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# (2-)Drinfel’d Double and (2-)BF Theory

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

###### Abstract

The gauge symmetry and shift/translational symmetry of a 3D BF action, which
are associated to a pair of dual Lie algebras, can be combined to form the
Drinfel’d double. This combined symmetry is the gauge symmetry of the Chern-
Simons action which is equivalent to the BF action, up to some boundary term.
We show that something similar happens in 4D when considering a 2-BF action
(aka BFCG action), whose symmetries are specified in terms of a pair of dual
strict Lie 2-algebras (ie. crossed-modules). Combining these symmetries gives
rise to a 2-Drinfel’d double which becomes the gauge symmetry structure of a
4D BF theory, up to a boundary term. Concretely, we show how using 2-gauge
transformations based on dual crossed-modules, the notion of 2-Drinfel’d
double defined in [1] appears. We also discuss how, similarly to the Lie
algebra case, the symmetric contribution of the $r$-matrix of the 2-Drinfel’d
double can be interpreted as a quadratic 2-Casimir, which allows to recover
the notion of duality.

###### Contents

  1. 1 Introduction
  2. 2 Recovering the Drinfel’d double as a gauge symmetry from dual gauge theories
    1. 2.1 Dual gauge theories
    2. 2.2 The Drinfel’d double as gauge symmetry
    3. 2.3 3D topological action invariant under the Drinfel’d double
  3. 3 Lie algebra crossed-module and its dual
    1. 3.1 Lie algebra crossed-module.
    2. 3.2 Dual crossed-module and Lie bialgebra crossed-modules.
    3. 3.3 Coadjoint representations
  4. 4 Recovering the 2-Drinfel’d double as a 2-gauge symmetry from dual 2-gauge theories
    1. 4.1 Dual 2-gauge theories
    2. 4.2 The 2-Drinfel’d double as 2-gauge symmetry
      1. 4.2.1 Combined 1-gauge transformations
      2. 4.2.2 Combined 2-gauge transformations
      3. 4.2.3 Mixed 1- and 2-gauge transformations
      4. 4.2.4 The 2-Manin triple
    3. 4.3 Strict Lie 2-bialgebras and 2-Manin rigidity
  5. 5 4D topological action invariant under the 2-Drinfel’d double
    1. 5.1 BF theory and interacting 2-BF theory.
    2. 5.2 Pairings on the 2-Drinfel’d double; the BF-BB theory.
  6. 6 The 2-graded classical $r$-matrix and the quadratic 2-Casimirs
    1. 6.1 The 2-graded classical $r$-matrix
    2. 6.2 Quadratic 2-Casimirs
    3. 6.3 Explicit constructions of the quadratic 2-Casimirs for some crossed-modules
      1. 6.3.1 2-Casimirs of the trivial canonical 2-algebra.
      2. 6.3.2 2-Casimirs of the skeletal suspension 2-algebra.
      3. 6.3.3 2-Casimirs of the 2-Drinfel’d double
  7. 7 Conclusions
  8. A Proofs of generalized (2-)covariance

##  1 Introduction

Quantum groups, or Hopf algebras, and their category of representations,
provide the symmetry structure to discuss topological quantum field theories
in three dimensions [2, 3, 4, 5]. Among these, the specific example of the
Drinfel’d double [6, 7] plays a special role. It is the relevant quantum group
to discuss the quantization of three dimensional gravity [8, 9]. It is the
symmetry inherent to the Kitaev model [10, 11] which could be used to build
quantum computers (see also [12] for some other symmetry than the double).
Typically it is built out of a pair of Hopf algebras that are dual to each
other.

The deformed notion of symmetry encoded in a quantum group can be derived [13]
or guessed [14, 15] from an action, such as the Chern-Simons action and
(equivalently in some cases) the BF action. When using an action as starting
point, the key step to see quantum groups structure to appear involves a
discretization procedure to recover some type of (generalized) lattice gauge
theory. At the classical level, this generates Poisson-Lie groups which are
the classical analogue of the notion of quantum group [16]. Poisson-Lie groups
are Lie groups equipped with a Poisson bracket which is compatible with the
group structure. In certain cases, the Poisson bracket is characterized by an
object called the $r$-matrix, which upon quantization becomes the quantum
$R$-matrix. The Drinfel’d double is an example of Poisson-Lie group. Looking
at the infinitesimal picture, ie. looking at the Lie algebras, we have the so-
called classical (Drinfel’d) double construction (also called Manin triple):
we have a pair of bialgebras which are dual to each other, and the cocycle of
one is dual to the Lie bracket of the other [17, 16]. These cocycles are also
the infinitesimal limit of the Poisson brackets compatible with the relevant
group structure. The symmetric part of the $r$-matrix is given in terms of the
quadratic Casimir of the Lie algebra behind the Drinfel’d double. This Casimir
encodes the duality structure.

Upon discretization of the BF theory, the Drinfel’d double as a Poisson-Lie
group arises naturally as the structure of the symmetry [13]. This means in
particular that one can expect that at the level of the action, somehow the
classical double should be present in the structure of the gauge symmetries.
This is kind of an understood fact, but here we will show explicitly how this
works as it will be useful for the generalization to higher-gauge symmetries.
We will also emphasize the different formulation of the action, either
invariant under the full double, which would be the Chern-Simons formulation,
or in terms of the pair of components of the double, which would amount to the
BF formulation.

What happens when we deal with four dimensional topological quantum field
theories expressed in terms of BF-type theories [18]? As discussed in
different places, when we go up in dimension, we go up in the categorical
ladder [3, 19, 4]. For the four dimensional case, we expect to use the notion
of quantum 2-groups (see [20] for a proposal), where a 2-group is a
categorification of the notion of group [21]. Note that there are several
options to define the notion of quantum 2-groups. Here we will follow the
intuition given by gauge theory and the associated topological actions one can
construct. The definition of a 2-group in terms of a crossed-module is likely
the most suited to discuss gauge theory generalizations [22, 23, 24, 25, 26].
It is not however the best one to discuss the notion of dualization, which is
better defined in terms of Lie 2-algebras [27, 1]. Indeed this latter
formulation, while equivalent, emphasizes better what needs to be dualized —
in particular, the notion of grading plays a key role.

While a general general framework for the notion of quantum 2-group is
missing, the notion of Poisson-Lie 2-group has been introduced [28] and one
can expect that one can provide more specific examples of Poisson-Lie 2-groups
when dealing with discretization of topological actions. Just as in 3D, there
are 2 main types of action for building out a topological model, the BF type
[18], which is just the higher dimensional version of the 3D one, or the BFCG
(aka 2-BF) actions [29, 30]. The former is usually interpreted as having gauge
symmetries and shift symmetries, which can be ultimately packaged in terms of
a 2-gauge symmetry [26]. The latter one is explicitly built on 2-gauge
symmetries, as well as some shift symmetries [29, 30]. Similarly to Chern-
Simons and BF theory, some examples of 2-BF models on 2-groups have been
repackaged into a new BF action where the variables are grouped together [31].

These similarities between the 3D and 4D cases beg the question, of whether a
notion of 2-Drinfel’d classical double might be at play when dealing with
2-gauge theories. Roughly speaking, we would expect a 2-Drinfel’d classical
double to consist in a pair of crossed-modules (more precisely strict Lie
2-bialgebras [1, 32]) that are dual in the appropriate sense to each other,
and such that the cocycle information of one would specify the structure of
the other. This object has already been defined in Ref. [1]. Here we would
like to use the gauge theoretic formulation and re-derive the notion of
2-Drinfel’d classical double. This is one of the main results of the paper. We
will then show that the 2-BF theories can be repackaged into a "larger" BF
theory with symmetry specified by the 2-Drinfel’d double. This will mean
conversely that 4D BF theories can be specified in terms of 2-BF theories, and
that there is an underlying 2-gauge symmetry at play. This has some
implications in terms of the discretization of these theories; see for example
[33] in the case of the Poincaré 2-group. As a specific case, we will show
that the standard Lorentz $BF$ theory can also be seen as a kind of 2-BF
theory, in terms of a 3D (quantum) Poincaré 2-group. Ultimately, this type of
considerations could be relevant to discuss quantum gravity models, since 4D
gravity can be seen as a constrained topological theory [34, 35, 36, 37].
Indeed the choice of symmetry structure influences the construction of the
quantum states which are built from the representations of the symmetry
structure. For example111Note that the choice of symmetry structure already
appears in 3D, where one can use the Drinfel’d double
$\mathfrak{g}\bowtie\mathfrak{g}^{*}$ as the main symmetry structure, as in
the Fock-Rosly approach [14, 15], and associated combinatorial quantization
[38, 39], or the gauge symmetry in terms of $\mathfrak{g}$ or
$\mathfrak{g}^{*}$ [40]., if we deal with the BF theory, we would construct
the discrete/quantum theory using representation of the gauge symmetries. On
the other hand, if we use instead the 2-BF formulation in terms of 2-gauge, we
would use the theory of representations of the crossed-module. Since we are
talking about the same theory, namely $BF$ theory, the quantum $BF$ amplitude
defined in terms of group representations [36] or 2-group representations
should be equivalent, just like in 3D where the Chern-Simons amplitude can be
related to the Turaev-Viro/Ponzano-Regge amplitudes [41, 42].

When defining the 2-Drinfel’d double, we can introduce a generalization of the
notion of $r$-matrix [1]. We show how something similar to the 3D case, namely
whether its symmetric component is encoding some notion of "Casimir" for the
Lie 2-algebra behind the 2-Drinfel’d double happens.

The article is organized as follows.

In Section 2, we illustrate our strategy with the standard Drinfel’d double.
We show how starting from a pair of dual gauge theories, we can recover the
Drinfel’d double. We recall then the standard construction to define a
topological action which is invariant under the Drinfel’d double and how it
can be broken down into components to recover the $BF$ formulation in terms of
the dual gauge theories.

In Section 3, we recall the definition of crossed-modules, how to recover a
dual crossed-module and the (co-)adjoint action of a crossed-module.

In Section 4, we provide one of the main results of the paper namely, we
reconstruct the notion of 2-Drinfeld double by considering dual 2-gauge
transformations. Considering that their order in which we apply them should
not matter we recover constraints on the dual crossed-modules which amount
exactly to the definition of the 2-Drinfeld double provided in [1].

Identifying the 2-Drinfel’d double symmetries allows in Section 5 to construct
an action invariant under such symmetries, which happens to be some 4D $BF$
theory type. We then explain how such $BF$ theory can also be seen as a pair
of dual 2-gauge theories. As a particular example we focus on Lorentz $BF$
theory. This is another of the highlights of the paper.

Finally, in Section 6, we discuss how the $r$-matrix structure of the
2-Drinfeld double also contains the information about the duality used to
defined the action, just like in the standard Drinfel’d double case.

##  2 Recovering the Drinfel’d double as a gauge symmetry from dual gauge
theories

In this section, we begin with a demonstration of our strategy to extract the
structure of the Drinfel’d double from gauge theory.

We consider a pair of dual gauge theories in the sense that their underlying
Lie algebras, $\mathfrak{g}$, $\mathfrak{g}^{*}$ are dual to each other.
Because of the duality, we can induce gauge transformations of $\mathfrak{g}$
on $\mathfrak{g}^{*}$ and vice-versa, through the co-adjoint representations.
We can then perform gauge transformations of $\mathfrak{g}$ and
$\mathfrak{g}^{*}$ on a sector, say $\mathfrak{g}$ in different order.
Requiring that we get the same result, whatever the order put strong
constraints on the shape of the transformations. The constraints are
equivalent to the compatibility relations of a matched pair of Lie algebras
$\mathfrak{g}\bowtie\mathfrak{g}^{*}$ [43]. To have such compatibilitiy
relations means that we need to modify accordingly the notion of covariant
derivative in each sector $\mathfrak{g}$ and $\mathfrak{g}^{*}$. This
modification is what we called the antisymmetrization procedure. As a result,
the covariant derivative (and therefore the gauge transformation) in a given
different sector will then depend on both connections.

These new gauge transformations can be combined together, these are the
Drinfel’d double gauge transformations.

We will then discuss how we can construct a theory invariant under such theory
either in terms of connections based on the Drinfel’d double
$\mathfrak{g}\bowtie\mathfrak{g}^{*}$ as a gauge symmetry (Chern-Simons), or
based on the components $\mathfrak{g}$ and $\mathfrak{g}^{*}$ (BF theory).

###  2.1 Dual gauge theories

##### Gauge theory

Let $P\rightarrow M$ denote a principal $G$-bundle over a $d$-dimensional
manifold $M$. It is standard lore that the $G$-connection
$A\in\Omega^{1}(M)\otimes\mathfrak{g}$ and its associated curvature

$F=dA+\frac{1}{2}[A\wedge A]\in\Omega^{2}(M)\otimes\mathfrak{g}$

undergoes the following gauge transformations

$A\rightarrow A^{\lambda}=A+d_{A}\lambda,\qquad F\rightarrow
F^{\lambda}=F+[F,\lambda]$ (2.1)

parameterized by a 0-form $\lambda\in\Omega^{0}(M)\otimes\mathfrak{g}$, where
$\mathfrak{g}=\operatorname{Lie}G$ is the Lie algebra of $G$.

##### Dual gauge theory

We can construct another principal bundle, $P^{\prime}\rightarrow M$ which is
a $G^{*}$-bundle over the $d$-dimensional manifold $M$. For this we introduce
$\mathfrak{g}^{*}$ the dual vector space to $\mathfrak{g}$. $\mathfrak{g}^{*}$
can also be seen as a Lie algebra. To this aim, we introduce a Lie bracket
$[\cdot,\cdot]_{*}$ on $\mathfrak{g}^{*}$. This can be accomplished by the
choice of a 1-cochain

$\psi:\mathfrak{g}\rightarrow\mathfrak{g}^{2\wedge}$

such that $[\cdot,\cdot]_{*}$ is defined as

$[g,g^{\prime}]_{*}(X)=(g\wedge
g^{\prime})(\psi(X)),\qquad\forall\leavevmode\nobreak\
g,g^{\prime}\in\mathfrak{g}^{*},X\in\mathfrak{g}.$

The Jacobi identity for the bracket is equivalent to the cocycle condition
[17] which is expressed in term of the coadjoint action
$\operatorname{ad}^{*}$ (see below)

$d\psi(X,X^{\prime})=\psi([X,X^{\prime}])-D(\operatorname{ad}^{*}_{X})\psi(X^{\prime})+D(\operatorname{ad}^{*}_{X^{\prime}})\psi(X)=0,$
(2.2)

where $D(X)=X\otimes 1+1\otimes X$ is the comultiplication map. As such,
$\psi\in Z^{1}(\mathfrak{g},\mathfrak{g}^{2\wedge})$ is a Lie algebra
1-cocycle, and $(\mathfrak{g};\psi)$ a Lie bialgebra [17]. $G^{*}$ would then
be the Lie group associated to $\mathfrak{g}^{*}$.

######  Remark 2.1.

The bialgebra property is symmetric under dualization: if $\mathfrak{g}$ is a
Lie bialgebra, so is its dual $\mathfrak{g}^{*}$, which is equipped with a
1-cocycle $\psi^{*}\in C^{1}(\mathfrak{g}^{*},(\mathfrak{g}^{*})^{2\wedge})$
dual to the Lie bracket of $\mathfrak{g}$. The situation can be summarized
below

Cocycle… is dual
to…Bracket${\mathfrak{g}:}$${\psi}$${{[\cdot,\cdot]}}$${\mathfrak{g}^{*}:}$${\psi^{*}}$${{[\cdot,\cdot]_{*}}}$

Hence to make the construction, we could also start for $G^{*}$ and its Lie
algebra $\mathfrak{g}^{*}$ instead of $G$ and its Lie algebra $\mathfrak{g}$.

Let us now turn to the gauge structure generated by the dual space
$\mathfrak{g}^{*}$. If we let $B\in\Omega^{1}(M)\otimes\mathfrak{g}^{*}$
denote a $G^{*}$-connection and

$\tilde{F}=dB+\frac{1}{2}[B\wedge B]_{*}$

its curvature, then we would also have the following gauge transformation

$B\rightarrow
B^{\tilde{\lambda}}=B+d_{B}\tilde{\lambda},\qquad\tilde{F}\rightarrow\tilde{F}^{\tilde{\lambda}}=\tilde{F}+[\tilde{F},\tilde{\lambda}]_{*}$
(2.3)

parameterized by a 0-form gauge parameter
$\tilde{\lambda}\in\Omega^{0}(M)\otimes\mathfrak{g}^{*}$.

##### Coadjoint representations.

Because the Lie algebras $\mathfrak{g}$ and $\mathfrak{g}^{*}$ are dual to
each other, we can dualize their respective adjoint actions, which are
respectively

$\operatorname{ad}_{X}(X^{\prime})=[X,X^{\prime}],\qquad\mathfrak{ad}_{g}(g^{\prime})=[g,g^{\prime}]_{*},$

where $X,X^{\prime}\in\mathfrak{g},g,g^{\prime}\in\mathfrak{g}^{*}$. By
dualizing with respect to the canonical evaluation pairing $(g,X)=g(X)$, a
pair of coadjoint representations are induced,

$\operatorname{ad}^{*}:\mathfrak{g}\rightarrow\operatorname{End}\mathfrak{g}^{*},\qquad\mathfrak{ad}^{*}:\mathfrak{g}^{*}\rightarrow\operatorname{End}\mathfrak{g},$

defined by

$\operatorname{ad}_{X}^{*}g(X^{\prime})=-g(\operatorname{ad}_{X}X^{\prime}),\qquad
g^{\prime}(\mathfrak{ad}_{g}^{*}X)=-\mathfrak{ad}_{g}g^{\prime}(X),$

for which the evaluation pairing is by definition invariant. This is then
inducing an invariant symmetric non-degenerate bilinear form [17]

$\langle\langle
X+g,X^{\prime}+g^{\prime}\rangle\rangle=g(X^{\prime})+g^{\prime}(X)$ (2.4)

on the direct sum $\mathfrak{g}\oplus\mathfrak{g}^{*}$. The connections $A,B$
admit a transformation under $\tilde{\lambda},\lambda$, respectively, through
these codajoint representations

$A\rightarrow A+\mathfrak{ad}_{\tilde{\lambda}}^{*}A,\qquad B\rightarrow
B+\operatorname{ad}_{\lambda}^{*}B.$ (2.5)

To summarize, because we have a pair of dual Lie algebras, we can introduce a
gauge transformation of one sector say $\mathfrak{g}$ on the dual one
$\mathfrak{g}^{*}$ and vice-versa. Indeed, the transformations Eqs. (2.1),
(2.3), (2.5) allow us to study, in each $\mathfrak{g}$\- or
$\mathfrak{g}^{*}$-sector, the gauge transformations parameterized by both
$\lambda$ and $\tilde{\lambda}$. In the following, we shall leverage this
observation to derive compatibility conditions on the coadjoint
representations $\operatorname{ad}^{*},\mathfrak{ad}^{*}$ in order for the
direct sum $\mathfrak{g}\oplus\mathfrak{g}^{*}$ to form a Manin triple, which
is an equivalent description of the Drinfel’d double [16].

###  2.2 The Drinfel’d double as gauge symmetry

Let us now consider how the connections $A,B$ transform under both
$\lambda,\tilde{\lambda}$. Since the sectors $\mathfrak{g}$ and
$\mathfrak{g}^{*}$ are acting on each other, the gauge parameters will
themselves be subject of gauge transformations. For example, the pure-gauge
$B=d\tilde{\lambda}$ transforms non-trivially under $\lambda$; similarly for
the pure-gauge $A=d\lambda$ under $\tilde{\lambda}$:

$\displaystyle d\tilde{\lambda}$ $\displaystyle\rightarrow$ $\displaystyle
d\tilde{\lambda}+\operatorname{ad}_{\lambda}^{*}d\tilde{\lambda}=d\tilde{\lambda}+d(\mathfrak{ad}_{\tilde{\lambda}}^{*}\lambda)-\mathfrak{ad}_{d\tilde{\lambda}}^{*}\lambda$
$\displaystyle d\lambda$ $\displaystyle\rightarrow$ $\displaystyle
d\lambda+\mathfrak{ad}_{\tilde{\lambda}}^{*}d\lambda=d\lambda+d(\operatorname{ad}_{\lambda}^{*}\tilde{\lambda})-\operatorname{ad}_{d\lambda}^{*}\tilde{\lambda},$

where we have used the Leibniz rule.

Now if we assume, in the first equation, that $\tilde{\lambda}$ is slowly-
varying (so that $\lambda d\tilde{\lambda}$ is small), then this induces a
transformation of the gauge parameter $\lambda$ itself

$\lambda\rightarrow\lambda+\mathfrak{ad}_{\tilde{\lambda}}^{*}\lambda+o(\lambda
d\tilde{\lambda}).$

Similarly, if we assume $\lambda$ to be slowly-varying, then we have

$\tilde{\lambda}\rightarrow\tilde{\lambda}+\operatorname{ad}_{\lambda}^{*}\tilde{\lambda}+o(\tilde{\lambda}d\lambda)$

in the dual sector.

In the following, we denote by $\boldsymbol{\lambda}=\lambda+\tilde{\lambda}$
the combined gauge parameter, and a direction of an arrow over it indicates
the order at which the gauge transformation parameterized by which parameter
is performed. For instance, $\overrightarrow{\boldsymbol{\lambda}}$ indicates
that we perform a gauge transformation by $\lambda$ first then
$\tilde{\lambda}$, and vice versa for $\overleftarrow{\boldsymbol{\lambda}}$.

