# Gauging the Gauge
and Anomaly Resolution

Hank Chen<EMAIL_ADDRESS>Department of Applied Mathematics, University
of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada, N2L 3G1
Florian Girelli<EMAIL_ADDRESS>Department of Applied Mathematics,
University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada,
N2L 3G1

###### Abstract

In this paper, we explore the algebraic and geometric structures that arise
from a procedure we dub "gauging the gauge", which involves the promotion of a
certain global, coordinate independent symmetry to a local one. By gauging the
global 1-form shift symmetry in a gauge theory, we demonstrate that the
structure of a Lie algebra crossed-module and its associated 2-gauge theory
arises. Moreover, performing this procedure once again on a 2-gauge theory
generates a 3-gauge theory, based on Lie algebra 2-crossed-modules. As such,
the physical procedure of "gauging the gauge" can be understood mathematically
as a categorification. Applications of such higher-gauge structures are
considered, including gravity, higher-energy physics and condensed matter
theory. Of particular interest is the mechanism of anomaly resolution, in
which one introduces a higher-gauge structure to absorb curvature excitations.
This mechanism has been shown to allow one to consistently gauge an anomalous
background symmetry in QFT .

###### Contents

  1. 0 Introduction
  2. 1 Gauging the 0-gauge
  3. 2 Gauging the 1-gauge
    1. 2.1 Shifting the connection
    2. 2.2 Lie algebra crossed-modules
    3. 2.3 Curvatures and Bianchi identities
      1. 2.3.1 Fake-curvature
      2. 2.3.2 2-curvature and 2-Bianchi identity
      3. 2.3.3 1-Bianchi anomaly and weak 2-Lie algebras
    4. 2.4 Gauge transformations
      1. 2.4.1 1- and 2-gauge transformations
      2. 2.4.2 2-curvature anomaly and the first descendant
    5. 2.5 Applications
      1. 2.5.1 2-BF theory
      2. 2.5.2 3D gravity
      3. 2.5.3 Anomaly resolution: monopole electrodynamics
      4. 2.5.4 4D topological orders: quasistring defects and surface linking
  4. 3 Gauging the 2-gauge
    1. 3.1 From shifting the 2-connection to a Lie algebra 2-crossed module
      1. 3.1.1 Shifting the 2-connection
      2. 3.1.2 3-curvature and 3-Bianchi identity
      3. 3.1.3 Lie algebra 2-crossed-module
    2. 3.2 Gauge transformations and descent equation
      1. 3.2.1 Gauge transformations
      2. 3.2.2 3-curvature anomaly and its first descendant
      3. 3.2.3 2-curvature anomaly and first descendant as 3-gauge data
    3. 3.3 Applications
      1. 3.3.1 3-BF theory
      2. 3.3.2 Anomaly resolution: 2-monopoles and the string 2-crossed-module
  5. 4 Conclusion
  6. A Classification of Lie algebra crossed-modules
    1. A.1 Lie algebra cohomology
    2. A.2 Theorem of Gerstenhaber
    3. A.3 The Postnikov class
  7. B 2-bundle homomorphisms
    1. B.1 Loop model of the weak string 2-algebra

##  0 Introduction

Understanding the symmetries of a physical system is tantamount to
constructing a mathematical model that describes its properties. From
condensed matter to higher-energy physics, models such as Ginzburg-Landau
theory of phase transitions and the Standard Model, are based upon this fact.
In fact, sometimes a handle on symmetry is all we have in more esoteric areas
of theoretical physics, such as string theory and quantum gravity, as no
sufficient experimental data and/or techniques are yet available.

In general, there are two notions of symmetry in a physical system: global and
local. Global symmetries act uniformly upon the entire system, while the
action of the local symmetry depends on the particular physical configuration
in $X$. Of course, the latter is the more general notion, and a global
symmetry can be made local by the procedure of gauging: we promote the
constant group elements to depend on points in $X$. Because of the locality
requirement for the symmetry, the notion of derivative is typically not
transforming covariantly unless one adds some compensating term, a connection.
In a sense, the notion of connection encodes the break down of the covariance
of the transformation of the derivative.

The structure of physical systems with local symmetries is described
geometrically by a principal bundle $P\rightarrow X$ [1]; solutions to the
equations of motion determine configurations on $P$ that represent a class of
physical configurations on $X$ under the action of transformations by $G$.
Many local geometric quantities on $P$, such as connection and curvature, are
important for understanding the physical system.

Moreover, recent developments in condensed matter theory [2, 3, 4] had also
shown that, aside from symmetry, topology plays a central role in the
characterization of physical systems as well. In particular, the topology of
the principal bundle $P\rightarrow X$ determines the presence of defects in
the physical system, which can alter its physical properties in a drastic,
non-perturbative manner [5, 6]. Also from a quantum gravity point of view,
understanding topology can be relevant. Indeed, one might need to sum over all
the possible topological configurations to build the transition amplitudes. In
fact, for example, 3d gravity is essentially about the topology of spacetime,
since it does not have local degrees of freedom.

It is important to note that there are different notions of "topological
features". We can characterize the topology of the manifold of interest (such
as spacetime), the topology of the principal bundle in the case of a gauge
theory, which is related to the cohomological features of the gauge symmetry.
The notion of magnetic monopole is related to the topology and geometry of the
principal bundle. Such topological features in the gauge symmetry structure
appear for example in quantum field theory, where they manifest as
obstructions to defining the path integral [4, 7]. Typically, all such non-
trivial topological features can be cast as anomalies, as due to their
presence, some quantities (such as $d_{A}F$, or the 1- or 2-curvature) are
non-trivial.

Giving a physical system, one typically tries to identify the symmetries in
order to characterize the physical conserved quantities or the quantum theory
(built as representations of this symmetry). Conversely, one can construct new
symmetry structures and then try to identify some physical systems associated
to them. One way to proceed is to categorify the notion of symmetries. It
consists in using category theory tools to build new types of symmetries [8].
For example the notion of 2-group comes up naturally when considering higher-
dimensional homotopy types [9, 10]. This approach relies on beautiful
elaborate mathematics which might leave the interested reader wondering what
are the physical motivations behind it. One goal of this paper is to argue
that this categorification process can be seen as a gauging principle so that
dealing with categorified local symmetries here is nothing else than gauging
(a gauge). Hence a $(n+1)$-gauge theory can be interpreted as a gauged
$n$-gauge theory. This idea is not new to the expert in categorical
symmetries, but we try to make this point in a pedagogical way, using mostly
basic tools of gauge theory and exploring how and when there could be some
generalizations.

Once we have in hand some new symmetries, one needs to identify systems where
such symmetries are realized. It has been known since some times that
categorified symmetries are the natural structure to probe the notion of
topology, thanks to the categorical ladder [11, 12]. For example the moduli
space of flat connection can be used to characterize topological features of a
3d manifold [13, 14]. One can expect the moduli space of flat 2-connections
(ie. with a categorified gauge symmetry) to probe some topological features of
a 4d manifold [15, 16, 17]. These categorified symmetries have been explored
from the physics perspective, mainly in the context of condensed matter [18,
19, 20] or string theory [21, 22]. In this paper we would like to highlight
the fact that a another physical application of these categorified symmetries
is to reproduce some of the feature of the topological anomalies we discussed
above, without any such singular structure. For example, it is well-known that
a magnetic monopole corresponds to a singular structure in the principal
bundle, namely the Bianchi identity is violated, due to the fact that
$d^{2}\neq 0$. Recent work has pointed out that such magnetic monopole physics
can be recovered using a 2-gauge theory, without a violation of the Bianchi
identity [23, 24, 25]. We show that something similar exists one more level
up: we can consider a violation of the 2-Bianchi identity and absorb it in a
non-singular 3-gauge theory. This is one of our new results.

The article is organized as follows.

In section 1, we recall quickly what is the usual notion of gauging. Then in
section 2, we proceed to gauge the gauge by making local, in some sense, a
hidden shift symmetry for the 1-curvature. The lack of covariance is encoded
in a 2-connection. To have a general structure, the notion of crossed module,
equivalent to a strict 2-group, is naturally introduced. We then discuss the
different possible generalizations of the gauge symmetry structure. In
particular, we describe how the structures of a weak Lie 2-algebra [26, 8]
manifests when the (1-)Bianchi identity is relaxed. We also point out how
having a specific non-zero 2-curvature can be related to the topological
properties of the crossed module, encoded by the so-called Postnikov class
[18]. We discuss then how the gauge symmetry structure needs to be adapted to
account for such case.

While the results are strictly not new in this section, we made an effort to
have an original presentation using a language that would be accessible to a
more physically minded reader.

In section 2.5, we discuss some applications of 2-gauge theory. The main
example is a topological theory, 2-BF theory [27, 28], which can be seen in
some sense as a sort of BF theory. According to the space-time dimension,
there are different types of applications. For example, in 3D, BF theory (ie.
in particular 3d gravity) can be seen as a specific (2-)gauge fixed 2-BF
theory. This had not been pinpointed before, to the best of our knowledge, and
it could provide some interesting new directions to explore 3d gravity. In 4D,
2-BF theory is somehow the canonical topological theory. Its symmetry
structure is associated to a 2-Drinfeld double and one would expect that
topological excitations (string like or point like) should be encoded in terms
of representations of 2-Drinfeld double, a direct generalization of the
Drinfeld double in the 3d case. In 5D, it was known that 2-gauge symmetries
could be related to the $w_{2}$ and $w_{3}$ Stiefel-Whitney classes [29, 30].
We highlight how this can be done through a 2-BF theory, which was not
emphasized previously to the best of our knowledge.

Finally, we also discuss how the magnetic monopole can be recovered from
considering a 2-gauge theory, with some interesting mobility conditions on the
currents [25]. This is illustrating the notion of anomaly resolution, that is
we exchange a system with non-trivial topological feature (in this case the a
non-trivial topology for the principal bundle) for a system with an extra
gauge symmetry structure which allows to reproduce the physics of the non-
trivial topological feature [23, 24]. A closely related notion of anomaly
inflow [31], and its relation to higher symmetries [32], has also been studied
recently in the high-energy physics literature.

In section 3, we gauge the 2-gauge, to obtain a 3-gauge theory. The gauging
follows the same step as in the 2-gauge case. We introduce some more general
shift transformation, which do not leave the derivative (of the 2-connection)
covariant. The lack of covariance is encoded in a 3-connection. We highlight
where some constraints, such as the 1-Bianchi identity, the Peiffer condition,
can be weakened. We construct the 3-gauge transformations given in terms of a
2-crossed module. As a direct generalization of the crossed module case, we
discuss how the presence of a non-trivial Postnikov class for the 2-crossed
module is associated with a non-zero 3-curvature and non-trivial 1-gauge
transformations.

In section 3.3, we discuss some applications when dealing with a 3-gauge
theory. We recall some of the main features of a 3-BF theory, following [33].
We point out that we would expect to have a 3-Drinfeld double symmetry at play
in this case (for a 5d spacetime). With the string Lie 2-algebra [34, 35] as
an explicit example, we then study a categorified analogue of the results in
[23, 24]: that an anomalous 2-group symmetry gauges into a non-anomalous mixed
0-, 1- and 2-form symmetry governed by a 3-group. Moreover, the subsequent
3-Yang-Mills theory $S_{\text{3YM}}$ and the associated 3-conservation laws
yield new and interesting higher-mobility constraints that have not appeared
previously, as far as we know.

The Appendices provide some mathematical background and results. In Appendix
A, we describe in full detail the classification of strict Lie 2-algebras/Lie
algebra crossed-modules by a degree-3 cohomology class [36], called the
Postnikov class. In Appendix B, we explore what the Postnikov class says about
our 2-gauge theories, and we also describe the relationship between the
Postnikov class and a similar quantity, called the Jacobiator, in a weak Lie
2-algebra [34, 35].

##  1 Gauging the 0-gauge

In the following, we first review in a pedestrian way the notion of gauging a
global symmetry. This is standard material, for which one can find many
introductions (eg. [1]).

Let $X$ denote a $d$-dimensional smooth manifold admitting an action by a Lie
group $G$. Consider a (smooth) function $\phi\in C^{\infty}(X)$ transforming
under a representation $\pi:G\rightarrow\operatorname{GL}(V)$ of the group $G$
for some vector space $V$, that is $\phi\in C^{\infty}(X)\otimes V$, namely
$\phi$ lies in the algebra of $V$-valued smooth functions on $X$.

Note that $\pi$ is an homomorphism, and the field $\phi$ transforms as

$\phi(x)\rightarrow\pi(g)\phi(x),\qquad g\in G.$

If $g\in G$ is not a $G$-valued function of $X$, then the derivative $d\phi$
transforms covariantly,

$d\phi\rightarrow d(\pi(g)\phi)=\pi(g)d\phi,$

and $G$ encodes a (global) 0-gauge symmetry.

We can promote $g$ to be a $G$-valued function of $X$ itself, such that we
still have the transformation law

$\phi(x)\rightarrow\pi({g(x)})\phi(x)\equiv g(x)\cdot\phi(x)=\phi^{\prime}.$

In this case we are dealing with a principal bundle with fiber $G$ and base
$X$.

The Leibniz rule for the exterior derivative $d$ dictates that111Note that for
notational simplicity we will not indicate $\pi$ anymore. The representation
$\pi$ of $G$ lifts to a representation of its Lie algebra
$\operatorname{Lie}G=\mathfrak{g}$. We will also omit $\pi$ in this case.

$d\phi\rightarrow g(d+g^{-1}dg)\cdot\phi.$

As such it is not $d\phi$ that transforms covariantly, but the covariant
derivative $\nabla\phi\equiv(d+gdg^{-1})\phi$. Indeed, we can introduce the
connection $A=gdg^{-1}\in\Omega^{1}(X)\otimes\mathfrak{g}$, to compensate for
the lack of covariance,

$gA\phi=d\phi^{\prime}-gd\phi\rightarrow A=g^{-1}dg.$ (1.1)

Notice that this connection has a natural invariance symmetry under the left
translation for all $h\in G$ constant (ie. $dh=0$).

$(hg)^{-1}d(hg)=g^{-1}dg$ (1.2)

This is the well-known fact that this is a left-invariant form.

Given the covariant derivative $\nabla=d+g^{-1}dg$, its associated curvature

$\operatorname{cur}\nabla=[\nabla,\nabla]=d(g^{-1}dg)+(g^{-1}dg)\wedge(g^{-1}dg)=0$

vanishes, where we have used the identity
$d(1)=d(g^{-1}g)=(dg^{-1})g+g^{-1}dg=0$. This means that the connection
$A=g^{-1}dg$ is flat.

##### The 0-form symmetry and 1-gauge transformations.

The connection 1-form in an arbitrary gauge,
$A\in\Omega^{1}(X)\otimes\mathfrak{g}$ and the associated curvature 2-form
$\operatorname{cur}A=F=d_{A}A=dA+\frac{1}{2}[A\wedge A]$ transform as

$A\rightarrow A^{g}=g^{-1}Ag+g^{-1}dg,\qquad F\rightarrow F^{g}=g^{-1}Fg.$
(1.3)

Expressing $g=\exp\lambda\approx 1+\lambda$ in terms of the infinitesimal
gauge parameter $\lambda\in\Omega^{0}(X)\otimes\mathfrak{g}$, we achieve the
(infinitesimal) (1-)gauge transformation laws

$\displaystyle A$ $\displaystyle\rightarrow$ $\displaystyle
A^{\lambda}=A+[A,\lambda]+d\lambda\equiv A+d_{A}\lambda,$ $\displaystyle F$
$\displaystyle\rightarrow$ $\displaystyle F^{\lambda}=F+[F,\lambda].$

They endow the bundle $P\rightarrow X$ with a  0-form gauge symmetry
parameterized by $\lambda$.

The Bianchi identity reads $d_{A}F=dF+[A\wedge F]=0$, which holds in general
for any principal $G$-bundle with connection $A$. Since $F$ transforms
covariantly, $d_{A}F$ also transforms covariantly

$d_{A}F\rightarrow d_{A^{\lambda}}F^{\lambda}=d_{A}F+[d_{A}F,\lambda].$

It is possible (and consistent) to achieve a 1-curvature anomaly $F=\sigma\neq
0$, as long as $\sigma\in\Omega^{2}(X)\otimes\mathfrak{g}$ satisfies
$d_{A}\sigma=0$, and transforms covariantly $\sigma\rightarrow g^{-1}\sigma
g$.

##### Global 1-form symmetry.

What we have recalled here is that, by gauging the global symmetry understood
as a "0-gauge" symmetry, we obtain an ordinary 1-gauge bundle $P\rightarrow X$
that is flat. However, one may notice that the curvature 2-form $F=d_{A}A$ has
a hidden symmetry in the presence of a non-trivial center $Z(\mathfrak{g})$.
This symmetry is given by

$A\rightarrow A+\alpha,$ (1.4)

where $\alpha$ is a closed 1-form valued in the center $Z(\mathfrak{g})$ of
the Lie algebra $\mathfrak{g}$, that is $\alpha\in\Omega^{1}_{0}(X)\otimes
Z(\mathfrak{g})$. As such the above gauge structure in fact manifests a
"1-form symmetry" parameterized by $\alpha$, on top of the pre-existing
1-gauge 0-form symmetry parameterized by $\lambda$. This 1-form symmetry is
affecting the connection $A$ but not its curvature.

##  2 Gauging the 1-gauge

In the 1-gauge case, we have highlighted two different types of invariance,
one specified by a left multiplication, in (1.2), the other one by a 1-form
shift in (1.4). It is natural to ask what happens when we gauge each symmetry,
ie. we make them non-constant. For the former, making $h$ non-constant amounts
to just another gauge transformation, so there is nothing new to be gained.
The latter is more interesting, as it leads to some new structures.

