Conversely, one can always "shrink the cylinder" in a trivial (open) bordism $\Gamma:\Sigma\rightarrow\Sigma'$ to give a homotopy equivalence $\Sigma\simeq\Sigma'$ (relative boundary). One can then ask if these are the only bordisms that do such a thing; for instance, are $h$-cobordisms $\Gamma:\Sigma\rightarrow\Sigma'$, where the inclusions $\Sigma,\Sigma'\hookrightarrow\Gamma$ induce homotopy equivalences, homotopic to the trivial bordism, and hence also give rise to homotopy equivalences? This is the so-called $h$-cobordism theorem <cit.>, and it turned out to be an extremely difficult problem. At dimension three, it is equivalent to the Poincaré conjecture.

Let us first setup the geometry. Let $\Gamma: \Sigma\rightarrow\Sigma'$ denote a bordism between the two surfaces $\Sigma_0,\Sigma_1$, based respectively at $(m_0,0),(m_1,1)\in Y$. The statement of bordism invariance is then given by the condition $V_{\partial\Gamma}=1$. The boundary $\partial\Gamma$ has three pieces: the "bottom cap" $\Sigma_0=\Sigma \times \{0\}$, the "top cap" $\Sigma_1=\Sigma'\times\{1\}$ and the "cylinder tube" $C\cong \partial\Sigma \times\gamma$, hence we must have

\begin{equation*}
V_{\overline{\Sigma}_1} V_{C}V_{\Sigma_0} = V^{-1}_{\Sigma_1} V_C V_{\Sigma_0} =1,
\end{equation*}

where $\overline{\Sigma}$ denotes the orientation reversal of $\Sigma$. See fig. <ref>.

Given a generic timelike curve $\gamma$ connecting the base points $(m_0,0),(m'_0,1)$ of the surfaces $\Sigma_0,\Sigma_1$, the idea is then to transport $V_{\overline{\Sigma}'\times \{1\}}$ down to the $u=0$ slice using a whiskering (<ref>):

\begin{equation*}
V_{\overline{\Sigma}'\times\{0\}} = W_{\gamma}^{-1}\rhd V_{\overline{\Sigma}_1}.
\end{equation*}

The above condition then translates into

\begin{equation}\label{stationarity}
(W_\gamma^{-1}\rhd V_{{\Sigma}'}^{-1})V_C V_\Sigma =1.
\end{equation}

This is the statement of bordism invariance.

A diagram of a bordism $\Gamma:\Sigma_0\rightarrow \Sigma_1$ and its boundary components $\overline{\Sigma}_1,\Sigma_0,C$.

Let $\Omega^\text{fr;o}_{2}(Y)_\text{/bdy}$ denote the set (Abelian group under disjoint union) of equivalence classes of surfaces in $Y$ with the same boundary, subject to the relation $\Sigma\sim\Sigma'$ iff there exists a framed open bordism $\Gamma$ between them as in Definition <ref>. We now prove the following.

If the chirality $\mathrm{d}\ell$ aligns with the direction $\mathrm{d}u$ of the foliation $\Phi$, then the 2-holonomy $V$ arising from the 3d theory (<ref>) descends to a map $\Omega^\text{fr;o}_2(Y)_\text{/bdy}\rightarrow\mathsf{H}$ on framed open bordism classes.

Given the covariance of (<ref>), we can WLOG choose the chirality $\mathrm{d}\ell = \mathrm{d}x_3$ along the third coordinate. By hypothesis, this identifies $\mathrm{d}x_3$ with the time direction $\mathrm{d}u$ and $\partial_\ell = \partial_u$.

Now recall from the proof of Proposition <ref> that the Lax 2-connection $(L,H)$ we obtain from $S_{3d}$ has $L_3=0, H_1=0, H_2=0$ when $\mathrm{d}\ell=\mathrm{d}x_3$. Since we have identified the third coordinate with $u\in\mathbb{R}$, this gives

\begin{equation*}
L_u = 0,\qquad \iota_{\partial_u}H=0.
\end{equation*}

The fact that $L_u=0$ means that the $W_\gamma=1$ is trivial along timelike curves $\gamma$, and the fact that $\iota_{\partial_u}H=0$ means that $V_C=1$ is trivial on surfaces $C$ whose normal $\hat n$ is spacelike — such as the cyinder tube piece $C$ of the surface $\partial\Gamma$. This reduces the left-hand side of (<ref>) down to

\begin{equation*}
V_{\Sigma'}^{-1}V_\Sigma=V_{\overline{\Sigma}'}V_\Sigma = V_{\overline{\Sigma}'\# \Sigma},
\end{equation*}

where we have used the fact that $V$ respects the gluing of surfaces (cf. <ref>). However, now that $\Sigma'$ and $\Sigma$ share the same time-slice $u=0$, the surface $\overline{\Sigma}'\#\Sigma$ bounds a 3-submanifold $S$ which is a homotopy relative boundary between $\Sigma'$ and $\Sigma$. We then have

$$V_{\overline{\Sigma}'\#\Sigma} = V_{\partial S}=1$$

by Theorem <ref>. This proves (<ref>).