##### Combined gauge transformation.

With this, we may then use Eqs. (2.1), (2.5) to compute the transformation law

$\displaystyle\overrightarrow{\boldsymbol{\lambda}}$ $\displaystyle:$
$\displaystyle A\xrightarrow{\lambda}A+d_{A}\lambda$
$\displaystyle\xrightarrow{\tilde{\lambda}}A+\mathfrak{ad}_{\tilde{\lambda}}^{*}A+d_{A}\lambda+\mathfrak{ad}_{\tilde{\lambda}}^{*}d\lambda$
$\displaystyle\qquad+[\mathfrak{ad}_{\tilde{\lambda}}^{*}A,\lambda]+[A,\mathfrak{ad}_{\tilde{\lambda}}^{*}\lambda]+[\mathfrak{ad}_{\tilde{\lambda}}^{*}A,\mathfrak{ad}_{\tilde{\lambda}}^{*}\lambda],$

while

$\displaystyle\overleftarrow{\boldsymbol{\lambda}}$ $\displaystyle:$
$\displaystyle
A\xrightarrow{\tilde{\lambda}}A+\mathfrak{ad}_{\tilde{\lambda}}^{*}A$
$\displaystyle\xrightarrow{\lambda}A+d_{A}\lambda+\mathfrak{ad}_{\tilde{\lambda}}^{*}A+\mathfrak{ad}_{\tilde{\lambda}}^{*}d\lambda$
$\displaystyle\qquad+\mathfrak{ad}_{\tilde{\lambda}}^{*}[A,\lambda]+\mathfrak{ad}_{\operatorname{ad}_{\lambda}^{*}\tilde{\lambda}}^{*}A+\mathfrak{ad}_{\operatorname{ad}_{\lambda}^{*}\tilde{\lambda}}^{*}d_{A}\lambda.$

If we neglect quadratic terms in the small gauge parameter $\lambda$, we
obtain the difference

$[\mathfrak{ad}_{\tilde{\lambda}}^{*}A,\lambda]+[A,\mathfrak{ad}^{*}_{\tilde{\lambda}}\lambda]-\mathfrak{ad}_{\tilde{\lambda}}^{*}[A,\lambda]-\mathfrak{ad}_{\operatorname{ad}_{\lambda}^{*}\tilde{\lambda}}^{*}A$

between the two transformations. This quantity is in fact undesirable, and
forcing it to vanish yields the following compatibility condition

$\mathfrak{ad}_{\tilde{\lambda}}^{*}[A,\lambda]=[\mathfrak{ad}_{\tilde{\lambda}}^{*}A,\lambda]+[A,\mathfrak{ad}^{*}_{\tilde{\lambda}}\lambda]-\mathfrak{ad}_{\operatorname{ad}_{\lambda}^{*}\tilde{\lambda}}^{*}A+\mathfrak{ad}_{\operatorname{ad}_{A}^{*}\tilde{\lambda}}^{*}\lambda,$
(2.6)

where we have explicitly antisymmetrized the last term
$-\mathfrak{ad}_{\operatorname{ad}_{\lambda}^{*}\tilde{\lambda}}^{*}A$ under
an interchange of $A,\lambda$ by adding a term
$\mathfrak{ad}_{\operatorname{ad}_{A}^{*}\tilde{\lambda}}^{*}\lambda$ to the
difference. This antisymmetrization procedure has major implications about the
gauge structure of the theory — we shall explain this in full detail in the
next section. This condition Eq. (2.6) ensures that the gauge transformation
by $\boldsymbol{\lambda}$ on $A$ is unambiguous — namely does not depend on
the order of the action of $\lambda,\tilde{\lambda}$.

On the dual side, we may argue analogously by exchanging the original and the
dual sectors:

$\lambda\leftrightsquigarrow\tilde{\lambda},\qquad\operatorname{ad}^{*}\leftrightsquigarrow\mathfrak{ad}^{*},\qquad
A\leftrightsquigarrow B.$

More explicitly, we compute from (2.3), (2.5) that

$\displaystyle\overrightarrow{\boldsymbol{\lambda}}$ $\displaystyle:$
$\displaystyle B\xrightarrow{\lambda}B+\operatorname{ad}_{\lambda}^{*}B$
$\displaystyle\xrightarrow{\tilde{\lambda}}B+d_{B}\tilde{\lambda}+\operatorname{ad}_{\lambda}^{*}B+\operatorname{ad}_{\lambda}^{*}d\tilde{\lambda}$
$\displaystyle\qquad+\operatorname{ad}_{\lambda}^{*}[B,\lambda]_{*}+\operatorname{ad}_{\mathfrak{ad}_{\tilde{\lambda}}^{*}\lambda}^{*}B+\operatorname{ad}_{\mathfrak{ad}_{\tilde{\lambda}}^{*}\lambda}^{*}d_{B}\tilde{\lambda},$

while

$\displaystyle\overleftarrow{\boldsymbol{\lambda}}$ $\displaystyle:$
$\displaystyle B\xrightarrow{\tilde{\lambda}}B+d_{B}\tilde{\lambda}$
$\displaystyle\xrightarrow{\lambda}B+\operatorname{ad}_{\lambda}^{*}B+d_{b}\tilde{\lambda}+\operatorname{ad}_{\lambda}^{*}d\tilde{\lambda}$
$\displaystyle\qquad+[\operatorname{ad}_{\lambda}^{*}B,\tilde{\lambda}]_{*}+[B,\operatorname{ad}_{\lambda}^{*}\tilde{\lambda}]_{*}+[\operatorname{ad}_{\lambda}^{*}B,\operatorname{ad}_{\lambda}^{*}\tilde{\lambda}]_{*}.$

Neglecting the term of order $\tilde{\lambda}^{2}$, we obtain the difference

$[\operatorname{ad}_{\lambda}^{*}B,\tilde{\lambda}]_{*}+[B,\operatorname{ad}^{*}_{\lambda}\tilde{\lambda}]_{*}-\operatorname{ad}_{\lambda}^{*}[B,\tilde{\lambda}]_{*}-\operatorname{ad}_{\mathfrak{ad}_{\tilde{\lambda}}^{*}\lambda}^{*}B,$

and hence the compatibility condition upon antisymmetrizing with respect to
$B,\tilde{\lambda}$,

$\operatorname{ad}_{\lambda}^{*}[B,\tilde{\lambda}]_{*}=[\operatorname{ad}_{\lambda}^{*}B,\tilde{\lambda}]_{*}+[B,\operatorname{ad}^{*}_{\lambda}\tilde{\lambda}]_{*}-\operatorname{ad}_{\mathfrak{ad}_{\tilde{\lambda}}^{*}\lambda}^{*}B+\operatorname{ad}_{\mathfrak{ad}_{B}^{*}\lambda}^{*}\tilde{\lambda},$
(2.7)

which is dual to Eq. (2.6).

##### The antisymmetrization procedure.

One crucial point to note is that, to impose the compatibility conditions Eqs.
(2.6), (2.7), we must introduces the terms
$\mathfrak{ad}_{\operatorname{ad}_{A}^{*}\tilde{\lambda}}^{*}\lambda,\operatorname{ad}_{\mathfrak{ad}_{B}^{*}\lambda}^{*}\tilde{\lambda}$
in order to preserve the explicit antisymmetry in $A,\lambda$ and
$B,\tilde{\lambda}$.

These additional terms, however, cannot just be introduced from thin air. We
must induce it from a modification of the gauge transformations we have
assumed in Eqs. (2.1), (2.5). In particular, we include the terms
$\mathfrak{ad}_{B}^{*}\lambda,\operatorname{ad}_{A}^{*}\tilde{\lambda}$ into
how $A,B$ transform respectively under $\lambda,\tilde{\lambda}$:

$A\rightarrow A+d_{A}\lambda-\mathfrak{ad}_{B}^{*}\lambda,\qquad B\rightarrow
B+d_{B}\tilde{\lambda}-\operatorname{ad}_{A}^{*}\tilde{\lambda}.$ (2.8)

We call this the antisymmetrization procedure of the gauge structure, and it
is a key technique that we shall use extensively in the rest of this paper.

It is simple to see why these extra terms are necessary. By construction, the
combined gauge transformation $\boldsymbol{\lambda}$ in either order reads

$\displaystyle\overrightarrow{\boldsymbol{\lambda}}:A\rightarrow\dots-\mathfrak{ad}_{B}^{*}\lambda+\mathfrak{ad}_{\operatorname{ad}_{A}^{*}\tilde{\lambda}}^{*}\lambda,$
$\displaystyle\overleftarrow{\boldsymbol{\lambda}}:A\rightarrow\dots-\mathfrak{ad}_{B}^{*}\lambda,$
$\displaystyle\overrightarrow{\boldsymbol{\lambda}}:B\rightarrow\dots-\operatorname{ad}_{A}^{*}\tilde{\lambda},$
$\displaystyle\overleftarrow{\boldsymbol{\lambda}}:B\rightarrow\dots-\operatorname{ad}_{A}^{*}\tilde{\lambda}+\operatorname{ad}_{\mathfrak{ad}_{B}^{*}\lambda}^{*}\tilde{\lambda},$

where "$\dots$" indicates terms that we have already seen above. Their
difference then manifests the desired terms

$\mathfrak{ad}_{\operatorname{ad}_{A}^{*}\tilde{\lambda}}^{*}\lambda+\operatorname{ad}_{\mathfrak{ad}_{B}^{*}\lambda}^{*}\tilde{\lambda}$

upon performing the antisymmetrization procedure.

However, Eq. (2.8) implies that the $\mathfrak{g}$\- and $\mathfrak{g}^{*}$
connections contribute both to define the proper notion of covariant
derivative: the gauge transformation for the connection $A$ depends on the
dual connection $B$, and vice versa. As such, they should come together to
define a combined connection

${\bf A}=A+B\in\Omega^{1}(M)\otimes(\mathfrak{g}\oplus\mathfrak{g}^{*})$

valued in the direct sum $\mathfrak{g}\oplus\mathfrak{g}^{*}$, upon which the
combined gauge transformation $\boldsymbol{\lambda}=\lambda+\tilde{\lambda}$
acts. This is a key observation that shall finally allow us to extract the
compatibility conditions in a Manin triple.

Note that the antisymmetrization procedure introduces a modified notion of
covariant derivative for the matched pair
$\mathfrak{g}\bowtie\mathfrak{g}^{*}$,

$\displaystyle D_{\bf A}\lambda$
$\displaystyle=d_{A}\lambda-\mathfrak{ad}_{B}^{*}\lambda$ $\displaystyle
D_{\bf A}\tilde{\lambda}$
$\displaystyle=d_{B}\tilde{\lambda}-\operatorname{ad}_{A}^{*}\tilde{\lambda}.$
(2.9)

which had also been discussed in [13, 44].

##### The Manin triple.

From the above computations, we have derived the following result:

######  Theorem 2.1.

The combined gauge transformation (including antisymmetrized terms induced
from Eq. (2.8))

${\bf A}\rightarrow{\bf A}^{\boldsymbol{\lambda}}={\bf A}+(D_{\bf
A}\lambda+\mathfrak{ad}_{\tilde{\lambda}}^{*}A)+(D_{\bf
A}\tilde{\lambda}+\operatorname{ad}_{\lambda}^{*}B)$ (2.10)

is unambiguous (namely independent of the order of applying
$\lambda,\tilde{\lambda}$) modulo terms of order
$o(\lambda^{2})+o(\tilde{\lambda}^{2})+o(\lambda
d\tilde{\lambda})+o(\tilde{\lambda}d\lambda)$ iff the compatibility conditions
Eqs. (2.6), (2.7) are satisfied.

If we distill Eqs. (2.6), (2.7) in terms of the Lie algebra values:

$\displaystyle\mathfrak{ad}_{g}^{*}[X,X^{\prime}]=[\mathfrak{ad}_{g}^{*}X,X^{\prime}]+[X,\mathfrak{ad}^{*}_{g}X^{\prime}]-\mathfrak{ad}_{\operatorname{ad}_{X^{\prime}}^{*}g}^{*}X+\mathfrak{ad}_{\operatorname{ad}_{X}^{*}g}^{*}X^{\prime},$
$\displaystyle\operatorname{ad}_{X}^{*}[g,g^{\prime}]_{*}=[\operatorname{ad}_{X}^{*}g,g^{\prime}]_{*}+[g,\operatorname{ad}^{*}_{X}g^{\prime}]_{*}-\operatorname{ad}_{\mathfrak{ad}_{g^{\prime}}^{*}X}^{*}g+\operatorname{ad}_{\mathfrak{ad}_{g}^{*}X}^{*}g^{\prime}$
(2.11)

for each $X,X^{\prime}\in\mathfrak{g},g,g^{\prime}\in\mathfrak{g}^{*}$, then
we in fact recover the necessary and sufficient conditions for the coadjoint
representations $\operatorname{ad}^{*},\mathfrak{ad}^{*}$ to come together and
form the standard Manin triple [16]

$\mathfrak{d}=\mathfrak{g}\leavevmode\nobreak\
_{\operatorname{ad}^{*}}\bowtie_{\mathfrak{ad}^{*}}\mathfrak{g}^{*}$

equipped with the standard Lie bracket

$\boldsymbol{[}X+g,X^{\prime}+g^{\prime}\boldsymbol{]}=[X,X^{\prime}]+[g,g^{\prime}]_{*}+\operatorname{ad}_{X}^{*}g^{\prime}+\mathfrak{ad}_{g}^{*}X-\mathfrak{ad}_{g^{\prime}}^{*}X^{\prime}-\operatorname{ad}_{X^{\prime}}^{*}g.$
(2.12)

It is known that this double Lie algebra structure $\mathfrak{d}$ integrates
to a Poisson-Lie group $D$ [17], and hence serves as a model for the Drinfel’d
double $\mathfrak{d}=D(\mathfrak{g})$.

##### Manin Rigidity

We have demonstrated that the gauge theory built on the direct sum
$\mathfrak{g}\oplus\mathfrak{g}^{*}$ of the Lie bialgebra $\mathfrak{g}$ and
its dual $\mathfrak{g}^{*}$ manifests canonically the structure of a Manin
triple if and only if the combined gauge transformation under
$\boldsymbol{\lambda}=\lambda+\tilde{\lambda}$ is unambiguous. In particular,
the naturally induced coadjoint representations
$\operatorname{ad}^{*},\mathfrak{ad}^{*}$ of the Lie bialgebra and its dual on
each other must necessarily satisfy Eq. (2.11).

Now if we consider generic algebra action $\mathfrak{g}\rhd\mathfrak{g}^{*}$
and back-action $\mathfrak{g}\lhd\mathfrak{g}^{*}$, equipped with the same
non-degenerate invariant bilinear form Eq. (2.4), we can prove the following
gauge-enforced rigidity result:

######  Corollary 2.1.

The induced combined gauge transformations
$\boldsymbol{\lambda}=\lambda+\tilde{\lambda}$ is unambiguous if and only if
the double

$\mathfrak{d}^{\prime}=\mathfrak{g}\bowtie\mathfrak{g}^{*}$

is isomorphic to the standard Manin triple
$\mathfrak{d}=\mathfrak{g}\leavevmode\nobreak\
_{\operatorname{ad}^{*}}\bowtie_{\mathfrak{ad}^{*}}\mathfrak{g}$.

###### Proof.

The same argument as in Section 2.2, leading up to Theorem 2.1, goes through
for the generic action/back-action $\rhd,\lhd$, such that the compatibility
condition Eq. (2.11) must hold, with $\operatorname{ad}^{*}$ replaced by
$\rhd$ and $\mathfrak{ad}^{*}$ replaced by $\lhd$. The induced Lie bracket
must then take the same form as Eq. (2.12), which forces
$\mathfrak{d}^{\prime}\cong\mathfrak{d}$. ∎

The rigidity of the Manin triple is well-known, but here we have demonstrated
by gauge theoretic considerations.

We shall generalize our method in the following to a 2-gauge structure, based
on a Lie algebra crossed-module $\mathfrak{g}$. We shall seek to extract
compatibility conditions for the (strict) coadjoint representations necessary
for the notion of a 2-Manin triple.

###  2.3 3D topological action invariant under the Drinfel’d double

The fact that the Manin triple $\mathfrak{d}$ forms a Lie algebra implies that
we can do gauge theory on it based on the standard bracket
$[\cdot,\cdot]_{\mathfrak{d}}=\boldsymbol{[}\cdot,\cdot\boldsymbol{]}$ in Eq.
(2.12). In particular, given the combined connection ${\bf A}$ we can write a
covariant curvature quantity

$\displaystyle{\bf F}$ $\displaystyle=$ $\displaystyle d{\bf
A}+\frac{1}{2}\boldsymbol{[}{\bf A}\wedge{\bf A}\boldsymbol{]}$ (2.13)
$\displaystyle=$ $\displaystyle(dA+\frac{1}{2}[A\wedge
A]+\mathfrak{ad}_{B}^{*}(\wedge A))+(dB+\frac{1}{2}[B\wedge
B]_{*}+\operatorname{ad}_{A}^{*}(\wedge B))$ $\displaystyle\equiv$
$\displaystyle\bar{F}+\bar{F}^{*},$

called the combined curvature, which has also appeared in Ref. [44]. The
covariance of ${\bf F}$ is manifest; for an explicit proof, see Appendix A.

##### Chern-Simons theory and interacting BF theory.

Given the combined curvature ${\bf F}$, one may then assemble a gauge-
invariant action and study its dynamics. Using the natural invariant non-
degenerate symmetric bilinear form Eq. (2.4) on $\mathfrak{d}$, we begin with
the Chern-Simons action

$S_{\text{CS}}[{\bf A}]=\frac{1}{2}\int_{M}\langle\langle{\bf F}\wedge{\bf
F}\rangle\rangle=\int_{M}\langle\bar{F}^{*}\wedge\bar{F}\rangle,$ (2.14)

where $M$ is a 4-dimensional manifold. It can be clearly seen that, due to the
covariance of the combined curvature ${\bf F}$, the action Eq. (2.14) is
invariant under the Drinfel’d double
$\mathfrak{d}=\mathfrak{g}\bowtie\mathfrak{g}^{*}$ gauge transformations.

If we write down the components of $\bar{F},\bar{F}^{*}$ explicitly, then one
may compute

$\displaystyle\langle\bar{F}^{*}\wedge\bar{F}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2}d\left[\langle B\wedge(dA+\frac{1}{2}[A\wedge
A])\rangle+\frac{1}{2}\langle[B\wedge B]_{*}\wedge A\rangle\right]$
$\displaystyle+\frac{1}{4}\langle[B\wedge B]_{*}\wedge[A\wedge
A]\rangle+\langle\operatorname{ad}_{A}^{*}(\wedge
B)\wedge\mathfrak{ad}_{B}^{*}(\wedge A)\rangle,$

where the equality stands up to a boundary term. By writing the final term as

$\langle\operatorname{ad}_{A}^{*}(\wedge B)\wedge\mathfrak{ad}_{B}^{*}(\wedge
A)\rangle=-\frac{1}{2}(\langle B\wedge[A\wedge\mathfrak{ad}_{B}^{*}(\wedge
A)]\rangle+\langle[B\wedge\operatorname{ad}_{A}^{*}(\wedge B)]_{*}\wedge
A\rangle),$

we can make use of Eq. (2.11) to have

$[A\wedge\mathfrak{ad}_{B}^{*}(\wedge
A)]=-\frac{1}{4}\mathfrak{ad}_{B}^{*}(\wedge[A\wedge
A]),\qquad[B\wedge\operatorname{ad}_{A}^{*}(\wedge
B)]_{*}=-\frac{1}{4}\operatorname{ad}_{A}^{*}(\wedge[B\wedge B]_{*}),$

which yields

$\displaystyle\langle\operatorname{ad}_{A}^{*}(\wedge
B)\wedge\mathfrak{ad}_{B}^{*}(\wedge A)\rangle$ $\displaystyle=$
$\displaystyle-\frac{1}{8}(\langle\mathfrak{ad}_{B}(\wedge B)\wedge[A\wedge
A]\rangle+\langle[B\wedge B]_{*}\wedge\operatorname{ad}_{A}^{*}(\wedge
A)\rangle)$ $\displaystyle=$ $\displaystyle-\frac{1}{4}\langle[B\wedge
B]_{*}\wedge[A\wedge A]\rangle.$

This allows us to write the Chern-Simons action as

$S_{\text{CS}}[{\bf A}]=\frac{1}{2}\int_{Z}\langle
B\wedge(dA+\frac{1}{2}[A\wedge A])\rangle+\frac{1}{2}\langle[B\wedge
B]_{*}\wedge A\rangle,$

where $Z=\partial M$ is the 3-dimensional boundary of $M$. If we recall the
ordinary curvature 2-form

$F=dA+\frac{1}{2}[A\wedge A],$

then we can rewrite

$S_{\text{CS}}[{\bf A}]=\frac{1}{2}\int_{Z}\langle B\wedge
F\rangle-\frac{1}{2}\langle B\wedge\mathfrak{ad}_{B}^{*}(\wedge
A)\rangle=\frac{1}{2}\int_{Z}\langle B\wedge
F\rangle+\langle\frac{1}{2}[B\wedge B]_{*}\wedge A\rangle,$

which is in fact a rewritting (up to a boundary term) of the standard BF
theory with a volume term [13].

If we vary $B$, then we achieve the flatness condition

$\delta_{A}S_{\text{CS}}=0\implies\bar{F}=dA+\frac{1}{2}[A\wedge
A]-\mathfrak{ad}_{B}^{*}(\wedge A)=0$

for the $\mathfrak{g}$-sector of the combined curvature Eq. (2.13). On the
other hand, if we vary $A$, then we obtain the flatness condition

$\delta_{B}S_{\text{CS}}=0\implies\bar{F}^{*}=dB+\frac{1}{2}[B\wedge
B]_{*}-\operatorname{ad}_{A}^{*}(\wedge B)=0$

for the $\mathfrak{g}^{*}$-sector of the combined curvature.