Relaxing the condition that $\alpha$ in (1.4) is constant and valued in the
center $Z(\mathfrak{g})$ will be called "gauging the 1-form gauge". So we
allow $\alpha\rightarrow a$ to become a generic 1-form
$a\in\Omega^{1}(X)\otimes\mathfrak{g}$ that has non-trivial coordinate
dependence on $X$, similar to the gauging procedure for the global/0-gauge
symmetry.

###  2.1 Shifting the connection

Typically, one may a priori take a gauge bundle $P\rightarrow X$ with the non-
trivial curvature $F=\sigma\neq 0$, then study the associated gauge theory.
Alternatively, we may perform a particular 1-form shift such that
$F\rightarrow F^{\prime}$ is transformed to a non-trivial value.

Indeed, under a generic 1-form shift.

$A\rightarrow A^{\prime}=A+a,$

we see that the curvature transforms accordingly as

$F\rightarrow F^{\prime}=d_{A^{\prime}}A^{\prime}=F+d_{A}a+\frac{1}{2}[a\wedge
a]=F+d_{A}a+\frac{1}{2}[a\wedge a].$ (2.1)

In the gauge where $A=0$, we just have

$F^{\prime}=da+\frac{1}{2}[a\wedge a],$

which is the curvature of $a$ considered as a $G$-connection. As such we may
shift the curvature to any value from zero, which serves as the central key
fact for anomaly resolution discussed later. Usually, the "gauging" story ends
here, and we deal with an arbitrary curvature associated to the connection in
a particular 1-form gauge $A=a$.

However, the above also shows that, by considering the 1-form shift as a
higher-form gauge symmetry, the (1-)curvature quantity $F$ is a gauge datum,
the notion of curvature is gauge dependent. We have then a pair of gauge
structures, one encoded in $g$ which in a sense encodes the arbitrariness of
the frame we deal with, and one encoded in $a$, which encodes the
arbitrariness of the curvature.

One can realize that the transformation (2.1) can be seen as lack of
covariance of the curvature 2-form under the arbitrary shift, analogous to the
one of the derivative of the field $\phi$ under $\pi(g)$. To amend for the
lack of covariance, we introduced a non-zero connection $A=gdg^{-1}$ in (1.1).

Hence in a similar manner, to amend for the lack of covariance of the
curvature under the arbitrary shift, we introduce a 2-form gauge connection
$\Sigma\in\Omega^{2}(X)\otimes\mathfrak{g}$ such that, in the gauge where
$A=0$

$\Sigma\equiv(F^{\prime}-F)=F^{\prime}=da+\frac{1}{2}[a\wedge a].$ (2.2)

If we define the curvature of $\Sigma$, as the 2-curvature,

$K=d_{A}\Sigma,$

then we see that by the Bianchi identity

$d_{A}\Sigma=d_{A}F=0,$

so that this 2-connection is flat. Indeed as we shall see later, this
2-connection $\Sigma=da+\frac{1}{2}[a\wedge a]$ is a "pure 2-gauge", analogous
to the flat pure 1-gauge $A=gdg^{-1}$ obtained from gauging the 0-gauge.

The construction so far is restrictive, in a sense since we focus on a
2-connection with value in the same Lie algebra $\mathfrak{g}$. It seems
natural to make it valued in some other Lie algebra $\mathfrak{h}$, together
with a map $t:\mathfrak{h}\rightarrow\mathfrak{g}$ (a homomorphism of Lie
algebras), which plays in a sense the same role as the representation $\pi$
when we dealt with a regular 1-gauge. The most natural notion to use is that
of a Lie 2-algebra [8]. There are different notions of it. The first we are
interested in is the notion of strict Lie 2-algebra, which can be equivalently
viewed as a Lie algebra crossed-module [36]. The crossed-module formulation is
most convenient to discuss the notion of 2-gauge theory. We shall also see how
the notion of a weak Lie 2-algebra can be relevant in this setting.

###  2.2 Lie algebra crossed-modules

We first define the notion of crossed-module and then discuss the fields
relevant to build a 2-gauge theory.

######  Definition 2.1.

Consider a pair of Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$, such that
there is an action of $\mathfrak{g}$ on $\mathfrak{h}$ noted $\rhd$. We also
introduce $t$ a Lie algebra homomorphism such that we have 2-term complex

$\mathfrak{G}:\mathfrak{h}\xrightarrow{t}\mathfrak{g},$ (2.3)

The 2-term algebra complex (2.3) is a  Lie algebra crossed-module if the
action and the $t$-map satisfy

  1. 1.

the Peiffer conditions, respectively the $\mathfrak{g}$-equivariance of $t$
and the Peiffer identity,

$t(x\rhd y)=[x,t(y)],\qquad t(y)\rhd y^{\prime}=[y,y^{\prime}],$

and

  2. 2.

the 2-Jacobi identities

$\displaystyle[[x,x^{\prime}],x^{\prime\prime}]+[[x^{\prime},x^{\prime\prime}],x]+[[x^{\prime\prime},x],x^{\prime}]=0,$
$\displaystyle x\rhd(x^{\prime}\rhd y)-x^{\prime}\rhd(x\rhd
y)-[x,x^{\prime}]\rhd y=0,$

for each $x,x^{\prime},x^{\prime\prime}\in\mathfrak{g}$ and
$y,y^{\prime}\in\mathfrak{h}$.

An important consequence of the Peiffer identity is that
$\operatorname{ker}t\subset Z(\mathfrak{h})$ is contained in the centre of
$\mathfrak{h}$.

The $\mathfrak{g}$-equivariance property of $t$ can be summarized by the
following diagram

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where $\operatorname{Der}\mathfrak{g}$ denotes the space of derivations on the
Lie algebra $\mathfrak{g}$, and the dashed arrow denotes an action.

Let us consider now the relevant connections: the 1-connection $A$ is valued
in $\mathfrak{g}$, while the 2-connection $\Sigma$ is valued in
$\mathfrak{h}$. As we will see in section 2.3.1, $t$ is a Lie algebra
homomorphism that allows us to connect fields valued in $\mathfrak{h}$ to ones
valued in $\mathfrak{g}$. This action $\rhd$ can be viewed in a sense as the
gauge transformations induced by $\mathfrak{g}$ on the fields/2-gauge
parameters with value in $\mathfrak{h}$. This will be discussed in section
2.4.

The covariant derivative we will use is still $d_{A}$, ie. it is defined in
terms of the 1-connection $A$. We will therefore use the action to define the
covariant derivative of a form with value in $\mathfrak{h}$. Taking an
arbitrary $\mathfrak{h}$-valued n-form $S\in\Omega^{n}(X)\otimes\mathfrak{h}$,
we introduce the wedge product $\wedge^{\rhd}$ between a 1-form and and
n-form,

$\wedge^{\rhd}:(\Omega^{1}(X)\otimes\mathfrak{g})\otimes(\Omega^{n}(X)\otimes\mathfrak{h})\xrightarrow{\wedge}\Omega^{n+1}(X)\otimes(\mathfrak{g}\otimes\mathfrak{h})\xrightarrow{\rhd}\Omega^{n+1}(X)\otimes\mathfrak{h}.$

This allows to define the covariant derivative of
$S\in\Omega^{n}(X)\otimes\mathfrak{h}$,

$d_{A}S\equiv dS+A\wedge^{\rhd}S.$

Putting together the differential $d_{A}\cdot=d\cdot+A\wedge^{\rhd}\cdot$ on
$\Omega^{n}(X)\otimes\mathfrak{h}$ with the t-map, and using the
$\mathfrak{g}$-equivariance222We have $t(A\wedge^{\rhd}S)=[A\wedge t(S)]$.
implies that the covariant derivative $d_{A}$ on $\mathfrak{h}$-valued forms
is mapped under $t$ to the covariant differential $d_{A}$ on
$\mathfrak{g}$-valued forms. This can be expressed compactly as

$td_{A}=d_{A}t.$ (2.5)

######  Remark 2.1.

It is well-known that Lie algebra crossed-modules $\mathfrak{G}$ are
equivalent to strict Lie 2-algebras [36, 37]. Following the mathematical
literature, we call the strict Lie 2-algebra skeletal if $t=0$, and trivial if
$t=\operatorname{id}$.

###  2.3 Curvatures and Bianchi identities

Given the general 2-Lie algebra framework, we explore the different notions of
curvatures that appear. First we have the notion of fake flatness which
relates the 2-connection to the 1-curvature up to the t-map. We then express
the properties of the 2-curvature and highlight it also satisfies a type of
Bianchi identity. Finally, we discuss how the one kind of violation of the
1-Bianchi identity can be recast in terms of a 2-gauge theory based on a weak
2-Lie algebra.

####  2.3.1 Fake-curvature

When using the crossed module formalism, the relation between the 2-connection
and the curvature we introduced in (2.2) can be rewritten as

$t(\Sigma)=F^{\prime}=da+a\wedge a,$

with $\Sigma=dL+{\frac{1}{2}}[L\wedge L]$, provided that $t(L)=a$. In fact
(2.2) can be readily obtained if $\mathfrak{h}=\mathfrak{g}$ and the t map is
the identity. Hence the construction in (2.2) can be seen as an example of a
2-gauge theory based on the identity crossed-module.

The relation (2.2) can also be interpreted as a generalized notion of
curvature

$\displaystyle\mathcal{F}=F^{\prime}-t(\Sigma),$

which is known as fake-curvature. The condition in which it is constrained to
be zero,

$\displaystyle\mathcal{F}=F^{\prime}-t(\Sigma)=0,$ (2.6)

is known as the fake-flatness condition. A naïve notion of "2-parallel
transport" serves as a geometric motivation for imposing (2.6) [38], but we
need not assume it at the infinitesimal level based on a Lie algebra crossed-
module/strict Lie 2-algebra. We will see nevertheless that such condition can
also appear when we consider 1- or 2-gauge transformations in section 2.4.

######  Remark 2.2.

It is possible to define a notion of higher-parallel transport without fake-
flatness $\mathcal{F}\neq 0$, which would move us into the realm of adjusted
2-parallel transport [34]. We shall not consider this in detail here.

As mentioned previously, we note that (2.6) can be interpreted as sourcing the
curvature with $t(\Sigma)$, allowing us to break away from a flat
1-connection. We will come back to this interpretation in section 2.5.

####  2.3.2 2-curvature and 2-Bianchi identity

The 2-curvature is defined as the tensor
$K=d_{A}\Sigma\in\Omega^{3}(X)\otimes\mathfrak{h}$. When the 2-connection is
pure 2-gauge $\Sigma=dL+\frac{1}{2}[L\wedge L]$, we have as expected $K=0$,

$\displaystyle d_{A}\Sigma=d^{2}L+\frac{1}{2}d[L\wedge
L]+t(L)\rhd(dL+\frac{1}{2}[L\wedge L])=0$ (2.7)

where for simplicity we picked the 1-gauge where $A=t(L)$ and we used that
$d^{2}=0$, the Peiffer identity and the Jacobi identity for $\mathfrak{h}$.

One may insert a 2-curvature anomaly $\kappa\neq 0$, $K=\kappa$ to go away
from the pure 2-gauge case. We will study this in section 2.4.2. As we are
going to show, $K$ is valued in $\operatorname{ker}t$, and so must $\kappa$.
Indeed, for any 2-connection, as a consequence of the fake-flatness condition
and the 1-Bianchi identity, the 2-curvature must be valued in
$\operatorname{ker}t\subset\mathfrak{h}$.

$t(K)=t(d_{A}\Sigma)=d_{A}t(\Sigma)=d_{A}F=0.$ (2.8)

As a consequence of the Bianchi identity, we have that
$d_{A}K\in\operatorname{ker}t$.

On the other hand, by the graded Leibniz rule, the 2-curvature $K$ satisfies

$d_{A}K=d_{A}(d_{A}\Sigma)=F\wedge^{\rhd}\Sigma=t(\Sigma)\wedge^{\rhd}\Sigma=[\Sigma\wedge\Sigma]|_{\operatorname{ker}t},$

where we used the Peiffer conditions. Note that since $d_{A}K$ is valued in
$\operatorname{ker}t$, we should project the commutator $[\Sigma\wedge\Sigma]$
to $\operatorname{ker}t$. However, since $\Sigma$ is a 2-form and
$[\cdot,\cdot]=(t\cdot)\rhd\cdot$ is skew-symmetric, this term vanishes and
hence we achieve the 2-Bianchi identity

$d_{A}K=0.$ (2.9)

We shall discuss in section 3 how such identity can be weakened.

####  2.3.3 1-Bianchi anomaly and weak 2-Lie algebras

Now suppose we forgo the 1-Bianchi identity, then $K$ needs not be valued in
$\operatorname{ker}t$.

$\displaystyle tK=d_{A}F$ $\displaystyle=dF+[A,F]=d^{2}A+\frac{1}{2}d[A\wedge
A]+[A\wedge dA]+\frac{1}{2}[A\wedge[A\wedge A]]$
$\displaystyle=d^{2}A+\frac{1}{2}[A\wedge[A\wedge A]]\neq 0,$

where we used that $d[A\wedge A]=[dA\wedge A]-[A\wedge dA]=-2[A\wedge dA]$.
There are two different ways to do this, one is to let $d^{2}A\neq 0$
(globally), in which case we have a monopole. The other way is if the second
term is non-vanishing, which occurs when we let go of the Jacobi identity on
$\mathfrak{g}$. In this case, $\mathfrak{g}$ is strictly speaking no longer a
Lie algebra; however, we shall see that the following structure we shall
derive can also be applied to the case where $\mathfrak{g}$ is a Lie algebra,
but $t=0$ must be identically zero.

######  Remark 2.3.

The two ways in which the 1-Bianchi identity is violated are distinct. The
violation of the Jacobi identity $[A\wedge[A\wedge A]]$ is of an algebraic
nature, and hence introduces non-trivial modifications to our Lie 2-algebra
structure; we shall focus on this case in the following. On the other hand,
the monopole case $d^{2}A\neq 0$ is of differential geometric nature, which
indicates a non-trivial topology of the 1-gauge theory. We shall discuss how
this 1-gauge topological feature can be treated using a 2-gauge formalism,
without a violation of the Bianchi identity in section 2.5.3. This will be an
example of the notion of anomaly resolution.

By relinquishing the Jacobi identity, we may write this term as a contribution
to $K$ by lifting it along $t$ up to $\mathfrak{h}$. In other words, we
introduce a skew-trilinear map — called appropriately the Jacobiator —
satisfying

$\mu:\mathfrak{g}^{\wedge
3}\rightarrow\mathfrak{h},\qquad\frac{1}{3!}t\mu(A,A,A)=[A\wedge[A\wedge A]],$
(2.10)

such that the modified 2-curvature reads [34]

$K=d_{A}\Sigma-\frac{1}{3!}\mu(A,A,A)=0.$ (2.11)

Since the term $\mu(A,A,A)$ arises due to the failure of the 1-Bianchi
identity, we call it the 1-Bianchi anomaly. Note $\mu$ only appears for non-
Abelian $\mathfrak{g}$. We note that the 1-gauge transformations need to be
carefully analyzed in this case as $\mu(A,A,A)$ will not be a tensor. We
discuss this in section 2.4.

This map $\mu$ is in fact precisely the homotopy map of a weak [26, 34] or a
semistrict [39] Lie 2-algebra. More precisely, the homotopy map is a trilinear
skew-symmetric map $\mu:\mathfrak{g}^{\wedge 3}\rightarrow\mathfrak{h}$
satisfying

$\displaystyle[x,[x^{\prime},x^{\prime\prime}]]+[x^{\prime},[x^{\prime\prime},x]]+[x^{\prime\prime},[x,x^{\prime}]]=t\mu(x,x^{\prime},x^{\prime\prime}),$
$\displaystyle x\rhd(x^{\prime}\rhd y)-x^{\prime}\rhd(x\rhd
y)-[x,x^{\prime}]\rhd y=\mu(x,x^{\prime},t(y))$ (2.12)

for each $x,x^{\prime},x^{\prime\prime}\in\mathfrak{g}$ and
$y\in\mathfrak{h}$. Indeed, (2.10) is equivalent to the first of these
conditions.

An important additional property that $\mu$ must satisfy is its
$\mathfrak{g}$-equivariance:

$x\rhd\mu(x_{1},x_{2},x_{3})=\mu([x,x_{1}],x_{2},x_{3})+\mu(x_{1},[x,x_{2}],x_{3})+\mu(x_{1},x_{2},[x,x_{3}]),$
(2.13)

as such we can compute

$\displaystyle d_{A}\mu(A,A,A)$ $\displaystyle=$ $\displaystyle
d(\mu(A,A,A))+A\wedge^{\rhd}\mu(A,A,A)\qquad\text{$\mathfrak{g}$-equivariance
and Leibniz rule}$ $\displaystyle=$
$\displaystyle\big{(}3\mu(dA,A,A))+\frac{3}{2}\mu([A,A],A,A)\big{)}\qquad\text{Trilinearity
of $\mu$}$ $\displaystyle=$ $\displaystyle 3\mu(F,A,A),$

where $\circlearrowright$ denotes a summation over cyclic permutations. The
factor of $\frac{3}{2}$ appears in the second line due to the fact that
$\mu([A,A],A,A)$ is symmetric under an exchange of the first argument $[A,A]$
and the last two arguments $A,A$. This gives rise to the modified 2-Bianchi
identity

$d_{A}K=F\wedge^{\rhd}\Sigma-\frac{1}{2}\mu(F,A,A)=0,$

which has also appeared in the context of the gauge theory based on a weak Lie
2-algebra [34].