When the two directionalities misalign, however, such as in the case of the Chern-Simons/matter theory (<ref>), then the 2-holonomies are merely homotopy invariants relative boundary, but are not framed open bordism invariants. From the QFT literature <cit.>, bordism invariants define non-perturbative anomalies; a similar sentiment is echoed in <cit.> (or Remark <ref>) for the 4d Crane-Yetter-Broda TQFT. there is therefore no perturbative degrees-of-freedom in $S_{3d}$ when the chirality and foliation align. An observation Remark <ref> that we shall make in the next section, where we study the homotopy Lie algebra structure of the currents, may provide an explanation for this phenomenon.

§ TOPOLOGICAL-HOLOMORPHIC 2+1D CURRENT ALGEBRA

Let us now investigate in more detail the algebra of currents in the boundary 3d action $S_{3d}$ given in (<ref>). Recall that in 4d Chern-Simons theory with the disorder operator $\omega=z^{-1}\rd z$ and chiral boundary conditions, one obtains the two-dimensional Wess-Zumino-Witten CFT <cit.>.  In particular, the fields  of the theory form the affine Kac-Moody current algebra. Here, we seek to characterize the 3-dimensional homotopical analogue of these currents.

We begin by recalling the fields and currents in our boundary theory. In general, we have the fields $k = \mathrm{d} gg^{-1}\in\Omega^1(Y)\otimes \g,\Theta\in\Omega^1(Y)\otimes \h$ and the following currents

\begin{equation*}
L = -k -\mu_1(\Theta^g),\qquad H = g\rhd (\rd\Theta - \Theta\wedge \Theta).
\end{equation*}

However, once we fix the Lagrangian $\mathcal{L}_\ell$ with the chirality vector $\vec\ell$, these currents are split up into two parts that are localized at the puncture $z=0$, with their dynamics governed by the boundary theory (<ref>). To write down the currents, we first introduce a (not necessarily intergrable) subbundle $T^*_\ell Y\subset T^*Y$ of covectors on $Y$ tangent to $\mathrm{d}\ell$, defined by the projection map $\operatorname{proj}_\ell:T^*Y\rightarrow T^*Y_\ell$ that maps $a\mapsto \operatorname{proj}_\ell a = (\vec a\cdot \vec \ell)\vec \ell \cdot \mathrm{d}\ell$.  Since $Y$ is 3d, there is a linear isomorphism between 1-forms and 2-forms on $Y$, and we will use this fact to also define the projection map $\operatorname{proj}_\ell:\Omega^2(Y) \rightarrow \Omega^2_\ell(Y)$ on 2-forms. We can therefore express our currents as

\begin{equation}\label{currentsdefinition}
L_\perp = -k_\perp - \mu_1(g\rhd \Theta_\perp)\,,\qquad   H_\ell = g\rhd (\operatorname{tor}\Theta_\perp - \frac{1}{2}[\Theta_\perp \times\Theta_\perp])\,,
\end{equation}

where $a_\perp = a - \operatorname{proj}_\ell a$ is the component of the 1- (or 2-)form $a$ normal to $\mathrm{d}\ell$, and $\operatorname{tor}a = \partial_\perp\times a$ denotes the scalar vorticity field of a 2d vector. The equations of motion they satisfy are given by

\begin{equation}\label{eom}
\partial_{\ell} L_\perp =0\,,\qquad \partial_{\ell} H_\ell  = 0\,,
\end{equation}

as verified in <ref>. We call the tuple $J = (L_\perp,H_\ell)$ a 2-current. Since $\Theta_\ell$ does not at all appear in (<ref>), we see that $\Theta = \Theta_\perp$, and hence we shall omit the subscript of "$\perp$" on $\Theta$ in the following.

§.§ The dual 2-currents

Now it is natural to expect the existence of a dual 2-current $\tilde J$ which satisfies an analogous set of fake- and 2-flatness equations of motion, and which can be constructed from $J$. This is in parallel with the case of the 2d Wess-Zumino-Witten theory, in which the right- and left-moving chiral currents satisfy dual equations of motion <cit.>. We will show that this is also the case for our theory (<ref>).

We being by constructing the currents $\tilde J = (\tilde L_\ell,\tilde H_\perp)$, which are given by

\begin{equation}\label{currentsdefinition1}
\tilde L_\ell = m_\ell,\qquad \tilde H_\perp = \partial_\ell \star_2\Theta + \mu_2(m_\ell,\star_2\Theta),
\end{equation}

where $m = g^{-1}\mathrm{d} g$ is the left Maurer-Cartan form and $\star_2 \Theta$ is the 2d Hodge star operator on $\Theta\in\Omega^1_\ell(Y)$. Let us now prove the following.

