)^{[0]}\times_{\hat{\kappa}}\text{U}(1)^{[1]}$
can still suffer from an ’t Hooft anomaly. Recall that, when $\hat{\kappa}=0$,
the anomaly polynomial for our system of Weyl fermions is

$\Phi_{6}=\frac{\mathcal{A}_{3}}{6}c_{1}(F)^{3}-\frac{\mathcal{A}_{\text{mixed}}}{24}p_{1}(R)c_{1}(F),$
(5.9)

where $c_{1}(F)=F/2\pi$ is the first Chern class of the $\text{U}(1)^{[0]}$
bundle, and $p_{1}(R)=\frac{1}{8\pi^{2}}\text{Tr}\,R\wedge R$ is the first
Pontryagin class of the tangent bundle, where $R$ is the curvature 2-form.
Before we continue, it is important to discuss the role of these anomaly
coefficients in the (co)bordism context, still for the case $\hat{\kappa}=0$.
Naïvely, one might think that the cubic anomaly coefficient $\mathcal{A}_{3}$
and the mixed $\text{U}(1)$-gravitational anomaly coefficient
$\mathcal{A}_{\text{mixed}}$ are the two integers that classify anomalies
according to cobordism, i.e. that $\mathcal{A}_{3}$ and
$\mathcal{A}_{\text{mixed}}$ can be chosen as generators of the group
$\operatorname*{Hom}\left(\Omega^{\text{Spin}}_{6}\left(B\text{U}(1)\right),\mathbb{Z}\right)\cong\mathbb{Z}\times\mathbb{Z}$.
However, this is not the correct identification because $\mathcal{A}_{3}$ and
$\mathcal{A}_{\text{mixed}}$ are not quite independent. It can be shown that

$\alpha_{1}:=c_{1}(F)^{3}\qquad\text{and}\qquad\alpha_{2}:=\frac{c_{1}(F)}{6}\left(c_{1}(F)^{2}-\frac{1}{4}p_{1}(R)\right)$
(5.10)

are independent integral cohomology classes.181818That $\alpha_{1}$ is
integral is evident from the definition of $c_{1}(F)$. To see that
$\alpha_{2}$ is integral, we observe that it is the anomaly polynomial for a
Weyl fermion with charge $+1$, and then apply the Atiyah–Singer index theorem.
In terms of these basis generators, we can write $\Phi_{6}$ as

$\Phi_{6}=\frac{1}{6}\left(\mathcal{A}_{3}-\mathcal{A}_{\text{mixed}}\right)\alpha_{1}+\mathcal{A}_{\text{mixed}}\alpha_{2}.$
(5.11)

Again, by the Atiyah–Singer index theorem, $\Phi_{6}$ must be integral. Since
$\alpha_{1}$ and $\alpha_{2}$ are integral basis generators, we can deduce
that the two integers $(r,s)$ that label
$\operatorname*{Hom}\left(\Omega^{\text{Spin}}_{6}\left(B\text{U}(1)\right),\mathbb{Z}\right)$
are

$(r,s)=\left(\frac{1}{6}\left(\mathcal{A}_{3}-\mathcal{A}_{\text{mixed}}\right),\mathcal{A}_{\text{mixed}}\right).$
(5.12)

Continuing, let’s switch the Postnikov class $\hat{\kappa}$ back on, and
couple the background 2-form gauge field $B^{(2)}$ to the $\text{U}(1)^{[1]}$
1-form symmetry. We are free to add a Green–Schwarz counter-term

$S_{\text{GS}}=\frac{in}{2\pi}\int_{M_{4}}B^{(2)}\wedge F,\qquad
n\in\mathbb{Z}$ (5.13)

to the action. Under the 2-group transformation (5.5), the effective action
shifts if the anomaly coefficients $\mathcal{A}_{3}$ and
$\mathcal{A}_{\text{mixed}}$ are non-zero, by

$\delta
S=\text{i}\frac{\mathcal{A}_{3}+6n\hat{\kappa}}{6}\int_{M_{4}}\lambda^{(0)}\frac{F\wedge
F}{4\pi^{2}}-\text{i}\frac{\mathcal{A}_{\text{mixed}}}{24}\int_{M_{4}}\lambda^{(0)}p_{1}(R).$
(5.14)

Since one may choose the counterterm coefficient $n$ to be any integer, one
sees that $\mathcal{A}_{3}$ is only well-defined modulo $6\hat{\kappa}$
Cordova:2018cvg . It follows that the integer
$r=\frac{1}{6}\left(\mathcal{A}_{3}-\mathcal{A}_{\text{mixed}}\right)$ that we
claim classifies the anomaly is not really valued in $\mathbb{Z}$, being well-
defined only modulo $|\hat{\kappa}|$. This corresponds to a global anomaly in
the 2-group symmetry, that is valued in the cyclic group

$\mathbb{Z}/m,\qquad m=|\hat{\kappa}|.$ (5.15)

We emphasize that, even though $\mathcal{A}_{3}$ is well-defined modulo
$6\hat{\kappa}$ (as was derived in Cordova:2018cvg ), the discrete group that
classifies the global anomaly has order $|\hat{\kappa}|$ (and not
$6|\hat{\kappa}|$). The mixed gravitational anomaly remains a
$\mathbb{Z}$-valued local anomaly.

###  5.2 From the bordism perspective

In this Section we show how the ’t Hooft anomalies afflicting the 2-group
global symmetry, that we have just described, can be precisely understood
using cobordism. To do so, we need to compute the generalized cohomology
groups

$H^{6}_{I\mathbb{Z}}\left(MT\left(\text{Spin}\times|\mathbb{G}|\right)\right),\qquad|\mathbb{G}|=U(1)^{[0]}\times_{\hat{\kappa}}U(1)^{[1]}\,,$
(5.16)

which are built from $\Omega_{5}^{\text{Spin}}(B|\mathbb{G}|)$ and
$\Omega_{6}^{\text{Spin}}(B|\mathbb{G}|)$, for each value of the Postnikov
class $\hat{\kappa}$. These abelian groups detect and classify all possible
anomalies for this 2-group symmetry type.