######  Remark 2.4.

Notice that if the weak 2-algebra is skeletal, namely $t=0$, there is no
violation to the Jacobi identity in the component $\mathfrak{g}$. An example
is the skeletal model string 2-algebra $\mathfrak{string}_{k}(\mathfrak{g})$
of a simple Lie algebra $\mathfrak{g}$ [34, 35], where $k\in\mathbb{Z}$ is
called the level. The 2-algebra structure is given by $t=0,\rhd=0$, and the
Jacobiator is $\mu=k\omega$, where $\omega$ is the fundamental 3-cocycle

$\omega=\langle\cdot,[\cdot,\cdot]\rangle\in Z^{3}(\mathfrak{g},\mathbb{R}).$

This is one of the most commonly-seen weak 2-algebras in the physics
literature. The bundle gerbe associated to the string 2-algebra describes the
string structure appearing in string theory [21, 22].

###  2.4 Gauge transformations

In this section, we review the different transformations we can perform and
the inherited compatibility conditions.

####  2.4.1 1- and 2-gauge transformations

##### 1-gauge transformations.

In order to preserve the fake flatness condition, we derive the
transformations of $\Sigma$ and then $K$, from the transformation of the
curvature 2-form (1.3).

$\displaystyle F$ $\displaystyle\rightarrow$ $\displaystyle
F^{\lambda}=F+[F,\lambda]\Rightarrow t(\Sigma)\rightarrow
t(\Sigma)+[t(\Sigma),\lambda]=t(\Sigma)-t(\lambda\rhd\Sigma)$
$\displaystyle\Sigma$ $\displaystyle\rightarrow$
$\displaystyle\Sigma-\lambda\rhd\Sigma$ $\displaystyle K=d_{A}\Sigma$
$\displaystyle\rightarrow$ $\displaystyle K-\lambda\rhd K,$ (2.14)

where $\lambda\in\Omega^{0}(X)\otimes\mathfrak{g}$.

Now suppose the underlying Lie 2-algebra is weak, with $\mu\neq 0$. We shall
see that, provided $\Sigma$ acquires an additional term [34]

$\Sigma\rightarrow\Sigma^{\lambda}=\Sigma-\lambda\rhd\Sigma-\frac{1}{2}\mu(\lambda,A,A)$
(2.15)

under 1-gauge transformation, then we preserve the covariance of the
2-curvature under the 1-gauge transformations,

$K\rightarrow K^{\lambda}=K-\lambda\rhd K+\mu(\lambda,A,\mathcal{F}).$

Indeed, working with the modified 2-curvature (2.11), we have from the
definition (2.12),

$-A\wedge^{\rhd}(\lambda\rhd\Sigma)+[A,\lambda]\wedge^{\rhd}\Sigma=-\mu(A,\lambda,t\Sigma)-\lambda\rhd(A\wedge^{\rhd}\Sigma)=\mu(\lambda,A,t\Sigma)-\lambda\rhd(A\wedge^{\rhd}\Sigma).$
(2.16)

On the other hand, we have by the $\mathfrak{g}$-equivariance of $\mu$,
(2.13), that

$\displaystyle\mu(d_{A}\lambda,A,A)$ $\displaystyle=$
$\displaystyle\mu(d\lambda,A,A)-\frac{1}{2}(\mu([\lambda,A],A,A)-\mu([A,\lambda],A,A))$
$\displaystyle=$ $\displaystyle
d(\mu(\lambda,A,A))+2\mu(\lambda,A,dA)+\frac{1}{2}(\frac{2}{3}\lambda\rhd\mu(A,A,A)+2\mu(\lambda,A,[A\wedge
A])-A\wedge^{\rhd}\mu(\lambda,A,A))$ $\displaystyle=$ $\displaystyle
2\mu(\lambda,A,F)+\frac{1}{3}\lambda\rhd\mu(A,A,A)-d_{A}\mu(\lambda,A,A).$

There are three such terms, hence we have

$\frac{1}{3!}\mu(A,A,A)\rightarrow\frac{1}{3!}\mu(A,A,A)+\mu(\lambda,A,F)+\frac{1}{3!}\lambda\rhd\mu(A,A,A)-\frac{1}{2}d_{A}\mu(\lambda,A,A)+o(\lambda^{2})$

modulo terms of higher order in $\lambda$. These terms precisely cancel the
$d_{A}\mu(\lambda,A,A)$ term in the 1-gauge transformation of $K$, as desired.

##### 2-gauge transformations.

The shift of the 1-connection parameterized by $L$ such that $a=t(L)$ is
interpreted as the 2-gauge transformation. Indeed, the 2-connection $\Sigma$
was introduced such that the 1-form shift $A\rightarrow A^{\prime}=A+t(L)$ in
the 1-connection was interpreted as a (2-)gauge symmetry.

Given the 2-form connection $\Sigma$ undergoes a corresponding 2-gauge
transformation,

$\displaystyle\Sigma\rightarrow\Sigma^{\prime}=\Sigma+d_{A}L+\frac{1}{2}[L\wedge
L],$ (2.17)

parameterized by a 1-form $L\in\Omega^{1}(X)\otimes\mathfrak{h}$, we see that
the fake-curvature $\mathcal{F}=F-t\Sigma$ is kept invariant, as desired. The
2-curvature is covariant under this 1-form shift transformation since, with
$A^{\prime}=A+t(L)$,

$\displaystyle K\rightarrow K^{\prime}$ $\displaystyle=$ $\displaystyle
d_{A^{\prime}}\Sigma^{\prime}=d_{A}\Sigma+t(L)\wedge^{\rhd}\Sigma+d_{A+t(L)}(d_{A}L+\frac{1}{2}[L\wedge
L])$ (2.18) $\displaystyle=$ $\displaystyle
K+[L\wedge\Sigma]+F\wedge^{\rhd}L+\frac{1}{2}d_{A}[L\wedge
L]+t(L)\wedge^{\rhd}d_{A}L+\frac{1}{2}t(L)\wedge^{\rhd}[L\wedge L]$
$\displaystyle=$ $\displaystyle
K-t\Sigma\wedge^{\rhd}L+F\wedge^{\rhd}L+\frac{1}{2}d_{A}[L\wedge L]+[L\wedge
d_{A}L]+\frac{1}{4}[L\wedge[L\wedge L]]$ $\displaystyle=$ $\displaystyle
K+{\cal F}\wedge^{\rhd}L\sim K$

where we used extensively the Peiffer conditions, and the Jacobi identity for
the cubic term in $L$. Note $K$ is invariant on-shell of the fake-flatness
condition $\mathcal{F}=0$.

Now let us consider how the modified 2-curvature $K$ (2.11) transforms in the
weak case $\mu\neq 0$. We seek to pick out terms in the computation of (2.18)
that implicitly uses the 2-Jacobi identities. All such terms occur in the
quantity

$d_{A+t(L)}(d_{A}L+\frac{1}{2}[L\wedge L]),$

which can be organized into three parts:

$o(L):\leavevmode\nobreak\ d_{A}d_{A}L,\qquad o(L^{2}):\leavevmode\nobreak\
d_{tL}d_{A}L+\frac{1}{2}d_{A}[L\wedge L],\qquad o(L^{3}):\leavevmode\nobreak\
\frac{1}{2}tL\wedge^{\rhd}[L\wedge L].$

Consider first the term linear in $L$, which gives

$d_{A}d_{A}L=(dA)\wedge^{\rhd}L+A\wedge^{\rhd}(A\wedge^{\rhd}L)=F\wedge^{\rhd}L+\frac{1}{2}\mu(A,A,tL)$

by using (2.12). The additional $\mu$-term here is compensated precisely by
the linear $o(L)$-terms in the 2-gauge transformation of $\mu(A,A,A)$:

$\frac{1}{3!}\mu(A,A,A)\rightarrow\frac{1}{3!}\mu(A,A,A)+\frac{1}{2}\mu(A,A,tL)+o(L^{2}).$

Next we look at the terms quadratic in $L$. This gives

$d_{tL}d_{A}L+\frac{1}{2}d_{A}[L\wedge L]=\frac{1}{2}A\wedge^{\rhd}[L\wedge
L]+[L\wedge(A\wedge^{\rhd}L)]=\mu(A,tL,tL)$

via (2.12), which is compensated precisely by the $o(L^{2})$-terms in the
transformation

$\frac{1}{3!}\mu(A,A,A)\rightarrow\frac{1}{3!}\mu(A,A,A)+\frac{1}{2}\mu(A,A,tL)+\frac{3!}{3!}\mu(A,tL,tL)+o(L^{3}).$

Finally, the cubic term is

$tL\wedge^{\rhd}[L\wedge L]=tL\wedge^{\rhd}[L\wedge L]=[L\wedge[L\wedge
L]]=\frac{1}{3!}\mu(tL,tL,tL),$

which is compensated by the $o(L^{3})$-term in the transformation

$\frac{1}{3!}\mu(A,A,A)\rightarrow\frac{1}{3!}\mu(A,A,A)+\frac{1}{2}\mu(A,A,tL)+\mu(A,tL,tL)+\frac{1}{3!}\mu(tL,tL,tL).$

As such, we see that the modified 2-curvature (2.11) follows also the 2-gauge
transform law (2.18).

##### Compatibility between 1- and 2-gauge transformations.

The shift has to be compatible with the 1-gauge transformation, so that the
new curvature transforms covariantly,

$\displaystyle A\rightarrow A^{\prime}=A+a\rightarrow
A^{\prime}+d_{A^{\prime}}\lambda\Rightarrow a=t(L)\rightarrow
a+[a,\lambda]=t(L)+[t(L),\lambda]$ (2.19) $\displaystyle L\rightarrow
L-\lambda\rhd L$ (2.20)

where we used the Peiffer conditions, as always. It is interesting to note
that 1-gauge $(\lambda,0)$ and 2-gauge $(0,L)$ transformations do not commute.
Through straightforward computations in the strict case $\mu=0$ [18, 28, 33],
we see that

$[(\lambda,0),(0,L)]=(0,\lambda\rhd L),$ (2.21)

so 2-gauge transformations in general form a semidirect product [40, 28]

$\operatorname{Gau}_{2}=(\Omega^{1}(X)\otimes\mathfrak{h})\rtimes(\Omega^{0}(X)\otimes\mathfrak{g})$

defined by (2.21).

It is possible to perform the same kinematical analysis for the weak case,
where $\mu\neq 0$. However, here the commutator between 2-gauge
transformations read [34]

$[(\lambda,L),(\lambda^{\prime},L^{\prime})]=(0,\lambda\rhd
L^{\prime}-\lambda^{\prime}\rhd
L)+(0,\mu(A,\lambda,\lambda^{\prime}))+\mu(\mathcal{F},\lambda,\lambda^{\prime}).$
(2.22)

This is a major issue, because the additional term
$\mu(\mathcal{F},\lambda,\lambda^{\prime})$ is not a gauge transformation —
the 2-gauge algebra $\operatorname{Gau}_{2}$ fails to close unless the fake
curvature condition $\mathcal{F}=0$ is always satisfied! This is one of the
motivations for the theory of adjusted parallel transport in [34]. Of course,
when $\mu=0$, we have a set of compatible gauge transformations, even if
possibly ${\cal F}\neq 0$.

Generally, we also have a "higher gauge transformation" on the 2-gauge
parameter $L\rightarrow L+d_{A}\ell$, where
$\ell\in\Omega^{0}(X)\otimes\mathfrak{h}$. If we take the two 2-gauge
parameters $L,L^{\prime}=L+d_{A}\ell$, and define

$\displaystyle\Sigma^{\prime}=\Sigma+d_{A}L+\frac{1}{2}[L\wedge L],$
$\displaystyle\Sigma^{\prime\prime}=\Sigma+d_{A}L^{\prime}+\frac{1}{2}[L^{\prime}\wedge
L^{\prime}]$ $\displaystyle A^{\prime}=A+tL,$ $\displaystyle
A^{\prime\prime}=A+tL^{\prime}=A+t(L+d_{A}\ell),$

then we have

$\displaystyle\Sigma^{\prime\prime}-\Sigma^{\prime}$ $\displaystyle=$
$\displaystyle
F\wedge^{\rhd}\ell+[L,d_{A}\ell]+\frac{1}{2}[d_{A}\ell,d_{A}\ell],$
$\displaystyle F^{\prime\prime}-F^{\prime}$ $\displaystyle=$
$\displaystyle[F,t(\ell)]+[tL,t(d_{A}\ell)]+\frac{1}{2}[t(d_{A}\ell),t(d_{A}\ell)].$

By the Peiffer conditions, we see that the two 2-gauge transformations
$L,L^{\prime}=L+d_{A}\ell$ act identically on the fake-curvature
$\mathcal{F}=F-t\Sigma$ [18, 41]. The computation (2.18) then implies that the
2-curvature $K$ is invariant on-shell of fake-flatness $\mathcal{F}=0$ under
both $L,L^{\prime}$. Because of this, the study of such higher gauge
transformation is not necessary in the context of higher-BF theories [28].

####  2.4.2 2-curvature anomaly and the first descendant

Recall from (2.18) that the 2-curvature $K$ is covariant under a 2-gauge
transformation. To introduce a 2-curvature anomaly $\kappa$ into the theory,
we require the anomaly equation of motion (EOM) $K=\kappa$ to transform
covariantly, identically to how $K$ transforms. On-shell of fake-flatness
$\mathcal{F}=0$, then, $\kappa=\kappa(A,\Sigma)$ must be 2-gauge invariant.
Now since under a 2-gauge transformation, $\Sigma$ shifts by an arbitrary
element in $\mathfrak{h}$ and hence $\kappa$ must be a constant as a function
of $\mathfrak{h}$. Regardless, it can still depend on
$\operatorname{coker}t=\mathfrak{g}/\operatorname{im}t$, as 2-gauge invariance
implies that it is shift invariant: $\kappa(A)=\kappa(A+tL)$.

We are going to see that this particular form of the 2-curvature anomaly
$\kappa(A)$ is in fact related to the topological classification of the
underlying Lie 2-algebra. Moreover, we shall see that the presence of this
particular $\kappa(A)$ will twist the gauge transformations in the 2-gauge
theory, such that the 2-curvature anomaly EOM $K=\kappa(A)$ transforms
covariantly. We will then explain how such specific curvature anomaly can be
related to the cohomological properties of the crossed module.

##### Twisting gauge transformations.

Suppose the 1-form connection $A$ transforms in the usual manner. We begin by
considering the large333”Large” means that the gauge parameter $g$ is group-
valued, not infinitesimal, and hence can have global topological character.
(twisted) 1-gauge transformation of the 2-connection $\Sigma$.

$\displaystyle\Sigma\rightarrow\Sigma^{g}=g\rhd\Sigma+\zeta(A,g).$

The $\zeta$ contribution is determined so that the 2-curvature anomaly
equation $K=\kappa$ is compatible with the twisted gauged transformations.
Indeed, given $\mathfrak{g}$-equivariance of the $\mathfrak{h}$-valued forms,
we wish for the expression

$d_{A^{g}}\Sigma^{g}=d_{A^{g}}(g\rhd\Sigma+\zeta(A,g))=g\rhd(d_{A}\Sigma)+d_{A^{g}}\zeta(A,g)=\kappa(A^{g}).$
(2.23)

This implies that we must have the following descent equation

$d_{A^{g}}\zeta(A,g)=\kappa(A^{g})-g\rhd\kappa(A),$ (2.24)

and the $\operatorname{ker}t$-valued 2-form $\zeta(A,g)$ solutions of this
equation is called the first descendant of $\kappa$. This provides a
differential equation which allows to express $\kappa$ in terms of $\zeta$. We
shall see this formalism in action in section 3.3.2.

Infinitesimally, we may expand $g=\exp\lambda\approx 1+\lambda$ in terms of
the 1-gauge parameter $\lambda\in\Omega^{1}\otimes\mathfrak{g}$. This gives a
Taylor expansion

$\zeta(A,g)\approx\zeta(A,1)+[d\zeta(A,1)]\cdot\lambda\equiv\zeta(A,1)+\zeta(A,\lambda),$

from which we may collect terms up to first order in $\lambda$ and rewrite
(2.24) as

$d_{A}\zeta(A,1)=\kappa(A),\qquad
d_{A^{\lambda}}\zeta(A,\lambda)=\kappa(A^{\lambda})-\lambda\rhd\kappa(A).$
(2.25)

Since $\zeta(A,\lambda)$ is valued in $\operatorname{ker}t$, it does not
conflict with the fake curvature condition,

$t(\Sigma)\rightarrow
t(\Sigma)^{\lambda}=t(\Sigma)-t(\lambda\rhd\Sigma)=t(\Sigma)+[t(\Sigma),\lambda],\qquad
F\rightarrow F+[F,\lambda],$

where we have used the Peiffer conditions.

Of course, if $\kappa=0$, then the first descendant $\zeta(A,\lambda)$ can be
chosen to vanish, in which case we reproduce the covariance of the 2-curvature
$K$,

$\bar{K}=K\rightarrow K-\lambda\rhd K.$

Conversely, $\zeta(A,\lambda)$ necessarily occurs in the presence of the
2-curvature anomaly $\kappa(A)$, and the descent equation (2.24) is the key
property that guarantees the 1-gauge covariance of the equation of motion
$K=\kappa$.

######  Remark 2.5.

One may also view $\zeta$ as a particular twist in the 1-gauge transformation
of $\Sigma$, which "inserts" the 2-curvature anomaly $\kappa$; this
perspective will be useful for anomaly resolution in section 3.3.2.