##### Pairings on the Drinfel’d double; non-standard Chern-Simons theory.

The form of the Chern-Simons action Eq. (2.14) depends on the choice of the
bilinear form $\langle\langle\cdot,\cdot\rangle\rangle$ on $\mathfrak{d}$.
From the natural choice Eq. (2.4) given by the evaluation pairing
$(\cdot,\cdot):\mathfrak{g}^{*}\otimes\mathfrak{g}\rightarrow\mathbb{R}$, the
above result yields an interacting BF theory. However, more general choices
are possible.

Given a bialgebra $(\mathfrak{g},\psi)$, we can form the structure of a Manin
triple on the direct sum $\mathfrak{d}\cong\mathfrak{g}\oplus\mathfrak{g}^{*}$
by endowing it with a symmetric non-degenerate invariant bilinear form
$\langle\langle\cdot,\cdot\rangle\rangle:\mathfrak{d}^{2\otimes}\rightarrow\mathbb{R}$,
with respect to which $\mathfrak{d}^{*}\cong\mathfrak{d}$ is self-dual as
bialgebras. It is clear that the natural evaluation pairing Eq. (2.4) is
merely one component of $\langle\langle\cdot,\cdot\rangle\rangle$. The other
component is given by the alternative pairing [45]

$\langle\langle X+g,X^{\prime}+g^{\prime}\rangle\rangle^{\prime}=\langle
X,X^{\prime}\rangle+\langle g,g^{\prime}\rangle$

that is "diagonal" in the direct sum $\mathfrak{g}\oplus\mathfrak{g}^{*}$. It
is invariant in the sense that

$\displaystyle\langle\operatorname{ad}_{X}X^{\prime},X^{\prime\prime}\rangle+\langle
X^{\prime},\operatorname{ad}_{X}X^{\prime\prime}\rangle=0,$
$\displaystyle\langle\mathfrak{ad}_{g}g^{\prime},g^{\prime\prime}\rangle+\langle
g^{\prime},\mathfrak{ad}_{g}g^{\prime\prime}\rangle=0$
$\displaystyle\langle\mathfrak{ad}_{g}^{*}X^{\prime},X^{\prime\prime}\rangle+\langle
X^{\prime},\mathfrak{ad}_{g}^{*}X^{\prime\prime}\rangle=0,$
$\displaystyle\langle\operatorname{ad}_{X}^{*}g^{\prime},g^{\prime\prime}\rangle+\langle
g^{\prime},\operatorname{ad}_{X}^{*}g^{\prime\prime}\rangle=0$

for each $X,X^{\prime},X^{\prime\prime}\in\mathfrak{g}$ and
$g,g^{\prime},g^{\prime\prime}\in\mathfrak{g}^{*}$. Such a pairing
$\langle\cdot,\cdot\rangle$ exists if, for instance, $\mathfrak{g}$ is
semisimple, whence $\langle\cdot,\cdot\rangle$ is the Killing form.

The Drinfel’d double $\mathfrak{d}=\mathfrak{g}\bowtie\mathfrak{g}^{*}$ has
the same structure Eq. (2.12) as previously derived when equipped with this
non-standard bilinear form $\langle\langle\cdot,\cdot\rangle\rangle^{\prime}$,
but the Chern-Simons action Eq. (2.14) now reads

$S_{\text{CS}}^{\prime}[{\bf A}]=\frac{1}{2}\int_{M}\langle\langle{\bf
F}\wedge{\bf
F}\rangle\rangle^{\prime}=\frac{1}{2}\int_{M}\langle\bar{F}^{*}\wedge\bar{F}^{*}\rangle+\langle\bar{F}\wedge\bar{F}\rangle.$

Standard results states that the Chern-Simons polynomial $\langle\langle{\bf
F}\wedge{\bf F}\rangle\rangle^{\prime}$ is exact, hence
$S^{\prime}_{\text{CS}}[{\bf A}]$ reduces to the boundary action

$S_{\text{CS}}^{\prime}[{\bf A}]=\frac{1}{2}\int_{Z}\langle\langle{\bf
A}\wedge d{\bf A}\rangle\rangle^{\prime}+\frac{1}{3}\langle\langle{\bf
A}\wedge\boldsymbol{[}{\bf A}\wedge{\bf
A}\boldsymbol{]}\rangle\rangle^{\prime}.$

From Eq. (2.13), we have

$\frac{1}{2}\boldsymbol{[}{\bf A}\wedge{\bf
A}\boldsymbol{]}=(\frac{1}{2}[A\wedge A]+\mathfrak{ad}_{B}^{*}(\wedge
A))+(\frac{1}{2}[B\wedge B]_{*}+\operatorname{ad}_{A}^{*}(\wedge B)),$

hence we see that

$\displaystyle S_{\text{CS}}^{\prime}[{\bf A}]$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{Z}\langle A\wedge(dA+\frac{1}{3}[A\wedge
A])\rangle+\langle B\wedge(dB+\frac{1}{3}[B\wedge B]_{*})\rangle$
$\displaystyle+\int_{Z}\langle A\wedge\mathfrak{ad}_{B}^{*}(\wedge
A)\rangle+\langle B\wedge\operatorname{ad}_{A}^{*}(\wedge B)\rangle$
$\displaystyle\equiv$
$\displaystyle\frac{1}{2}(S_{\text{CS}}[A]+S_{\text{CS}}[B])+\int_{Z}\langle
A\wedge\mathfrak{ad}_{B}^{*}(\wedge A)\rangle+\langle
B\wedge\operatorname{ad}_{A}^{*}(\wedge B)\rangle,$

where we have defined the usual Chern-Simons action
$S_{\text{CS}}[A],S_{\text{CS}}[B]$ in each of the individual
$\mathfrak{g},\mathfrak{g}^{*}$-sectors. The rest are interaction terms.

One may consider the Chern-Simons theory $S_{\text{CS}}^{\prime\prime}$ given
by a linear combination of the alternative pairing with the canonical one

$\langle\langle\cdot,\cdot\rangle\rangle^{\prime\prime}=\alpha\langle\langle\cdot,\cdot\rangle\rangle+\beta\langle\langle\cdot\,\cdot\rangle\rangle^{\prime},$

such that the non-degeneracy requires the real parameters $\alpha,\beta$ to
satisfy $\alpha^{2}+\beta^{2}\neq 0$. We would then have

$S_{\text{CS}}^{\prime\prime}[{\bf A}]=\alpha S_{\text{CS}}[{\bf A}]+\beta
S_{\text{CS}}^{\prime}[{\bf A}].$

Such a theory had been considered in detail in [45].

##  3 Lie algebra crossed-module and its dual

In this section, we construct a pair of dual Lie algebra crossed-modules, just
like we defined a pair of dual Lie algebras. We also discuss the motion of
adjoint and co-adjoint representations in this set up.

###  3.1 Lie algebra crossed-module.

We first review the notion of a Lie algebra crossed-module. Let
$\mathfrak{g}_{0},\mathfrak{g}_{-1}$ be a pair of Lie algebras, a Lie algebra
crossed-module is given in terms of a map

$t:\mathfrak{g}_{-1}\rightarrow\mathfrak{g}_{0},$

called the "$t$-map" or the differential, together with an action $\rhd$ of
$\mathfrak{g}_{0}$ on $\mathfrak{g}_{-1}$ such that the Pfeiffer identities

$t(X\rhd Y)=[X,tY],\qquad tY\rhd
Y^{\prime}=[Y,Y^{\prime}],\qquad\forall\leavevmode\nobreak\
X\in\mathfrak{g}_{0},\leavevmode\nobreak\ Y,Y^{\prime}\in\mathfrak{g}_{-1}$
(3.1)

are satisfied, in addition to the 2-Jacobi identities

$\displaystyle[X,[X^{\prime},X^{\prime\prime}]]+[X^{\prime},[X^{\prime\prime},X]]+[X^{\prime\prime},[X,X^{\prime}]]=0,$
$\displaystyle X\rhd(X^{\prime}\rhd Y)-X^{\prime}\rhd(X\rhd
Y)-[X,X^{\prime}]\rhd Y=0$ (3.2)

for all $X,X^{\prime},X^{\prime\prime}\in\mathfrak{g}_{0}$ and
$Y\in\mathfrak{g}_{-1}$. Notice here that, by the second Pfeiffer identity Eq.
(3.1), the Lie algebra bracket $[\cdot,\cdot]$ on $\mathfrak{g}_{-1}$ is
completely determined by the structures $t,\rhd$. A Lie algebra crossed-module
is also called a strict Lie 2-algebra.

######  Definition 3.1.

We call a Lie algebra crossed-module trivial if $t=\operatorname{id}$, and
skeletal if $t=0$ [46, 32, 28, 1].

We shall denote a Lie algebra crossed-module by
$\mathfrak{g}=(\mathfrak{g}_{-1}\xrightarrow{t}\mathfrak{g}_{0},\rhd,[\cdot,\cdot])$,
where $[\cdot,\cdot]$ is the Lie bracket on $\mathfrak{g}_{0}$. This is indeed
the minimal information one needs since, the Lie bracket of
$\mathfrak{g}_{-1}$ is determined from the second Pfeiffer identity in terms
of the action and $t$-map. The subscripts on the Lie algebras are called the
degrees of the grading in $\mathfrak{g}$. It is a very useful tool for keeping
track of the crossed-module structures.

Notice that there is a natural adjoint representation of $\mathfrak{g}$ on
itself. Since we have two pieces, $\mathfrak{g}_{0}$ and $\mathfrak{g}_{-1}$,
we can split this adjoint representation into two components, one coming from
$\mathfrak{g}_{0}$ and one from $\mathfrak{g}_{-1}$. Indeed, we have the
bracket of $\mathfrak{g}_{0}$ on itself but also the action $\rhd$ of
$\mathfrak{g}_{0}$ on $\mathfrak{g}_{-1}$. The adjoint representation of the
$\mathfrak{g}_{-1}$ sector is also specified by the crossed-module action
$\rhd$. This is denoted by

$\operatorname{ad}=(\operatorname{ad}_{0},\operatorname{ad}_{-1}):\mathfrak{g}\rightarrow\operatorname{End}\mathfrak{g},$
(3.3)

which comes in graded components
$\operatorname{ad}_{0},\operatorname{ad}_{-1}$ defined by

$\operatorname{ad}_{0}(X)=(\operatorname{ad}_{X}\equiv[X,\cdot]\,,\,\chi_{X}\equiv
X\rhd\cdot)\in\operatorname{End}(\mathfrak{g}_{0}\oplus\mathfrak{g}_{-1}),\qquad\operatorname{ad}_{-1}(Y)=\cdot\rhd
Y\in\operatorname{Hom}(\mathfrak{g}_{0},\mathfrak{g}_{-1})$

for each $X\in\mathfrak{g}_{0},Y\in\mathfrak{g}_{-1}$.

######  Example 3.1.

In general, there are two ways of associating a Lie algebra $\mathfrak{l}$ to
a Lie algebra crossed-module.

  1. 1.

The canonical embedding: trivial crossed-module: for any 1-algebra
$\mathfrak{l}$, we define the crossed-module

$\operatorname{id}_{\mathfrak{l}}=(\mathfrak{g}_{-1}\xrightarrow{\operatorname{id}}\mathfrak{g}_{0},\rhd,[\cdot,\cdot]),\qquad\mathfrak{g}_{-1}=\mathfrak{g}_{0}=\mathfrak{l}$

whose $t$-map is the identity. The Pfeiffer identities Eq. (3.1) force the
action $\rhd=[\cdot,\cdot]$ to be the adjoint action. We call the association
$\mathfrak{l}\mapsto\operatorname{id}_{\mathfrak{l}}$ the canonical embedding,
and in fact embeds the category of Lie algebras into the category of Lie
algebra crossed-modules [1, 47].

  2. 2.

The suspension embedding: skeletal crossed-module: given the semidirect
product $V\rtimes\mathfrak{u}$, where $\mathfrak{u}$ is a Lie algebra acting
on the (abelian) Lie algebra $V$, we can associate a skeletal crossed-module
structure to $V\rtimes\mathfrak{u}$,

$\mathfrak{g}^{0}=(\mathfrak{g}_{-1}\xrightarrow{0}\mathfrak{g}_{0},\rhd,[\cdot,\cdot]),\qquad\mathfrak{g}_{-1}=V,\leavevmode\nobreak\
\mathfrak{g}_{0}=\mathfrak{u}$

to it. The Pfeiffer identities Eq. (3.1) enforce $\mathfrak{g}_{-1}=V$ to be
Abelian (ie. a vector space), but otherwise imposes no constraints on the
action $\rhd$. Provided $V$ is Abelian, the crossed-module structure of
$\mathfrak{g}^{0}$ coincides with the Lie bracket of $V\rtimes\mathfrak{u}$

$[X+Y,X^{\prime}+Y^{\prime}]=[X,X]+X\rhd Y^{\prime}-X^{\prime}\rhd Y,\qquad
X,X^{\prime}\in\mathfrak{g},\leavevmode\nobreak\ Y,Y^{\prime}\in V.$ (3.4)

Aside from the grading (ie. degree information) inherent in the crossed-module
$\mathfrak{g}^{0}$, there is no distinction between the algebraic structures
of $\mathfrak{g}^{0}$ and $V\rtimes\mathfrak{u}$. We call the association
$V\rtimes\mathfrak{u}\mapsto\mathfrak{g}^{0}$ the suspension embedding.

###  3.2 Dual crossed-module and Lie bialgebra crossed-modules.

In the following, we shall also need the notion of a dual Lie algebra crossed-
module. As we mentioned already, we only need to dualize the minimal data
encoding the crossed-module structure.

A crossed-module is defined in terms of the Lie algebra $\mathfrak{g}_{0}$,
the space $\mathfrak{g}_{1}$, the action of $\mathfrak{g}_{0}$ on
$\mathfrak{g}_{-1}$ and the $t$-map. The Lie algebra structure is inherited
from the $t$-map and the action. Canonically, the duality relation would be
between opposite grading. This means that if $\mathfrak{g}_{0}$ (resp.
$\mathfrak{g}_{-1}$) has degree 0 (resp. $-1$), then $\mathfrak{g}_{0}^{*}$
will have degree $-1$ (resp. $0$).

Hence the dual crossed-module would be characterized by the Lie algebra
$\mathfrak{g}^{*}_{-1}$ of degree 0, the (vector) space $\mathfrak{g}_{0}^{*}$
which would become a Lie algebra if we specify the action $\rhd^{*}$ and the
dual $t$-map. This means that to define the dual crossed-module, we need to
define a cocycle on $\mathfrak{g}_{-1}$ to define the degree 0 Lie algebra
$\mathfrak{g}^{*}_{-1}$. We need a cocycle which would give the action
$\rhd^{*}$. Finally, dualizing the $t$-map will provide the right $t$-map
between $\mathfrak{g}_{0}^{*}$ and $\mathfrak{g}_{-1}$. The Pfeiffer
identities for the dual structures are then translated in terms of the cocycle
properties.

More explicitly, the dual structures are induced by a pair of Lie algebra
cochains

$\delta_{-1}:\mathfrak{g}_{-1}\rightarrow\mathfrak{g}_{-1}\wedge\mathfrak{g}_{-1},\qquad\delta_{0}:\mathfrak{g}_{0}\rightarrow(\mathfrak{g}_{-1}\otimes\mathfrak{g}_{0})\oplus(\mathfrak{g}_{0}\otimes\mathfrak{g}_{-1})$
(3.5)

such that we induce the crossed-module structures $[\cdot,\cdot]_{*},\rhd^{*}$
in the following way

$[f,f^{\prime}]_{*}(Y)=(f\wedge
f^{\prime})(\delta_{-1}Y),\qquad(f\rhd^{*}g)(X)=(f\wedge g)(\delta_{0}X)$

for each $X\in\mathfrak{g}_{0},Y\in\mathfrak{g}_{-1}$ and
$g\in\mathfrak{g}_{0}^{*},f,f^{\prime}\in\mathfrak{g}_{-1}^{*}$ in the dual.
Here, duality is once again taken with respect to the canonical evaluation
pairing

$(f+g,X+Y)=g(X)+f(Y),$

which induces the natural invariant non-degenerate symmetric bilinear form [1]

$\langle\langle(X+f)+(Y+g),(X^{\prime}+f^{\prime})+(Y^{\prime}+g^{\prime})\rangle\rangle=g(X^{\prime})+g^{\prime}(X)+f(Y^{\prime})+f^{\prime}(Y)$
(3.6)

on the direct sum
$(\mathfrak{g}_{0}\oplus\mathfrak{g}_{-1}^{*})\oplus(\mathfrak{g}_{-1}\oplus\mathfrak{g}_{0}^{*})$
as a vector space, where we collected together the terms of the same degree.

We now introduce a map
$\tilde{t}:\mathfrak{g}_{0}^{*}\rightarrow\mathfrak{g}_{-1}^{*}$ such that the
dual analogue of the Pfeiffer identities Eq. (3.1)

$\tilde{t}(f\rhd^{*}g)=[f,\tilde{t}g]_{*},\qquad\tilde{t}g\rhd^{*}g^{\prime}=[g,g^{\prime}]_{*},\qquad\forall\leavevmode\nobreak\
f\in\mathfrak{g}_{-1}^{*},\leavevmode\nobreak\
g,g^{\prime}\in\mathfrak{g}_{0}^{*}$

are satisfied. When written in terms of the 2-cochains
$(\delta_{-1},\delta_{0})$, they are equivalent to222To see this, we for
instance evaluate $\delta_{0}\tilde{t}^{*}$ to yield $(f\wedge
g)(\delta_{0}\tilde{t}^{*}Y)=(f\rhd^{*}g)(\tilde{t}^{*}(Y))=(\tilde{t}(f\rhd^{*}g))(Y)$,
while $(f\wedge g)((\tilde{t}^{*}\otimes
1+1\otimes\tilde{t}^{*})\delta_{-1}(Y))=(f\wedge\tilde{t}g)(\delta_{-1}Y)=[f,\tilde{t}g]_{*}(Y)$.

$\delta_{0}\tilde{t}^{*}=(\tilde{t}^{*}\otimes
1+1\otimes\tilde{t}^{*})\delta_{-1},\qquad(\tilde{t}^{*}\otimes
1-1\otimes\tilde{t}^{*})\delta_{0}=0,$ (3.7)

where $\tilde{t}^{*}:\mathfrak{g}_{-1}\rightarrow\mathfrak{g}_{0}$ is the dual
map of $\tilde{t}$. Notice that, once again, the dual bracket
$[\cdot,\cdot]_{*}$ on $\mathfrak{g}_{0}^{*}$ is determined completely by the
structures $\tilde{t},\rhd^{*}$ from the second Pfeiffer identity.

Now in addition to the Pfeiffer identities, we must enforce the analogues of
the 2-Jacobi identities Eq. (3.2). These read

$\displaystyle[f,[f^{\prime},f^{\prime\prime}]_{*}]_{*}+[f^{\prime},[f^{\prime\prime},f]_{*}]_{*}+[f^{\prime\prime},[f,f^{\prime}]_{*}]_{*}=0,$
$\displaystyle
f\rhd^{*}(f^{\prime}\rhd^{*}g)-f^{\prime}\rhd^{*}(f\rhd^{*}g)-[f,f^{\prime}]_{*}\rhd^{*}g=0,$
(3.8)

where
$f,f^{\prime},f^{\prime\prime}\in\mathfrak{g}_{-1}^{*},g\in\mathfrak{g}_{0}^{*}$,
which is equivalent to the equivariance of the structures
$\rhd^{*},[\cdot,\cdot]_{*}$ under the coadjoint representation333Notice the
action $\rhd^{*}$ in the dual crossed-module $\mathfrak{g}^{*}[1]$ is distinct
from the coaction $\chi^{*}$ of $\mathfrak{g}_{0}$ on $\mathfrak{g}^{*}_{-1}$.
$\operatorname{ad}^{*}=(\operatorname{ad}^{*}_{0},\operatorname{ad}_{-1}^{*})$
obtained by dualizing Eq. (3.3):

$\displaystyle\operatorname{ad}_{X}^{*}(f\rhd^{*}g)$ $\displaystyle=$
$\displaystyle(\chi_{X}^{*}f)\rhd^{*}g-f\rhd^{*}(\operatorname{ad}_{X}^{*}g),$
$\displaystyle\chi_{X}^{*}[f,f^{\prime}]*$ $\displaystyle=$
$\displaystyle[\chi_{X}^{*}f,f^{\prime}]+[f,\chi_{X}^{*}f^{\prime}],$
$\displaystyle\operatorname{ad}_{-1}^{*}(Y)[f,f^{\prime}]$ $\displaystyle=$
$\displaystyle
f\rhd^{*}\operatorname{ad}_{-1}^{*}(Y)(f^{\prime})-f^{\prime}\rhd^{*}\operatorname{ad}_{-1}^{*}(Y)(f).$

If written in terms of the 2-cochain $(\delta_{-1},\delta_{0})$, we obtain
explicitly the compatibility relations

$\displaystyle\delta_{0}([X,X^{\prime}])$ $\displaystyle=$
$\displaystyle(X\rhd\otimes
1+1\otimes\operatorname{ad}_{X})\delta_{0}(X^{\prime})-(X^{\prime}\rhd\otimes
1+1\otimes\operatorname{ad}_{X^{\prime}})\delta_{0}(X),$
$\displaystyle\delta_{-1}(X\rhd Y)$ $\displaystyle=$
$\displaystyle(X\rhd\otimes 1+1\otimes X\rhd)\delta_{-1}(Y)+\delta_{0}(X)(\rhd
Y\otimes 1+1\otimes\rhd Y).$ (3.9)

If Eqs. (3.7) and (3.9) are satisfied, then
$\mathfrak{g}^{*}[1]=(\mathfrak{g}_{0}^{*}\xrightarrow{\tilde{t}}\mathfrak{g}_{-1}^{*},[\cdot,\cdot]_{*},\rhd^{*})$
is a bona fide crossed-module444In keeping with the mathematical literature,
we have kept the notation ”$[1]$”, which indicates a degree-shift in the
grading.. We call such a 2-cochain $(\delta_{-1},\delta_{0})$ a 2-cocycle, and
$(\mathfrak{g};\delta_{-1},\delta_{0})$ a Lie bialgebra crossed-module.