To compute these bordism groups, we follow the general strategy outlined in
§3.2. We first apply the cohomological Serre spectral sequence to the
fibration

$K(\mathbb{Z},3)\to B|\mathbb{G}|\to K(\mathbb{Z},2)$ (5.17)

to calculate the cohomology of $B|\mathbb{G}|$ in relevant low degrees. Then
we will convert the result into homology groups, which are fed into the
Atiyah–Hirzebruch spectral sequence for the fibration $\text{pt}\to
B|\mathbb{G}|\to B|\mathbb{G}|$, for which the second page is
$E^{2}_{p,q}=H_{p}\left(B|\mathbb{G}|;\Omega^{\text{Spin}}_{q}(\text{pt})\right)$,
to compute the spin bordism.

So, to begin, the $E_{2}$ page of the cohomological Serre spectral sequence
for the fibration (5.17) is given by

$E_{2}^{p,q}=H^{p}\big{(}K(\mathbb{Z},2);H^{q}(K(\mathbb{Z},3);\mathbb{Z})\big{)}.$
(5.18)

Using
$H^{\bullet}(K(\mathbb{Z},3);\mathbb{Z})\cong\\{\mathbb{Z},0,0,\mathbb{Z},0,0,\mathbb{Z}/2,0,\mathbb{Z}/3,\mathbb{Z}/2,\ldots\\}$,
given in Table 3 of Appendix A.1, we can construct the $E_{2}$ page as shown
in the left-hand side of Fig. 2. In fact, what is shown there is the $E_{4}$
page, since the entries are sparse enough that there are no non-trivial
differentials in the region we are interested in until page $E_{4}$.

Figure 2: Fourth and fifth pages of the Serre spectral sequence to compute the
integral cohomology of the fibration $K(\mathbb{Z},3)\to B|\mathbb{G}|\to
K(\mathbb{Z},2)$.

The differentials $\alpha$, $\beta$, and $\gamma$ shown on the left-hand
diagram of Fig. 2 are linear in the Postnikov class
$\hat{\kappa}\in\mathbb{Z}$ (see the Appendix of Ref. Kapustin:2013uxa ). More
precisely, we write

$H^{3}(K(\mathbb{Z},3);\mathbb{Z})\cong\operatorname*{Hom}(\mathbb{Z}^{[1]},\mathbb{Z})\cong\mathbb{Z}$

using the universal coefficient theorem and the fact that
$K(\mathbb{Z},3)=B^{2}\text{U}(1)^{[1]}=B^{3}\mathbb{Z}^{[1]}$. Here, we
include the superscript to emphasise that this $\mathbb{Z}$ comes from our
$\text{U}(1)$ 1-form symmetry part. From this, we can write the entry
$E^{0,3}_{4}$ in the Serre spectral sequence as
$\operatorname*{Hom}(\mathbb{Z}^{[1]},\mathbb{Z})$. Then, the differential
$\alpha$ is given by ‘contraction’ (adopting the terminology of
Kapustin:2013uxa ) with the Postnikov class

$\begin{split}\alpha:\operatorname*{Hom}(\mathbb{Z}^{[1]},\mathbb{Z})&\to
H^{4}(B\text{U}(1)^{[0]};\mathbb{Z})\\\ x&\mapsto
x\circ\hat{\kappa}\end{split}$ (5.19)

where we make use of the fact that the Postnikov class is a cohomology class

$\hat{\kappa}\in H^{3}(B\text{U}(1)^{[0]},\text{U}(1)^{[1]})\cong
H^{4}(B\text{U}(1)^{[0]};\mathbb{Z}^{[1]})\;.$

Similar arguments apply for the differentials $\beta$ and $\gamma$. Thus,
$\alpha$, $\beta$, and $\gamma$ map $1$ to $\pm\hat{\kappa}$, resulting in the
$E_{5}$ page as shown, where $m:=|\hat{\kappa}|$. We can then read off the
integral cohomology groups to be

$H^{\bullet}(B|\mathbb{G}|;\mathbb{Z})\cong\\{\mathbb{Z},0,\mathbb{Z},0,\mathbb{Z}/m,0,e(\mathbb{Z}/2,\mathbb{Z}/m),0,e(\mathbb{Z}/6,\mathbb{Z}/m),\ldots\\}\,$
(5.20)

where the notation $e(A,B)$ denotes an extension of $A$ by $B$, viz. a group
that fits in the short exact sequence $B\hookrightarrow
e(A,B)\twoheadrightarrow A$. We compute the mod 2 cohomology by the same
method in Appendix A.2.

Heuristically, one can also argue for the form of $\alpha$ and $\beta$ as
follows (c.f. Appendix B.6 of Ref. Lee:2020ewl ). Let us start with a
representative of a generator of the cohomology group
$H^{3}(K(\mathbb{Z},3),\mathbb{Z})\cong\mathbb{Z}$, and ask what it becomes
when $K(\mathbb{Z},3)$ is the fibre of $B|\mathbb{G}|$. Given the
normalisation condition (5.3), the 3-form

$\tilde{h}:=\text{d}B^{(2)}/2\pi$ (5.21)

represents the generator of
$H^{3}(K(\mathbb{Z},3);\mathbb{Z})\cong\mathbb{Z}$. This is true when
$K(\mathbb{Z},3)$ stands on its own. However, when we pass to the 2-group
$\mathbb{G}$ and take $K(\mathbb{Z},3)$ to be the fibre of $B|\mathbb{G}|$,
then $\tilde{h}$ is not gauge-invariant under (5.5) for a general Postnikov
class, and cannot be a representative of any cohomology class for
$B|\mathbb{G}|$. We can remedy this by modifying the definition of $\tilde{h}$
to

$h:=\tilde{h}-\frac{\hat{\kappa}}{2\pi}A\wedge F.$ (5.22)

The trade off is that the gauge-invariant $h$ is no longer closed; instead

$\text{d}h=-\frac{\hat{\kappa}}{4\pi^{2}}F\wedge F=-\hat{\kappa}c_{1}\cup
c_{1},$ (5.23)

where $c_{1}:=\frac{F}{2\pi}$ is the first Chern class of the $U(1)^{[0]}$
bundle, and $c_{1}\cup c_{1}$ can be taken as the generator of
$H^{4}(K(\mathbb{Z},2);\mathbb{Z})$. The 2-group relation (5.23) implies that
both the differentials $\alpha$ and $\beta$ in Fig. 2 map $1$ to
$-\hat{\kappa}$.191919The extra minus sign comes from the convention used to
define the Postnikov class.