Now we need to see that twisting the 1-gauge transformations by $\zeta$ does
not spoil the compatibility with the 2-gauge transformations. Toward this, we
perform a 2-gauge transformation on the descent equation (2.25). The terms
that we acquire proportional to $L$ are, by the Peiffer identity,

$\displaystyle tL\wedge^{\rhd}\zeta(A,\lambda)$ $\displaystyle=$
$\displaystyle-t\zeta(A,\lambda)\wedge^{\rhd}L=0,$
$\displaystyle{[tL\wedge\lambda]\wedge^{\rhd}\zeta(A,\lambda)}$
$\displaystyle=$ $\displaystyle-t(\lambda\rhd
L)\wedge^{\rhd}\zeta(A,\lambda)=(t\zeta(A,\lambda))\wedge^{\rhd}(\lambda\rhd
L)=0,$

where we have used the fact that $\zeta$ is valued in $\operatorname{ker}t$.
If we choose $\zeta(\cdot,\lambda)$ as a function on
$\operatorname{coker}t=\mathfrak{g}/\operatorname{im}t$ like $\kappa$, then
(2.25) is shift-invariant. This proves our desired claim.

##### The Postnikov class.

The anomaly $\kappa(A)$ has a cohomological interpretation. Indeed,
$\kappa=\kappa(A)\in\Omega^{3}(X)\otimes\operatorname{ker}t$ can be
interpreted as a Lie algebra 3-cocycle $\kappa\in
Z^{3}(\operatorname{coker}t,\operatorname{ker}t)$.

######  Definition 2.2.

We call the cohomology class $[\kappa]\in
H^{3}(\operatorname{coker}t,\operatorname{ker}t)$ the Postnikov class of the
crossed-module $\mathfrak{G}$. A crossed-module/strict 2-algebra
$\mathfrak{G}$ is called non-trivial if $[\kappa]\neq 0$.

$[\kappa]$ classifies the crossed-module $\mathfrak{G}$ up to elementary
equivalence [36, 35]. Weak Lie 2-algebras are classified by the same data [42,
39]. We give further details about the Postnikov class in Appendix A.3.

Notice that the function $\kappa$ is only required to be a Lie algebra
3-cocycle, and hence is not necessarily covariantly closed. This means that,
in the presence of $\kappa(A)$, the 2-Bianchi identity (2.9) can in fact be
violated, due to the 2-curvature anomaly EOM $K=\kappa(A)$ giving
$d_{A}K=d_{A}\kappa(A)\neq 0$. We shall see an example of this in section
3.3.2.

The astute reader may have noticed a close parallel between the Postnikov
anomaly $\kappa(A)$ and the Bianchi anomaly $\mu(A,A,A)$. They both define an
anomaly of the 2-flatness condition, and the resulting 2-curvature quantity
$K$ have identical gauge transformation properties.

For $t\neq 0$, the two structures are actually different. Indeed, the
1-Bianchi anomaly $\mu(A,A,A)$ is not invariant under the 1-form shift
symmetry $A\rightarrow A+tL$, while $\kappa$ by hypothesis is. This speaks to
the fact that, unlike their strict counterparts, weak Lie 2-algebras and non-
trivial Lie algebra crossed-modules are not equivalent when $t\neq 0$. Indeed,
the component $\mathfrak{g}$ in a weak 2-algebra is not a Lie algebra, as the
2-Jacobi identities (2.12) do not hold. The quantity
$\frac{1}{2}\mu(\lambda,A,A)$ that appeared in (2.15), which seems to serve as
the first descendant of $\mu(A,A,A)$, does not satisfy the descent equation
(2.25).

On the other hand, it is known that a non-trivial Lie algebra crossed-module
$\mathfrak{g}=\mathfrak{g}_{-1}\xrightarrow{t}\mathfrak{g}_{0}$ is classified,
up to elementary equivalence, by precisely the data of a skeletal weak Lie
2-algebra $\mathfrak{s}=(\mathfrak{n}\oplus V,\kappa)$ [8, 35], where
$\mathfrak{n}=\operatorname{coker}t$ and $\operatorname{ker}t=V$. Here, the
Postnikov class $\kappa$ plays the role of the homotopy map for the 2-term
graded Lie algebra $V\xrightarrow{0}\mathfrak{n}$. Indeed, as $\mathfrak{s}$
is skeletal, there is no violation of the 2-Jacobi identities. Therefore, one
may see a weak skeletal Lie 2-algebra as a non-trivial Lie algebra crossed-
module.

######  Remark 2.6.

It was proven [35] that the skeletal, weak, string 2-algebra
$\mathfrak{string}_{k}(\mathfrak{g})$ has an alternative description in terms
of a non-trivial crossed-module, called the loop model $\mathfrak{l}_{k}$,
whose Postnikov class $[\kappa]\in H^{3}(\mathfrak{g},\mathbb{R})$ is
represented in the 2-gauge theory [34] by its $S^{1}$-transgression [21, 43]
to $[\chi]\in H^{2}(\Omega\mathfrak{g},\mathbb{R})$; see section 3.3.2 and
Appendix B.1.

The non-trivial crossed-module formulation has the distinct advantage that the
2-gauge theory it defines is free of the problems plaguing that of a weak
2-algebra, such as the lack of closure of gauge transformations in (2.22).
This is precisely because of the descent equation satisfied by the first
descendant $\zeta(A,\lambda)$ of $\kappa(A)$, which ensures that the 2-gauge
structure closes and is consistent [34], even in the presence of a non-trivial
Postnikov class [18]. We shall see this point in action in sections 2.5.4 and
3.3.2.

###  2.5 Applications

In this section, we discuss concrete examples of 2-gauge structures that arise
naturally from physical applications.

####  2.5.1 2-BF theory

The simplest action to consider is an action for which the fake-flatness and
the flat 2-curvature are obtained as equations of motion (EOMs), so such an
action is topological. By analogy to the BF case, we would call this action a
2-BF action [28, 33].

We mention briefly here that the partition function of the 2-BF model can be
discretized. This gives rise to the Yetter model [44, 40, 45], in which the
2-connections $(A,\Sigma)$ are assigned to a triangulation as
"$\mathfrak{G}$-valued colourings" [18, 16]. It has been used to describe
topological phases protected by higher-form global symmetries [18, 19, 30].

##### Action and EOMs.

Let $X$ be a manifold of dimension $d$ and let us fix a Lie algebra crossed-
module $\mathfrak{G}=\mathfrak{h}\xrightarrow{t}\mathfrak{g}$. Let
$\mathfrak{g}^{*}$ denote the dual space of linear functionals on
$\mathfrak{g}$, and similarly let $\mathfrak{h}^{*}$ denote the dual space of
$\mathfrak{h}$. We denote by $\langle\cdot,\cdot\rangle$ the duality pairing
for them.

We begin by introducing Lagrange multipliers
$B\in\Omega^{d-2}\otimes\mathfrak{g}^{*},\,C\in\Omega^{d-3}\otimes\mathfrak{h}^{*}$
which implements the aforementioned flatness conditions. The 2-BF action, also
called the BFCG action [28, 33], in the absence of 2-curvature anomalies is

$S_{\text{2BF}}[A,\Sigma]=\int_{X}\langle
B\wedge\mathcal{F}(A,\Sigma)\rangle+\langle
C\wedge\mathcal{G}(A,\Sigma)\rangle,$ (2.26)

where $\mathcal{F}(A,\Sigma)=F-t(\Sigma)$ and
$\mathcal{G}(A,\Sigma)=K=d_{A}\Sigma$. For $d<3$, the 2-$BF$ theory reduces to
a $BF$ theory, since the dual field $C$ does not exist.

The first half of the EOMs are

$\delta B\Rightarrow\mathcal{F}=F-t(\Sigma)=0,\qquad\delta
C\Rightarrow\mathcal{G}=d_{A}\Sigma=0,$

which implement precisely the fake curvature and 2-flatness conditions,
respectively. On the other hand, we also have the option to vary $A$ and
$\Sigma$. These must be done more carefully: we first introduce a map
$\Delta:\mathfrak{h}\wedge\mathfrak{h}^{*}\rightarrow\mathfrak{g}^{*}$ dual to
the crossed-module action:

$\langle
C\wedge(A\wedge^{\rhd}\Sigma)\rangle=-\langle\Delta(C\wedge\Sigma)\wedge
A\rangle.$

Second, we define the map $t^{*}:\mathfrak{g}^{*}\rightarrow\mathfrak{h}^{*}$
dual (with respect to the pairings $\langle\cdot,\cdot\rangle$) to the
crossed-module map $t:\mathfrak{h}\rightarrow\mathfrak{g}$, and write

$\langle B\wedge t(\Sigma)\rangle=\langle t^{*}(B)\wedge\Sigma\rangle.$

We also introduce the dual of the action and adjoint representation,

$\langle y,x\rhd y^{\prime}\rangle=-\langle
x\rhd^{*}y,y^{\prime}\rangle,\qquad\langle
x^{\prime},[x,x^{\prime\prime}]\rangle=-\langle[x,x^{\prime}]^{*},x^{\prime\prime}\rangle,$

for all
$y\in\mathfrak{h},y^{\prime}\in\mathfrak{h}^{*},x\in\mathfrak{g},x^{\prime}\in\mathfrak{g}^{*}$.

These yield

$\delta A\Rightarrow dB+[A\wedge
B]^{*}-\Delta(C\wedge\Sigma)=0,\qquad\delta\Sigma\Rightarrow
t^{*}B+dC+A\wedge^{\rhd^{*}}C=0.$

If we define the quantities

$\tilde{F}\equiv d_{A}C=dC+A\wedge^{\rhd^{*}}C,\qquad\tilde{K}\equiv
d_{A}B=dB+[A\wedge B]^{*},$

we see that these sets of EOMs read

$\tilde{F}=t^{*}(B),\qquad\tilde{K}=\Delta(C\wedge\Sigma),$ (2.27)

the first of which looks like a fake-flatness condition for the dual fields.
This suggests that $B,C$ should be treated as a 2-connection as well, valued
in a Lie algebra crossed-module
$t^{*}:\mathfrak{g}^{*}\rightarrow\mathfrak{h}^{*}$.

Indeed, this is precisely the dual Lie algebra crossed-module

$\mathfrak{G}^{*}[1]=(t^{*}:\mathfrak{g}^{*}\rightarrow\mathfrak{h}^{*},\tilde{\rhd}),$

whose graded Lie algebra structure is induced by the choice of a Lie algebra
2-cochain444More concretely, we have  $\langle[f,f^{\prime}],y\rangle=\langle
f\wedge f^{\prime},\delta_{-1}(y)\rangle,\qquad\langle
f\tilde{\rhd}g,x\rangle=\langle f\wedge g,\delta_{-1}(x)\rangle,$ where
$f,f^{\prime}\in\mathfrak{h}^{*},g\in\mathfrak{g}^{*}$.

$\delta_{-1}:\mathfrak{h}\rightarrow\mathfrak{h}^{\wedge
2},\qquad\delta_{0}:\mathfrak{g}\rightarrow\mathfrak{h}\wedge\mathfrak{g}.$

When dealing with the action (2.26), the dual Lie 2-algebra is Abelian with
trivial 2-cochain $(\delta_{-1},\delta_{0})=0$, and hence is the same thing as
a 2-vector space [46]. For more general details on these algebraic objects, we
refer the reader to the mathematical literature [37, 26, 47].

######  Remark 2.7.

Dualizing the crossed-module map $t:\mathfrak{h}\rightarrow\mathfrak{g}$ leads
to $t^{*}:\mathfrak{g}^{*}\rightarrow\mathfrak{h}^{*}$, hence the dual Lie
2-algebra $\mathfrak{G}^{*}[1]$ comes with a shift $[1]$ in the grading of the
underlying vector spaces. This is a small subtlety in the mathematical
notation that we shall keep in order to be consistent with the literature.

##### Symmetries of the action.

It was shown in [28] (see also [33]) that the 2-$BF$ action (2.26) is
preserved under the operations

$\displaystyle\lambda:\begin{cases}{\cal F}\rightarrow{\cal F}^{\lambda}={\cal
F}+[{\cal F},\lambda]\\\ {\cal G}\rightarrow{\cal G}^{\lambda}={\cal
G}+\lambda\rhd{\cal G}\end{cases},$ $\displaystyle L:\begin{cases}{\cal
F}\rightarrow{\cal F}^{L}={\cal F}\\\ {\cal G}\rightarrow{\cal G}^{L}={\cal
G}+{\cal F}\wedge^{\rhd}L\end{cases},$ (2.28)
$\displaystyle\lambda:\begin{cases}B\rightarrow
B^{\lambda}=B+[\lambda,B]^{*}\\\ C\rightarrow
C^{\lambda}=C+\lambda\rhd^{*}C\end{cases},$ $\displaystyle
L:\begin{cases}B\rightarrow B^{L}=B+\Delta(C\wedge L)\\\ C\rightarrow
C^{L}=C\end{cases},$ (2.29)

where we recognize the transformations of ${\cal F}$ and ${\cal G}$ we
obtained in section 2.4. Notice ${\cal G}^{L}$ is invariant only on-shell of
the fake curvature condition ${\cal F}=0$, which we had assumed in (2.18).

Algebraically, this implies that the 2-gauge group
$\operatorname{Gau}_{2}=(\Omega^{1}(X)\otimes\mathfrak{h})\rtimes(\Omega^{0}(X)\otimes\mathfrak{g})$
acts naturally on the dual fields $B,C$. In other words, the original
2-algebra $\mathfrak{G}$ has a natural action on the dual 2-algebra
$\mathfrak{G}^{*}[1]$ induced by the data $\rhd^{*},\Delta$ emergent form the
dual EOMs (2.27). These actions define a strict coadjoint representation [37]
of the 2-algebra $\mathfrak{G}$ on its dual $\mathfrak{G}^{*}[1]$.

######  Remark 2.8.

In general, the dual Lie 2-algebra $\mathfrak{G}^{*}[1]$ can be non-Abelian
and define its own gauge sector. The corresponding gauge parameters
$(\tilde{\lambda},\tilde{L})\in\mathfrak{G}^{*}[1]$ transforms the dual fields
$(C,B)$ as

$\tilde{\lambda}:\begin{cases}C\rightarrow
C^{\tilde{\lambda}}=C+d_{C}\tilde{\lambda}\\\ B\rightarrow
B^{\tilde{\lambda}}=B+\tilde{\lambda}\rhd^{*}B\end{cases},\qquad\tilde{L}:\begin{cases}C\rightarrow
C^{\tilde{L}}=C+\tilde{t}\tilde{L}\\\ B\rightarrow
B^{\tilde{L}}=B+d_{C}\tilde{L}+\frac{1}{2}[\tilde{L}\wedge\tilde{L}]_{*}\end{cases}.$

If there is a non-trivial back-action of $\mathfrak{G}^{*}[1]$ on
$\mathfrak{G}$, then $(A,\Sigma)$ would transform under
$(\tilde{\lambda},\tilde{L})$ as well, analogous to how $(C,B)$ transforms
under $(\lambda,L)$ in (2.29). If certain coherence conditions are satisfied
between these actions, then the pair $(\mathfrak{G},\mathfrak{G}^{*}[1])$
defines a 2-Manin triple

$\mathfrak{D}=\mathfrak{G}\leavevmode\nobreak\
_{\operatorname{ad}^{*}}\bowtie_{\mathfrak{ad}^{*}}\mathfrak{G}^{*}[1],$

which serves as a model for a "2-Drinfel’d double" [37, 47] — a categorified
notion of the classical Drinfel’d double
$\mathfrak{d}=\mathfrak{g}\bowtie\mathfrak{g}^{*}$ for a Lie algebra
$\mathfrak{g}$ [48]. For a more detailed study and analysis, see [47].

####  2.5.2 3D gravity

3D gravity is topological, as there are no propagating local degrees of
freedom. In the Einstein-Cartan formalism, the Einstein equations take the
shape [49, 50] ($4\pi G=1$)

$\displaystyle F_{I}\,=\,T_{I},$ (2.30)

where $F$ is the curvature of the spin-connection $A$, and $I$ is the internal
$\mathfrak{su}(2)$-index (we chose the Euclidian signature). $T_{I}$ is the
stress-energy tensor555Typically in the usual framework,
$T_{I}=\frac{\delta{S}_{\text{matter}}}{\delta e^{I}}$, where
${S}_{\text{matter}}$ is the action for the matter degrees of freedom., which
could include the cosmological constant contribution. By assumption, $F$
satisfies the Bianchi identity and hence $T$ is conserved. Following our
discussion from the previous sections, it seems natural to interpret the
stress energy contribution as some form of curvature excitation and as such it
would fit within the scheme of 2-gauge theory. In particular we would identify
the stress-energy tensor with the 2-connection up to the t-map.

$\displaystyle F_{I}\,=\,T_{I}=t_{I}(\Sigma).$ (2.31)

Note however that the stress-energy tensor is actually fixed, there is no
shift symmetry. Hence one would expect to recover 3d gravity as some kind of
2-gauge fixed theory. Indeed, starting from the 2-$BF$ action, and fixing the
shape of the 2-connection allows to recover 3d gravity coupled with a particle
(seen as a topological defect) or with a non-zero cosmological constant.

First let us fix the crossed module to be the (infinitesimal) identity crossed
module

$\mathfrak{I}_{\mathfrak{su}(2)}=(t=\operatorname{id}:\mathfrak{su}(2)\rightarrow\mathfrak{su}(2),\rhd=\operatorname{ad}),$

to discuss 3d gravity in the Euclidean signature. We will note $J_{I}$ the
generators of $\mathfrak{su}(2)$. We now consider the 2-$BF$ action based on
$\mathfrak{I}_{\mathfrak{su}(2)}$ with $X$ a 3d manifold.