######  Remark 3.1.

Similar to bialgebras, the bialgebra crossed-module structure is symmetric
under dualization: if $\mathfrak{g}$ is a Lie bialgebra crossed-module, then
so is its dual $\mathfrak{g}^{*}[1]$, which is equipped with the 2-cocycle

$\delta_{-1}^{*}:\mathfrak{g}_{0}^{*}\rightarrow\mathfrak{g}_{0}^{*}\wedge\mathfrak{g}_{0}^{*},\qquad\delta_{0}^{*}:\mathfrak{g}_{-1}^{*}\rightarrow(\mathfrak{g}_{-1}^{*}\otimes\mathfrak{g}_{0}^{*})\oplus(\mathfrak{g}_{0}^{*}\otimes\mathfrak{g}_{-1}^{*})$

dual to the crossed-module structure $([\cdot,\cdot],\rhd)$ of $\mathfrak{g}$,
satisfying analogues of Eqs. (3.7) and (3.9). However, this means that
$\tilde{t}^{*}$ must be related to the original crossed-module map $t$; in
fact, we must have $\tilde{t}=t^{*}$ in a bialgebra crossed-module [32, 1].
The situation is summarized below:

2-Cocyclecomponents… are dual to
…ActionBracket${\mathfrak{g}:}$${\delta_{-1}}$${\delta_{0}}$${\rhd}$${{[\cdot,\cdot]}}$${{\mathfrak{g}^{*}[1]:}}$${\delta_{-1}^{*}}$${\delta_{0}^{*}}$${\rhd^{*}}$${{[\cdot,\cdot]_{*}}}$$\scriptstyle{\tilde{t}=t^{*}}$

######  Example 3.2.

Consider the examples given in Example 3.1.

  1. 1.

The canonical bicrossed-module: It is clear that any Lie 2-bialgebra structure
on $\operatorname{id}_{\mathfrak{l}}$ collapses to just the data on the Lie
algebra $\mathfrak{l}$: indeed, Eq. (3.7) implies that the 2-cocycle
$(\delta_{-1},\delta_{0})$ is merely two copies of the usual Lie algebra
1-cocycle $\psi\in Z^{1}(\mathfrak{l},\mathfrak{l}^{2\wedge})$.

Due to the relation $\tilde{t}=t^{*}=\operatorname{id}$, the dual 2-algebra
$\operatorname{id}_{\mathfrak{l}}^{*}[1]$ is also trivial, whose crossed-
module structure is determined by the 2-cocycle
$(\delta_{-1},\delta_{0})=(\psi,\psi)$. Moreover, the coherence condition Eq.
(3.7) implies that the dual bracket $[\cdot,\cdot]_{*}$ induced by $\psi$
constitutes the crossed-module structures in
$\operatorname{id}_{\mathfrak{l}}^{*}[1]$.

As 2-bialgebras, we thus have
$\operatorname{id}_{\mathfrak{l}}^{*}[1]=\operatorname{id}_{\mathfrak{l}^{*}}$;
namely duality commutes with the canonical embedding. We call
$(\operatorname{id}_{\mathfrak{l}};\psi,\psi)$ the canonical bialgebra
crossed-module associated to the bialgebra $(\mathfrak{l};\psi)$.

  2. 2.

The suspension bialgebra crossed-module: Given the semidirect product
$V\rtimes\mathfrak{u}$, taking the dual in general "flips" the action
$(V\rtimes\mathfrak{u})^{*}=\mathfrak{u}^{*}\rtimes V^{*}$. Take a 1-cocycle
$\psi\in Z^{1}(V\rtimes\mathfrak{u},(V\rtimes\mathfrak{u})^{2\wedge})$ on the
semidirect product such that

$\displaystyle((f+g)\wedge(f^{\prime}+g^{\prime}))(\psi(X+Y))$
$\displaystyle=$ $\displaystyle[f+g,f^{\prime}+g^{\prime}]_{*}(X+Y)$
$\displaystyle=$
$\displaystyle[f,f^{\prime}]_{*}(Y)+[g,g^{\prime}]_{*}(X)+f\rhd^{*}g^{\prime}(X)-f^{\prime}\rhd^{*}g(X)$

for each $f,f^{\prime}\in V^{*},g,g^{\prime}\in\mathfrak{u}^{*}$ and
$X\in\mathfrak{u},Y\in V$, we see that components of the 2-cochain
$(\delta_{-1},\delta_{0})$ in fact make an appearance:

$[f,f^{\prime}](Y)=(f\wedge
f^{\prime})(\delta_{-1}Y),\qquad(f\rhd^{*}g)(X)=(f\wedge g)(\delta_{0}X).$

However, the term involving $[g,g^{\prime}]_{*}$ is missing, which suggests
that $\mathfrak{u}^{*}$ must be Abelian. This is consistent with the fact that
the $t$-map $\tilde{t}=t^{*}=0$ on the dual crossed-module
$(\mathfrak{g}^{0})^{*}[1]$ is also trivial. When this is the case, Eq. (3.7)
becomes vacuous while Eq. (3.9) is equivalent to the cocycle condition for
$\psi$. As a consequence, we have the following proposition.

######  Proposition 3.1.

If the 1-cocycle $\psi|_{\mathfrak{u}}$ restricted to $\mathfrak{u}\subset
V\rtimes\mathfrak{u}$ has trivial image in $\mathfrak{u}^{2\wedge}$ (hence
$\mathfrak{u}^{*}$ is Abelian), then $\psi=\delta_{-1}+\delta_{0}$ determines
a 2-cocycle $(\delta_{-1},\delta_{0})$ for its skeletal suspension
$\mathfrak{g}^{0}$.

This implies that $(\mathfrak{g}^{*})^{0}=(\mathfrak{g}^{0})^{*}[1]$; namely
the duality commutes with the suspension embedding, provided the hypothesis of
the proposition is true. We call $(\mathfrak{g}^{0};\delta_{-1},\delta_{0})$
the suspension bialgebra crossed-module associated to the bialgebra
$(V\rtimes\mathfrak{u};\psi)$.

Similar to the original crossed-module, there is also a natural adjoint
representation of $\mathfrak{g}^{*}[1]$ on itself. This is denoted

$\mathfrak{ad}=(\mathfrak{ad}_{0},\mathfrak{ad}_{-1}):\mathfrak{g}^{*}[1]\rightarrow\operatorname{End}\mathfrak{g}^{*}[1].$
(3.10)

It also comes in graded components given by

$\mathfrak{ad}_{0}(f)=([f,\cdot]_{*}\equiv\mathfrak{ad}_{f}\,,\,f\rhd\cdot\equiv\eta_{f})\in\operatorname{End}(\mathfrak{g}_{-1}^{*}\oplus\mathfrak{g}_{0}^{*}),\qquad\mathfrak{ad}_{-1}(g)=\cdot\rhd^{*}g\in\operatorname{Hom}(\mathfrak{g}_{-1}^{*},\mathfrak{g}_{0}^{*}),$

where $f\in\mathfrak{g}_{-1}^{*},g\in\mathfrak{g}_{0}^{*}$.

###  3.3 Coadjoint representations

By dualizing the adjoint representations Eqs. (3.3) and (3.10), we obtain the
coadjoint representations

$\displaystyle(\operatorname{ad}^{*}_{0},\operatorname{ad}^{*}_{-1}):\mathfrak{g}\rightarrow\operatorname{End}\mathfrak{g}^{*}[1],$
$\displaystyle(\mathfrak{ad}_{0}^{*},\mathfrak{ad}_{-1}^{*}):\mathfrak{g}^{*}[1]\rightarrow\operatorname{End}\mathfrak{g},$
$\displaystyle\operatorname{ad}_{0}^{*}=(\operatorname{ad}^{*},\chi^{*}):\mathfrak{g}_{0}\rightarrow\operatorname{End}(\mathfrak{g}_{0}^{*}\oplus\mathfrak{g}_{-1}^{*}),$
$\displaystyle\mathfrak{ad}_{0}^{*}=(\mathfrak{ad}^{*},\eta^{*}):\mathfrak{g}_{-1}^{*}\rightarrow\operatorname{End}(\mathfrak{g}_{-1}\oplus\mathfrak{g}_{0}),$
$\displaystyle\operatorname{ad}_{-1}^{*}\equiv\Delta:\mathfrak{g}_{-1}\rightarrow\operatorname{Hom}(\mathfrak{g}_{-1}^{*},\mathfrak{g}_{0}^{*}),$
$\displaystyle\mathfrak{ad}_{-1}^{*}\equiv\tilde{\Delta}:\mathfrak{g}_{0}^{*}\rightarrow\operatorname{Hom}(\mathfrak{g}_{0},\mathfrak{g}_{-1}).$
(3.11)

Explicitly for each $X,X^{\prime}\in\mathfrak{g}_{0},Y\in\mathfrak{g}_{-1}$
and $g,g^{\prime}\in\mathfrak{g}_{0}^{*},f\in\mathfrak{g}_{-1}^{*}$, they are
defined in graded components by

$\displaystyle(\operatorname{ad}^{*}_{X}g)(X^{\prime})=-g([X,X^{\prime}]),$
$\displaystyle f^{\prime}(\mathfrak{ad}^{*}_{f}Y)=-[f,f^{\prime}]_{*}(Y),$
$\displaystyle(\chi_{X}^{*}f)(Y)=-f(X\rhd Y),$ $\displaystyle
g(\eta_{f}^{*}X)=-(f\rhd^{*}g)(X),$ $\displaystyle(\Delta_{Y}(f))(X)=-f(X\rhd
Y),$ $\displaystyle f(\tilde{\Delta}_{g}(X))=-(f\rhd^{*}g)(X).$

It can then be seen that the canonical evaluation pairing
$(f+g,X+Y)=g(X)+f(Y)$ — and hence its induced pairing Eq. (3.6) — is by
definition invariant under the coadjoint representations Eq. (3.11). The first
Pfeiffer identities
$t\chi=\operatorname{ad}t,\tilde{t}\eta=\mathfrak{ad}\tilde{t}$ then lead to

$\displaystyle\chi^{*}_{X}t^{*}=t^{*}\operatorname{ad}^{*}_{X},$
$\displaystyle\eta^{*}_{f}\tilde{t}^{*}=\tilde{t}^{*}\mathfrak{ad}^{*}_{f},$
(3.12)
$\displaystyle\Delta_{Y}\circ\tilde{t}=\operatorname{ad}_{\tilde{t}^{*}Y}^{*},$
$\displaystyle\tilde{\Delta}_{g}\circ t=\mathfrak{ad}_{t^{*}g}^{*}.$ (3.13)

If $\operatorname{ad}^{*},\mathfrak{ad}^{*}$ satisfy Eqs. (3.12), (3.13), then
Eq. (3.11) define strict coadjoint representations of $\mathfrak{g}$ and
$\mathfrak{g}^{*}[1]$ on each other. These shall play a very important role in
the following.

##  4 Recovering the 2-Drinfel’d double as a 2-gauge symmetry from dual
2-gauge theories

Similarly to the 1-gauge case, we are going to define a pair of 2-gauge
theories, based on dual crossed-modules. Because they are dual, we will be
able to define an action of the dual 2-gauge on the 2-gauge data and vice-
versa. Then demanding that the order does not matter will put constraints
which are the higher gauge equivalent of the matched pair of Lie algebras.
Since now we have to deal with both 1- and 2-gauge transformations, there are
more compatibility rules to consider.

###  4.1 Dual 2-gauge theories

Given the principal 2-bundle $\mathcal{P}\rightarrow M$ with structure 2-group
$\mathcal{G}$, it is equipped a 2-connection $(A,\Sigma)$ consisting of a
$\mathfrak{g}_{0}$-valued 1-form $A$ and $\mathfrak{g}_{-1}$-valued 2-form
$\Sigma$.

The 2-gauge (infinitesimal) transformations are given by [22, 25, 48] (recall
$[\cdot,\cdot]=\operatorname{ad}$)

$\lambda:\begin{cases}A\rightarrow A^{\lambda}=A+d_{A}\lambda\\\
\Sigma\rightarrow\Sigma^{\lambda}=\Sigma+\lambda\rhd\Sigma\end{cases},\qquad
L:\begin{cases}A\rightarrow A^{L}=A+tL\\\
\Sigma\rightarrow\Sigma^{L}=\Sigma+d_{A}L+\frac{1}{2}[L\wedge L]\end{cases},$

where $d_{A}\lambda=d\lambda+[A,\lambda]$ and
$\lambda\in\Omega^{0}(M)\otimes\mathfrak{g}_{0}$ and
$L\in\Omega^{1}(M)\otimes\mathfrak{g}_{-1}$ are gauge parameters. Given the
following quantities

$\mathcal{F}=F-t\Sigma,\qquad\mathcal{G}=d_{A}\Sigma$ (4.1)

where $F=dA+\frac{1}{2}[A\wedge A]$ is the curvature of $A$, one can then
compute that they in fact transform covariantly [30, 49]:

$\lambda:\begin{cases}\mathcal{F}\rightarrow\mathcal{F}^{\lambda}=\mathcal{F}+[\lambda,\mathcal{F}]\\\
\mathcal{G}\rightarrow\mathcal{G}^{\lambda}=\mathcal{G}+\lambda\rhd\mathcal{G}\end{cases},\qquad
L:\begin{cases}\mathcal{F}\rightarrow\mathcal{F}^{L}=\mathcal{F}\\\
\mathcal{G}\rightarrow\mathcal{G}^{L}=\mathcal{G}+\mathcal{F}\wedge^{\rhd}L\end{cases}.$
(4.2)

The quantity $\mathcal{F}$ is known as fake-flatness, while $\mathcal{G}$ is
the 2-curvature.

##### Dual 2-gauge structure.

Let us consider now a dual principal 2-bundle $\mathcal{P}^{*}\rightarrow X$,
which is equipped with a dual 2-connection $(C,B)$ valued in the strict
2-algebra $\mathfrak{g}^{*}[1]$. We thus have the corresponding 2-gauge
transformations (recall $[\cdot,\cdot]_{*}=\mathfrak{ad}$)

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

where $d_{C}\tilde{\lambda}=d\tilde{\lambda}+[C,\tilde{\lambda}]_{*}$ and
$\tilde{\lambda}\in\Omega^{0}(M)\otimes\mathfrak{g}_{-1}^{*}$ and
$\tilde{L}\in\Omega^{1}(M)\otimes\mathfrak{g}_{0}^{*}$ are gauge parameters.

The fake-flatness and 2-curvature quantities are given by

$\tilde{\mathcal{F}}=\tilde{F}-\tilde{t}B,\qquad\tilde{\mathcal{G}}=d_{C}B,$
(4.3)

where $\tilde{F}=d_{C}C=dC+\frac{1}{2}[C\wedge C]_{*}$ is the curvature of
$C$. One may compute that

$\tilde{\lambda}:\begin{cases}\tilde{\mathcal{F}}\rightarrow\tilde{\mathcal{F}}^{\tilde{\lambda}}=\tilde{\mathcal{F}}+[\tilde{\lambda},\mathcal{F}]_{*}\\\
\tilde{\mathcal{G}}\rightarrow\tilde{\mathcal{G}}^{\tilde{\lambda}}=\tilde{\mathcal{G}}+\tilde{\lambda}\rhd^{*}\mathcal{G}\end{cases},\qquad\tilde{L}:\begin{cases}\tilde{\mathcal{F}}\rightarrow\tilde{\mathcal{F}}^{\tilde{L}}=\tilde{\mathcal{F}}\\\
\tilde{\mathcal{G}}\rightarrow\tilde{\mathcal{G}}^{\tilde{L}}=\tilde{\mathcal{G}}+\tilde{\mathcal{F}}\wedge^{\rhd^{*}}\tilde{L}\end{cases},$
(4.4)

form which we see that the quantities
$\tilde{\mathcal{F}},\tilde{\mathcal{G}}$ are indeed covariant.

##### The coadjoint action and back-action.

Now recall that the crossed-module $\mathfrak{g}$ and its dual
$\mathfrak{g}^{*}[1]$ define natural representations on each other given in
Eq. (3.11). With these, the 2-connections $(A,\Sigma)$ shall transform under
the dual gauge parameters $\tilde{\lambda},\tilde{L}$, as follows:

$\displaystyle\lambda:\begin{cases}B\rightarrow
B^{\lambda}=B+\operatorname{ad}_{\lambda}^{*}B\\\ C\rightarrow
C^{\lambda}=C+\chi_{\lambda}^{*}C\end{cases},$ $\displaystyle
L:\begin{cases}B\rightarrow B^{L}=B+\Delta(C\wedge L)\\\ C\rightarrow
C^{L}=C\end{cases},$ (4.5)

and similarly for the dual 2-connections $(C,B)$ as

$\displaystyle\tilde{\lambda}:\begin{cases}\Sigma\rightarrow\Sigma^{\tilde{\lambda}}=\Sigma+\mathfrak{ad}^{*}_{\tilde{\lambda}}\Sigma\\\
A\rightarrow A^{\tilde{\lambda}}=A+\eta_{\tilde{\lambda}}^{*}A\end{cases},$
$\displaystyle\tilde{L}:\begin{cases}\Sigma\rightarrow\Sigma^{\tilde{L}}=\Sigma+\tilde{\Delta}(A\wedge\tilde{L})\\\
A\rightarrow A^{\tilde{L}}=A\end{cases}.$ (4.6)

These transformations have also appeared previously in the study of 2-BF
theories [30, 49], which are also sometimes known as "$BFCG$-theory" [50].

As in Section 2, we posit that the gauge transformations given in Eqs. (4.2),
(4.4), (4.5), (4.6) should allow us to derive conditions such that the direct
sum $\mathfrak{g}\oplus\mathfrak{g}^{*}[1]$ forms the 2-Manin triple

$\mathfrak{d}=\mathfrak{g}\leavevmode\nobreak\
_{\operatorname{ad}^{*}}\bowtie_{\mathfrak{ad}^{*}}\mathfrak{g}^{*}[1].$

In fact, we shall recover Eqs. (3.12), (3.13) as well, purely from the
considerations of the 2-gauge symmetries present in each of the 2-algebra
sectors.

###  4.2 The 2-Drinfel’d double as 2-gauge symmetry

We follow the same strategy as in Section 2.2, and use Eqs. (4.2), (4.4),
(4.5), (4.6) to perform the gauge transformations.

We begin by defining the combined 2-gauge parameter by

$\boldsymbol{\lambda}=\lambda+\tilde{\lambda}\in\Omega^{0}(M)\otimes(\mathfrak{g}_{0}\oplus\mathfrak{g}_{-1}^{*}),\qquad{\bf
L}=L+\tilde{L}\in\Omega^{1}(M)\otimes(\mathfrak{g}_{-1}\oplus\mathfrak{g}_{0}^{*}).$

We grouped them so that they have the same degree, as forms but also in terms
of the grading.

We are going to consider the different combinations we can make in different
order and impose that the order does not matter. This will impose a set of
constraints which will be equivalent to the definition of the 2-Drinfel’d
double.

####  4.2.1 Combined 1-gauge transformations

As in Section 2.2, the 1-connection $A$, say, transforms not only under
$\lambda$ but also $\tilde{\lambda}$, and similarly for $C$. By considering
pure gauges $A=d\lambda,C=d\tilde{\lambda}$, the induced gauge transformations
are

$\lambda\rightarrow\lambda+\eta_{\tilde{\lambda}}^{*}\lambda+o(\lambda
d\tilde{\lambda}),\qquad\tilde{\lambda}\rightarrow\tilde{\lambda}+\chi_{\lambda}^{*}\tilde{\lambda}+o(\tilde{\lambda}d\lambda),$

modulo variations in the 1-gauge parameters.

##### 1-gauge transformations of the 1-form connections.

We can follow an analogous treatment as previously, but with
$\mathfrak{ad}^{*},\operatorname{ad}^{*}$ replaced with $\eta^{*},\chi^{*}$ in
accordance with Eqs. (4.2), (4.4), (4.5), (4.6). Running the same arguments as
in Section 2.2, we introduce here $C$-dependent gauge transformations on $A$
and vice versa, in order to properly antisymmetrize the compatibility
conditions.

$A\rightarrow A+d_{A}\lambda-\eta_{C}^{*}\lambda,\qquad C\rightarrow
C+d_{C}\tilde{\lambda}-\chi_{A}^{*}\tilde{\lambda}.$

This leads us to consider the combined 1-connection

${\bf
A}=A+C\in\Omega^{1}(M)\otimes(\mathfrak{g}_{0}\bowtie\mathfrak{g}_{-1}^{*})$

valued in the degree-$0$ components of the direct sum
$\mathfrak{g}\bowtie\mathfrak{g}^{*}[1]$.