From this cohomological starting point, we can proceed to compute the spin
bordism groups. We find it is helpful to split the discussion into the cases
where $m$ is even or odd, for which the bordism group calculations are tackled
using different tricks. As a warm up, we first consider the simplest case
where $m$ is zero, corresponding to a 0-form and 1-form symmetry that do not
mix.

####  5.2.1 Zero Postnikov class

We first consider the trivial toric 2-group where $B|\mathbb{G}|$ is a simply
a product space $K(\mathbb{Z},2)\times K(\mathbb{Z},3)$. As $B|\mathbb{G}|$ is
a product, we can use the Künneth theorem to determine the homology groups of
$K(\mathbb{Z},2)\times K(\mathbb{Z},3)$ from the homology groups of each
factor, given by Eq. (A.2) and Table 3 in Appendix A.1. We obtain

$H_{\bullet}(K(\mathbb{Z},2)\times
K(\mathbb{Z},3);\mathbb{Z})\cong\left\\{\mathbb{Z},0,\mathbb{Z},\mathbb{Z},\mathbb{Z},\mathbb{Z}\times\mathbb{Z}/2,\mathbb{Z},\mathbb{Z}\times\mathbb{Z}/6,\mathbb{Z}\times\mathbb{Z}/2,\ldots\right\\}$
(5.24)

and construct the second page of the AHSS as shown in Fig. 3, with non-trivial
differentials on the $E^{2}$ page indicated by coloured arrows.

Figure 3: The $E^{2}$ and $E^{3}$ pages for the Atiyah–Hirzebruch spectral
sequence for $\Omega^{\text{Spin}}_{\bullet}(K(\mathbb{Z},2)\times
K(\mathbb{Z},3))$.

The differentials on the zeroth and the first rows are the composition
$\widetilde{\text{Sq}^{2}}\circ\rho_{2}$ and $\widetilde{\text{Sq}^{2}}$,
respectively, where $\rho_{2}$ is reduction modulo 2. To compute the action of
these differentials, we need to know how the Steenrod squares act on the mod 2
cohomology ring of the product $K(\mathbb{Z},2)\times K(\mathbb{Z},3)$. From
the mod 2 cohomology rings of $K(\mathbb{Z},2)$ and $K(\mathbb{Z},3)$ given by
(A.3) and (A.4), we obtain

$H^{\bullet}(K(\mathbb{Z},2)\times
K(\mathbb{Z},3);\mathbb{Z}/2)\cong\mathbb{Z}/2[c_{1},\tau_{3},\text{~{}Sq}^{2}\tau_{3},\text{~{}Sq}^{4}\text{~{}Sq}^{2}\tau_{3},\ldots],$
(5.25)

where $c_{1}$ and $\tau_{3}$ are the unique generators in degree $2$ and $3$,
respectively. In our diagram, the red maps correspond to
$\text{Sq}^{2}c_{1}=c_{1}^{2}$, the blue maps to $\text{Sq}^{2}\tau_{3}$ being
a generator of $H^{\bullet}(K(\mathbb{Z},3);\mathbb{Z}/2)$, and the magenta
maps to $\text{Sq}^{2}(c_{1}\tau_{3})=c_{1}^{2}\tau_{3}$. The differential
depicted in the $E^{3}$ page from $E_{7,0}^{3}$ in the diagram must be non-
trivial by comparing $\Omega^{\text{Spin}}_{6}(K(\mathbb{Z},2)\times
K(\mathbb{Z},3))$ with the result computed with the Adams spectral sequence.
We can then read off the bordism groups in lower degrees, which we collect
below in Table 1. We piece together the cobordism group

$H_{I\mathbb{Z}}^{6}(MT(\text{Spin}\times\text{U}(1)\times
B\text{U}(1)))\cong\mathbb{Z}\times\mathbb{Z},$ (5.26)

which classifies anomalies for this symmetry type. The pair of
$\mathbb{Z}$-valued local anomalies anomalies just corrresponds to the usual
cubic $c_{1}^{3}$ and mixed gravitational $c_{1}p_{1}$ anomalies associated
with the $\text{U}(1)$ 0-form symmetry. The presence of the 1-form symmetry
here plays no role in anomaly cancellation for this dimension.

####  5.2.2 Even Postnikov class

Now we turn to the case where $m=|\hat{\kappa}|$ is a non-zero even integer.
Continuing from the cohomology calculation above, summarised in Eq. (5.20),
there are two options for the extension $e(\mathbb{Z}/2,\mathbb{Z}/m)$; either
the trivial extension $\mathbb{Z}/2\times\mathbb{Z}/m$ or the non-trivial one
$e(\mathbb{Z}/2,\mathbb{Z}/m)\cong\mathbb{Z}/(2m)$, which are non-isomorphic.
By comparing the mod 2 cohomology calculated from applying the universal
coefficient theorem to (5.20) and the one calculated directly from the Serre
spectral sequence, one can show that the correct extension is the direct
product $\mathbb{Z}/2\times\mathbb{Z}/m$. The integral homology for
$B|\mathbb{G}|$ is then

$\displaystyle H_{\bullet}\left(B|\mathbb{G}|;\mathbb{Z}\right)$
$\displaystyle\cong\left\\{\mathbb{Z},0,\mathbb{Z},\mathbb{Z}/m,0,\mathbb{Z}/m\times\mathbb{Z}/2,0,e(\mathbb{Z}/6,\mathbb{Z}/m),\ldots\right\\}$
(5.27)

and the mod 2 homology is

$H_{\bullet}(B|\mathbb{G}|;\mathbb{Z}/2)\cong\left\\{\mathbb{Z}/2,0,\mathbb{Z}/2,\mathbb{Z}/2,\mathbb{Z}/2,\mathbb{Z}/2\times\mathbb{Z}/2,\ldots\right\\}\,,$
(5.28)

The $E^{2}$ page of the AHSS is then given by Fig. 4. We observe that there
are non-vanishing differentials already on the $E^{2}$ page (which will not be
the case when we turn to the case of odd $m$). These non-trivial $E^{2}$
differentials make the spectral sequence easier to compute, as follows.