$S_{\text{2BF}}[A,\Sigma,B,C]=\int_{X}\langle B\wedge(F-\Sigma)\rangle+\langle
C\wedge d_{A}\Sigma\rangle,$ (2.32)

and discuss the different values $\Sigma$ can take. The fake flatness
condition, which is one of EOMs is then

$F=\Sigma.$ (2.33)

The first obvious value is to 2-gauge fix $\Sigma$ to $\Sigma=0$, which would
amounts to recover pure (Euclidian) gravity

$S_{\text{2BF}}[A,\Sigma=0,B,C]=S_{\text{BF}}[B,A]=\int_{X}\langle B\wedge
F\rangle,$ (2.34)

with the 1-form $B$ interpreted as the frame field. Plugging back $\Sigma=0$
in (2.33) allows to recover the 3d vacuum Einstein equation.

The next interesting value is to pick a 1-gauge where we 2-gauge fix $\Sigma$
to be $\Sigma=mJ_{3}\delta_{W}^{(2)}(x)$, where $\delta^{(2)}(x)$ is the
densitized Dirac delta function, localizing $\Sigma$ on a worldline $W$. $m$
is the mass of the defect. In an arbitrary 1-gauge, parameterized by $g$, we
have then $\Sigma^{I}=p^{I}\delta_{W}^{(2)}(x)=\Sigma^{I}_{p}$, with
$p^{I}=mg^{-1}J_{3}g$ interpreted as the momentum of the defect.

With such value, the 2-BF action (2.32) becomes

$S_{\text{2BF}}[A,\Sigma^{I}_{p},B,C]=\int_{X}\langle B\wedge
F\rangle-\int_{W}\langle B,p\rangle+\int_{W}\langle C\wedge d_{A}p\rangle,$
(2.35)

where we recognize the standard action of gravity coupled with a particle
[51], supplemented by a term encoding the conservation of momentum. Plugging
back $\Sigma^{I}=p^{I}\delta_{W}^{(2)}(x)$ in (2.33) allows to recover the 3d
Einstein equation in the presence of a point-like particle.

Such construction could be extended to the discrete case where one could
analyze how the Yetter amplitude can provide the Ponzano-Regge amplitude
coupled to a particle. This will be explored elsewhere.

Finally, in general as a 2-form on a 3d manifold with value in
$\mathfrak{su}(2)$, there exists a covector $\tilde{e}$ and an arbitrary
constant rescaling $\lambda$ such that

$\displaystyle\Sigma=\frac{\lambda}{2}[\tilde{e}\wedge\tilde{e}]=\Sigma_{\lambda}.$
(2.36)

Plugging back this value of $\Sigma$ in (2.33) would resemble very much
Einstein equation in the presence of a cosmological constant $\lambda$, if
$\tilde{e}$ was identified as the frame field $B$. Let us therefore write
without loss of generality
$\Sigma=\frac{\lambda}{2}[\tilde{e}\wedge\tilde{e}]$ and impose that
$\tilde{e}=B$. At the level of the 2-BF action, we have therefore

$S_{\text{2BF}}[A,\Sigma_{\lambda},B=\tilde{e},C]=\int_{X}\langle
B\wedge(F-\frac{\lambda}{2}[\tilde{e}\wedge\tilde{e}])\rangle-\lambda\langle[C\wedge\tilde{e}]\wedge
d_{A}\tilde{e}\rangle+\langle\phi,\tilde{e}-B\rangle,$ (2.37)

where $\phi$ is a Lagrange multiplier. The Palatini formulation of 3d gravity
involves $e_{\mu}^{I},\,A_{\mu}^{I}$, ie. 18 variables. In the 2-BF action, we
have additional 12 variables from $B_{\mu}^{I}$ and $C^{I}$. To recover the
Palatini formulation, we go on-shell of 2-flatness $d_{A}\Sigma=0$ and impose
the constraint

$\displaystyle B=\tilde{e},$ (2.38)

which allows to reduce on-shell to the usual number of variables and 3d
gravitational action

$\displaystyle S_{\text{2BF}}[A,\Sigma_{\lambda},B,C]$ $\displaystyle\approx$
$\displaystyle
S_{\text{gr}}[A,\tilde{e}]=\int_{X}\langle\tilde{e}\wedge(F-\frac{\lambda}{3}[\tilde{e}\wedge\tilde{e}]\rangle.$
(2.39)

The dual EOMs (2.27) state that

$d_{A}C=B,\qquad d_{A}B=d_{A}\tilde{e}=[\Sigma,C],$

namely the coframe $B$ is covariantly exact (which can always be achieved
locally [52]) and the torsion $T=d_{A}B$ is given by
$[C,\Sigma]=\frac{\lambda}{2}[C,[\tilde{e}\wedge\tilde{e}]]$. Of course,
torsion-freeness $T=0$ requires that this quantity must vanish. In order to
see this, we use the Jacobi identity such that

$\frac{1}{2}[C,[\tilde{e}\wedge\tilde{e}]]=[\tilde{e}\wedge[\tilde{e},C]].$

Now as $C$ is a 0-form, we have $[C,C]=0$, hence on-shell of the duel EOM
$d_{A}C=B$ we have

$0=d_{A}[C,C]=2[d_{A}C,C]=2[B,C]=2[\tilde{e},C],$

and we indeed have torsion-freeness $d_{A}B=0$.

######  Remark 2.9.

In the 2-gauge formalism (2.37), we can say that we have a pair of (co-)frame
fields, $B$ and $e$. The constraint to recover canonical 3d gravity (2.39) is
to identify them $B=e$ via (2.38). This is an analogue of the simplicity
constraint in 4d gravity: there, one starts with 4d $BF$ action based
essentially on a pair of frame fields, then impose the simplicity constraint
that identifies them [53, 54, 55]. It would then be interesting to see how
this construction gives rise to the notion of quantum groups, upon imposing
the simplicity constraints.

####  2.5.3 Anomaly resolution: monopole electrodynamics

For this example, we combine ideas from [23, 24, 25] and introduce the
magnetic monopole in the context of 2-gauge theory. Let $X$ denote a closed
oriented smooth 4d manifold, which is not spin.

Consider on $X$ a principal $U(1)$-gauge bundle, whose curvature 2-form $F$
has non-trivial flux across a 2-surface $S\subset X$. This can be interpreted
as a violation of the Bianchi identity $dF\neq 0$: on the 3-surface $V$
spanned by $S\subset X$, we have

$\int_{S}F=\int_{V}dF\neq 0$

by Stokes’s theorem. Physically speaking, $S$ encloses a  magnetic monopole,
whose current $j_{m}$ is given by the EOM $dF=\ast j_{m}\neq 0$. In the usual
manner, we define

$q_{m}=\frac{1}{2\pi}\int_{V}\ast j_{m}=\frac{1}{2\pi}\int_{S}F$ (2.40)

as the magnetic charge enclosed by the bounding 3-surface $V\subset X$.

######  Remark 2.10.

The historical motivation for studying such an "anomalous" Maxwell’s theory is
that $\text{QED}_{3+1}$ suffers from a perturbative chiral anomaly in the
presence of a single spin-$\frac{1}{2}$ Weyl fermion. The counterterm for this
anomaly is the Abelian Chern-Simons action [7]

$\mathcal{A}[A]=\frac{1}{4\pi}\int_{X}F\wedge F;$

indeed, by an integration by parts, we can introduce this Chern-Simons term
$\mathcal{A}[A]=-\frac{1}{4\pi}\int_{X}\ast j_{m}\wedge A$ with a monopole
current $j_{m}$, if we go on-shell of the EOM $\ast j_{m}=dF$. The Nielsen-
Ninomiya theorem states that, on the lattice, this anomaly cannot be removed
without introducing a Weyl fermion of opposite chirality.

##### Anomalous gauge transformations.

Recall that a non-trivial monopole current $j_{m}\neq 0$ implies that
$d^{2}A=dF=\ast j_{m}\neq 0$. This type of failure of the Bianchi identity
implies that the $U(1)$ connection $A$ acquires a non-trivial holonomy about
some closed (timelike) 1-cycle $l\subset X$, called the monopole worldline.

To treat this problem, we "fatten" (ie. take a small tubular neighborhood
around) $l$ and excise it away from $X\rightsquigarrow X^{\prime}$ [2]. This
yields new 4-manifold $X^{\prime}$, which has a boundary $\partial
X^{\prime}\cong S^{2}\times l$; see Fig. 1 (Left). The $U(1)$-connection $A$
is now regular on $X^{\prime}$, but its gauge transformation comes with an
anomalous component [3, 56]

$A\rightarrow A+d\lambda_{0}+d\lambda_{1},\qquad d^{2}\lambda_{0}=0,\quad
d^{2}\lambda_{1}\neq 0.$

The anomalous component is required such that, by integrating over the
2-sphere $S^{2}\subset S^{2}\times l$ enclosing the monopole, we achieve

${q}_{m}=\frac{1}{4\pi}\int_{S^{2}}F\rightarrow\frac{1}{4\pi}\int_{S^{2}}F+d^{2}\lambda_{0}+d^{2}\lambda_{1}=\frac{1}{2\pi}\int_{S^{1}}d\lambda_{1}\neq
0$

a non-trivial monopole charge, where $S^{1}=H_{+}\cap H_{-}\hookrightarrow
S^{2}$ is the equator, and we have used Stokes’s theorem on each of the
patches666On a 3d Cauchy slice containing $V$, the equation $d\lambda_{1}=\ast
dA=\ast F$ is known as the (Abelian) monopole equation [57]. $H_{\pm}$
covering $S^{2}$. Historically, this anomalous $U(1)$ gauge theory also plays
an important role in the vortex-driven 2d Kosterlitz-Thouless phase transition
[3].

Conversely, if the $U(1)$ gauge symmetry $A\rightarrow A+d\lambda_{0}$ is non-
anomalous, then there can be no monopole charge ${q}_{m}=0$ and the magnetic
current must vanish

${q}_{m}=\frac{1}{4\pi}\int_{S^{2}\times l}dF=\frac{1}{4\pi}\int_{S^{2}\times
l}\ast j_{m}=0.$

The construction of a non-trivial monopole charge $q_{m}$ from patching the
anomalous gauge transformation $d\lambda_{1}$ (as a Čech cocycle) across the
equator of a sphere is called clutching [58]. More generally, this casts the
monopole charge $q_{m}$ as an element in a differential cohomology of $X$
[59].

Figure 1: The clutching construction for the monopole. Left: The excised
spacetime 4-manifold $X^{\prime}$ and its boundary $\partial X^{\prime}\cong
S\times l$. Right: The singularity structure of $\lambda_{1}$ on the
equatorial slice $S^{1}$ across $S\cong S^{2}$. If the monopole charge $q_{m}$
is non-trivial, then a branch cut appears.

Any given monopole current $j_{m}$ can thus be inserted by designing the
singularity of $\lambda_{1}$; see Fig. 1 (Right). It computes the winding
number

$\int_{S^{1}}d\lambda_{1}=\lambda_{1}(1)-\lambda_{1}(0)\in 2\pi\mathbb{Z},$

hence the quantized monopole charge $q_{m}\in\mathbb{Z}$ is fixed by
topology777This accounts for the Dirac monopole quantization
${q}_{m}\in\mathbb{Z}$ in 4D, or the quantization of the vortex charge in 2d
[3].. In fact, this winding number is precisely the first Chern number
$\int_{S}c_{1}(F)$, where the first Chern class $c_{1}(F)=\frac{1}{2\pi}[F]\in
H^{2}(X,\mathbb{Z})$ topologically classifies the $U(1)$-bundle on $X$ up to
isomorphism. More details can be found in [2].

######  Remark 2.11.

Given two $U(1)$-bundles $P,P^{\prime}\rightarrow X$ with monopole charges
$q_{m},q_{m}^{\prime}$, the tensor bundle $P\otimes P^{\prime}$ has the sum
$q_{m}+q_{m}^{\prime}$ as its monopole charge; this is precisely the additive
property of characteristic classes [58]

$c_{1}(P\otimes P^{\prime})=c_{1}(P)+c_{1}(P^{\prime}),$

and describes the process of monopole fusion. By the clutching construction,
we can identify monopole defects with branch points of $L=d\lambda_{1}$ along
the equator $S^{1}\subset S^{2}$. As multiple monopoles time-evolve and fuse,
a graph is traced out on the cylinder $S^{1}\times l$. The fusion algebra of
the Hilbert space of such graph states is known as the (2+1)D Ocneanu’s tube
algebra [60, 61, 62].

If we write $d\lambda_{1}=L$ as a generic 1-form, then we see that the
anomalous $U(1)$ gauge theory acquires a shift $A\rightarrow A+L$, generating
a "$U(1)_{1}$ 1-form symmetry" [23, 24]. We can then follow our previous
construction and introduce a 2-form $\Sigma$ as in 2.2.

##### Resolving the monopole anomaly; 2-gauge structure.

One may treat monopole Maxwell’s theory as a compact $U(1)$ gauge theory
equipped with a quantized Gauss law

$\frac{1}{2\pi}\int_{S}F=0\mod\mathbb{Z},$

which forces $A\sim A+2\pi$ to be defined only modulo $2\pi\mathbb{Z}$ [56].
Alternatively, however, introducing a 2-gauge structure is in fact the most
consistent way to treat the monopole [23, 24].

The idea is to insert a 2-form gauge field $\Sigma$ that absorbs the flux of
the magnetic monopole. This is accomplished by the crucial monopole property

$\int_{S}\Sigma=\int_{S}F,\qquad\forall\text{ closed 2-surfaces }S\subset
X^{\prime},$ (2.41)

which states that the quantized monopole charge ${q}_{m}\in\mathbb{Z}$ is
matched by the 2-form gauge field $\Sigma$. However, the monopole condition
(2.41) does not imply the fake-flatness $\Sigma=F$ on the nose, but only up to
closed 2-forms. We shall abuse notation and write such closed 2-forms as $dL$,
which can in general have a non-trivial integral over the surface $S$.

The curvature $F$ transforms as $F\rightarrow F+dL$ under this 2-gauge/1-form
shift symmetry, which can in fact change the monopole charge if $L$ has non-
trivial periods

$\frac{1}{2\pi}\int_{S^{2}}dL$

on the boundary 2-sphere $S^{2}\subset S^{2}\times l=\partial X^{\prime}$. In
order to absorb this ambiguity, we must force the 2-form $\Sigma$ to transform
as

$\Sigma\rightarrow\Sigma+dL.$ (2.42)

On the other hand, recall that the $U(1)$-connection $A$, as well as the gauge
parameter $\lambda=\lambda_{0}$, are all regular on the excised 4-manifold
$X^{\prime}$, and hence admit an extension into the whole of $X$. As such $F$
— and hence $\Sigma$ — is invariant under a 1-gauge transformation.

Thus the anomalous $U(1)$-gauge theory achieves a mixed 0-form/1-form
$U(1)_{1}\times U(1)$ symmetry, governed by a trivial 2-group

$\mathfrak{I}_{U(1)}=(t=\operatorname{id}:U(1)_{1}\rightarrow U(1),\rhd=1).$
(2.43)

The action $\rhd=1$ at the group level implies $\rhd=0$ at the algebra level,
which allows us to write the transformation law (2.42) as

$\Sigma\rightarrow\Sigma+dL+\lambda\rhd\Sigma=\Sigma+dL;$

we have the invariant 2-curvature and the higher Bianchi identity

$K=d\Sigma\rightarrow d\Sigma+d^{2}L=K,\qquad dK=0.$

The fake-flatness $F=t\Sigma=\Sigma$ then encodes the monopole condition
(2.41), and the 2-curvature $K=\ast j_{m}\neq 0$ can be used to encode the
monopole current $j_{m}$ without assuming a violation of Bianchi identity
$dF\neq 0$.

##### Relation to Green-Schwarz anomaly cancellation.

What we have demonstrated is that, to resolve an anomalous 0-form symmetry, we
must introduce a 2-group structure with mixed 0- and 1-form symmetry. This is
precisely the idea leveraged in [23] in order to implement the Green-Schwarz
mechanism of anomaly cancellation in QFT. We describe this procedure briefly
in the following.

Consider a field theory with background $U(1)\times U(1)^{\prime}$-symmetry,
in which the first copy of $U(1)$ is anomalous in the sense that its
associated curvature $F$ has a monopole defect as described above. Here, we
use a prime to indicate the other non-anomalous copy $U(1)^{\prime}$, with
associated curvature $F^{\prime}$. The anomaly polynomial, which appears under
a $U(1)$ transformation of the partition function, takes the form

$-\int F\wedge F^{\prime}.$ (2.44)

To cancel this anomaly, we introduce precisely the structure of the 2-group
$\mathfrak{I}_{U(1)}$ as above to resolve the monopole anomaly.

By taking the monopole current $\ast j_{m}=dF$ as a source for the 2-curvature
anomaly, such that $K=\kappa(A)=dF$, we can solve easily the descent equation
(2.25) $\zeta(A,\lambda)=F$. Suppose we source the 2-form connection $\Sigma$
with a 2-form current $J$, and impose the dynamical EOM $F^{\prime}=\ast J$ in
the non-anomalous sector, then, performing the modified gauge transformation
$\Sigma\rightarrow\Sigma+\zeta(A,\lambda)=\Sigma+F$ on the sourcing term gives

$\int_{X}\Sigma\wedge\ast J\rightarrow\int_{X}\Sigma\wedge\ast
J+\int_{X}F\wedge\ast J\sim\int_{X}\Sigma\wedge\ast J+\int_{X}F\wedge
F^{\prime},$

which cancels exactly the mixed anomaly (2.44) in the partition function. This
mechanism allows a consistent gauging of the 0-form symmetry $U(1)\times
U(1)^{\prime}$; for more details, see [23, 24]. Note that the underlying
2-group structure is given by

$t:U(1)_{1}\rightarrow U(1)\times U(1)^{\prime},\qquad t(z)=(z,0),$

where the first factor $\operatorname{id}:U(1)_{1}\rightarrow U(1)$ is the
trivial 2-group $\mathfrak{I}_{U(1)}$ (2.43).