Moreover, this antisymmetrization procedure introduces the modified covariant
derivatives

$D_{\bf A}^{(0)}\lambda=d_{A}\lambda-\eta_{C}^{*}\lambda,\qquad D_{\bf
A}^{(0)}\tilde{\lambda}=d_{C}\tilde{\lambda}-\chi_{A}^{*}\tilde{\lambda},$
(4.7)

for the combined 1-connection ${\bf A}=A+C$, in analogy with Eq. (2.9). We
therefore have the analogous result:

######  Lemma 4.1.

The gauge transformation (including antisymmetrized terms)

${\bf A}\rightarrow{\bf A}^{\boldsymbol{\lambda}}={\bf A}+(D_{\bf
A}^{(0)}\lambda+\eta_{\tilde{\lambda}}^{*}A)+(D_{\bf
A}^{(0)}\tilde{\lambda}\chi_{\lambda}^{*}C)$

is unambiguous (independent of the order at which $\lambda,\tilde{\lambda}$
acts) modulo terms of order
$o(\lambda^{2})+o(\tilde{\lambda}^{2})+o(\tilde{\lambda}d\lambda)+o(\lambda
d\tilde{\lambda})$ iff the first compatibility conditions

$\displaystyle\eta^{*}_{\tilde{\lambda}}[A,\lambda]$ $\displaystyle=$
$\displaystyle[\eta^{*}_{\tilde{\lambda}}A,\lambda]+[A,\eta^{*}_{\tilde{\lambda}}\lambda]-\eta_{\chi_{A}^{*}\tilde{\lambda}}^{*}\lambda+\eta_{\chi_{\lambda}^{*}\tilde{\lambda}}^{*}A,$
$\displaystyle\chi^{*}_{\lambda}[C,\tilde{\lambda}]_{*}$ $\displaystyle=$
$\displaystyle[\chi_{\lambda}^{*}C,\tilde{\lambda}]_{*}+[C,\chi_{\lambda}^{*}\tilde{\lambda}]_{*}-\chi^{*}_{\eta_{C}^{*}\lambda}\tilde{\lambda}+\chi^{*}_{\eta_{\tilde{\lambda}}^{*}\lambda}C$
(4.8)

are satisfied.

##### 1-gauge transformations of the 2-form connections.

Less trivially, the 2-connection $\boldsymbol{\Sigma}$ also transforms under
the 1-gauge $\lambda+\tilde{\lambda}$. Consider first $\Sigma$. To study how
it transforms, we compute the action of consecutive transformations in
$\lambda$ and $\tilde{\lambda}$, with Eqs. (4.2), (4.6)

$\displaystyle\overrightarrow{\boldsymbol{\lambda}}$ $\displaystyle:$
$\displaystyle\Sigma\xrightarrow{\lambda}\Sigma+\lambda\rhd\Sigma$
$\displaystyle\xrightarrow{\tilde{\lambda}}\Sigma+\mathfrak{ad}_{\tilde{\lambda}}^{*}\Sigma+\lambda\rhd\Sigma$
$\displaystyle\qquad+\lambda\rhd\mathfrak{ad}_{\tilde{\lambda}}^{*}\Sigma+(\eta_{\tilde{\lambda}}^{*}\lambda)\rhd\Sigma+\underbrace{(\eta_{\tilde{\lambda}}^{*}\lambda)\rhd(\mathfrak{ad}_{\tilde{\lambda}}^{*}\Sigma)}_{\sim
o(\tilde{\lambda}^{2})},$

while the other order gives

$\displaystyle\overleftarrow{\boldsymbol{\lambda}}$ $\displaystyle:$
$\displaystyle\Sigma\xrightarrow{\tilde{\lambda}}\Sigma+\mathfrak{ad}_{\tilde{\lambda}}^{*}\Sigma$
$\displaystyle\xrightarrow{\lambda}\Sigma+\lambda\rhd\Sigma+\mathfrak{ad}_{\tilde{\lambda}}^{*}\Sigma$
$\displaystyle\qquad\mathfrak{ad}_{\tilde{\lambda}}^{*}(\lambda\rhd\Sigma)+\mathfrak{ad}_{\chi_{\lambda}^{*}\tilde{\lambda}}^{*}\Sigma+\underbrace{\mathfrak{ad}_{\chi_{\lambda}^{*}\tilde{\lambda}}^{*}(\lambda\rhd\Sigma)}_{\sim
o(\lambda^{2})}.$

The difference between them, modulo quadratic terms in the small gauge
parameters $\lambda,\tilde{\lambda}$, then reads

$\mathfrak{ad}_{\tilde{\lambda}}^{*}(\lambda\rhd\Sigma)+\mathfrak{ad}_{\chi_{\lambda}^{*}\tilde{\lambda}}^{*}\Sigma-\lambda\rhd\mathfrak{ad}_{\tilde{\lambda}}^{*}\Sigma-(\eta_{\tilde{\lambda}}^{*}\lambda)\rhd\Sigma,$
(4.9)

and we must force it to vanish.

This argument can be repeated for the dual sector, with the replacements

$\Sigma\leftrightsquigarrow
B,\qquad\chi^{*}\leftrightsquigarrow\eta^{*},\qquad\mathfrak{ad}^{*}\leftrightsquigarrow\operatorname{ad}^{*},\qquad\Delta\leftrightsquigarrow\tilde{\Delta}.$

More precisely, we see from Eqs. (4.4), (4.5) that $B$ transforms as

$\displaystyle\overrightarrow{\boldsymbol{\lambda}}$ $\displaystyle:$
$\displaystyle\Sigma\xrightarrow{\lambda}B+\operatorname{ad}_{\tilde{\lambda}}^{*}B$
$\displaystyle\xrightarrow{\tilde{\lambda}}B+\tilde{\lambda}\rhd^{*}B+\operatorname{ad}_{\lambda}^{*}B$
$\displaystyle\qquad\operatorname{ad}_{\lambda}^{*}(\tilde{\lambda}\rhd^{*}B)+\operatorname{ad}_{\eta_{\tilde{\lambda}}^{*}\lambda}^{*}B+\underbrace{\mathfrak{ad}_{\chi_{\lambda}^{*}\tilde{\lambda}}^{*}(\tilde{\lambda}\rhd^{*}B)}_{\sim
o(\tilde{\lambda}^{2})},$

while

$\displaystyle\overleftarrow{\boldsymbol{\lambda}}$ $\displaystyle:$
$\displaystyle B\xrightarrow{\tilde{\lambda}}B+\tilde{\lambda}\rhd^{*}B$
$\displaystyle\xrightarrow{\lambda}B+\operatorname{ad}_{\lambda}^{*}B+\tilde{\lambda}\rhd^{*}B$
$\displaystyle\qquad+\tilde{\lambda}\rhd^{*}\operatorname{ad}_{\lambda}^{*}B+(\chi_{\lambda}^{*}\tilde{\lambda})\rhd^{*}B+\underbrace{(\chi_{\lambda}^{*}\tilde{\lambda})\rhd^{*}(\operatorname{ad}_{\lambda}^{*}B)}_{\sim
o(\lambda^{2})}.$

Their difference modulo $\lambda^{2},\tilde{\lambda}^{2}$ then reads

$\operatorname{ad}_{\lambda}^{*}(\tilde{\lambda}\rhd^{*}B)+\operatorname{ad}_{\eta_{\tilde{\lambda}}^{*}\lambda}^{*}B-\tilde{\lambda}\rhd^{*}\operatorname{ad}_{\lambda}^{*}B-(\chi_{\lambda}^{*}\tilde{\lambda})\rhd^{*}B,$
(4.10)

and we also require this to vanish.

##### Antisymmetrization procedure for the 2-form connection.

Similar to the 1-gauge case, we must antisymmetrize the last term
$\mathfrak{ad}^{*}_{\chi_{\lambda}^{*}\tilde{\lambda}}\Sigma$ with respect to
$\lambda,\Sigma$ on Eq. (4.9), and similarly with respect to
$\tilde{\lambda},B$ on Eq. (4.9). As such, we must modify how the
2-connections $\Sigma,B$ transform under the 1-gauge parameters
$\lambda,\tilde{\lambda}$, respectively:

$\Sigma\rightarrow\Sigma+\lambda\rhd\Sigma-\mathfrak{ad}_{B}^{*}\lambda,\qquad
B\rightarrow
B+\tilde{\lambda}\rhd^{*}B-\operatorname{ad}_{\Sigma}^{*}\tilde{\lambda}.$
(4.11)

In contrast to the 1-group case, however, it is a bit subtle to see why such
terms are necessary.

The subtlety here is that the coadjoint action in the degree-(-1) sectors
$\mathfrak{g}_{-1},\mathfrak{g}_{0}^{*}$ are determined by the second Pfeiffer
identity, whence terms such as
$\mathfrak{ad}_{B}^{*}\lambda,\operatorname{ad}_{\Sigma}^{*}\tilde{\lambda}$
can be written in terms of the maps $\Delta,\tilde{\Delta}$ by Eq. (3.13). To
be explicit, the second Pfeiffer identity Eq. (3.1) gives

$\operatorname{ad}_{Y}Y^{\prime}=[Y,Y^{\prime}]=tY\rhd Y^{\prime},$

whence pairing against $f\in\mathfrak{g}^{*}_{-1}$ gives

$(\operatorname{ad}_{Y}^{*}f)(Y^{\prime})=f([Y^{\prime},Y])=f(tY^{\prime}\rhd
Y)=(-\Delta_{Y}(f))(tY^{\prime})$

for any $Y,Y^{\prime}\in\mathfrak{g}_{-1}$. Similarly in the dual sector, we
have

$g^{\prime}(\mathfrak{ad}_{g}^{*}X)=(\tilde{t}g^{\prime})(-\tilde{\Delta}_{g}(X))$

for each $g,g^{\prime}\in\mathfrak{g}_{0}^{*}$ and $X\in\mathfrak{g}_{0}$.
This allows us to rewrite

$\mathfrak{ad}_{B}^{*}\lambda=-\tilde{\Delta}_{B}(\lambda),\qquad\operatorname{ad}_{\Sigma}^{*}\tilde{\lambda}=-\Delta_{\Sigma}(\tilde{\lambda}).$
(4.12)

Now, if we compute from Eq. (4.11) that

$\displaystyle\overrightarrow{\boldsymbol{\lambda}}:\Sigma\rightarrow\dots-\mathfrak{ad}_{B}^{*}\lambda+\mathfrak{ad}_{\operatorname{ad}_{\Sigma}^{*}\tilde{\lambda}}^{*}\lambda,$
$\displaystyle\overleftarrow{\boldsymbol{\lambda}}:\Sigma\rightarrow\dots-\mathfrak{ad}_{B}^{*}\lambda,$
$\displaystyle\overrightarrow{\boldsymbol{\lambda}}:B\rightarrow\dots-\operatorname{ad}_{\Sigma}^{*}\tilde{\lambda},$
$\displaystyle\overleftarrow{\boldsymbol{\lambda}}:B\rightarrow\dots-\operatorname{ad}_{\Sigma}^{*}\tilde{\lambda}+\operatorname{ad}_{\mathfrak{ad}_{\Sigma}^{*}\tilde{\lambda}},$

where "$\dots$" indicate terms we have encountered already previously. Then,
by Eq. (4.12), the difference

$\mathfrak{ad}_{\operatorname{ad}_{\Sigma}^{*}\tilde{\lambda}}^{*}\lambda+\operatorname{ad}_{\mathfrak{ad}_{\Sigma}^{*}\tilde{\lambda}}=\tilde{\Delta}_{\Delta_{\Sigma}(\tilde{\lambda})}(\lambda)+\Delta_{\tilde{\Delta}_{B}(\lambda)}(\tilde{\lambda})$

in fact yields the desired antisymmetrization terms. Appending these terms to
Eqs. (4.9), (4.10) leads to the desired compatibility conditions

$\displaystyle\mathfrak{ad}_{\tilde{\lambda}}^{*}(\lambda\rhd\Sigma)$
$\displaystyle=$
$\displaystyle\lambda\rhd(\mathfrak{ad}^{*}_{\tilde{\lambda}}\Sigma)+(\eta_{\tilde{\lambda}}^{*}\lambda)\rhd\Sigma-\mathfrak{ad}^{*}_{\chi_{\lambda}^{*}\tilde{\lambda}}\Sigma+\tilde{\Delta}_{\Delta_{\Sigma}(\tilde{\lambda})}(\lambda),$
(4.13)
$\displaystyle\operatorname{ad}_{\lambda}^{*}(\tilde{\lambda}\rhd^{*}B)$
$\displaystyle=$
$\displaystyle\tilde{\lambda}\rhd^{*}(\operatorname{ad}_{\lambda}^{*}B)+(\chi_{\lambda}^{*}\tilde{\lambda})\rhd^{*}B-\operatorname{ad}^{*}_{\eta_{\tilde{\lambda}}^{*}\lambda}B+\Delta_{\tilde{\Delta}_{B}(\lambda)}(\tilde{\lambda}).$
(4.14)

The $B$-dependent 1-gauge transformation on $\Sigma$ (and vice versa) suggest
we should define a combined 2-connection

$\boldsymbol{\Sigma}=\Sigma+B\in\Omega^{2}(M)\otimes(\mathfrak{g}_{-1}\oplus\mathfrak{g}_{0}^{*})$

valued in the degree-$(-1)$ components of the direct sum
$\mathfrak{g}\oplus\mathfrak{g}^{*}[1]$.

Moreover, a modified covariant action is also introduced by the combined
2-connection $\boldsymbol{\Sigma}=\Sigma+B$ via the antisymmetrization
procedure:

$\displaystyle D_{\boldsymbol{\Sigma}}^{(0)}\lambda$ $\displaystyle=$
$\displaystyle\lambda\rhd\Sigma-\mathfrak{ad}_{B}^{*}\lambda=\operatorname{ad}_{-1}(\Sigma)(\lambda)-\mathfrak{ad}_{B}^{*}\lambda,$
$\displaystyle D_{\boldsymbol{\Sigma}}^{(0)}\tilde{\lambda}$ $\displaystyle=$
$\displaystyle\tilde{\lambda}\rhd^{*}B-\operatorname{ad}_{\Sigma}^{*}\tilde{\lambda}=\mathfrak{ad}_{-1}(B)(\tilde{\lambda})-\operatorname{ad}_{\Sigma}^{*}\tilde{\lambda}.$
(4.15)

This yields the following result:

######  Lemma 4.2.

The 1-gauge transformation (including antisymmetrized terms in Eq. (4.11))

$\boldsymbol{\Sigma}\rightarrow\boldsymbol{\Sigma}^{\boldsymbol{\lambda}}=\boldsymbol{\Sigma}+(D_{\boldsymbol{\Sigma}}^{(0)}\lambda+\mathfrak{ad}_{\tilde{\lambda}}^{*}\Sigma)+(D_{\boldsymbol{\Sigma}}^{(0)}\tilde{\lambda}+\operatorname{ad}_{\lambda}^{*}B)$

is unambiguous modulo terms of the order
$o(\lambda^{2})+o(\tilde{\lambda}^{2})+o(\tilde{\lambda}d\lambda)+o(\lambda
d\tilde{\lambda})$ iff the second compatibility conditions Eqs. (4.13), (4.14)
are satisfied.

####  4.2.2 Combined 2-gauge transformations

Let us now study the 2-gauge transformations under the 2-gauge parameter ${\bf
L}=L+\tilde{L}$. As $A,C$ do not transform respectively under $\tilde{L},L$,
we see that there is no ambiguity in ${\bf A}$ under ${\bf L}$:

${\bf A}\rightarrow{\bf A}^{\bf L}={\bf A}+tL+\tilde{t}\tilde{L}={\bf A}+T{\bf
L},$

where we introduced the total $t$-map $T=t\otimes 1+1\otimes\tilde{t}$. On the
other hand, the 2-connections $\Sigma,B$ do transform under $\tilde{L}$ and
$L$ respectively, but only as shifts by $\tilde{\Delta}_{\tilde{L}}(\wedge
A),\Delta_{L}(\wedge C)$, respectively, that do not depend on $\Sigma,B$
themselves. Therefore, there is no induced transformations of $L$ and
$\tilde{L}$ on each other, and terms of cubic order
$o(L^{2}\tilde{L})+o(\tilde{L}^{2}L)$ do not occur. Consequently, the
quadratic terms $L^{2},\tilde{L}^{2}$ do not effect our results, and we shall
neglect them in the following.

Now we consider the 2-gauge shift transformations on the 2-connections
$\Sigma,B$, and demand the difference in the different order to be zero.
Similar to before, this leads to a modification of the 2-gauge transformation,
because we must antisymmetrize the terms in these differences with respect to
$\tilde{L},A$ and $L,C$, respectively. This can be accomplished by inserting
the terms $-\operatorname{ad}_{A}^{*}\tilde{L}$ into how $B$ transforms under
$\tilde{L}$ in Eq. (4.5). Similarly, we insert the term
$-\mathfrak{ad}_{C}^{*}L$ into how $\Sigma$ transforms under $L$ in Eq. (4.2).
In other words, we introduce the 2-gauge covariant derivatives

$D^{(1)}_{\bf A}L=d_{A}L-\mathfrak{ad}_{C}^{*}(\wedge L),\qquad D^{(1)}_{\bf
A}\tilde{L}=d_{C}\tilde{L}-\operatorname{ad}_{A}^{*}(\wedge\tilde{L})$ (4.16)

of the matched pair $\mathfrak{g}_{-1}\bowtie\mathfrak{g}_{0}^{*}$ in
degree-(-1).

We can check explicitly now that the order of the transformations does not
matter provided we have some consistency conditions that we shall now derive.
We compute directly with Eqs. (4.2), (4.4) and the modified 2-gauge covariant
derivatives Eq. (4.16):

$\displaystyle\overrightarrow{\bf L}$ $\displaystyle:$
$\displaystyle\Sigma\xrightarrow{L}\Sigma+d_{A}L-\mathfrak{ad}_{C}^{*}(\wedge
L)$
$\displaystyle\xrightarrow{\tilde{L}}\Sigma+d_{A}L+\tilde{\Delta}_{\tilde{L}}(\wedge
A)-\mathfrak{ad}_{C}^{*}(\wedge L)$
$\displaystyle\qquad-\mathfrak{ad}_{\tilde{t}\tilde{L}}^{*}(\wedge L).$
$\displaystyle\overleftarrow{\bf L}$ $\displaystyle:$
$\displaystyle\Sigma\xrightarrow{\tilde{L}}\Sigma+\tilde{\Delta}_{\tilde{L}}(\wedge
A)$
$\displaystyle\xrightarrow{L}\Sigma+d_{A}L+\tilde{\Delta}_{\tilde{L}}(\wedge
A)-\mathfrak{ad}_{C}^{*}(\wedge L)$
$\displaystyle\qquad+\tilde{\Delta}_{\tilde{L}}(\wedge tL).$

Similarly, we may compute with Eqs.(4.5), (4.6) that

$\displaystyle\overleftarrow{\bf L}$ $\displaystyle:$ $\displaystyle
B\xrightarrow{\tilde{L}}B+d_{C}\tilde{L}-\operatorname{ad}_{A}^{*}(\wedge\tilde{L})$
$\displaystyle\xrightarrow{L}B+d_{C}\tilde{L}+\Delta_{L}(\wedge
C)-\operatorname{ad}_{A}^{*}(\wedge\tilde{L})$
$\displaystyle\qquad-\operatorname{ad}_{tL}^{*}(\wedge\tilde{L}).$
$\displaystyle\overrightarrow{\bf L}$ $\displaystyle:$ $\displaystyle
B\xrightarrow{L}B+\Delta_{L}(\wedge C)$
$\displaystyle\xrightarrow{\tilde{L}}B+d_{C}\tilde{L}+\Delta_{L}(\wedge
C)-\operatorname{ad}_{A}^{*}(\wedge\tilde{L})$
$\displaystyle\qquad+\Delta_{L}(\wedge\tilde{t}\tilde{L}).$

Now the difference in these transformations reads

$(\Delta_{L}(\wedge\tilde{t}\tilde{L})+\operatorname{ad}_{tL}^{*}(\wedge\tilde{L}))+(\tilde{\Delta}_{\tilde{L}}(\wedge
tL)+\mathfrak{ad}_{\tilde{t}\tilde{L}}^{*}(\wedge L)),$

from which we see that the condition $\tilde{t}=t^{*}$ implies

$(\Delta_{L}(\wedge\tilde{t}\tilde{L})-\operatorname{ad}_{\tilde{t}^{*}L}^{*}(\wedge\tilde{L}))+(\tilde{\Delta}_{\tilde{L}}(\wedge
tL)-\mathfrak{ad}_{t^{*}\tilde{L}}^{*}(\wedge L))=0,$

which vanishes by Eq. (3.13).

We therefore have the following result:

######  Lemma 4.3.

The combined gauge transformation (including antisymmetrized terms)

$\boldsymbol{\Sigma}\xrightarrow{\bf L}\boldsymbol{\Sigma}^{\bf
L}=\boldsymbol{\Sigma}+(D_{\bf A}^{(1)}L+\tilde{\Delta}_{\tilde{L}}(\wedge
A))+(D_{\bf A}^{(1)}\tilde{L}+\Delta_{L}(\wedge C))$

is unambiguous exactly iff $\tilde{t}=t^{*}$ and
$\Delta=\operatorname{ad}_{-1}^{*},\tilde{\Delta}=\mathfrak{ad}_{-1}^{*}$
satisfy Eq. (3.13).

####  4.2.3 Mixed 1- and 2-gauge transformations

Let us now study the combined gauge transformations under the mixed parameter
$(\tilde{\lambda},L)$, and dually under $(\lambda,\tilde{L})$. Toward this, we
must study how these gauge parameters interact.

First, as $C$ is invariant under $L$, so is the pure-gauge 1-connection
$C=d\tilde{\lambda}$ and hence there are no terms involving the action of $L$
on $\tilde{\lambda}$. On the other hand, a pure-gauge 2-connection
$\Sigma=d_{A}L$ admits a gauge transformation

$d_{A}L\rightarrow d_{A}L+\mathfrak{ad}_{\tilde{\lambda}}^{*}d_{A}L.$ (4.17)

By Leibniz rule, we see that we have

$dL\rightarrow
dL+\mathfrak{ad}_{\tilde{\lambda}}^{*}dL=dL+d(\mathfrak{ad}_{\tilde{\lambda}}^{*}L)-\mathfrak{ad}_{d\tilde{\lambda}}^{*}\wedge
L,$

hence this induces an action of $\tilde{\lambda}$ on $L$:

$L\rightarrow L+\mathfrak{ad}_{\tilde{\lambda}}^{*}L+o(d\tilde{\lambda}L)$

modulo terms proportional to $d\tilde{\lambda}$.