Figure 4:  The $E^{2}$ page of the AHSS for the fibration $\text{pt}\to
B|\mathbb{G}|\to B|\mathbb{G}|$ when $\mathbb{G}$ is the 2-group
$\text{U}(1)^{[0]}\times_{\hat{\kappa}}\text{U}(1)^{[1]}$ with
$m:=|\hat{\kappa}|$ even.

The differentials $\alpha$ and $\beta$ on the second page are duals
$\widetilde{\text{~{}Sq}^{2}}$ of the Steenrod square $\text{~{}Sq}^{2}$,
while the differential $\gamma$ is a mod 2 reduction followed by the dual of
$\mathrm{Sq}^{2}$ teichner1993signature . The Steenrod square dual to $\beta$
sends a unique generator in $H^{2}(B|\mathbb{G}|;\mathbb{Z}/2)$ to a unique
generator in $H^{4}(B|\mathbb{G}|;\mathbb{Z}/2)$ (c.f. Appendix A.2), so
$\beta$ must be the non-trivial map from $\mathbb{Z}/2$ to $\mathbb{Z}/2$,
killing off both factors. Similarly, the Steenrod square dual to $\alpha$ acts
on the unique generator of $H^{3}(B|\mathbb{G}|;\mathbb{Z}/2)$ that comes from
the generator $\tau_{3}\in H^{3}(K(\mathbb{Z},3);\mathbb{Z}/2)$ of the fibre,
which we will also label by $\tau_{3}$. The image is a generator of
$H^{5}(B|\mathbb{G}|;\mathbb{Z}/2)$ that comes from $\text{Sq}^{2}\tau_{3}$ of
$H^{5}(K(\mathbb{Z},3);\mathbb{Z}/2)$. It generates the $\mathbb{Z}/2$ factor
in the mod 2 cohomology that is a reduction from the $\mathbb{Z}/2$ factor in
the integral cohomology, and not the $\mathbb{Z}/m$ factor since it is the
$\mathbb{Z}/2$ factor that arise from the fibre’s contribution. Therefore,
both $\alpha$ and $\gamma$ must be non-trivial, with
$\ker\gamma\cong\mathbb{Z}/m$, as indicated in the $E^{3}$ page in Fig. 4.

Before continuing, we pause here to emphasize the importance of using (spin)
cobordism, rather than just cohomology, to study ’t Hooft anomalies in these
theories. At the level of cohomology, the existence of a non-trivial
$\mathbb{Z}/2$-valued cohomology class $\text{Sq}^{2}\tau_{3}\in
H^{5}(K(\mathbb{Z},3);\mathbb{Z}/2)$ might suggest there is a non-trivial SPT
phase on 5-manifolds $M_{5}$, or equivalently a non-trivial anomaly theory for
the corresponding 4d theory, with partition function

$Z=\exp\left(\pi\text{i}\int_{M_{5}}\text{Sq}^{2}\tau_{3}\right),$ (5.29)

where $\text{Sq}^{2}\tau_{3}$ is pulled back to $M_{5}$. But, one can use Wu’s
relation to trade the Steenrod square operation for a Stiefel–Whitney class
Wu1950 , viz.

$Z=\exp\left(\pi\text{i}\int_{M_{5}}w_{2}(TM_{5})\cup\tau_{3}\right).$ (5.30)

If we now restrict to $M_{5}$ being a spin manifold, then $w_{2}(TM_{5})$ is
trivial and we immediately learn that this SPT phase is trivial. Of course,
this ‘trivialisation’ of the SPT phase corresponding to the cohomology class
in $H^{5}(K(\mathbb{Z},3);\mathbb{Z}/2)$ is automatically captured by the
spectral sequence computation for spin bordism, by the non-triviality of the
map $\gamma$ on turning from the second to the third page of the AHSS.

Continuing with the bordism computation, we now find that the entries in the
range $p+q\leq 5$ stabilise on this $E^{3}$ page, whence we can read off the
spin bordism groups for 2-group symmetry
$\mathbb{G}=\text{U}(1)^{[0]}\times_{\hat{\kappa}}\text{U}(1)^{[1]}$ with even
Postnikov class $\hat{\kappa}$ up to degree-5, which are listed in Table 1.
There is still an undetermined group extension in
$\Omega^{\text{Spin}}_{3}(B|\mathbb{G}|)$ but we will argue from the physics
point of view in Section 6 that it must be the non-trivial extension
$\mathbb{Z}/(2m)$. In summary, our results are those given in Table 1.

$i$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$
---|---|---|---|---|---|---|---
$\Omega^{\text{Spin}}_{i}(B|\mathbb{G}|)$, $\hat{\kappa}\neq 0$ | $\mathbb{Z}$ | $\mathbb{Z}/2$ | $\mathbb{Z}\times\mathbb{Z}/2$ | $\mathbb{Z}/(2|\hat{\kappa}|)$ | $\mathbb{Z}$ | $\mathbb{Z}/|\hat{\kappa}|$ | $\mathbb{Z}\times\text{Tors}$
$\Omega^{\text{Spin}}_{i}(B|\mathbb{G}|)$, $\hat{\kappa}=0$ | $\mathbb{Z}$ | $\mathbb{Z}/2$ | $\mathbb{Z}\times\mathbb{Z}/2$ | $\mathbb{Z}$ | $\mathbb{Z}\times\mathbb{Z}$ | $\mathbb{Z}$ | $\mathbb{Z}\times\mathbb{Z}\times\text{Tors}$
Table 1: Spin bordism groups for the twisted 2-group symmetry
$\mathbb{G}=\text{U}(1)^{[0]}\times_{\hat{\kappa}}\text{U}(1)^{[1]}$. Here,
Tors denotes an undetermined pure torsion part. For comparison, we also
include the bordism groups in the case $\hat{\kappa}=0$, which corresponds to
a direct product of 0-form and 1-form symmetries. One can thus see how, for
both 2d and 4d quantum field theories, local anomalies in the untwisted case
morph into global anomalies when the Postnikov class $\hat{\kappa}$ is turned
on, thanks to a Green–Schwarz-like mechanism.