##### 2-Yang-Mills theory; 2-conservation law and mobility restriction.

We now utilize the above concept of monopole anomaly resolution in order to
study an anomaly-free version of monopole electrodynamics. By anomaly-free, we
mean that the $U(1)$-bundle $P\rightarrow X$ under consideration has trivial
first Chern class.

Such a bundle hosts no monopole anomaly, and the Bianchi identity $dF=0$ is
satisfied everywhere. In order to introduce a monopole charge, we intend to
design a 2-form $\Sigma$ with a non-trivial quantized period,

$0\neq q_{m}=\frac{1}{2\pi}\int_{S}\Sigma\in\mathbb{Z},\qquad\Sigma=dL,$
(2.45)

(where the 1-form $L$ has a branch cut as shown in Fig. 1) from an action
principle. Recall that, by the clutching construction, the value of the
monopole charge is fixed by the singularity structure of $L$.

The goal is therefore to construct an action of a 2-gauge theory that
describes the electrodynamics of regular anomaly-free Maxwell’s theory, as
well as the monopole configuration of the 2-form connection $\Sigma=dL$. We
begin by forming the manifestly invariant quantity $\mathcal{F}=F-\Sigma$
under (regular) 1-form shift symmetry888The prime is to distinguish
$L^{\prime}$ from the $L$ chosen in (2.45). Here we consider regular
$L^{\prime}$, free of branch-cuts. $A\rightarrow A+L^{\prime}$, and write down
the Abelian 2-Yang-Mills theory [63]

$S_{\text{2YM}}[A,\Sigma]=\int_{X}\ast\mathcal{F}\wedge\mathcal{F}=\int_{X}\ast(F-\Sigma)\wedge(F-\Sigma),$
(2.46)

which has also appeared in the study of topological orders protected by
subsystem symmetry [25]. By varying the 2-connection $\Sigma$, we obtain the
EOM $\ast(F-\Sigma)=0$, which is nothing but fake-flatness
$\mathcal{F}=F-\Sigma=0$. However, in order to have non-trivial monopoles
charges, we must consider the off-shell configurations $F\neq\Sigma$. This is
because fake-flatness kills the monopole $K=d\Sigma=dF=0$ by the Bianchi
identity. Note that we do not expect any issue despite a violation of the fake
flatness condition since the theory is Abelian.

In order to have a non-trivial monopole configuration, we must therefore
source the 2-connection $\Sigma$. We do something more general here and source
both the 1- and 2-connections $A,\Sigma$ individually, with 1- and 2-form
currents $j_{e},J$, respectively. This inserts the following terms

$S_{\text{2cur}}=\int_{X}\ast j_{e}\wedge A+\int_{X}\ast J\wedge\Sigma$ (2.47)

into $S_{\text{2YM}}$. Intuitively, the 2-form current $J$ should be related
in some way to the monopole current $\ast j_{m}$; indeed, upon a variation of
$\Sigma$, the sourced action (2.46) together with current contribution (2.47)
leads to the EOM

$\ast(F-\Sigma)=\ast J\implies d\Sigma=-dJ,$

where we have used the Bianchi identity $dF=0$. By definition, the pure-gauge
2-connection $\Sigma=dL$ has quantized period given by the monopole charge

$q_{m}=\frac{1}{2\pi}\int_{S}\Sigma=\frac{1}{2\pi}\int_{S\times
l}d\Sigma=-\frac{1}{2\pi}\int_{S\times l}dJ,$

and combined with the definition (2.40) leads to

$dJ=-\ast j_{m}.$

This identifies the flux $dJ$ of the 2-form current $J$ with precisely the
monopole current. This makes sense, as $J$ sources the 2-connection $\Sigma$
that introduces the monopole.

######  Remark 2.12.

We emphasize here that the 2-Yang-Mills theory (2.46) is anomaly-free, meaning
that it does not have any monopole currents $j_{m}$ as the Bianchi identity
$dF=0$ is satisfied. Indeed, the main point of the construction is to source
the monopole charge with a 2-form current $J$ without introducing anomalies.

Now to derive the conservation laws of the currents $j_{e},J$, we make gauge
transformations on $S_{\text{2YM}}+S_{\text{2cur}}$. A (regular) 0-gauge
transformation $A\rightarrow A+d\lambda$ leads to

$S_{\text{2cur}}\rightarrow S_{\text{2cur}}+\int_{X}\ast j_{e}\wedge
d\lambda=S_{\text{2cur}}-\int_{X}\lambda(d\ast j_{e}),$

which accounts for the conservation $d\ast j_{e}=0$ of the electric current.
Suppose now we make the 1-form shift transformation $A\rightarrow
A+L^{\prime},\Sigma\rightarrow\Sigma+dL^{\prime}$. The action $S_{\text{2YM}}$
remains invariant, but the sourcing terms acquire

$S_{\text{2cur}}\rightarrow S_{\text{2cur}}+\int_{X}\ast j_{e}\wedge
L^{\prime}+\ast J\wedge dL^{\prime}=S_{\text{2cur}}+\int_{X}(\ast j_{e}-d\ast
J)\wedge L^{\prime},$

which implies the 2-conservation law

$d\ast J=\ast j_{e}.$ (2.48)

This implies that the conservation $d\ast J=0$ of the 2-form current $J$
occurs only if $j_{e}=0$ — the 2-form current $J$ is conserved only if
isolated charges are immobile, precisely like a dipole. Indeed, (2.48) is also
known as a dipole conservation law [25].

This mobility restriction is similar to that for fractons [64, 65]. There had
been effort to describe (3+1)D fracton models in the continuum with foliated
BF-type field theories [66].

####  2.5.4 4D topological orders: quasistring defects and surface linking

We have seen in the monopole case above that certain curvature anomalies can
be used to represent topological invariants of $X$. There, the first Chern
class $c_{1}\in H^{2}(X,\mathbb{Z})$ classifying complex line bundles on $X$
can be represented by the curvature 2-form $F$ through Chern-Weil theory [58,
67].

The topological invariants of $X$ of particular interest are the Stiefel-
Whitney classes $w\in H^{\bullet}(X,\mathbb{Z}_{2})$ of the tangent bundle
$TX\rightarrow X$, which classifies the framing of $X$ [58, 67]. Our goal in
this section is to leverage the structures of a 2-group in order to insert
higher-dimensional topological anomalies. Since we shall only be interested in
the topological defects of the theory, we assume our structure 2-group
$\mathfrak{G}=(V\xrightarrow{0}N,\rhd)$ is skeletal with $t=0$.

##### Inserting higher-form topological defects.

Let $X$ be a framed 5-manifold with boundary $Y=\partial X$. Given a skeletal
2-group $\mathfrak{G}=(0:V\rightarrow N,\rhd)$, its associated 2-gauge theory
encodes the following fake-flatness and 2-flatness conditions,

$\mathcal{F}=F=0,\qquad\mathcal{G}=K-\kappa(A)=0,$

such that the 1- and 2-connections $(A,\Sigma)$ are in a sense "decoupled".
Excitations of the theory can be inserted separately by modifying these EOMs
[19]

$\mathcal{F}=f_{2}\in
C^{2}(X,\mathbb{R})\otimes\mathfrak{n},\qquad\mathcal{G}=g_{3}\in
C^{3}(X,\mathbb{R})\otimes V,$

where $C^{\bullet}(X,\mathbb{R})$ denote the complex of $\mathbb{R}$-valued
differential cochains on $X$. The worldvolumes of the excitations in $Y$ are
determined by the restrictions of the cochains $f_{2},g_{3}$ to $Y$ via
Poincaré duality [58]

$\operatorname{PD}:C^{n}(Y,\mathbb{R})\rightarrow C_{4-n}(Y,\mathbb{R}).$

More explicitly, if $\iota:Y\hookrightarrow X$ is the inclusion of the 4d
boundary, then $\operatorname{PD}(\iota^{*}g_{3})$ is a 1-cycle (a worldline)
on $Y$ [25]. Similarly, the 2-cycle $\operatorname{PD}(\iota^{*}f_{2})$ can be
interpreted as the worldsheet of a string-like excitation [19, 30].

One of the key points demonstrated in section 2.5.3 (and later in section
3.3.2) is that the characteristic classes contribute as "anomalous"
topological excitations, or defects, of the theory. As such our goal is to
construct a 2-gauge structure hosting the Stiefel-Whitney classes
$w_{2},w_{3}$ as defects. Following [18, 19], we work directly with flat
discrete 2-connections that exhibit the Stiefel-Whitney classes as topological
defects.

The construction of gauge fields that capture topological defects gives rise
to an invertible topological quantum field theory (TQFT), such as Yetter
theory [40] or Dijkgraaf-Witten theory [18, 62]. These objects make an
appearance in high-energy physics [68, 24] and condensed matter physics [69,
20], as it is common lore [6] that anomalies in QFTs are in a very general
sense topological.

######  Remark 2.13.

One can expect that topological features are relevant in quantum gravity.
Indeed, if we accept that quantum gravitational fluctuations allow for
topology change, then one way to keep track of them is to use a 2-group
structure, through a 2-connections with non-trivial topological
configurations, such as those that we shall construct below. Another
interesting direction would be to formulate the 2-group field theory [70]
which generates topological amplitudes to non-trivial (ie. non-zero Postnikov
class) crossed-modules. We shall leave this to a future work.

##### Discrete flat 2-connections.

Recall that flat $G$-connections on $Y$ can be uniquely assigned, up to
homotopy, through the choice of a classifying map $f:X\rightarrow BG$, where
$BG$ is the classifying space of $G$ [58]. A similar situation occurs for
2-groups; in the following, we shall focus on the discrete skeletal 2-group
$\mathcal{D}(\mathbb{Z}_{2})\equiv(\mathbb{Z}_{2}\xrightarrow{0}\mathbb{Z}_{2},\rhd=0)$.

The classifying space $B\mathcal{D}(\mathbb{Z}_{2})$ can be constructed from a
Postnikov tower [18, 19, 9], such that a discrete flat
$\mathcal{D}(\mathbb{Z}_{2})$-connection can be assigned onto $X$ through the
choice of a classifying map [18, 19, 71]

$f:X\rightarrow B\mathcal{D}(\mathbb{Z}_{2}),$

up to homotopy. Now a simple computation in group cohomology [72] yields a
non-trivial generator $[\kappa]\in
H^{3}(\mathbb{Z}_{2},\mathbb{Z}_{2})\cong\mathbb{Z}_{2}$, which allows us to
define

$w_{3}=f^{*}[\kappa]$ (2.49)

as the pullback by $f$. This equation uniquely determines $f$ up to homotopy
[18].

The discrete 2-gauge structure we aim for should then have as EOMs on the
boundary $Y=\partial X$:

$F=dA=0,\qquad K=f^{*}\kappa=\kappa(A)=w_{3},$

where $(A,\Sigma)$ denotes a flat $\mathcal{D}(\mathbb{Z}_{2})$-connection
defined by the classifying map $f$. We have thus defined a discrete
2-connection that realizes the third Stiefel-Whitney class $w_{3}$ as a
2-curvature anomaly.

These EOMs can be recovered from a 2-BF type action based on the 2-group
$\mathcal{D}(\mathbb{Z}_{2})$. For this, we introduce the Lagrange multipliers
$b\in C^{2}(Y,\mathbb{Z}_{2}),c\in C^{1}(Y,\mathbb{Z}_{2})$ on the boundary
$Y=\partial X$, and we recover a 2-BF action similar to (2.26),

$S_{\text{2BF}}=\frac{1}{2}\int_{Y}b\cup
F+c\cup(K-f^{*}\kappa)=\frac{1}{2}\int_{Y}b\cup dA+c\cup(d\Sigma-w_{3}),$

where $\cup$ is the cup product on $\mathbb{Z}_{2}$-valued cochains [58].

We would like to introduce the second Stiefel-Whitney class $w_{2}$ in a
similar way, but this time associated to the 1-curvature $F$. One cannot do
that directly since 2-group $\mathcal{D}$ is skeletal and so by construction
we have $F=0$. We can, however, insert it as a dual 1-curvature anomaly, that
is as a source from the 1-gauge theory inherited from the dual 2-group
$\mathcal{D}^{*}$ gauge theory. By appending a term

$\frac{1}{2}\int_{Y}w_{2}\cup\Sigma$

to the 2-BF theory $S_{\text{2BF}}$, we achieve the EOM upon a variation of
$\Sigma$:

$\tilde{F}=dc=w_{2},$

which can be interpreted as an anomaly in the dual curvature (2.27). After
going on-shell of $F=0$, the boundary action then reads [30]

$S_{\text{2BF}}\sim S_{\partial}=\frac{1}{2}\int_{Y}c\cup(d\Sigma-
w_{3})+w_{2}\cup\Sigma.$ (2.50)

Notice that we have not assumed integral lifts — ie. an integral cohomology
class $c_{2}\in H^{2}(X,\mathbb{Z})$ such that $c_{2}\mod 2=w_{2}$ — exist for
either $w_{2}$ or $w_{3}$. Indeed, an integral lift exists for $w_{2}$ only
when $w_{3}=0$ [29]!

Due to the closure $dw_{n}=0$ of the Stiefel-Whitney classes, we have

$\frac{1}{2}d\left(c\cup(d\Sigma-
w_{3})+w_{2}\cup\Sigma\right)=\frac{1}{2}(dc\cup w_{3}+w_{2}\cup d\Sigma)\sim
w_{2}\cup w_{3},$

where we have used the EOMs for $c$ and $\Sigma$. This means that
$S_{\partial}$ can be interpreted as the boundary action of a bulk 5d
symmetry-protected topological (SPT) phase

$\mathcal{C}_{\text{5d}}=\int_{X}w_{2}\cup w_{3},$ (2.51)

protected by the global 2-group symmetry $\mathcal{D}(\mathbb{Z}_{2})$.
Conversely, we may begin with the action (2.51), and interpret the cochains
$c,\Sigma$ and their EOMs as trivializations for $w_{2},w_{3}$ on the boundary
$Y=\partial X$ [29, 30].

######  Remark 2.14.

A topological order is symmetry-protected if it describes a gapped phase that
is topologically trivial when the symmetry is ignored [73]. In general, an
invertible topological order can be interpreted as being hosted on the
boundary of a bulk symmetry protected topological (SPT) order999This is the
holographic bulk-boundary correspondence, which states that a $d$-dimensional
(possibly anomalous) topological order $\mathcal{C}$ determines uniquely an
anomaly-free order $\mathcal{D}$ in $(d+1)$-dimensions [20]., through the
mechanism of anomaly inflow/resolution [31, 29].

The above general formalism of using a 2-group structure to capture higher-
degree topological invariants has appeared in [18, 19]. The action (2.50) and
topological order (2.51) specifically has also been studied in [30], and we
shall follow this reference and provide a brief summary of its interesting
properties in the remainder of this section., The new insight we provided here
is the understanding that the EOM $dc=w_{2}$ appears as a dual EOM for the
associated 2-$BF$ theory, which implies that the boundary action (2.50) is
characterized by an underlying 2-Drinfel’d double associated to
$\mathcal{D}(\mathbb{Z}_{2})$; see Remark 2.8. This shall be made explicit in
an upcoming work by the authors.

##### Framed submanifolds; the fermionic quasistring order.

The order (2.51), $\mathcal{C}_{\text{5d}}$, hosts on $Y$ a (closed)
"magnetic" quasistring described by $\Sigma$, and an "electric" dual
quasiaprticle described by $c$; we denote by $l^{2},l$ their worldvolumes,
respectively. To understand what this means geometrically, we recall that a
framing is equivalent to a trivialization of the normal bundle [58], and that
the Stiefel-Whitney classes $w_{n}$ keep track of the twists in the framing of
$(n-1)$-dimensional embedded submanifolds [30]; see Fig 2.

Figure 2: Twists in the framing structure: the "push-offs" [30]
$l^{\prime},(l^{2})^{\prime}$ of the worldvolumes $l,l^{2}$ along their
framings are shown in red. Left: The quasiparticle described by $c$, and a
twist of its 1-dimensional worldline $l$. Right: The quasistring described by
$\Sigma$, and a twist in its 2-dimensional worldsheet $l^{2}$.

$\Sigma,c$ are fields associated to $w_{3},w_{2}$, hence they "detect"
respectively, via their values $\pm 1$ on $1,2$-dimensional submanifolds
$l,l^{2}\subset X$, twists in the framings of $l,l^{2}$. Those that exhibit
this twisting are interpreted as the worldvolumes of the quasi-
particle/string.

######  Remark 2.15.

Many topological orders, such as the 5d quasistring order
$\mathcal{C}_{\text{5d}}$ (2.51) here, are cobordism invariants [29, 31, 30,
6]. This means, in particular, that they all vanish
$\mathcal{C}_{\text{5d}}=0$ on bounding 5-manifolds. Bordism invariants are
elements of the (framed) bordism group $\Omega^{O}_{\ast}$ [74, 69]. They
constitute the non-perturbative part of the anomalies that appear in QFTs [6,
31].

##### The $w_{2}w_{3}$ gravitational anomaly.

One of the most interesting properties of the order $\mathcal{C}_{\text{5d}}$
given in (2.51) is that it detects a gravitational anomaly101010Namely, an
anomaly under diffeomorphism, aside from the chiral anomaly due to time-
reversal symmetry $\mathbb{Z}_{2}^{T}$. [29, 30]: taking $Y=\mathbb{C}P^{2}$
and $X_{\varphi}=Y\times_{\varphi}S^{1}$ and the diffeomorphism
$\varphi:(z_{1},z_{2})\mapsto(\bar{z}_{1},\bar{z}_{2})$, where the $z_{i}$ are
coordinates on $\mathbb{C}P^{2}$, one has $\mathcal{C}_{\text{5d}}=1\mod 2$,
which evaluates to a non-trivial anomaly

$\mathcal{Z}(Y)=(-1)^{\mathcal{C}_{\text{5d}}}=\exp\left(i\pi\int_{X_{\varphi}}w_{2}\cup
w_{3}\right)=-1$ (2.52)

associated to the diffeomorphism $\varphi$.