On the dual side, we similarly have no action of $\tilde{L}$ on $\lambda$, as
$A$ is invariant under $\tilde{L}$. On the other hand, we have

$d\tilde{L}\rightarrow
d\tilde{L}+\operatorname{ad}_{\lambda}^{*}d\tilde{L}=d\tilde{L}+d(\operatorname{ad}_{\lambda}^{*}\tilde{L})-\operatorname{ad}_{d\lambda}^{*}\wedge\tilde{L},$

which induces the action of $\lambda$ on $\tilde{L}$:

$\tilde{L}\rightarrow\tilde{L}+\operatorname{ad}_{\lambda}^{*}L+o(d\lambda\tilde{L})$

modulo terms of order $o(d\lambda\tilde{L})$.

##### On 1-connections.

With this acquired, we now consider how the 1-connections transform under
these mixed gauge transformations. First as $\tilde{L}$ acts trivially on $A$,
and similarly $L$ acts trivially on $C$, we see that there is in fact no
ambiguity

$A\xrightarrow{(\lambda,\tilde{L})}A+d_{A}\lambda,\qquad
C\xrightarrow{(\tilde{\lambda},L)}C+d_{C}\tilde{\lambda}.$

On the other hand, we see that we have

$\displaystyle\overrightarrow{(\tilde{\lambda},L)}$ $\displaystyle:$
$\displaystyle
A\xrightarrow{\tilde{\lambda}}A+\eta_{\tilde{\lambda}}^{*}A\xrightarrow{L}A+tL+\eta_{\tilde{\lambda}}^{*}A+\eta_{\tilde{\lambda}}^{*}tL,$
$\displaystyle\overleftarrow{(\tilde{\lambda},L)}$ $\displaystyle:$
$\displaystyle
A\xrightarrow{L}A+tL\xrightarrow{\tilde{\lambda}}A+\eta_{\tilde{\lambda}}^{*}A+tL+t\mathfrak{ad}_{\tilde{\lambda}}^{*}L,$

and similarly

$\displaystyle\overrightarrow{(\lambda,\tilde{L})}$ $\displaystyle:$
$\displaystyle
C\xrightarrow{\lambda}C+\chi_{\lambda}^{*}C\xrightarrow{\tilde{L}}C+\tilde{t}\tilde{L}+\chi_{\lambda}^{*}\tilde{t}\tilde{L},$
$\displaystyle\overleftarrow{(\lambda,\tilde{L})}$ $\displaystyle:$
$\displaystyle
C\xrightarrow{\tilde{L}}C+\tilde{t}\tilde{L}\xrightarrow{\lambda}C+\chi_{\lambda}^{*}C+\tilde{t}\tilde{L}+\tilde{t}\operatorname{ad}_{\lambda}^{*}L.$

The difference in these gauge transformations is

$(\eta_{\tilde{\lambda}}^{*}tL-t\mathfrak{ad}_{\tilde{\lambda}}^{*}L)+(\chi_{\lambda}^{*}\tilde{t}\tilde{L}-\tilde{t}\operatorname{ad}_{\lambda}^{*}\tilde{L}),$

and the condition for them to vanish — with $\tilde{t}=t^{*}$ understood — is
in fact nothing but the Pfeiffer identities

$\eta_{\tilde{\lambda}}^{*}\tilde{t}^{*}L=\tilde{t}^{*}\mathfrak{ad}_{\tilde{\lambda}}^{*}L,\qquad\chi_{\lambda}^{*}t^{*}\tilde{L}=t^{*}\operatorname{ad}_{\lambda}^{*}\tilde{L}$

in Eq. (3.12). In other words, we have the following result:

######  Lemma 4.4.

The mixed gauge transformations

$A\xrightarrow{(\tilde{\lambda},L)}A+tL+\mathfrak{ad}_{\tilde{\lambda}}^{*}A+\mathfrak{ad}_{\tilde{\lambda}}^{*}tL,\qquad
C\xrightarrow{(\lambda,\tilde{L})}C+\tilde{t}\tilde{L}+\operatorname{ad}_{\lambda}^{*}C+\operatorname{ad}_{\lambda}^{*}\tilde{t}\tilde{L}$

are unambiguous modulo the order
$o(d\lambda\tilde{L})+o(d\tilde{\lambda}L)+o(\tilde{\lambda}^{2})+o(\lambda^{2})$
iff $\tilde{t}=t^{*}$ and
$\operatorname{ad}_{0}^{*}=(\operatorname{ad}^{*},\chi^{*}),\mathfrak{ad}_{0}^{*}=(\mathfrak{ad}^{*},\eta^{*})$
satisfy Eq. (3.12).

We therefore see that Lemmas 4.3, 4.4 make Eqs. (3.11) define bona fide strict
coadjoint representations, as anticipated.

##### On 2-connections.

We now turn to the mixed gauge transformations of the 2-connections
$\Sigma,B$. Let us begin with $(\lambda,\tilde{L})$. First, we have

$\displaystyle\overrightarrow{(\lambda,\tilde{L})}$ $\displaystyle:$
$\displaystyle\Sigma\xrightarrow{\lambda}\Sigma+\lambda\rhd\Sigma\xrightarrow{\tilde{L}}\Sigma+\lambda\rhd\Sigma+\tilde{\Delta}_{\tilde{L}}(\wedge
A)$ $\displaystyle\qquad+\lambda\rhd\tilde{\Delta}_{\tilde{L}}(\wedge A),$
$\displaystyle\overleftarrow{(\lambda,\tilde{L})}$ $\displaystyle:$
$\displaystyle\Sigma\xrightarrow{\tilde{L}}\Sigma+\tilde{\Delta}_{\tilde{L}}(\wedge
A)$
$\displaystyle\xrightarrow{\lambda}\Sigma+\lambda\rhd\Sigma+\tilde{\Delta}_{\tilde{L}}(\wedge
A)$
$\displaystyle\qquad+\tilde{\Delta}_{\operatorname{ad}_{\lambda}^{*}\tilde{L}}(\wedge
A)+\tilde{\Delta}_{\tilde{L}}(\wedge
d_{A}\lambda)+\underbrace{\tilde{\Delta}_{\operatorname{ad}_{\lambda}^{*}\tilde{L}}(\wedge
d_{A}\lambda)}_{\sim o(\lambda^{2})}.$

Modulo terms of order $o(d\lambda\tilde{L})+o(\lambda^{2})$, the difference is
the quantity

$\tilde{\Delta}_{\tilde{L}}(\wedge[A,\lambda])+\tilde{\Delta}_{\operatorname{ad}_{\lambda}^{*}\tilde{L}}(\wedge
A)-\lambda\rhd\tilde{\Delta}_{\tilde{L}}(\wedge A).$

In order to force this quantity to vanish, the last two terms must be
antisymmetrized with respect to $A$ and $\lambda$. This can be accomplished by
the covariant derivative Eq. (4.16), from which we acquire the terms

$-\tilde{\Delta}_{\operatorname{ad}_{A}^{*}(\wedge\tilde{L})}(\lambda)+A\wedge^{\rhd}\tilde{\Delta}_{\tilde{L}}(\lambda),$

and achieve the compatibility condition

$\tilde{\Delta}_{\tilde{L}}(\wedge[A,\lambda])=\lambda\rhd\tilde{\Delta}_{\tilde{L}}(\wedge
A)-A\wedge^{\rhd}\tilde{\Delta}_{\tilde{L}}(\lambda)-\tilde{\Delta}_{\operatorname{ad}_{\lambda}^{*}\tilde{L}}(\wedge
A)+\tilde{\Delta}_{\operatorname{ad}_{A}^{*}(\wedge\tilde{L})}(\lambda).$
(4.18)

Next, we consider the dual transformation $(\tilde{\lambda},L)$ on the
2-connection $B$. We notice that the above argument can be run through
analogously; indeed, we have

$\displaystyle\overrightarrow{(\tilde{\lambda},L)}$ $\displaystyle:$
$\displaystyle
B\xrightarrow{\tilde{\lambda}}B+\tilde{\lambda}\rhd^{*}B\xrightarrow{L}B+\tilde{\lambda}\rhd^{*}B+\Delta_{L}(\wedge
C)$ $\displaystyle\qquad+\tilde{\lambda}\rhd^{*}\Delta_{L}(\wedge C),$
$\displaystyle\overleftarrow{(\tilde{\lambda},L)}$ $\displaystyle:$
$\displaystyle B\xrightarrow{L}B+\Delta_{L}(\wedge C)$
$\displaystyle\xrightarrow{\tilde{\lambda}}B+\tilde{\lambda}\rhd^{*}B+\Delta_{L}(\wedge
C)$ $\displaystyle\qquad+\Delta_{\mathfrak{ad}_{\tilde{\lambda}}^{*}L}(\wedge
C)+\Delta_{L}(\wedge
d_{C}\tilde{\lambda})+\underbrace{\Delta_{\mathfrak{ad}_{\tilde{\lambda}}^{*}L}(\wedge
d_{C}\tilde{\lambda})}_{\sim o(\tilde{\lambda}^{2})}.$

Modulo terms of order $o(d\tilde{\lambda}L)+o(\tilde{\lambda}^{2})$, the
difference reads

$\Delta_{L}(\wedge[C,\tilde{\lambda}]_{*})+\Delta_{\mathfrak{ad}_{\tilde{\lambda}}^{*}L}(\wedge
C)-\tilde{\lambda}\rhd^{*}\Delta_{L}(\wedge C).$

Antisymmetrizing the last two terms with the covariant derivative Eq. (4.15),
and setting this difference to zero yields the dual set of Eq. (4.18):

$\Delta_{L}(\wedge[C,\tilde{\lambda}]_{*})=\tilde{\lambda}\rhd^{*}\Delta_{L}(\wedge
C)-C\wedge^{\rhd^{*}}\Delta_{L}(\tilde{\lambda})-\Delta_{\mathfrak{ad}_{\tilde{\lambda}}^{*}L}(\wedge
C)+\Delta_{\mathfrak{ad}_{C}^{*}(\wedge L)}(\tilde{\lambda}).$ (4.19)

Finally, the question now is what would one yield by considering the mixed
gauge $(\tilde{\lambda},L)$ on $\Sigma$, or dually the mixed gauge
$(\lambda,\tilde{L})$ on $B$? We in fact do not get anything new; indeed, we
directly compute

$\displaystyle\overrightarrow{(\tilde{\lambda},L)}$ $\displaystyle:$
$\displaystyle\Sigma\xrightarrow{\tilde{\lambda}}\Sigma+\mathfrak{ad}_{\tilde{\lambda}}^{*}\Sigma$
$\displaystyle\xrightarrow{L}\Sigma+d_{A}L+-\mathfrak{ad}_{C}^{*}(\wedge
L)+\mathfrak{ad}_{\tilde{\lambda}}^{*}\Sigma$
$\displaystyle\qquad+\mathfrak{ad}_{\tilde{\lambda}}^{*}d_{A}L$
$\displaystyle\overleftarrow{(\tilde{\lambda},L)}$ $\displaystyle:$
$\displaystyle\Sigma\xrightarrow{L}\Sigma+d_{A}L-\mathfrak{ad}_{C}^{*}(\wedge
L)$
$\displaystyle\xrightarrow{\tilde{\lambda}}\Sigma+d_{A}L+\mathfrak{ad}_{\tilde{\lambda}}^{*}\Sigma-\mathfrak{ad}_{C}^{*}(\wedge
L)$
$\displaystyle\qquad+d_{A}\mathfrak{ad}_{\tilde{\lambda}}^{*}L(\eta_{\tilde{\lambda}}^{*}A)\wedge^{\rhd}L+$
$\displaystyle\qquad+\underbrace{(\eta_{\tilde{\lambda}}^{*}A)\wedge^{\rhd}\mathfrak{ad}_{\tilde{\lambda}}^{*}L}_{o(\tilde{\lambda}^{2})}.$

By neglecting terms of order $o(d\tilde{\lambda}L)+o(\tilde{\lambda}^{2})$,
the difference reads

$\mathfrak{ad}_{\tilde{\lambda}}^{*}(A\wedge^{\rhd}L)-A\wedge^{\rhd}\mathfrak{ad}_{\tilde{\lambda}}^{*}L-(\eta_{\tilde{\lambda}}^{*}A)\wedge^{\rhd}L.$

Setting this quantity to zero requires us to antisymmetrize the last two terms
with respect to $A$ and $L$.

To do this, we introduce the terms in the modified covariant derivative Eq.
(4.15). This yields the compatibility condition

$\mathfrak{ad}_{\tilde{\lambda}}^{*}(A\wedge^{\rhd}L)=A\wedge^{\rhd}\mathfrak{ad}_{\tilde{\lambda}}^{*}L-L\wedge^{\rhd}\mathfrak{ad}_{\tilde{\lambda}}^{*}A-(\eta_{\tilde{\lambda}}^{*}A)\wedge^{\rhd}L+(\eta_{\tilde{\lambda}}^{*}L)\wedge^{\rhd}A.$

Dually, the difference in how $B$ transforms under $(\lambda,\tilde{L})$ gives
rise to the compatibility condition

$\operatorname{ad}_{\lambda}^{*}(C\wedge^{\rhd^{*}}\tilde{L})=C\wedge^{\rhd^{*}}\operatorname{ad}_{\lambda}^{*}\tilde{L}-\tilde{L}\wedge^{\rhd^{*}}\operatorname{ad}_{\lambda}^{*}C-(\chi_{\lambda}^{*}C)\wedge^{\rhd^{*}}\tilde{L}+(\chi_{\lambda}^{*}\tilde{L})\wedge^{\rhd^{*}}C.$

These conditions bear striking resemblance to the second compatibility
conditions Eqs. (4.13), (4.14); they in fact lead to the same condition (cf.
Eq. (4.21)).

####  4.2.4 The 2-Manin triple

We have derived the following result:

######  Theorem 4.1.

The combined 2-gauge transformations (including antisymmetrized terms) of the
combined 2-connection $({\bf A},\boldsymbol{\Sigma})=(A+C,\Sigma+B)$

$\displaystyle\boldsymbol{\lambda}$ $\displaystyle:$
$\displaystyle\begin{cases}{\bf A}\rightarrow{\bf
A}^{\boldsymbol{\lambda}}={\bf A}+(D_{\bf
A}^{(0)}\lambda+\eta_{\tilde{\lambda}}^{*}A)+(D_{\bf
A}^{(0)}\tilde{\lambda}+\chi_{\lambda}^{*}C)\\\
\boldsymbol{\Sigma}\rightarrow\boldsymbol{\Sigma}^{\boldsymbol{\lambda}}=\boldsymbol{\Sigma}+(D_{\boldsymbol{\Sigma}}^{(0)}\lambda+\mathfrak{ad}_{\tilde{\lambda}}^{*}\Sigma)+(D_{\boldsymbol{\Sigma}}^{(0)}\tilde{\lambda}+\operatorname{ad}_{\lambda}^{*}B)\end{cases},$
$\displaystyle{\bf L}$ $\displaystyle:$ $\displaystyle\begin{cases}{\bf
A}\rightarrow{\bf A}^{\bf L}={\bf A}+(tL)+(\tilde{t}\tilde{L})\\\
\boldsymbol{\Sigma}\rightarrow\boldsymbol{\Sigma}^{\bf
L}=\boldsymbol{\Sigma}+(D_{\bf A}^{(1)}L+\tilde{\Delta}_{\tilde{L}}(\wedge
A))+(D_{\bf A}^{(1)}\tilde{L}+\Delta_{L}(\wedge C))\end{cases},$

in terms of the modified covariant derivatives Eqs. (4.7), (4.15), (4.16), are
unambiguous (ie. independent of the order at which any of the gauge parameters
$\lambda,\tilde{\lambda},L,\tilde{L}$ act) modulo terms of the order

$o(d\lambda(\tilde{\lambda}+\tilde{L}))+o(d\tilde{\lambda}(\lambda+L))+o(d\tilde{L}L)+o(dL\tilde{L})+o(\lambda^{2})+o(\tilde{\lambda}^{2})$

iff

  1. 1.

the maps in Eq. (3.11),

$\operatorname{ad}^{*}:\mathfrak{g}\rightarrow\operatorname{End}\mathfrak{g}^{*}[1],\qquad\mathfrak{ad}^{*}:\mathfrak{g}^{*}[1]\rightarrow\operatorname{End}\mathfrak{g},$

satisfy Eqs. (3.12), (3.13) and hence define strict coadjoint representations,
and

  2. 2.

the compatibility conditions Eqs. (4.8), (4.13), (4.14), (4.18), (4.19) are
satisfied.

These are in fact the necessary and sufficient conditions derived in Ref. [1]
for a strict Lie 2-bialgebra $\mathfrak{g}$ to form the standard 2-Manin
triple555As a Drinfel’d double is equivalent to a Manin triple [16], by abuse
of language, we will also freely interchange the name 2-Drinfel’d double with
2-Manin triple.

$\mathfrak{d}=\mathfrak{g}\leavevmode\nobreak\
_{\operatorname{ad}^{*}}\bowtie_{\mathfrak{ad}^{*}}\mathfrak{g}^{*}[1]$

with its dual $\mathfrak{g}^{*}[1]$. Indeed, if we distill all of the above
compatibility conditions to their Lie algebra values, then we obtain666Note
certain signs are different due to the antisymmetry of the wedge product on
forms.

$\displaystyle\eta_{f}^{*}[X,X^{\prime}]$ $\displaystyle=$
$\displaystyle[\eta_{f}^{*}X,X^{\prime}]+[X,\eta_{f}^{*}X^{\prime}]-\eta_{\chi^{*}_{X}f}^{*}X^{\prime}+\eta_{\chi^{*}_{X^{\prime}}f}^{*}X,$
$\displaystyle\chi_{X}^{*}[f,f^{\prime}]_{*}$ $\displaystyle=$
$\displaystyle[\chi_{X}^{*}f,f^{\prime}]_{*}+[f,\chi_{X}^{*}f]_{*}-\chi_{\eta_{f}^{*}X}^{*}f^{\prime}+\chi_{\eta_{f^{\prime}}^{*}X}f,$
(4.20) $\displaystyle\mathfrak{ad}_{f}^{*}(X\rhd Y)$ $\displaystyle=$
$\displaystyle X\rhd(\mathfrak{ad}_{f}^{*}Y)+(\eta_{f}^{*}X)\rhd
Y-\mathfrak{ad}_{\chi_{X}^{*}f}Y+\tilde{\Delta}_{\Delta_{Y}(f)}(X),$
$\displaystyle\operatorname{ad}_{X}^{*}(f\rhd^{*}g)$ $\displaystyle=$
$\displaystyle
f\rhd^{*}(\operatorname{ad}_{X}^{*}g)+(\chi_{X}^{*}f)\rhd^{*}g-\operatorname{ad}_{\eta_{f}^{*}X}g+\Delta_{\tilde{\Delta}_{g}(X)}(f),$
(4.21) $\displaystyle\tilde{\Delta}_{g}([X,X^{\prime}])$ $\displaystyle=$
$\displaystyle
X\rhd\tilde{\Delta}_{g}(X^{\prime})+X^{\prime}\rhd\tilde{\Delta}_{g}(X)-\tilde{\Delta}_{\operatorname{ad}_{X}^{*}g}(X^{\prime})+\tilde{\Delta}_{\operatorname{ad}_{X^{\prime}}^{*}g}(X),$
$\displaystyle\Delta_{Y}([f,f^{\prime}]_{*})$ $\displaystyle=$ $\displaystyle
f\rhd^{*}\Delta_{Y}(f^{\prime})+f^{\prime}\rhd^{*}\Delta_{Y}(f)-\Delta_{\mathfrak{ad}_{f}^{*}Y}(f^{\prime})+\Delta_{\mathfrak{ad}_{f^{\prime}}^{*}Y}(f)$
(4.22)

for each
$X,X^{\prime}\in\mathfrak{g}_{0},f,f^{\prime}\in\mathfrak{g}_{-1}^{*},Y\in\mathfrak{g}_{-1},g\in\mathfrak{g}_{0}^{*}$.

Given the compatibility conditions Eqs. (4.20), (4.21), (4.22), the 2-Manin
triple $\mathfrak{d}$ forms Lie algebra crossed-module. The $t$-map is given
by

$T=t+\tilde{t}=t+t^{*}:\mathfrak{g}_{-1}\oplus\mathfrak{g}_{0}^{*}\rightarrow\mathfrak{g}_{0}\oplus\mathfrak{g}_{-1}^{*},$

and the crossed-module structures

$\displaystyle\boldsymbol{[}X+f,X^{\prime}+f^{\prime}\boldsymbol{]}$
$\displaystyle=$
$\displaystyle[X,X^{\prime}]+[f,f^{\prime}]_{*}+\chi_{X}^{*}f^{\prime}-\chi_{X^{\prime}}^{*}f+\eta_{f}^{*}X^{\prime}-\eta_{f^{\prime}}^{*}X,$
$\displaystyle(X+f)\leavevmode\nobreak\
\mathrlap{\triangleright}{\rhd}\leavevmode\nobreak\ (Y+g)$ $\displaystyle=$
$\displaystyle X\rhd
Y+f\rhd^{*}g+\tilde{\Delta}_{g}(X)-\operatorname{ad}_{X}^{*}g+\Delta_{Y}(f)-\mathfrak{ad}_{f}^{*}Y$
(4.23)

for each
$X,X^{\prime}\in\mathfrak{g}_{0},Y\in\mathfrak{g}_{-1},f,f^{\prime}\in\mathfrak{g}_{-1}^{*},g\in\mathfrak{g}_{0}^{*}$.
The degree-(-1) bracket

$\displaystyle\boldsymbol{[}Y+g,Y^{\prime}+g^{\prime}\boldsymbol{]}$
$\displaystyle=$ $\displaystyle T(Y+g)\leavevmode\nobreak\
\mathrlap{\triangleright}{\rhd}\leavevmode\nobreak\ (Y^{\prime}+g^{\prime})$
$\displaystyle=$
$\displaystyle[Y,Y^{\prime}]+[g,g^{\prime}]_{*}+\tilde{\Delta}_{g^{\prime}}(tY)-\operatorname{ad}_{tY}^{*}g^{\prime}+\Delta_{Y^{\prime}}(t^{*}g)-\mathfrak{ad}_{t^{*}g}^{*}Y^{\prime}$

is given by the second Pfeiffer identity from Eq. (4.23).