####  5.2.3 Odd Postnikov class

We now turn to the case where $m$ is odd, for which the spin bordism
computation turns out to be rather more difficult, technically. Firstly, when
$m$ is odd, the extension $e(\mathbb{Z}/2,\mathbb{Z}/m)$ is unambiguously
$\mathbb{Z}/m\times\mathbb{Z}/2$ (which is, for odd $m$, isomorphic to
$\mathbb{Z}/(2m)$). The integral homology of $B|\mathbb{G}|$ is as written in
Eq. (5.27), and the mod 2 homology groups are now

$\displaystyle H_{\bullet}\left(B|\mathbb{G}|;\mathbb{Z}/2\right)$
$\displaystyle\cong\left\\{\mathbb{Z}/2,0,\mathbb{Z}/2,0,0,\mathbb{Z}/2,\ldots\right\\}$
(5.31)

by the universal coefficient theorem. Now we feed these results into the AHSS
for the trivial point fibration $\text{pt}\to B|\mathbb{G}|\to B|\mathbb{G}|$.
The $E^{2}$ has no non-trivial differentials in the range of interest.

Figure 5: The $E^{3}$ page of the AHSS for the fibration $\text{pt}\to
B|\mathbb{G}|\to B|\mathbb{G}|$ when $\mathbb{G}$ is the 2-group
$\text{U}(1)^{[0]}\times_{\hat{\kappa}}\text{U}(1)^{[1]}$ with
$m:=|\hat{\kappa}|$ odd.

The $E^{3}$ page is shown in Fig. 5, with a single potentially non-trivial
differential in the range of interest, $\text{d}_{3}:E^{3}_{5,0}\to
E^{3}_{2,2}$. If this differential were trivial, then
$\Omega^{\text{Spin}}_{5}(B|\mathbb{G}|)$ would equal
$\mathbb{Z}/m\times\mathbb{Z}/2$, and
$\Omega^{\text{Spin}}_{4}(B|\mathbb{G}|)$ would equal
$\mathbb{Z}\times\mathbb{Z}/2$. The factor of $\mathbb{Z}/m$ in
$\Omega^{\text{Spin}}_{5}(B|\mathbb{G}|)$ would correspond to the global
anomaly that we described in the previous Section, which we know to be valued
in $\mathbb{Z}/m$ as in (5.15). The extra factor of $\mathbb{Z}/2$ would
presumably correspond to a further global anomaly that we have not seen so far
by physics arguments. On the other hand, if this differential were the non-
trivial map, then $\Omega^{\text{Spin}}_{5}(B|\mathbb{G}|)$ would equal
$\mathbb{Z}/m$, agreeing precisely with the physics account, and
$\Omega^{\text{Spin}}_{4}(B|\mathbb{G}|)$ would just equal $\mathbb{Z}$.

It turns out that this all-important $d_{3}$ differential is non-trivial. But
being a differential on the third page, there are no straightforwardly-
applicable formulae (analogous to the formulae in terms of Steenrod squares
that are available on the second page) that we can use to compute it directly.
Indeed, our usual AHSS plus ASS techniques are not sufficient to constrain
this differential. This differential can nonetheless be evaluated using
different arguments,202020We are very grateful to Arun Debray for sharing this
ingenious argument to compute this differential. Appendix C is written by Arun
Debray.  which we present in Appendix C. The gist of the argument is as
follows.

The central character is a long exact sequence in bordism groups,212121See
Appendix E of Ref Debray:2023yrs  for a related theorem. The proof of the
theorem used in this paper, and that of Ref Debray:2023yrs , will appear in
future work DebrayNEW . analogous to the Gysin long exact sequence in ordinary
homology, of the form

$\dots\to\Omega^{\text{Spin}}_{d}(S(V))\to\Omega^{\text{Spin}}_{d}(B|\mathbb{G}|)\to\Omega^{\text{Spin}}_{d-2}((B|\mathbb{G}|)^{V-2})\to\Omega^{\text{Spin}}_{d}(S(V))\to\dots$
(5.32)

Here $V\to B|\mathbb{G}|$ is the pullback bundle (which is rank 2) of the
tautological line bundle $L\to B\text{U}(1)^{[0]}$ along the quotient map
$q:B|\mathbb{G}|\to B\text{U}(1)^{[0]}$, $S(V)\to B|\mathbb{G}|$ is the
associated sphere bundle of $V$, and $(B|\mathbb{G}|)^{V-2}$ is the Thom
spectrum of the virtual bundle. This long exact sequence in bordism is
extremely powerful: by proving that the groups
$\Omega^{\text{Spin}}_{5}(S(V))$ and
$\Omega^{\text{Spin}}_{3}((B|\mathbb{G}|)^{V-2})$ both lack free and 2-torsion
summands, one learns that the group $\Omega^{\text{Spin}}_{5}(B|\mathbb{G}|)$
in the middle, which is our bordism group of interest, also lacks 2-torsion.
The $\mathbb{Z}/2$ factor discussed above is therefore absent (from which we
learn that it must be killed by the differential $d_{3}$ which is therefore
non-trivial).

Putting things together, we can thus extract all the spin bordism groups for
2-group symmetry
$\mathbb{G}=\text{U}(1)^{[0]}\times_{\hat{\kappa}}\text{U}(1)^{[1]}$, with odd
Postnikov class $\hat{\kappa}$, up to degree-5.