In fact, this mapping torus $X_{\varphi}=\mathbb{C}P^{2}\times_{\varphi}S^{1}$
generates the framed bordism group $\Omega^{O}_{5}$; in other words, any other
5-dimensional cobordism $X$ that evaluates to $-1$ in (2.52) is cobordant to
the mapping torus $X_{\varphi}$ of $Y=\mathbb{C}P^{2}$.

The partition function $\mathcal{Z}$ in (2.52) defines an invertible fermionic
topological quantum field theory (TQFT) $\mathcal{Z}$ [6, 31, 69] given by the
order $\mathcal{C}_{\text{5d}}$. In general, whether a TQFT is bosonic or
fermionic is determined by the self-braiding statistics of its defects [73,
19].

In dimensions $\geq 3$, point-like defects can be braided such that their
worldlines $l,l^{\prime}$ are linked, as shown in Fig. 2 (Left). This
procedure is encoded by a linking number $\operatorname{lk}(l,l^{\prime})$,
which changes by 1 upon a twist [74]. In dimensions $\geq 4$, one can braid
worldsheets $l^{2},(l^{2})^{\prime}$ with each other, as shown in Fig. 2
(Right). We also have a corresponding surface-linking number
$\operatorname{lk}(l^{2},(l^{2})^{\prime})$, which also changes by 1 upon a
twist [30].

The spin-TQFT $\mathcal{Z}$ exhibiting the gravitational anomaly given in
(2.52) defines a topological order with fermionic quasiparticle and
quasistring excitations [29, 30]. What this means is that the Wilson loop and
surface operators corresponding to these quasiparticles and quasistrings are
accompanifed by the following phases,

$(-1)^{\operatorname{lk}(l,l^{\prime})}\qquad(-1)^{\operatorname{lk}(l^{2},(l^{2})^{\prime})},$
(2.53)

in the quantum theory; if these phases are present, then the exictations are
bosonic.

##  3 Gauging the 2-gauge

Recall from section 2 that we noticed that the curvature $F$ was invariant
under a shift $\alpha$, a closed form with value in the center of
$\mathfrak{g}$. Generalizing this to an arbitrary shift led to the "gauging
the 1-gauge". This approach extends to the 2-gauge case. Indeed, given we have
a covariantly closed 2-form $\sigma$ such that $d_{A}\sigma=0$, we see that
the 2-curvature $K=d_{A}\Sigma$ is (strongly) invariant under the shift
$\Sigma\rightarrow\Sigma+\sigma$.

In the following, we shall gauge this global symmetry by taking this shift to
be an arbitrary 2-form $\sigma\in\Omega^{2}(X)\otimes\mathfrak{h}$. In this
sense, we will make the 2-curvature gauge datum.

###  3.1 From shifting the 2-connection to a Lie algebra 2-crossed module

####  3.1.1 Shifting the 2-connection

Consider a 2-connection shift $\Sigma\rightarrow\Sigma+\sigma$ by an arbitrary
2-form $\sigma$. The 2-curvature $K$ then transforms accordingly

$K\rightarrow K^{\prime}=d_{A}\Sigma+d_{A}\sigma,$

hence the 3-curvature $K\rightarrow K^{\prime}\neq K$ fails to be invariant,
even on-shell of the fake-flatness condition. To remedy this, we introduce a
3-form gauge field by

$\Gamma\equiv K^{\prime}-K=d_{A}\sigma.$ (3.1)

Following the same reasoning as previously, the shift we are performing does
not have to come from the same algebra $\Sigma$ is valued in. We consider a
2-form $\mathcal{L}\in\Omega^{2}(X)\otimes\mathfrak{i}$ valued in another Lie
algebra $\mathfrak{i}=\operatorname{Lie}I$, and replace (3.1) by a "pure
gauge" connection 3-form $\Gamma=d_{A}\mathcal{L}$.

Repeating the same steps as before, we may expect the existence of another
crossed-module
$\mathfrak{S}=(t^{\prime}:\mathfrak{i}\rightarrow\mathfrak{h},\bar{\rhd})$
such that the shift transformation on the 2-connection

$\Sigma\rightarrow\Sigma+t^{\prime}\mathcal{L},\qquad
t^{\prime}\mathcal{L}=\sigma$

becomes a "3-gauge transformation" parameterized by
$\mathcal{L}\in\Omega^{3}(X)\otimes\mathfrak{i}$. This yields yet another
invariant fake curvature quantity, called the 2-fake curvature

$\displaystyle\mathcal{G}\equiv K-t^{\prime}(\Gamma),$

and the associated 2-fake-flatness condition $K=t^{\prime}\Gamma$. Notice once
again that the 2-curvature anomaly $K=\kappa$ can now be absorbed as a part of
the 3-connection $\Gamma$ — we shall make this statement precise in section
3.2.2.

Since $K$ is valued in $\operatorname{ker}t$ on-shell of the fake-flatness
condition $\mathcal{F}=0$, enforcing also the 2-fake-flatness condition
implies $\operatorname{ker}t=\operatorname{im}t^{\prime}$, i.e. $t\circ
t^{\prime}=0$. We thus obtain an exact Lie algebra complex

$\mathscr{G}:\mathfrak{i}\xrightarrow{t^{\prime}}\mathfrak{h}\xrightarrow{t}\mathfrak{g},$
(3.2)

for which $t^{\prime}:\mathfrak{i}\rightarrow\mathfrak{h}$ is a crossed-
module. Here, $\mathfrak{g}$ acts on both $\mathfrak{h}$ and $\mathfrak{i}$,
denoted respectively by $\rhd$ and $\rhd^{\prime}$, under which the maps
$t,t^{\prime}$ are both $\mathfrak{g}$-equivariant — meaning that the
following diagram, which is a generalization of (2.4),

${\mathfrak{i}}$${\mathfrak{h}}$${\mathfrak{g}}$${\operatorname{Der}\mathfrak{i}}$${\operatorname{Der}\mathfrak{h}}$${\operatorname{Inn}\mathfrak{g}}$$\scriptstyle{t^{\prime}}$$\scriptstyle{t}$$\scriptstyle{\rhd}$$\scriptstyle{\rhd^{\prime}}$$\scriptstyle{t^{\prime}}$$\scriptstyle{t}$

commutes, where $\operatorname{Der}\mathfrak{g}$ denotes the space of
derivations of the Lie algebra $\mathfrak{g}$.

####  3.1.2 3-curvature and 3-Bianchi identity

##### 3-curvature.

Let us define the 3-curvature $T=d_{A}\Gamma$ in the naïve way. With the
exactness $\operatorname{im}t^{\prime}=\operatorname{ket}t$ of the complex
(3.2) in mind, as well as on-shell of the 2-fake-flatness condition
$t^{\prime}\Gamma=K$, we see that

$t^{\prime}(T)=d_{A}(t^{\prime}(\Gamma))=d_{A}K=((t\Sigma)\wedge^{\rhd}\Sigma)|_{\operatorname{ker}t}.$

If the Peiffer identity for the part $t:\mathfrak{h}\rightarrow\mathfrak{g}$
of (3.2) holds, then this term coincides with
$[\Sigma\wedge\Sigma]|_{\operatorname{ker}t}$ as computed in (2.11), which
vanishes. This is the 2-Bianchi identity.

However, we can also consider the case where the Peiffer identity does not
hold, so that $t:\mathfrak{h}\rightarrow\mathfrak{g}$ is no longer necessarily
a crossed-module, but merely a precrossed-module. This means that only the
first Peiffer condition (equivariance) is satisfied, and $(ty)\rhd y^{\prime}$
does not coincide with the bracket $[y,y^{\prime}]$ on $\mathfrak{h}$ as
$(t\cdot)\rhd\cdot$ is in general not skew-symmetric111111Indeed, the skew-
symmetry of this quantity is an axiom in the 2-algebra formulation that
defines the Lie bracket on $\mathfrak{h}$ [37]..

To treat the term $(t\Sigma)\wedge^{\rhd}\Sigma|_{\operatorname{ker}t}$, we
invoke the exactness of the complex (3.2) to write it as the image of some
quantity $\\{\Sigma\wedge\Sigma\\}_{\text{Pf}}$ under $t^{\prime}$. This
quantity is defined by the Peiffer lifting map
$\\{\cdot,\cdot\\}_{\text{Pf}}:\mathfrak{h}^{2\otimes}\rightarrow\mathfrak{i}$,
which satisfies

$t^{\prime}\\{y,y^{\prime}\\}_{\text{Pf}}=-((ty^{\prime})\rhd
y)|_{\operatorname{ker}t},\qquad\forall y,y^{\prime}\in\mathfrak{h}.$ (3.3)

In other words, $\\{\cdot,\cdot\\}_{\text{Pf}}$ lifts
$((t\cdot)\rhd\cdot)|_{\operatorname{ker}t}$ along $t^{\prime}$ up to
$\mathfrak{i}$. If we wish for the 3-curvature to be valued in
$\operatorname{ker}t^{\prime}\subset\mathfrak{i}$, similar to how the
2-curvature $K$ is valued in $\operatorname{ker}t\subset\mathfrak{h}$, we must
replace $T$ by the modified 3-curvature $\mathcal{H}$,

$\mathcal{H}=d_{A}\Gamma+Q_{\Sigma},\qquad
Q_{\Sigma}=\\{\Sigma\wedge\Sigma\\}_{\text{Pf}}.$ (3.4)

This extra quadratic term $Q_{\Sigma}$ is an artifact of forgoing the Peiffer
identity for $t:\mathfrak{h}\rightarrow\mathfrak{g}$ but, importantly, should
not itself be considered an anomaly. We shall elaborate more on this in
section 3.2.2.

##### 3-Bianchi identity.

Given the modified 3-curvature (3.4), we can assess what properties we should
consider to have if we want to have a 3-Bianchi identity to hold (on shell of
the fake-flatness condition $t\Sigma=F$),

$d_{A}\mathcal{H}=d_{A}(T+Q_{\Sigma})=0.$ (3.5)

We compute, assuming that there would be no violation of 1-Bianchi identity,

$\displaystyle d_{A}\mathcal{H}$ $\displaystyle=$ $\displaystyle
F\wedge^{\rhd^{\prime}}\Gamma+\\{K\wedge\Sigma\\}_{\text{Pf}}+\\{\Sigma\wedge
K\\}_{\text{Pf}}$ $\displaystyle=$ $\displaystyle
t\Sigma\wedge^{\rhd^{\prime}}\Gamma+\\{K\wedge\Sigma\\}_{\text{Pf}}+\\{\Sigma\wedge
K\\}_{\text{Pf}}$

Demanding that the last quantity is zero, on-shell of the 2-flatness
condition, imposes a relation between $\\{\cdot,\cdot\\}_{\text{Pf}}$, $t$,
$t^{\prime}$ and $\rhd^{\prime}$, which appears when we consider a Lie algebra
2-crossed-module. Such relation is one of the 2-Peiffer conditions, which we
shall see below.

######  Remark 3.1.

It would be possible to investigate the notion of a "weak 2-crossed module" by
violating the 1-Bianchi identity. This can be accomplished by relaxing the
Jacobi identity on $\mathfrak{g}$ in the definition of a 2-crossed-module. We
shall not delve on this here, however, as this requires much more elaboration.

####  3.1.3 Lie algebra 2-crossed-module

We are now ready to define what a 2-crossed-module is. Let
$\mathfrak{g},\mathfrak{h},\mathfrak{i}$ be Lie algebras.

######  Definition 3.1.

[33] The 3-term algebra complex $\mathscr{G}$ in (3.2), equipped with the
bilinear map $\\{\cdot,\cdot\\}_{\text{Pf}}:\mathfrak{h}^{\otimes
2}\rightarrow\mathfrak{i}$, is a 2-crossed module if and only if

  * •

the action $\mathfrak{h}\bar{\rhd}\mathfrak{i}$ defined by
$y\bar{\rhd}z\equiv-\\{t^{\prime}z,y\\}_{\text{Pf}}$ makes
$\mathfrak{S}=(t^{\prime}:\mathfrak{i}\rightarrow\mathfrak{h},\bar{\rhd})$
into a crossed-module,

  * •

the 2-Peiffer conditions are satisfied

$\displaystyle\\{t^{\prime}z,y\\}_{\text{Pf}}+\\{y,t^{\prime}z\\}_{\text{Pf}}$
$\displaystyle=$ $\displaystyle-ty\rhd^{\prime}z,$
$\displaystyle{[z,z^{\prime}]}$ $\displaystyle=$
$\displaystyle\\{t^{\prime}z,t^{\prime}z^{\prime}\\}_{\text{Pf}},$

for all $y\in\mathfrak{h}$ and $z,z^{\prime}\in\mathfrak{i}$,

  * •

the 3-Jacobi identities is satisfied

$\displaystyle\\{y,[y^{\prime},y^{\prime\prime}]\\}_{\text{Pf}}$
$\displaystyle=$
$\displaystyle\\{(y,y^{\prime})_{\text{Pf}},y^{\prime\prime}\\}_{\text{Pf}}-\\{(y,y^{\prime\prime})_{\text{Pf}},y^{\prime}\\}_{\text{Pf}},$
$\displaystyle\\{[y,y^{\prime}],y^{\prime\prime}\\}_{\text{Pf}}$
$\displaystyle=$ $\displaystyle\\{ty\rhd
y^{\prime},y^{\prime\prime}\\}_{\text{Pf}}-\\{ty^{\prime}\rhd
y,y^{\prime\prime}\\}_{\text{Pf}}$
$\displaystyle+\\{y^{\prime},(y,y^{\prime\prime})_{\text{Pf}}\\}_{\text{Pf}}-\\{y,(y^{\prime},y^{\prime\prime})_{\text{Pf}}\\}_{\text{Pf}}$

for all $y,y^{\prime},y^{\prime\prime}\in\mathfrak{h}$, where
$(\cdot,\cdot)_{\text{Pf}}\equiv t^{\prime}\\{\cdot,\cdot\\}_{\text{Pf}}$ is
the $\operatorname{im}t^{\prime}=\operatorname{ker}t$-valued Peiffer pairing,
and

  * •

the lifting condition (3.3), is satisfied. Both
$\\{\cdot,\cdot\\}_{\text{Pf}}$ and $(\cdot,\cdot)_{\text{Pf}}$ are
$\mathfrak{g}$-equivariant.

Note $\\{\cdot,\cdot\\}_{\text{Pf}}$ is in general not skew-symmetric.

It was proposed [33] that this 2-crossed-module serves as the structure of a
principal 3-gauge bundle $\mathscr{P}\rightarrow X$.

In this case, we have the 1-gauge connection $A$ with value in $\mathfrak{g}$,
the 2-connection $\Sigma$ with value in $\mathfrak{h}$ and the 3-connection
$\Gamma$ with value in $\mathfrak{i}$. The modified 3-curvature satisfies the
3-Bianchi identity thanks to the first of the 2-Peiffer conditions. We have
indeed that

$t\Sigma\wedge^{\rhd^{\prime}}\Gamma=-\\{t^{\prime}\Gamma,\Sigma\\}_{\text{Pf}}-\\{\Sigma,t^{\prime}\Gamma\\}_{\text{Pf}},$

which together with the 2-fake flatness insures that the 3-Bianchi is
satisfied. We now study the gauge transformation structure.

###  3.2 Gauge transformations and descent equation

We now turn to the 3-gauge transformations on the fields $(A,\Sigma,\Gamma)$
in question. Let us fix the notation

$\lambda\in\Omega^{0}(X)\otimes\mathfrak{g},\qquad
L\in\Omega^{1}(X)\otimes\mathfrak{h},\qquad\mathcal{L}\in\Omega^{2}(X)\otimes\mathfrak{i}$

for the 0-, 1- and 2-form gauge parameters in the theory. We shall derive the
3-gauge transformation rules, under the principle that the curvature
quantities

$\mathcal{F}=F-t\Sigma,\qquad\mathcal{G}=K-t^{\prime}\Gamma,\qquad\mathcal{H}=d_{A}\Gamma+Q_{\Sigma}$

transform covariantly. We shall recover the results given in [75].

####  3.2.1 Gauge transformations

We begin by assuming for simplicity that the 2-gauge sector, involving the
fields $(A,\Sigma)$, should retain the same transformation laws under the
2-gauge parameters $(\lambda,L)$ as we have derived in section 2.4.

##### 1-gauge transformations.

We utilize the action of $\mathfrak{g}$ on the two other Lie algebras
$\mathfrak{h},\mathfrak{i}$. Since in the 2-gauge context,
$K=d_{A}\Sigma\rightarrow K-\lambda\rhd K$ is covariant, the 3-form $\Gamma$
must also transform covariantly,

$t^{\prime}\Gamma\rightarrow t^{\prime}\Gamma-\lambda\rhd
t^{\prime}\Gamma=t^{\prime}\Gamma-t^{\prime}(\lambda\rhd^{\prime}\Gamma)\implies\Gamma\rightarrow\Gamma-\lambda\rhd^{\prime}\Gamma,$

in order to achieve the covariance of the 2-fake-curvature

$\mathcal{G}=K-t^{\prime}\Gamma\rightarrow\mathcal{G}^{\lambda}=\mathcal{G}-\lambda\rhd\mathcal{G}.$

Moreover, due to the $\mathfrak{g}$-equivariance of the Peiffer lifting map
$\\{\cdot,\cdot\\}_{\text{Pf}}$, the modified 3-curvature (3.4) also achieves
the covariant transformation

$\mathcal{H}=d_{A}\Gamma+Q_{\Sigma}\rightarrow\mathcal{H}^{\lambda}=\mathcal{H}-\lambda\rhd^{\prime}\mathcal{H},$
(3.6)

as desired.