Crucially, the canonical crossed-module structure on the 2-Manin triple
$\mathfrak{d}=\mathfrak{g}\bowtie\mathfrak{g}^{*}[1]$ is defined in accordance
with the grading. When written in terms of the components, we see that the
"double" presentation and its crossed-module structure are related

$(\mathfrak{g}_{-1},\mathfrak{g}_{0})\bowtie(\mathfrak{g}_{0}^{*},\mathfrak{g}_{-1}^{*})\leftrightsquigarrow(\mathfrak{g}_{-1},\mathfrak{g}_{0}^{*})\xrightarrow{T}(\mathfrak{g}_{0},\mathfrak{g}_{-1}^{*}),$

by an exchange $\mathfrak{g}_{0}\leftrightsquigarrow\mathfrak{g}^{*}$ of a
component at degree-0.

######  Example 4.1.

Consider the bialgebra crossed-modules
$(\operatorname{id}_{\mathfrak{l}};\psi,\psi)$ and
$(\mathfrak{g}^{0};\delta_{-1},\delta_{0})$ given in Example 3.2.

  1. 1.

Canonical 2-Manin triple: Fix the 1-cocycle $\psi=\delta_{-1}=\delta_{0}$ on
$\mathfrak{l}$. Due to the Pfeiffer identities, all of the components of Eq.
(3.11) collapse to merely the coadjoint representation:

$\chi^{*}=\operatorname{ad}^{*}=-\Delta,\qquad\eta^{*}=\mathfrak{ad}^{*}=-\tilde{\Delta}.$

Denote by $\mathfrak{1}_{\mathfrak{l}}$ the 2-Manin triple obtained from the
canonical bialgebra crossed-module
$(\operatorname{id}_{\mathfrak{l}};\psi,\psi)$ associated to the bialgebra
$(\mathfrak{l};\psi)$, then we see that the compatibility conditions Eqs.
(4.20), (4.21), (4.22) collapse to merely three copies of the compatibility
condition Eq. (2.11) for $\mathfrak{d}$.

This means that the bialgebra crossed-modules
$\mathfrak{1}_{\mathfrak{l}}=\operatorname{id}_{\mathfrak{d}}$ in fact
coincide, as they have the same crossed-module structures given by Eq. (4.23).
In essence, the canonical embedding commutes with the "Manin-ization"
$\mathfrak{l}\mapsto\mathfrak{d}$; this fact had also been noted in Ref. [1].
We call $\mathfrak{1}_{\mathfrak{l}}=\operatorname{id}_{\mathfrak{d}}$ the
canonical 2-Manin triple associated to the bialgebra $(\mathfrak{l};\psi)$.

  2. 2.

Suspension 2-Manin triple: Fix the 1-cocyce $\psi=\delta_{-1}+\delta_{0}$ on
$V\rtimes\mathfrak{u}$ as given in Proposition 3.1. Summing the crossed-module
structure Eq. (4.23) on the 2-Manin triple $\mathfrak{d}^{0}$ of the
suspension $\mathfrak{g}^{0}$ yields

$\displaystyle\boldsymbol{[}(X+Y)+(f+g),(X^{\prime}+Y^{\prime})+(f^{\prime}+g^{\prime})\boldsymbol{]}_{\mathfrak{d}}$
$\displaystyle=$
$\displaystyle\boldsymbol{[}X+f,X^{\prime}+f^{\prime}\boldsymbol{]}_{\mathfrak{d}^{0}}$
(4.24) $\displaystyle+(X+f)\leavevmode\nobreak\
\mathrlap{\triangleright}{\rhd}\leavevmode\nobreak\ (Y^{\prime}+g^{\prime})$
$\displaystyle-(X^{\prime}+f^{\prime})\leavevmode\nobreak\
\mathrlap{\triangleright}{\rhd}\leavevmode\nobreak\ (Y+g),$

where $X\in\mathfrak{u},g\in\mathfrak{u}^{*},Y\in V,f\in V^{*}$. This coicides
with the Lie bracket Eq. (2.12) on the Manin triple
$\mathfrak{d}=\mathfrak{d}(V\rtimes\mathfrak{u})$ of the semidirect product
$V\rtimes\mathfrak{u}$.

If we denote by $\mu\mathfrak{d}^{0}$ the 2-Manin triple $\mathfrak{d}^{0}$
with the grading ignored, then we see that $\mathfrak{d}=\mu\mathfrak{d}^{0}$
have identical algebraic structures. We call $\mathfrak{d}^{0}$ the suspension
2-Manin triple associated to the bialgebra $(V\rtimes\mathfrak{u};\psi)$.

It is clear from the Lie brackets Eq. (4.23) that the components of
$\mathfrak{d}$ with the same degree, namely
$\mathfrak{g}_{0}\bowtie\mathfrak{g}_{-1}^{*}$ and
$\mathfrak{g}_{-1}\bowtie\mathfrak{g}_{0}^{*}$, in fact themselves form
Drinfel’d doubles. In other words, a Lie bialgebra crossed-module
$(\mathfrak{g},\mathfrak{g}^{*}[1])$ defines two Lie bialgebras
$(\mathfrak{g}_{0},\mathfrak{g}_{-1}^{*}),(\mathfrak{g}_{-1},\mathfrak{g}_{0}^{*})$
[32, 1].

###  4.3 Strict Lie 2-bialgebras and 2-Manin rigidity

So far, the results we have obtained were all expressed in terms of Lie
algebra crossed-modules. However, it is well-known that Lie algebra crossed-
modules are equivalent to strict Lie 2-algebras [47, 32, 1]. We shall see that
all of our results carry over to the known 2-algebra scenario.

Recall a (strict) Lie 2-algebra is a two-term graded vector space
$\mathfrak{g}=\mathfrak{g}_{-1}\oplus\mathfrak{g}_{0}$ equipped with a
degree-1 differential $l_{1}:\mathfrak{g}_{-1}\rightarrow\mathfrak{g}_{0}$ and
a graded Lie bracket
$l_{2}=[\cdot,\cdot]:\mathfrak{g}_{i}\wedge\mathfrak{g}_{j}\rightarrow\mathfrak{g}_{i+j}$
(where $i,j=0,-1$), such that the Koszul relations

$\displaystyle[X,X^{\prime}]=-[X^{\prime},X],\qquad[X,Y]=-[Y,X],\qquad[Y,Y^{\prime}]=0,$
$\displaystyle
l_{1}[X,Y]=[X,l_{1}Y],\qquad[l_{1}Y,Y^{\prime}]=[Y,l_{1}Y^{\prime}],$
$\displaystyle{[[X,X^{\prime}],X^{\prime\prime}]}+[[X^{\prime},X^{\prime\prime}],X]+[[X^{\prime\prime},X],X^{\prime}]=0,$
$\displaystyle{[[X,X^{\prime}],Y]}+[[X^{\prime},Y],X]+[[Y,X],X^{\prime}]=0$

are satisfied for each $X,X^{\prime},X^{\prime\prime}\in\mathfrak{g}_{0}$ and
$Y,Y^{\prime}\in\mathfrak{g}_{-1}$.

If we redefine $l_{1}=t:\mathfrak{g}_{-1}\rightarrow\mathfrak{g}_{0}$, and put

$l_{2}|_{\Lambda^{2}\mathfrak{g}_{0}}=[\cdot,\cdot]_{\mathfrak{g}_{0}},\qquad
l_{2}|_{\mathfrak{g}_{0}\wedge\mathfrak{g}_{-1}}=\mathfrak{g}_{0}\rhd\mathfrak{g}_{-1},$
(4.25)

then we see that the above Koszul relations above reproduce precisely the
Pfeiffer identities Eq. (3.1), as well as the 2-Jacobi identities Eqs. (3.2).
Moreover, if the 2-cochain $(\delta_{-1},\delta_{0})$ Eq. (3.5) introduced in
Section 3 satisfy the coherence conditions Eqs. (3.7), (3.9), then
$(\delta_{-1},\delta_{0})$ is in fact a Lie 2-algebra 2-cocycle.

######  Proposition 4.1.

Lie algebra crossed-modules (resp. bialgebra crossed-modules) coincide with
strict Lie 2-algebras (resp. Lie 2-bialgebras) as defined in Ref. [1].

Now, given the natural coadjoint representations Eq. (3.11) of $\mathfrak{g}$
and its dual $\mathfrak{g}^{*}[1]$ on each other, the conditions Eqs. (4.20),
(4.21), (4.22) in fact coincide with those derived in Ref. [1]. These
conditions are necessary for $\mathfrak{g}\oplus\mathfrak{g}^{*}[1]$ to form a
matched pair of 2-algebras, hence our notion of the standard 2-Manin triple
coincides with that given in Ref. [1] as well.

##### 2-Manin rigidity.

Suppose, given a Lie 2-bialgebra $\mathfrak{g}$ and its dual
$\mathfrak{g}^{*}[1]$, we introduce a pair of generic graded action/back-
action

$\bar{\rhd}:\mathfrak{g}\rightarrow\operatorname{End}\mathfrak{g}^{*}[1],\qquad\bar{\lhd}:\mathfrak{g}^{*}[1]\rightarrow\operatorname{End}\mathfrak{g}.$

Then we in fact have the following gauge-enforced result

######  Corollary 4.1.

Given $\tilde{t}=t^{*}$, the induced combined 2-gauge transformations
$(\boldsymbol{\lambda},{\bf L})=(\lambda+\tilde{\lambda},L+\tilde{L})$ are
unambiguous up to the order given in Theorem 4.1 iff

  1. 1.

the action/back-action pair $\bar{\rhd},\bar{\lhd}$ form strict coadjoint
representations, and

  2. 2.

the double

$\mathfrak{d}^{\prime}=\mathfrak{g}\overline{\bowtie}\mathfrak{g}^{*}[1]$

is isomorphic to the standard 2-Manin triple
$\mathfrak{d}=\mathfrak{g}\leavevmode\nobreak\
_{\operatorname{ad}^{*}}\bowtie_{\mathfrak{ad}^{*}}\mathfrak{g}^{*}[1]$.

###### Proof.

The same argument as in Section 4.2, leading up to Theorem 4.1, goes through
for the generic action/back-action $\bar{\rhd},\bar{\lhd}$ such that, given
$\tilde{t}=t^{*}$,

  1. 1.

they satisfy the analogues of Eqs. (3.12) and (3.13), and

  2. 2.

they satisfy the analogues of Eqs. (4.8), (4.13), (4.14), (4.18), (4.19),

with $\operatorname{ad}$ replaced by $\bar{\rhd}$ and $\mathfrak{ad}$ replaced
by $\bar{\lhd}$. The induced crossed-module structure for
$\mathfrak{d}^{\prime}$ must then take the same form Eq. (4.23), which forces
$\mathfrak{d}^{\prime}\cong\mathfrak{d}$. ∎

This rigidity of 2-Manin triples was proven in Ref. [1], and here we have
demonstrated that this is enforced by gauge-theoretic considerations. In
contrast to Corollary 2.1, however, the 2-gauge theory not only enforces this
rigidity, but also the conditions Eqs. (3.12), (3.13) necessary for
$\bar{\rhd},\bar{\lhd}$ to form strict coadjoint representations.

##  5 4D topological action invariant under the 2-Drinfel’d double

As $\mathfrak{d}$ is itself a (strict) Lie 2-algebra, we can study the 2-gauge
theory associated to it. Here, we seek to construct curvature quantities that
are covariant under the combined 2-gauge transformation
$(\boldsymbol{\lambda},{\bf L})$, briefly 2-covariant. In analogy with Section
2.3, we can define a combined curvature that reads

$\displaystyle{\bf F}$ $\displaystyle=$ $\displaystyle d{\bf
A}+\frac{1}{2}\boldsymbol{[}{\bf A}\wedge{\bf A}\boldsymbol{]}$ (5.1)
$\displaystyle=$ $\displaystyle(dA+\frac{1}{2}[A\wedge A]+\eta_{C}^{*}(\wedge
A))$ $\displaystyle+(dC+\frac{1}{2}[C\wedge C]_{*}+\chi_{A}^{*}(\wedge C))$
$\displaystyle\equiv$ $\displaystyle\bar{F}+\bar{F}^{*},$

which yields the combined fake-flatness

$\boldsymbol{\mathcal{F}}={\bf
F}-T\boldsymbol{\Sigma}=(\bar{F}-t\Sigma)+(\bar{F}^{*}-t^{*}B)\equiv\bar{\mathcal{F}}+\bar{\mathcal{F}}^{*},$
(5.2)

by making use of the crossed-module map $T=t+\tilde{t}$ for $\mathfrak{d}$.
The bracket can be computed as

$\llbracket\boldsymbol{\mathcal{F}},\boldsymbol{\lambda}\rrbracket=([\bar{\mathcal{F}},\lambda]+\eta_{\tilde{\lambda}}^{*}\bar{\mathcal{F}}-\eta_{\bar{\mathcal{F}}^{*}}^{*}\lambda)+([\bar{\mathcal{F}}^{*},\tilde{\lambda}]_{*}+\chi_{\lambda}^{*}\bar{\mathcal{F}}^{*}-\chi_{\bar{\mathcal{F}}}^{*}\tilde{\lambda}).$

Analogously, we define the combined 2-curvature

$\displaystyle\boldsymbol{\mathcal{G}}$ $\displaystyle=$ $\displaystyle
d\boldsymbol{\Sigma}+{\bf
A}\wedge^{\mathrlap{\triangleright}{\rhd}}\boldsymbol{\Sigma}$ (5.3)
$\displaystyle=$
$\displaystyle(d\Sigma+A\wedge^{\rhd}\Sigma+\tilde{\Delta}_{B}(\wedge
A)-\mathfrak{ad}_{C}^{*}(\wedge\Sigma))$
$\displaystyle+(dB+C\wedge^{\rhd^{*}}+\Delta_{\Sigma}(\wedge
C)-\operatorname{ad}_{A}^{*}(\wedge B))$ $\displaystyle\equiv$
$\displaystyle\bar{\mathcal{G}}+\bar{\mathcal{G}}^{*}.$

The 2-covariance of $\boldsymbol{\mathcal{F}},\boldsymbol{\mathcal{G}}$ can
once again be inferred immediately from the crossed-module structure of the
standard 2-Manin triple $\mathfrak{d}$; for an explicit proof, see Appendix A.
It in fact allows us to rewrite the combined 2-gauge transformations in
Theorem 4.1 as

$\boldsymbol{\lambda}:\begin{cases}{\bf A}\rightarrow{\bf
A}^{\boldsymbol{\lambda}}={\bf A}+d_{\bf A}\boldsymbol{\lambda}\\\
\boldsymbol{\Sigma}\rightarrow\boldsymbol{\Sigma}^{\boldsymbol{\lambda}}=\boldsymbol{\Sigma}+\boldsymbol{\lambda}\leavevmode\nobreak\
\mathrlap{\triangleright}{\rhd}\leavevmode\nobreak\
\boldsymbol{\Sigma}\end{cases},\qquad{\bf L}:\begin{cases}{\bf
A}\rightarrow{\bf A}^{\bf L}={\bf A}+T{\bf L}\\\ \boldsymbol{\Sigma}^{\bf
L}=\boldsymbol{\Sigma}+d_{\bf A}{\bf L}\end{cases}.$ (5.4)

This is a very useful and compact way of organizing the combined 2-gauge
transformations.

######  Example 5.1.

Recall the canonical and suspension 2-Manin triples considered in Example 4.1.

  1. 1.

Pick a 1-cocycle $\psi$ on $\mathfrak{l}$. We demonstrate the gauge-theoretic
consequences of
$\mathfrak{1}_{\mathfrak{l}}=\operatorname{id}_{\mathfrak{d}}$. First, the
2-covariant quantities in the 2-Manin triple $\mathfrak{1}_{\mathfrak{l}}$ are
given by Eq. (5.2) and (5.3). Recalling that all components of Eq. (3.11)
collapse to merely the coadjoint representation $\operatorname{ad}^{*}$, the
sum of these 2-covariant quantities can in fact be written as

$\displaystyle\bar{\mathcal{G}}+\bar{\mathcal{G}}^{*}$ $\displaystyle=$
$\displaystyle d(\Sigma+B)+\boldsymbol{[}(A+C)\wedge(\Sigma+B)\boldsymbol{]},$
$\displaystyle\bar{\mathcal{F}}+\bar{\mathcal{F}}^{*}$ $\displaystyle=$
$\displaystyle
d(A+C)+\frac{1}{2}\boldsymbol{[}(A+C)\wedge(A+C)\boldsymbol{]}-(\Sigma+B),$

where $\boldsymbol{[}\cdot,\cdot\boldsymbol{]}$ denotes the bracket Eq. (2.12)
of the Manin triple $\mathfrak{d}$ of $\mathfrak{l}$. By interpreting
$(A+C,\Sigma+B)$ as the combined 2-connection on
$\operatorname{id}_{\mathfrak{d}}$, we see that $\mathfrak{1}_{\mathfrak{l}}$
and $\operatorname{id}_{\mathfrak{d}}$ host identical 2-covariant quantities,
which is consistent with the fact that
$\mathfrak{1}_{\mathfrak{l}}=\operatorname{id}_{\mathfrak{d}}$.

  2. 2.

Pick a 1-cocycle $\psi$ on $\mathfrak{h}=V\rtimes\mathfrak{u}$. As in Example
4.1, let $\mathfrak{d}(\mathfrak{h})$ denote the Drinfel’d double associated
to the bialgebra $(\mathfrak{h};\psi)$, and let $\mathfrak{d}^{0}$ denote the
2-Drinfel’d double associated to the skeletal suspension $\mathfrak{d}^{0}$ of
$\mathfrak{h}$. We demonstrate the gauge-theoretic consequences of
$\mathfrak{d}(\mathfrak{h})=\mu\mathfrak{d}^{0}$, where $\mu$ is the map that
forgets the grading in the crossed-module. As the form-degree of the fields
depend on the grading, there is a procedure that "promotes" the covariant
quantities on $\mathfrak{d}(\mathfrak{h})$ to 2-covariant quantities based on
$\mathfrak{d}^{0}$.

The components $\bar{F}\in\mathfrak{h},\bar{F}^{*}\in\mathfrak{h}^{*}$ in the
combined curvature ${\bf F}=\bar{F}+\bar{F}^{*}\in\mathfrak{d}(\mathfrak{h})$
in Eq. (2.13) can be written as

$\bar{F}=dA+\frac{1}{2}[A,A]+\mathfrak{ad}_{b}^{*}(\wedge
A),\qquad\bar{F}^{*}=db+\frac{1}{2}[b\wedge
b]_{*}+\operatorname{ad}_{A}^{*}(\wedge b).$

By decomposing into components of the semidirect product
$\mathfrak{h}=V\rtimes\mathfrak{u}$, we have

$\displaystyle\bar{F}$ $\displaystyle=$ $\displaystyle
dA|_{\mathfrak{u}}+\frac{1}{2}[A|_{\mathfrak{u}}\wedge
A|_{\mathfrak{u}}]+\eta_{b|_{V^{*}}}^{*}(\wedge A|_{\mathfrak{u}})$
$\displaystyle+dA|_{V}+A|_{\mathfrak{u}}\rhd
A|_{V}+\mathfrak{ad}_{b|_{V^{*}}}^{*}(\wedge
A|_{V})-\tilde{\Delta}_{b|_{\mathfrak{u}^{*}}}(\wedge A|_{\mathfrak{u}}),$
$\displaystyle\bar{F}^{*}$ $\displaystyle=$ $\displaystyle
db|_{V^{*}}+\frac{1}{2}[b|_{V^{*}}\wedge
b|_{V^{*}}]_{*}+\chi_{A|_{\mathfrak{u}}}^{*}(\wedge b|_{V^{*}})$ (5.5)
$\displaystyle+db|_{\mathfrak{u}^{*}}+b|_{V^{*}}\rhd^{*}b|_{\mathfrak{u}^{*}}+\operatorname{ad}_{A|_{\mathfrak{u}}}^{*}(\wedge
b|_{\mathfrak{u}^{*}})-\Delta_{A|_{V}}(\wedge b|_{V^{*}}),$

where we have defined, for each $X\in\mathfrak{u},Y\in V,f\in
V^{*},g\in\mathfrak{u}^{*}$, the notations

$(\chi_{X}^{*}f)(Y)=\Delta_{Y}(f)(X)=-f(X\rhd Y),\qquad
g(\eta^{*}_{f}X)=f(\tilde{\Delta}_{g}(X))=-(f\rhd^{*}g)(X),$

as per convention. This notation is convenient, as it allows us to identify

$\displaystyle A|_{\mathfrak{u}}=A^{0},$ $\displaystyle b|_{V^{*}}=C^{0}$
$\displaystyle\bar{F}|_{\mathfrak{u}}=\bar{F}^{0},$
$\displaystyle\bar{F}^{*}|_{V^{*}}=(\bar{F}^{0})^{*},$

where ${\bf F}^{0}=\bar{F}^{0}+(\bar{F}^{0})^{*}$ the 2-covariant 1-curvature
defined in Eq. (5.1).