####  5.2.4 The free part

When $m$ is non-trivial, regardless of its parity, we can also show that the
free part of $\Omega^{\text{Spin}}_{6}(B|\mathbb{G}|)$ is given by
$\mathbb{Z}$ coming from the entry $E^{r}_{2,4}$. First, observe that the free
part of the $\Omega^{\text{Spin}}_{6}(B|\mathbb{G}|)$, if there is one at all,
can only come from this entry as any other entry from the diagonal $p+q=6$ is
either trivial or pure torsion. Next, we need to show that $E^{\infty}_{2,4}$
is non-trivial. The only differential that could kill it along the way is
$d_{5}:E^{5}_{7,0}\to E^{5}_{2,4}$. But this differential must be trivial
because $E^{5}_{7,0}$ is pure torsion (as $E^{2}_{7,0}$, given by
$H_{7}(B|\mathbb{G}|;\mathbb{Z})\cong e(\mathbb{Z}/6,\mathbb{Z}/m)$ in (5.27),
is pure torsion), and $\operatorname*{Hom}(G,\mathbb{Z})=0$ if $G$ is pure
torsion. Therefore, there is one free factor in the diagonal
$E^{\infty}_{p,q}$ with $p+q=6$, and we obtain

$\operatorname*{Hom}\left(\Omega^{\text{Spin}}_{6}(B|\mathbb{G}|),\mathbb{Z}\right)\cong\mathbb{Z}.$
(5.33)

From these results, we piece together the cobordism group that classifies
anomalies for this global 2-group symmetry, valid for even and odd $m\neq 0$:

$\boxed{H^{6}_{I\mathbb{Z}}\left(MT\left(\text{Spin}\times|\mathbb{G}|\right)\right)\cong\mathbb{Z}/m\times\mathbb{Z}\,.}$
(5.34)

In terms of the anomaly coefficients, the space of anomaly theories is
classified by

$(r,s)=\left(\frac{1}{6}\left(\mathcal{A}_{3}-\mathcal{A}_{\mathrm{mixed}}\right)\text{~{}mod~{}}m,\mathcal{A}_{\mathrm{mixed}}\right)\in\mathbb{Z}/m\times\mathbb{Z}\,,\qquad
m=|\hat{\kappa}|\,,$ (5.35)

in agreement with the results of the previous Subsection. To reiterate, the
local anomaly associated with $c_{1}^{3}$ (i.e. whose coefficient is the sum
of $U(1)^{[0]}$ charges cubed) becomes a global anomaly when the 0-form and
1-form symmetries are fused into a 2-group defined so that the mixed ’t Hooft
anomaly ($\sim f\wedge F\wedge F$) vanishes. The mixed gravitational anomaly
associated with $p_{1}c_{1}$ remains a $\mathbb{Z}$-valued local anomaly.

The result for the spin bordism group in degree-3 in Table 1 is also
interesting, implying the presence of an novel global anomaly structure for
the corresponding symmetry in two dimensions. We discuss this from a physics
perspective in §6.

##  6 Abelian 2-group enhancement in two dimensions

In this Section, we consider 2d avatars of the 4d 2-group anomalies that we
have been discussing. In this lower-dimensional version we will find some
interesting differences.

In §5.2, we computed the spin bordism groups for the 2-group symmetry
$\mathbb{G}=\text{U}(1)^{[0]}\times_{\hat{\kappa}}\text{U}(1)^{[1]}$. In
particular, we obtained

$\Omega_{3}^{\text{Spin}}(B|\mathbb{G}|)\cong\mathbb{Z}/(2m)$

where $m=|\hat{\kappa}|$. With the fourth bordism group given by $\mathbb{Z}$,
we can put things together to get the cobordism group

$H^{4}_{I\mathbb{Z}}\left(M\left(\text{Spin}\times\mathbb{G}\right)\right)\cong\mathbb{Z}/(2m)\times\mathbb{Z}\,.$
(6.1)

This is similar to the result for $H^{6}_{I\mathbb{Z}}$ that was pertinent to
anomalies in 4d, except for the fact that the global anomaly is here ‘twice as
fine’ as the $\mathbb{Z}/m$ anomaly in 4d.222222The $\mathbb{Z}$-valued local
anomaly here is simply the gravitational anomaly associated with
$-\frac{1}{24}p_{1}(R)\text{Tr}\,_{\bf{R}}1\subset\Phi_{4}$ in the degree-4
anomaly polynomial, which can always be cancelled by adding neutral fermions
and so plays no further role in our discussion. That extra division by 2 will
correspond to a subtle new global anomaly associated with the spin structure,
which will be our main interest in this Section.

Before we discuss the global $\mathbb{Z}/(2m)$ anomaly in more depth, let us
first discuss the physics interpretation of a 2-group symmetry $\mathbb{G}$ in
two dimensions, which is rather different to the 4d case. Recall that in 4d,
the 1-form symmetry $\text{U}(1)^{[1]}$ was identified with the magnetic
1-form symmetry, with 2-form current proportional to $\star f$. But in general
$d\geq 3$ spacetime dimensions, the magnetic symmetry is a $(d-3)$-form
symmetry, and so there is no such symmetry in 2d. There is, however, a
trivially conserved ‘1-form symmetry’ $\text{U}(1)^{[1]}_{\text{top}}$ whose
2-form current is simply $j_{\text{top}}=\mathrm{vol}_{2}$, the volume form.
This symmetry does not act on any line operators in the theory and so one
should not think of it as a physical 1-form symmetry – but it will play a role
in what follows.

Now suppose there is a global 0-form symmetry $G^{[0]}$ with background gauge
field $A$ with curvature $F$. If we first cancel the pure gravitational
anomaly by adding neutral fermions, the local anomalies in $G^{[0]}$ are
captured by the degree-4 anomaly polynomial

$\Phi_{4}=\text{Tr}\,_{\bf{R}}\left(\frac{F\wedge F}{8\pi^{2}}\right)\,.$
(6.2)

The usual ’t Hooftian interpretation of $\Phi_{4}\neq 0$ would be that, if one
tries to gauge $G^{[0]}$, the anomaly ‘breaks’ the symmetry $G^{[0]}$ in the
quantum theory. However, Sharpe re-interprets $\Phi_{4}\neq 0$ as indicating a
weaker 2-group symmetry structure in the 2d quantum theory Sharpe:2015mja ,
corresponding to the extension
$B\text{U}(1)\hookrightarrow\mathbb{G}\twoheadrightarrow G^{[0]}$. In other
words, in 2d one can invoke the auxiliary $\text{U}(1)$ 1-form symmetry to
trade an anomaly in a 0-form symmetry for a 2-group structure. This is of
course just a simpler version of the 4d situation described in §5.1, in which
a mixed ’t Hooft anomaly corresponding to a term $\sim f\wedge F\wedge
F\supset\Phi_{6}$ was re-interpreted as signalling a weaker 2-group symmetry.