##### 2-gauge transformations.

For this case, we utilize (2.18) for the 2-gauge transformation law of $K$,
with the caveat that the Peiffer identity for
$t:\mathfrak{h}\rightarrow\mathfrak{g}$ is no longer necessarily satisfied. As
such, we write the $[L\wedge L]$ term in the 2-gauge transformation
$\Sigma^{L}$ as $tL\wedge L$. This yields

$K\rightarrow K^{L}=K+tL\wedge^{\rhd}\Sigma^{L}+F\wedge^{\rhd}L.$

This indicates the transformation property

$t^{\prime}\Gamma\rightarrow t^{\prime}\Gamma-
tL\wedge^{\rhd}\Sigma^{L}-t\Sigma\wedge^{\rhd}L,$

such that we achieve the desired covariance

$\mathcal{G}=K-t^{\prime}\Gamma\rightarrow\mathcal{G}^{L}=\mathcal{G}-\mathcal{F}\wedge^{\rhd}L.$
(3.7)

Using the lifting condition (3.3) for the Peiffer pairing
$(\cdot,\cdot)_{\text{Pf}}=t^{\prime}\\{\cdot,\cdot\\}_{\text{Pf}}$, we have

$-tL\wedge^{\rhd}\Sigma^{L}=(\Sigma^{L}\wedge
L)_{\text{Pf}}=t^{\prime}\\{\Sigma^{L}\wedge
L\\}_{\text{Pf}},\qquad-t\Sigma\wedge^{\rhd}L=(L\wedge\Sigma)_{\text{Pf}}=t^{\prime}\\{L\wedge\Sigma\\}_{\text{Pf}},$

which allows us to deduce the 2-gauge transformation

$\Gamma\rightarrow\Gamma^{L}=\Gamma+\\{\Sigma^{L}\wedge
L\\}_{\text{Pf}}+\\{L\wedge\Sigma\\}_{\text{Pf}}$

for $\Gamma$, provided the 1-Bianchi identity is satisfied (ie. $K$ is valued
in $\operatorname{ker}t=\operatorname{im}t^{\prime}$).

A direct computation with this 2-gauge transformation law for $\Sigma$ and
$\Gamma$ then produces [75]

$\mathcal{H}\rightarrow\mathcal{H}^{L}=\mathcal{H}+\\{\mathcal{G}^{L}\wedge
L\\}_{\text{Pf}}+\\{L\wedge\mathcal{G}\\}_{\text{Pf}}\sim\mathcal{H},$

which is indeed invariant on-shell of the 1- and 2-fake-flatness conditions
$\mathcal{F},\mathcal{G}=0$.

##### 3-gauge transformations.

We expect that a 3-gauge transformation should be parameterized by a 2-form,
$\mathcal{L}$, in $\mathfrak{i}$. Hence we naturally posit that $A$ is
invariant, $A\rightarrow A^{\mathcal{L}}=A$, and hence so is the 1-curvature
$F\rightarrow F^{\mathcal{L}}=F$.

For the remaining fields $\Sigma,\Gamma$, we follow the above gauging the
gauge argument and induce a shift transformation

$\Sigma\rightarrow\Sigma^{\mathcal{L}}=\Sigma+t^{\prime}\mathcal{L},\qquad\Gamma\rightarrow\Gamma^{\mathcal{L}}=\Gamma+d_{A}\mathcal{L}.$

Recall that the product of a $\mathfrak{g}$-valued form with a
$\mathfrak{i}$-valued form is performed through the action $\rhd^{\prime}$.

We now compute that the 2-curvature transforms as (note the
$\mathfrak{g}$-equivariance of $t^{\prime}$)

$K\rightarrow
K^{\mathcal{L}}=K+d_{A}t^{\prime}\mathcal{L}=K+t^{\prime}d_{A}\mathcal{L},$

which gives rise to the invariance of the 2-fake-curvature

$\mathcal{G}\rightarrow\mathcal{G}^{\mathcal{L}}=\mathcal{G}.$ (3.8)

Furthermore, as $tt^{\prime}=0$ and $F$ is unchanged, we see that the fake-
curvature $\mathcal{F}=F-t\Sigma$ is in fact also invariant under the 3-gauge.

Now performing a 3-gauge transformation on the modified 3-curvature
$\mathcal{H}$ gives

$\mathcal{H}\rightarrow\mathcal{H}^{\mathcal{L}}=\mathcal{H}+F\wedge^{\rhd^{\prime}}\mathcal{L}+\\{t^{\prime}\mathcal{L}\wedge\Sigma\\}_{\text{Pf}}+\\{\Sigma\wedge
t^{\prime}\mathcal{L}\\}_{\text{Pf}}+\\{t^{\prime}\mathcal{L}\wedge
t^{\prime}\mathcal{L}\\}_{\text{Pf}},$

for which we may employ the 2-Peiffer conditions (recall $\mathcal{L}$ is a
2-form)

$\\{t^{\prime}\mathcal{L}\wedge\Sigma\\}_{\text{Pf}}+\\{\Sigma\wedge
t^{\prime}\mathcal{L}\\}_{\text{Pf}}=-t\Sigma\wedge^{\rhd^{\prime}}\mathcal{L},\qquad\\{t^{\prime}\mathcal{L}\wedge
t^{\prime}\mathcal{L}\\}_{\text{Pf}}=[\mathcal{L}\wedge\mathcal{L}]=0$

to deduce the covariance

$\mathcal{H}\rightarrow\mathcal{H}^{\mathcal{L}}=\mathcal{H}+\mathcal{F}\wedge^{\rhd^{\prime}}\mathcal{L}\sim\mathcal{H}.$
(3.9)

The modified 3-curvature $\mathcal{H}$ is thus invariant on-shell of the fake-
flatness condition $\mathcal{F}=0$ [75] (which we recall is preserved by the
exactness $tt^{\prime}=0$ of the complex (3.2)).

In summary, we have the following 3-gauge transformations

$\displaystyle\lambda$ $\displaystyle:$
$\displaystyle\begin{cases}A\rightarrow A^{\lambda}=A+d_{A}\lambda\\\
\Sigma\rightarrow\Sigma^{\lambda}=\Sigma-\lambda\rhd\Sigma\\\
\Gamma\rightarrow\Gamma^{\lambda}=\Gamma-\lambda\rhd^{\prime}\Gamma\end{cases},$
$\displaystyle L$ $\displaystyle:$ $\displaystyle\begin{cases}A\rightarrow
A^{L}=A+tL\\\ \Sigma\rightarrow\Sigma^{L}=\Sigma+d_{A}L+\frac{1}{2}tL\wedge
L\\\ \Gamma\rightarrow\Gamma^{L}=\Gamma+\\{\Sigma^{L}\wedge
L\\}_{\text{Pf}}+\\{L\wedge\Sigma\\}_{\text{Pf}}\end{cases},$
$\displaystyle\mathcal{L}$ $\displaystyle:$
$\displaystyle\begin{cases}A\rightarrow A^{\mathcal{L}}=A\\\
\Sigma\rightarrow\Sigma^{\mathcal{L}}=\Sigma+t^{\prime}\mathcal{L}\\\
\Gamma\rightarrow\Gamma^{\mathcal{L}}=\Gamma+d_{A}\mathcal{L}\end{cases}$
(3.10)

that generate the gauge symmetry $\operatorname{Gau}_{3}$ of our 3-gauge
theory. This was also derived in [33, 75].

##### Compatibility between the 3-gauge transformations.

Importantly, it was noted in [33] that 2-gauge transformations $L$ do not
generate a subalgebra when the Peiffer bracket is not zero. Indeed, they
commute only up to a 3-gauge transformation,

$[(0,L_{1},0),(0,L_{2},0)]=(0,0,\mathcal{L}_{12}),\qquad\mathcal{L}_{12}=2(\\{L_{1}\wedge
L_{2}\\}_{\text{Pf}}-\\{L_{2}\wedge L_{1}\\}_{\text{Pf}}).$ (3.11)

On the other hand, the computations

$[(\lambda_{1},0,0),(\lambda_{2},0,0)]=([\lambda_{1},\lambda_{2}],0,0),\qquad[(0,0,\mathcal{L}_{1}),(0,0,\mathcal{L}_{2})]=0,$

as well as the obvious results

$[(\lambda,0,0),(0,L,0)]=(0,\lambda\rhd
L,0),\qquad[(\lambda,0,0),(0,0,\mathcal{L})]=(0,0,\lambda\rhd^{\prime}\mathcal{L}),$

allow us to completely characterize the 3-gauge group as

$\operatorname{Gau}_{3}=\mathscr{E}\rtimes(\Omega^{0}(X)\otimes\mathfrak{g}),\qquad
0\rightarrow\Omega^{2}(X)\otimes\mathfrak{i}\rightarrow\mathscr{E}\rightarrow\Omega^{1}(X)\otimes\mathfrak{h}\rightarrow
0.$

Here, $\mathscr{E}$ can be seen as a sort of central extension of the 2-gauge
by the 3-gauge according to (3.11).

####  3.2.2 3-curvature anomaly and its first descendant

The goal in this section is to study the anomaly $\tau$ of the modified
3-curvature $\mathcal{H}$, as well as derive conditions on its descendant. In
the absence of the 2-Bianchi anomaly, the modified 3-curvature $\mathcal{H}$
is valued in $\operatorname{ker}t^{\prime}$, and so must the anomaly $\tau$.
We shall see that, analogous to section 2.4.2, given that the 3-curvature
anomaly $\tau$ takes a particular form, then it is related to the classifying
cohomology class of the underlying 2-crossed-module $\mathscr{G}$ under
consideration.

We wish to insert $\tau$ which preserves the covariance of the anomaly EOM
$\mathcal{H}=\tau$ under the 3-gauge transformations (3.10). This tells us
that $\tau$ should transform convariantly under a 3-gauge transformation,
identically to how $\mathcal{H}$ transforms (3.9). Therefore, on-shell of the
fake-flatness condition $\mathcal{F}=0$, the 3-curvature anomaly $\tau$ should
be invariant under a 3-gauge transformation, meaning that it must be 2-shift-
invariant $\tau(A,\Sigma)=\tau(A,\Sigma+t^{\prime}\mathcal{L})$. As such,
$\tau$ can only be a function on $\mathfrak{g}$ and
$\operatorname{coker}t^{\prime}=\mathfrak{h}/\operatorname{im}t^{\prime}$.
Notice that the quadratic term depends only on
$\operatorname{coker}t^{\prime}$, as
$Q_{t\mathcal{L}}=[\mathcal{L}\wedge\mathcal{L}]=0$ by the 2-Peiffer
condition.

We now examine the conditions for which the 3-curvature anomaly EOM
$\mathcal{H}=\tau$ is covariant under 2-gauge transformations. Once again, the
covariance of $\mathcal{H}$ (3.2.1) implies that $\tau=\tau(A)$ cannot depend
on the 2-connection $\Sigma$, and must be shift-invariant
$\tau(A)=\tau(A+tL)$. This casts $\tau$ as a
$\operatorname{ker}t^{\prime}$-valued, degree-4 function of
$\operatorname{coker}t$, which is precisely the data of a 4-cocycle
representative of the Lie algebra cohomology class $[\tau]\in
H^{4}(\operatorname{coker}t,\operatorname{ker}t^{\prime})$ that classifies the
2-crossed-module $\mathscr{G}$ up to equivalence [36, 10, 9]; see also Remark
A.1 in Appendix A.3.

##### Twisted 1-gauge transformations.

Going back to the anomaly EOM $\mathcal{H}=\tau(A)$ for the modified
3-curvature, we have shown that $\tau$ is a function of
$\operatorname{coker}t=\mathfrak{g}/\operatorname{im}t$ valued in
$\operatorname{ker}t$. This puts us in an identical situation as the
2-curvature anomaly $\kappa(A)$.

With the 1-gauge transformations remaining, we suppose the 1- and 2-form
connections $(A,\Sigma)$ transform as usual, and define the first descendant
$\xi(A,\lambda)$ of $\tau(A)$ as a twisted 1-gauge transformation in the
3-connection satisfying the descent equation,

$\Gamma\rightarrow\Gamma+\lambda\rhd^{\prime}\Gamma+\xi(A,\lambda),\qquad
d_{A^{\lambda}}\xi(A,\lambda)=\tau(A^{\lambda})-\lambda\rhd^{\prime}\tau(A),$

such that the covariance of $\mathcal{H}$, (3.6), gives

$\mathcal{H}^{\lambda}=\mathcal{H}+\lambda\rhd^{\prime}\mathcal{H}=\tau(A^{\lambda}).$

Here, the first descendant of the 3-gauge
$\xi(A,\lambda)\in\Omega^{3}(X)\otimes\operatorname{ker}t^{\prime}$ is a
3-form, in contrast to the 2-form $\zeta(A,\lambda)$ encountered in section
2.4.2.

With $\tau$ and $\xi$ valued in $\operatorname{ker}t^{\prime}$, this twisted
gauge transformation does not conflict with the covariance of the modified
3-curvature $\mathcal{H}$ above. Moreover, due to the exactness
$tt^{\prime}=0$ of the complex (3.2), the 3-gauge descendant $\xi$ is
independent from any 2-curvature anomaly.

####  3.2.3 2-curvature anomaly and first descendant as 3-gauge data

It is more accurate to say that there is only the 3-gauge descendant here, as
the 2-curvature anomaly $K=\kappa(A)$ can be understood as a particular
3-connection via the 2-fake-flatness condition $t^{\prime}\Gamma=K$. Indeed,
as $K=d_{A}\Sigma$, we can consider $\Gamma=d_{A}\mathcal{L}(A)$ as a pure
3-gauge whose gauge parameter depends on the 1-connection $A$, with
$t^{\prime}=\operatorname{id}$ the identity.

Note that $\kappa=\kappa(A)$ can only depend on the 1-connection $A$, which is
unaffected by the 3-gauge transformation. Hence locally, we can perform a
3-gauge shift parametrized by $\mathcal{L}$ (which could depend on $A$) in
order to remove the 2-curvature anomaly,

$K-\kappa(A)\rightarrow K+d_{A}\mathcal{L}(A)-\kappa(A)=K,$

which introduces a gauge-fixing of the 3-connection to a pure gauge
$\Gamma=d_{A}\mathcal{L}(A)$. With this choice understood, we now make a
3-gauge transformation followed by a 1-gauge transformation on the
2-connection

$\Sigma\xrightarrow{\mathcal{L}}\Sigma+\mathcal{L}\xrightarrow{\lambda}\Sigma+\lambda\rhd\Sigma+\mathcal{L}+\mathcal{L}^{\lambda},$

where we have kept the transformation
$\mathcal{L}\xrightarrow{\lambda}\mathcal{L}+\mathcal{L}^{\lambda}$ implicit,
as $\mathcal{L}=\mathcal{L}(A)$ now depends on $A$.

In the opposite order, we have

$\Sigma\xrightarrow{\lambda}\Sigma+\lambda\rhd\Sigma\xrightarrow{\mathcal{L}}\Sigma+\mathcal{L}+\lambda\rhd(\Sigma+\mathcal{L})=\Sigma+\lambda\rhd\Sigma+\mathcal{L}+\lambda\rhd\mathcal{L}.$

The difference is the expression

$\mathcal{L}^{\lambda}-\lambda\rhd\mathcal{L},$

which upon taking the gauge-transformed covariant derivative $d_{A^{\lambda}}$
yields

$d_{A^{\lambda}}\mathcal{L}^{\lambda}-d_{A^{\lambda}}(\lambda\rhd\mathcal{L})=(d_{A}\mathcal{L})^{\lambda}-\lambda\rhd
d_{A}\mathcal{L}=\kappa(A^{\lambda})-\lambda\rhd\kappa(A).$

This is nothing but the descent equation (2.25) satisfied by the first
descendant $\zeta(A,\lambda)$ of the 2-curvature anomaly $\kappa(A)$, if we
take

$\zeta(A,\lambda)=\mathcal{L}^{\lambda}-\lambda\rhd\mathcal{L}.$

This leads to the identification with the commutator

$[(0,0,\mathcal{L}),(\lambda,0,0)]=(0,0,\zeta(A,\lambda)).$ (3.12)

In other words, the 2-curvature anomaly $\kappa(A)$ arising from a Postnikov
class can be absorbed by a pure 3-gauge, while its descendant can be absorbed
by the commutator (3.12), thereby embedding a non-trivial 2-gauge theory into
an anomaly-free 3-gauge theory. This is the spirit of anomaly resolution, and
we shall see this in action in section 3.3.2.

###  3.3 Applications

In this section, we discuss concrete examples in which 3-gauge structures
naturally arise.

####  3.3.1 3-BF theory

The simplest topological action to consider is once again an action
implementing merely the constraints — namely, the 1-, 2-fake-flatness and the
flat 3-curvature conditions — as equations of motion (EOM). Such a theory has
been studied in detail in [33], and we follow their treatment here as well.

##### Action and EOMs.

As previously, we fix a 2-crossed-module
$\mathscr{G}=\mathfrak{i}\xrightarrow{t^{\prime}}\mathfrak{h}\xrightarrow{t}\mathfrak{g}$,
and we introduce the dual spaces
$\mathfrak{i}^{*},\mathfrak{h}^{*},\mathfrak{g}^{*}$ of linear functionals on
respectively the Lie algebras $\mathfrak{i},\mathfrak{h},\mathfrak{g}$. We
denote their pairing forms collectively by $\langle\cdot,\cdot\rangle$. We
begin by introducing Lagrange multipliers