Now what of the other components $A|_{V},b|_{\mathfrak{u}^{*}}$ of the
connections? If we bump up the form-degrees of these quantities to 2-forms,
and rename them $\Sigma^{0},B^{0}$, respectively, then the curvature
quantities $\bar{F}|_{V},\bar{F}^{*}|_{\mathfrak{u}^{*}}$ become 3-forms. From
the formulas Eq. (5.5) above, these 3-forms are given by

$\displaystyle
d\Sigma^{0}+A|_{\mathfrak{u}}\rhd\Sigma^{0}+\mathfrak{ad}_{b|_{V^{*}}}^{*}(\wedge\Sigma^{0})-\tilde{\Delta}_{B^{0}}(\wedge
A|_{\mathfrak{u}})$ $\displaystyle=$ $\displaystyle
d\Sigma^{0}+A^{0}\rhd\Sigma^{0}+\mathfrak{ad}_{C^{0}}^{*}(\wedge\Sigma^{0})-\tilde{\Delta}_{B^{0}}(\wedge
A^{0}),$ $\displaystyle
dB^{0}+b_{V^{*}}\rhd^{*}B^{0}+\operatorname{ad}_{A|_{\mathfrak{u}}}^{*}(\wedge
B^{0})+-\Delta_{\Sigma^{0}}(\wedge b|_{V^{*}})$ $\displaystyle=$
$\displaystyle dB^{0}+C^{0}\rhd^{*}B^{0}+\operatorname{ad}_{A^{0}}^{*}(\wedge
B^{0})-\Delta_{\Sigma^{0}}(\wedge V^{0}),$

which is precisely the 2-covariant 2-curvature
$\boldsymbol{\mathcal{G}}^{0}=\bar{\mathcal{G}}^{0}+(\bar{\mathcal{G}}^{0})^{*}$
given in Eq. (5.3) for the 2-Drinfel’d double $\mathfrak{d}^{0}$, with the
combined 2-connection given by

$({\bf A}^{0},\boldsymbol{\Sigma}^{0})=(A^{0}+C^{0},\Sigma^{0}+B^{0}).$

In this sense, the suspension of the Drinfel’d double to make it a 2-Drinfel’d
double can be seen equivalently as considering the bumping in dimensions of
the forms $A|_{V},b|_{\mathfrak{u}^{*}}\rightsquigarrow\Sigma^{0},B^{0}$.

###  5.1 BF theory and interacting 2-BF theory.

Using the 2-covariant quantities
$\boldsymbol{\mathcal{F}},\boldsymbol{\mathcal{G}}$ in Eq. (5.2), (5.3), we
can consider a 2-gauge theory based on the 2-Manin triple
$\mathfrak{d}=\mathfrak{g}\bowtie\mathfrak{g}^{*}[1]$. First, given the
bilinear form Eq. (3.6) on $\mathfrak{d}$, we can write down the BF theory

$\displaystyle S_{\textbf{BF}}[{\bf A},\boldsymbol{\Sigma}]$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{M}\langle\langle\boldsymbol{\Sigma}\wedge\boldsymbol{\mathcal{F}}\rangle\rangle=\frac{1}{2}\int_{M}\langle
B\wedge\bar{\mathcal{F}}\rangle+\langle\bar{\mathcal{F}}^{*}\wedge\Sigma\rangle$
(5.6) $\displaystyle=$ $\displaystyle\frac{1}{2}\int_{M}\langle
B\wedge\bar{F}\rangle+\langle\bar{F}^{*}\wedge\Sigma\rangle-\left[\langle
B\wedge t\Sigma\rangle+\langle t^{*}B\wedge\Sigma\rangle\right]$

on a 4-dimensional manifold $M$, where $\bar{F}$ and $\bar{F}^{*}$ are defined
in Eq. (5.1).

Notice that the last two terms are in fact identical:

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

Moreover, by decomposing into components, each of the first two terms read
explicitly

$\displaystyle\langle B\wedge\bar{F}\rangle$ $\displaystyle=$
$\displaystyle\langle B\wedge F\rangle+\langle B\wedge\eta_{C}^{*}(\wedge
A)\rangle,$ $\displaystyle\langle\bar{F}^{*}\wedge\Sigma\rangle$
$\displaystyle=$ $\displaystyle\langle dC+\frac{1}{2}[C\wedge
C]_{*}+\chi_{A}^{*}(\wedge C)\wedge\Sigma\rangle,$

where $F=dA+\frac{1}{2}[A\wedge A]$ is the conventional 1-curvature on
$\mathfrak{g}_{0}$. An integration by parts (neglecting the boundary term
$d\langle C\wedge\Sigma\rangle$) yields

$\langle\bar{F}^{*}\wedge\Sigma\rangle=-\langle
C\wedge\mathcal{G}\rangle+\frac{1}{2}\langle[C\wedge
C]_{*}\wedge\Sigma\rangle$

in terms of the conventional 2-curvature
$\mathcal{G}=d\Sigma+A\wedge^{\rhd}\Sigma$ on $\mathfrak{g}_{-1}$. Thus we see
that the BF theory Eq. (5.6) can be written as

$\displaystyle S_{\textbf{BF}}[{\bf A},\boldsymbol{\Sigma}]$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{M}\langle
B\wedge\mathcal{F}^{\prime}\rangle-\langle C\wedge\mathcal{G}\rangle$
$\displaystyle+\frac{1}{2}\int_{M}\langle\frac{1}{2}[C\wedge
C]_{*}\wedge\Sigma\rangle+\langle B\wedge\eta_{C}^{*}(\wedge A)\rangle,$

which is a 2-BF theory with interaction terms, where
$\mathcal{F}^{\prime}=F-2t\Sigma$ is a kind of "doubled" fake-flatness.

The BF action is invariant under the 2-Drinfel’d double
$\mathfrak{d}=\mathfrak{g}\bowtie\mathfrak{g}^{*}[1]$. From Eq. (5.4) and from
invariance of the bilinear form $\langle\langle\cdot,\cdot\rangle\rangle$ we
have

$\displaystyle\langle\langle\boldsymbol{\lambda}\leavevmode\nobreak\
\mathrlap{\triangleright}{\rhd}\leavevmode\nobreak\
\boldsymbol{\Sigma}\wedge\boldsymbol{\mathcal{F}}\rangle\rangle=\langle\langle\boldsymbol{\Sigma}\wedge\llbracket\boldsymbol{\lambda},\boldsymbol{\mathcal{F}}\rrbracket\rangle\rangle,\qquad\langle\langle({\bf
A}\wedge^{\mathrlap{\triangleright}{\rhd}}{\bf
L})\wedge\boldsymbol{\mathcal{F}}\rangle\rangle=-\langle\langle{\bf
L}\wedge\llbracket{\bf
A}\wedge\boldsymbol{\mathcal{F}}\rrbracket\rangle\rangle.$

The first equation indicates that the integrand
$\langle\langle\boldsymbol{\Sigma}\wedge\boldsymbol{\mathcal{F}}\rangle\rangle$
is invariant under the 1-gauge. On the other hand, the second equation must be
supplemented by a term $\langle\langle d{\bf
L}\wedge\boldsymbol{\mathcal{F}}\rangle\rangle$, which coincides with
$\langle{\bf L}\wedge d\boldsymbol{\mathcal{F}}\rangle\rangle$ modulo a
boundary term. We thus have

$\langle\langle d_{\bf A}{\bf
L}\wedge\boldsymbol{\mathcal{F}}\rangle\rangle=-\langle\langle{\bf L}\wedge
d_{\bf A}\boldsymbol{\mathcal{F}}\rangle\rangle=0$

from the Bianchi identity $d_{\bf A}{\bf F}=0$. This is essentially the same
basic computation performed in Ref. [48].

By varying the field $B$, we receive the equation of motion (EOM)

$\delta_{B}S_{\textbf{BF}}=0\implies\bar{\mathcal{F}}^{\prime}=dA+\frac{1}{2}[A\wedge
A]+\eta_{C}^{*}(\wedge A)-2t\Sigma=0$

for the doubled version of the fake-flatness Eq. (5.2) in the
$\mathfrak{g}$-sector. To vary the field $C$, we recall that we may rewrite

$\langle B\wedge\eta_{C}^{*}(\wedge
A)\rangle=-\langle(C\wedge^{\rhd^{*}}B)\wedge A\rangle=\langle
C\wedge\tilde{\Delta}_{B}(\wedge A)\rangle,$ (5.7)

then we obtain the flatness EOM

$\delta_{C}S_{\textbf{BF}}=0\implies\bar{\mathcal{G}}^{\prime}=d\Sigma+A\wedge^{\rhd}\Sigma-\tilde{\Delta}_{B}(\wedge
A)+\mathfrak{ad}_{C}^{*}(\wedge\Sigma)=0$

for the $\mathfrak{g}$-sector of the combined 2-curvature Eq. (5.3), except
the signs of the latter two terms are reversed.

On the other hand, it is clear that if we vary the fields $(A,\Sigma)$, then
we obtain the dual counterparts of the above flatness EOMs

$\displaystyle\delta_{A}S_{\textbf{BF}}=0$ $\displaystyle\Rightarrow$
$\displaystyle\bar{\mathcal{G}}^{\prime*}=dB+C\wedge^{\rhd^{*}}B-\Delta_{\Sigma}(\wedge
C)+\operatorname{ad}_{A}^{*}(\wedge B)=0$
$\displaystyle\delta_{B}S_{\textbf{BF}}=0$ $\displaystyle\Rightarrow$
$\displaystyle\bar{\mathcal{F}}^{\prime*}=dC+\frac{1}{2}[C\wedge
C]_{*}+\chi_{A}^{*}(\wedge C)-2\tilde{t}B=0$

in the $\mathfrak{g}^{*}[1]$-sector. This means that the monster BF theory Eq.
(5.6) exhibits the 2-Drinfel’d double $\mathfrak{d}$ as symmetry, where the
bilinear form Eq. (3.6) is chosen.

######  Example 5.2.

As a specific example, let us consider the skeletal crossed-module
$(\mathbb{R}^{6}\xrightarrow{t=0}\mathfrak{so}(3,1),\rhd,[\cdot,\cdot]_{\mathfrak{so}(3,1)})$
which is the natural symmetry structure of a $\mathfrak{so}(3,1)$ $BF$ theory
[26]. It can be interpreted as a 2-Drinfel’d double if one uses the Iwasawa
decomposition of
$\mathfrak{so}(3,1)\approx\mathfrak{su}(2)\bowtie\mathfrak{an}_{2}$, and the
decomposition $\mathbb{R}^{6}\approx\mathbb{R}^{3}\times\mathbb{R}^{3}$. By
re-ordering appropriately the components as was discussed in Section 4.2, the
symmetries of the $\mathfrak{so}(3,1)$ $BF$ theory can also be seen as given
in terms of a pair of dual 2-gauge theories given in terms of the skeletal
crossed-modules
$\mathfrak{g}=(\mathbb{R}^{3}\xrightarrow{t=0}\mathfrak{su}(2),\rhd,[\cdot,\cdot]_{\mathfrak{su}(2)})$
and
$\mathfrak{g}^{*}[1]=(\mathbb{R}^{3}\xrightarrow{t^{*}=0}\mathfrak{an}_{2},\rhd^{*},[\cdot,\cdot]_{\mathfrak{an}_{2}})$.
Said otherwise, the $\mathfrak{so}(3,1)$ $BF$ theory can roughly be viewed as
the "Chern-Simons" formulation of a 2-gauge topological theory. Note that upon
integration of the different crossed-modules, we would recover non-trivial
Poisson 2-groups hence quantum 2-groups upon quantization, since the crossed-
modules are equipped with non-trivial cocycles.

Our construction emphasizes that there are hidden 2-gauge symmetries within
the well-known 4D $BF$ theories. This point was already made in the specific
case of the 2-Poincaré case [31]. Our construction has important consequences
for the discretization/quantization of the theory. Indeed, in 3D, quantizing
either the Chern-Simons formulation or the $BF$ formulation means we construct
our quantum states using different (quantum) groups. Of course, since it is
the same theory, it means that there are relations among the different
amplitudes: it is well-known that the Turaev-Viro amplitude can be related to
the Chern-Simons amplitude [41, 42]. If mathematically the two formulations
are related/equivalent, physically some might be more relevant than others.
For example, in the $BF$ formulation, one has access to the frame field hence
the gravitational field. The loop quantum gravity formulation for 3D gravity
relies on the $BF$ formulation [40]. In the Chern-Simons formulation, the
frame field is combined with the spin connection so that in this case it might
more difficult to address physical questions specifically about the frame
field.

We would like to point out that something similar would happen in 4D. One can
construct the quantum states of 4D BF theory as a gauge theory as it is now
standard [36]. The new aspect we want to emphasize is that there should also
be an equivalent description of the quantum amplitudes in terms of a 2-gauge
theory, presumably given in terms of the amplitude of the Yetter type [51]. A
first step in this direction was accomplished in the 2-Poincaré case, where it
was shown that the theory could be discretized as a standard lattice gauge
theory or instead as a 2-gauge theory [33]. Once again, one formulation might
be better suited than the other one according to the physical questions one
intends to ask.

###  5.2 Pairings on the 2-Drinfel’d double; the BF-BB theory.

Similar to the 1-gauge case, the natural bilinear form Eq. (3.6) was induced
from the canonical evaluation pairing $(f+g,X+Y)=g(X)+f(Y)$, where
$f\in\mathfrak{g}_{-1}^{*},g\in\mathfrak{g}_{0}^{*},Y,\in\mathfrak{g}_{-1},X\in\mathfrak{g}_{0}$.
In this section, we explore the possibility of introducing an alternative
pairing $\langle\langle\cdot,\cdot\rangle\rangle^{\prime}$ on the direct sum
$\mathfrak{g}\oplus\mathfrak{g}^{*}[1]$ of the crossed-modules.

We proceed in direct analogy with the 1-algebra case. Given a bialgebra
crossed-module $(\mathfrak{g};\delta_{-1},\delta_{0})$, we can endow a 2-Manin
triple structure on the direct sum
$\mathfrak{d}\cong\mathfrak{g}\oplus\mathfrak{g}_{-1}^{*}\cong(\mathfrak{g}_{-1}\oplus\mathfrak{g}_{0}^{*})\oplus(\mathfrak{g}_{0}\oplus\mathfrak{g}_{-1}^{*})$
by equipping it with a non-degenerate symmetric invariant bilinear form
$\langle\langle\cdot,\cdot\rangle\rangle$. In contrast to the 1-algebra case,
it can be seen that the natural evaluation pairing Eq. (3.6) was merely one of
four possible pairing components on $\mathfrak{d}$.

Specifically, given any
$(Y+g)+(X+f)\in(\mathfrak{g}_{-1}\oplus\mathfrak{g}_{0}^{*})\oplus(\mathfrak{g}_{0}\oplus\mathfrak{g}_{-1}^{*})$,
we have the following:

  1. 1.

Grading-inhomogeneous, referring to pairings across sectors of distinct
degrees. This case includes the natural blinear pairing Eq. (3.6),

$\langle\langle(Y+g)+(X+f),(Y^{\prime}+g^{\prime})+(X^{\prime}+f^{\prime})\rangle\rangle=g(X^{\prime})+g^{\prime}(X)+f(Y^{\prime})+f^{\prime}(Y)$
(5.8)

as well as the following

$\langle\langle(Y+g)+(X+f),(Y^{\prime}+g^{\prime})+(X^{\prime}+f^{\prime})\rangle\rangle^{\prime}=(\langle
Y,X^{\prime}\rangle+\langle f,g^{\prime}\rangle)+(\langle
X,Y^{\prime}\rangle+\langle g,f^{\prime}\rangle),$ (5.9)

where $\langle\cdot,\cdot\rangle$ denotes an "off-diagonal" invariant bilinear
form on the crossed-module
$\mathfrak{g}\cong\mathfrak{g}_{-1}\oplus\mathfrak{g}_{0}$ and
$\mathfrak{g}_{0}^{*}\oplus\mathfrak{g}_{-1}^{*}$. Non-degeneracy then
requires $\mathfrak{g}_{0}$ and $\mathfrak{g}_{-1}$ to have the same
dimension.

  2. 2.

Grading-homogeneous, referring to pairings across sectors of the same degree,
such as the following

$\langle\langle(Y+g)+(X+f),(Y^{\prime}+g^{\prime})+(X^{\prime}+f^{\prime})\rangle\rangle_{\text{hom}}=(\langle
Y,Y^{\prime}\rangle_{-1}+\langle g,g^{\prime}\rangle_{0})+(\langle
X,X^{\prime}\rangle_{0}+\langle f,f^{\prime}\rangle_{-1})$

induced by a "diagonal" invariant bilinear form
$\langle\cdot,\cdot\rangle_{-1.0}$ on the crossed-module
$\mathfrak{g}\cong\mathfrak{g}_{-1}\oplus\mathfrak{g}_{0}$. The other one is

$\langle\langle(Y+g)+(X+f),(Y^{\prime}+g^{\prime})+(X^{\prime}+f^{\prime})\rangle\rangle^{\prime}_{\text{hom}}=(\langle
Y,g^{\prime}\rangle_{1}+\langle X,f^{\prime}\rangle_{0})+(\langle
f,X^{\prime}\rangle_{0}+\langle g,Y^{\prime}\rangle_{1}),$

which is the degree-homogeneous version of the natural bilinear form Eq.
(3.6).

As usual, these alternative pairings satisfy certain invariance properties
similar to the natural pairing Eq. (3.6). For the alternative pairing
$\langle\langle\cdot,\cdot\rangle\rangle^{\prime}$ given in Eq. (5.9), for
instance, we must have

$\displaystyle\langle X\rhd Y,X^{\prime}\rangle+\langle
Y,\operatorname{ad}_{X}X^{\prime}\rangle=0,$ $\displaystyle\langle
f\rhd^{*}g,f^{\prime}\rangle+\langle g,\mathfrak{ad}_{f}f^{\prime}\rangle=0,$
$\displaystyle\langle\mathfrak{ad}_{f}^{*}Y,X\rangle+\langle
Y,\eta_{f}^{*}X\rangle=0,$
$\displaystyle\langle\operatorname{ad}_{X}^{*}g,f\rangle+\langle
g,\chi_{X}^{*}f^{\prime}\rangle=0$
$\displaystyle\langle\tilde{\Delta}_{g}(X),X^{\prime}\rangle+\langle
X,\tilde{\Delta}_{g}(X^{\prime})\rangle=0,$
$\displaystyle\langle\Delta_{Y}(f),f^{\prime}\rangle+\langle
f,\Delta_{Y}(f^{\prime})\rangle=0,$ (5.10)

where $\mathfrak{ad}_{g},\operatorname{ad}_{Y}$ are induced by the second
Pfeiffer identity Eq. (3.1).

The new ingredient here is, in addition to invariance, the following duality
condition777The natural pairing Eq. (3.6) satisfies this by definition, as
$(g,tY)=g(tY)=t^{*}g(Y)=(t^{*}g,Y)$. (note $T^{*}=T$ is self-dual)

$\langle\langle T\leavevmode\nobreak\ \cdot\leavevmode\nobreak\
,\leavevmode\nobreak\ \cdot\leavevmode\nobreak\
\rangle\rangle=\langle\langle\leavevmode\nobreak\ \cdot\leavevmode\nobreak\
,T\leavevmode\nobreak\ \cdot\leavevmode\nobreak\ \rangle\rangle$ (5.11)

against the crossed-module map $T$ of the 2-Manin triple
$\mathfrak{d}=\mathfrak{g}\bowtie\mathfrak{g}^{*}[1]$. For the alternative
pairings, this puts non-trivial conditions on the crossed-module map $t$. Take
the alternative pairing $\langle\langle\cdot,\cdot\rangle\rangle^{\prime}$
given in Eq. (5.9), say, then we must have

$\langle tY,Y^{\prime}\rangle=\langle Y,tY^{\prime}\rangle,\qquad\langle
t^{*}g,g^{\prime}\rangle=\langle g,t^{*}g^{\prime}\rangle.$ (5.12)

In other words, $t$ must be symmetric in the basis diagonalizing the bilinear
form $\langle\cdot,\cdot\rangle$.

Let consider again a BF theory type as in Eq. (5.6). It is clear that the
grading-homogeneous pairing types give a trivial theory: the 2-form
$\boldsymbol{\Sigma}$ has degree-(-1) while the 2-form
$\boldsymbol{\mathcal{F}}$ has degree-0, hence for this specific choice of
action, non-trivial pairings must be grading-inhomogeneous.

Focusing on the alternative pairing Eq. (5.9), the BF theory then reads

$\displaystyle S_{\textbf{BF}}^{\prime}[{\bf A},\boldsymbol{\Sigma}]$
$\displaystyle=$
$\displaystyle\frac{1}{2}\int_{M}\langle\langle\boldsymbol{\Sigma}\wedge\boldsymbol{\mathcal{F}}\rangle\rangle^{\prime}=\frac{1}{2}\int_{M}\langle
B\wedge\bar{\mathcal{F}}^{*}\rangle+\langle\Sigma\wedge\bar{\mathcal{F}}\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{2}\int_{M}\langle
B\wedge\bar{F}^{*}\rangle+\langle\Sigma\wedge\bar{F}\rangle-\left[\langle
B\wedge t^{*}B\rangle+\langle\Sigma\wedge t\Sigma\rangle\right]$
$\displaystyle=$