But in this Section we will see that, in the case that one needs a spin
structure to define the field theory, it is not always possible to completely
absorb the ’t Hooft anomaly associated with $\Phi_{4}$ by a well-defined
2-group structure. Rather, there can be a residual $\mathbb{Z}/2$-valued
anomaly left over, which one should interpret as an ’t Hooft anomaly in the
2-group symmetry itself.

###  6.1 ‘Spin structure anomalies’ in two dimensions

To see how this works, let us now be more precise and set
$G^{[0]}=\text{U}(1)^{[0]}$. Letting $\\{q_{i}\\}$ denote the
$\text{U}(1)^{[0]}$ charges of a set of (say, left-handed) chiral fermions,
the anomaly polynomial is

$\Phi_{4}=\frac{1}{2}\mathcal{A}_{2}\,c_{1}\cup
c_{1}\,,\qquad\mathcal{A}_{2}:=\sum_{i}q_{i}^{2}\,,$ (6.3)

where $c_{1}=F/2\pi$ is the first Chern class of the $\text{U}(1)^{[0]}$
bundle. By inflow, one can use the associated Chern–Simons form
$I_{3}=\frac{\mathcal{A}_{2}}{8\pi^{2}}A\wedge F$ to compute the variation of
the effective action under the background gauge transformation $A\to
A+d\lambda^{(0)}$,

$\delta
S_{0}=-\text{i}\frac{\mathcal{A}_{2}}{4\pi}\int_{M_{2}}\lambda^{(0)}F\,.$
(6.4)

Now, let us try to implement the philosophy above, and trivialise this anomaly
by promoting $\text{U}(1)^{[0]}$ to a 2-group global symmetry by fusing with
$\text{U}(1)^{[1]}_{\text{top}}$. We turn on a background 2-form gauge field
$B^{(2)}$ that couples to the trivially conserved current
$j_{\text{top}}=\mathrm{vol}_{2}$, via the coupling

$S_{\text{coupling}}=\text{i}\int_{M_{2}}\star
j_{\text{top}}B^{(2)}=\text{i}\int_{M_{2}}B^{(2)},$ (6.5)

because $\star\mathrm{vol}_{2}=1$.

We consider the $\text{U}(1)^{[0]}$ and $\text{U}(1)^{[1]}_{\text{top}}$
symmetries to be fused via the toric 2-group structure
$\mathbb{G}=\text{U}(1)^{[0]}\times_{\hat{\kappa}}U(1)^{[1]}_{\text{top}}$,
where $\hat{\kappa}\in\mathbb{Z}$ is the Postnikov class. This prescribes the
by-now-familiar transformation (5.5) on the background gauge fields, under
which

$\delta
S_{\text{coupling}}=\text{i}\frac{\hat{\kappa}}{2\pi}\int_{M_{2}}\lambda^{(0)}F\,.$
(6.6)

We see that the local anomaly associated with the $F\wedge F$ term in
$\Phi_{4}$ is completely removed iff

$\hat{\kappa}=\frac{\mathcal{A}_{2}}{2}.$ (6.7)

But, since the Postnikov class $\hat{\kappa}$ is neccessarily integral, this
is possible only when $\mathcal{A}_{2}$ is an even integer. Thus, there is an
order 2 anomaly remaining when $\mathcal{A}_{2}$ is odd.

This chimes perfectly with our computation of the bordism group
$\Omega_{3}^{\text{Spin}}(B|\mathbb{G}|)$ in §5.2, that we quoted at the
beginning of this Section, which implies that there is in general a mod
$2|\hat{\kappa}|$ global anomaly. If one imagines fixing the 2-group
structure, in other words fixing an integer $\hat{\kappa}$, then one can vary
the fermion content and ask whether there is a global ’t Hooft anomaly for the
2-group $\mathbb{G}$. We can think of the coupling term (6.5) as a
Green–Schwarz counterterm, which effectively shifts the anomaly coefficient by

$\mathcal{A}_{2}\to\mathcal{A}_{2}+2n\hat{\kappa}\,,\qquad n\in\mathbb{Z},$
(6.8)

and so the residual anomaly is clearly

$\mathcal{A}_{2}\text{~{}mod~{}}2m\,,\qquad m=|\hat{\kappa}|.$ (6.9)

This is the $\mathbb{Z}/(2m)$-valued global anomaly in the 2-group symmetry
that is detected by $\Omega_{3}^{\text{Spin}}(B|\mathbb{G}|)$. Even if we
choose the minimal value for $\hat{\kappa}$, a mod 2 anomaly persists iff

$\sum_{i}q_{i}^{2}=1\text{~{}mod~{}}2\implies N_{o}=1\text{~{}mod~{}}2\,,$
(6.10)

where $N_{o}$ is the number of fermions with odd $\text{U}(1)^{[0]}$ charges.

One can offer a different perspective on this anomaly by thinking about a
generator for $\Omega_{3}^{\text{Spin}}(B|\mathbb{G}|)$, which makes it more
transparent how this mod 2 anomaly is related to the requirement of a spin
structure. If we look back at the AHSS in Fig. 5, we see that the factor of
$\mathbb{Z}/2$ that ends up in $\Omega_{3}^{\text{Spin}}(B|\mathbb{G}|)$ comes
from the $E^{2}_{2,1}$ element, which stabilizes straight away from the
$E^{2}$ page. Since
$E^{2}_{2,1}=H_{2}\left(B|\mathbb{G}|;\Omega_{1}^{\text{Spin}}(\text{pt})\right)$,
this suggests a generator for this $\mathbb{Z}/2$ factor can be taken to be a
mapping torus

$M^{3}=S^{2}\times S^{1},$ (6.11)

with $c_{1}(F)\in 2\mathbb{Z}+1$ on the $S^{2}$ factor, corresponding to an
odd-charged monopole, and the non-bounding spin structure on the $S^{1}$,
corresponding to a non-trivial element of
$\Omega_{1}^{\text{Spin}}(\text{pt})\cong\mathbb{Z}/2$. This would suggest
that the transformation by $(-1)^{F}$, which counts fermion zero modes, is
anomalous on $S^{2}$ with an odd-charged monopole configuration for the
background gauge field.232323We do not believe, however, that this anomaly can
be detected from the torus (in this case, just a circle) Hilbert space, using
the methods set out in Ref. Delmastro:2021xox . This is because a system with
one Weyl fermion with unit $\text{U}(1)$ charge (together with a Weyl fermion
of opposite chirality to cancel the gravitational anomaly) will have an even
number of Majorana zero modes, which allows construction of a
$\mathbb{Z}/2$-graded Hilbert space even in the sector twisted by
$-1\in\text{U}(1)$. (The failure to construct a $\mathbb{Z}/2$-graded Hilbert
space is one sign of anomalies involving the spin structure, as studied in
Delmastro:2021xox .)

We can of course see this from an elementary calculation. For our set of left-
handed chiral fermions with charges $\\{q_{i}\\}$, the index of the Dirac
operator on $S^{2}$, which recall is $\mathrm{Ind}(i\not{D}_{2})=n_{L}-n_{R}$
the number of (LH minus RH) zero modes of the Dirac operator, is equal to

$\mathrm{Ind}(i\not{D}_{2})=\int_{S^{2}}\Phi_{2}=\int_{S^{2}}\text{Tr}\,_{\bf{R}}\left(\frac{F}{2\pi}\right)=c_{1}(F)\sum_{i}q_{i}$
(6.12)

by the 2d Atiyah–Singer index theorem. Thus, the total number of zero modes
$N:=n_{L}+n_{R}$, which is congruent to $n_{L}-n_{R}$ mod 2, satisfies

$N=c_{1}(F)\sum_{i}q_{i}\text{~{}mod~{}}2\,.$ (6.13)

Choosing $c_{1}(F)$ to be odd on $S^{2}$, as we are free to do, we see that
$(-1)^{F}$, which counts these zero modes, flips the sign of the partition
function when

$\sum_{i}q_{i}=1\text{~{}mod~{}}2\implies N_{o}=1\text{~{}mod~{}}2.$ (6.14)

For such fermion content, the 2-group symmetry and $(-1)^{F}$ are therefore
equivalently anomalous. (From the 2-group perspective, the anomalous
transformation corresponds simply to the element
$e^{\text{i}\pi}\in\text{U}(1)^{[0]}$.242424 The existence of this global
$\mathbb{Z}/2$-valued anomaly, despite the fact the anomalous $\text{U}(1)$
gauge transformation is clearly connected to the identity, further evidences
the fact that global anomalies do not require the existence of ‘large gauge
transformations’ captured by, say, $\pi_{d}(G)$ (when restricting the
spacetime topology to be a sphere). This disagreement between homotopy- and
bordism-based criteria for global anomalies was discussed in Ref.
Davighi:2020kok . In this case, the global anomaly arises because an otherwise
local anomaly is only incompletely dealt with by the Green–Schwarz mechanism.
)

This should be contrasted with the situation in four dimensions that we
examined in §5.1. In that case, the corresponding formula for the number of
fermion zero modes, on say $M_{4}=S^{4}$, is
$\mathrm{Ind}(i\not{D}_{4})=\frac{1}{2}c_{1}^{2}\sum_{i}q_{i}^{2}$. But, for a
spin 4-manifold, the Atiyah–Singer index theorem implies that $c_{1}^{2}$ is
an even integer, and moreover cancelling the local mixed gravitational anomaly
$\implies\sum_{i}q_{i}=0\implies\sum_{i}q_{i}^{2}=0\text{~{}mod~{}}2$. The
index and therefore the number of zero modes is then necessarily even, meaning
there is no analogous anomaly in $(-1)^{F}$. This is why the spin bordism
calculation exposes ‘only’ a mod $|\hat{\kappa}|$ anomaly in the 4d case, but
the finer mod $2|\hat{\kappa}|$ in 2d.252525A more mundane way to understand
this difference is therefore simply as a consequence of the normalisation of
the various characteristic classes on spin manifolds of the relevant
dimension.

##### Example: the 3-4-5-0 model.

To furnish an explicit example of a theory featuring this irremovable anomaly,
we consider the so-called “3-4-5-0 model”; that is, a $\text{U}(1)$ gauge
theory in 2d with two left-moving Weyl fermions $\psi_{1}$, $\psi_{2}$ with
charges $3$ and $4$, and two right-moving Weyl fermions $\chi_{1}$, $\chi_{2}$
with charges $5$ and $0$. The theory is free of both gauge and gravitational
anomalies, and so we can ask about its global symmetries. There is a somewhat
trivial $\text{U}(1)_{\chi_{2}}$ global symmetry rotating the neutral fermion
$\chi_{2}$ (with charge $1$). Naïvely, there are two further $\text{U}(1)$
global symmetries, call them $\text{U}(1)_{1}$ and $\text{U}(1)_{2}$, that act
on the remaining fermions, whose charges $Q_{1}$ and $Q_{2}$ can be chosen
independently in one basis as follows:

Field | $Q_{\text{gauge}}$ | $Q_{1}$ | $Q_{2}$
---|---|---|---
$\psi_{1}$ | $3$ | $3$ | $1$
$\psi_{2}$ | $4$ | $4$ | $3$
$\chi_{1}$ | $5$ | $5$ | $3$

One immediately sees that $Q_{1}$ coincides with the gauge charge. Thus, the
$\text{U}(1)_{1}$ transformation can be undone by a gauge transformation.
Thus, the correct global symmetry is a product
$\text{U}(1)_{2}\times\text{U}(1)_{\chi_{2}}$ with charges given by:

Field | $Q_{\text{gauge}}$ | $Q_{2}$ | $Q_{\chi_{2}}$
---|---|---|---
$\psi_{1}$ | $3$ | $1$ | $0$
$\psi_{2}$ | $4$ | $3$ | $0$
$\chi_{1}$ | $5$ | $3$ | $0$
$\chi_{2}$ | $0$ | $0$ | $1$