If $J = (L_\perp,H_\ell)$ satisfies (<ref>), then the dual currents $\tilde J = (\tilde L_\ell,\tilde H_\perp)$ satisfy

\begin{equation}\label{eom1}
\partial_\perp \tilde L_\ell = -\mu_1(\tilde H_\perp)\,,\qquad \partial_\perp\cdot \tilde H_\perp=0\,.
\end{equation}

Notice these equations (<ref>) are nothing but the fake and 2-flatness conditions for $\tilde J$.

This is a direct computation.

\begin{align*}
\partial_\perp \tilde L_\ell &= \partial_\perp (g^{-1}\partial_\ell g) = \operatorname{Ad}_g^{-1}\partial_\ell (\partial_\perp gg^{-1}) \\
&= -\operatorname{Ad}_g^{-1}\mu_1(\partial_\ell\Theta^g)) = -\mu_1(\mu_2(g^{-1}\partial_\ell g,\Theta) + \partial_\ell\Theta) = -\mu_1\tilde H_\perp,
\end{align*}

where we have used the equation $\partial_\ell L_\perp =0$ in (<ref>) in the second line. Next, we have

\begin{align*}
\partial_\perp\cdot \tilde H_\perp &= \partial_\perp \cdot (\partial_\ell \star_2 \Theta + \mu_2(g^{-1}\partial_\ell,\star_2\Theta)) \\
&=\partial_\ell (\partial_\perp\cdot\star_2 \Theta) + \mu_2(\partial_\perp(g^{-1}\partial_\ell g)\cdot \star_2\Theta) + \mu_2(g^{-1}\partial_\ell g,\partial_\perp\cdot \star_2\Theta) \\
&= \partial_\ell (\operatorname{tor} \Theta) - [H_\perp\cdot \star_2\Theta] + \mu_2(g^{-1}\partial_\ell g,\operatorname{tor}\Theta) \\
&= g^{-1}\rhd (\partial_\ell g\rhd (\operatorname{tor}\Theta)) - [\partial_\ell \star_2\Theta + \mu_2(g^{-1}\partial_\ell,\star_2\Theta)\cdot \star_2\Theta]\,,
\end{align*}

where in the third line we have used the fact that $a_\perp\cdot\star_2 b_\perp = a_\perp\times b_\perp$ and the first equation of (<ref>), and in the fourth line we have used the definition of $\tilde H_\perp$. Consider now just the second term: recall $\star_2^2=-1$ on 1-forms in 2d,

\begin{align*}
[\partial_\ell \star_2\Theta\cdot \star_2\Theta] &= \tfrac{1}{2}\partial_\ell [\star_2\Theta\cdot \star_2\Theta ] = \frac{1}{2}\partial_\ell [\Theta\times \Theta]\\
[\mu_2(g^{-1}\partial_\ell g \,,\star_2\Theta)\cdot \star_2\Theta] &= \tfrac{1}{2}\mu_2(g^{-1}\partial_\ell g,[\Theta\times \Theta])\,,
\end{align*}

where the first line follows from the product rule and the second line follows from the Jacobi identity. The sum of these terms then gives

\begin{equation*}
\frac{1}{2}g^{-1} \rhd\partial_\ell (g\rhd [\Theta\times \Theta])\,,
\end{equation*}

which combines with $ g^{-1}\rhd (\partial_\ell g\rhd (\operatorname{tor}\Theta))$ to give precisely

\begin{equation*}
g^{-1}\rhd (\partial_\ell g\rhd (\operatorname{tor}\Theta - \frac{1}{2}[\Theta\times \Theta])) = g^{-1}\rhd \partial_\ell H_\ell = 0
\end{equation*}

by (<ref>).

We call these sets equations of motion (<ref>), (<ref>) the 2-Lax equations for the 2-currents $J,\tilde J$. The goal now is to study the differential graded algebraic structures of these topological-holomorphic currents.

§.§ Fields as derived 2-group elements in D

Notice the 2-currents $J,\tilde J$, as far as their components are concerned, are just particular "splicing"/splitting of the full 1- and 2-forms $(L,H)$. Therefore, in order to characterize them, it suffices to characterize $(L,H)$ and then perform the slicing. We first introduce the covariant derivative $D = \rd - \mu_2(k,-)$ such that

\begin{equation*}
g\rhd \rd \Theta = D(g\rhd\Theta)\,.
\end{equation*}

In this way, depending on how the components of the currents are sliced up for $J,\tilde J$, the 2-Lax equations (<ref>), (<ref>) can be written entirely in terms of the fields $(g,\Theta^g)$. This ubiquitous appearance of $\Theta^g = g\rhd\Theta$ instead of the field $\Theta$ itself, both in the actions and the currents, is not a coincidence.

This can be explained in the context of the derived superfield formulation <cit.>. Recall the derived 2-group $\mathsf{D}\mathbb{G}$ as defined in <ref>; our fields $(g,\Theta^g)$ are degree-0 elements of the following graded algebra:

\begin{equation*}
\mathsf{D}\mathscr{L}^\bullet=\Omega^\bullet(Y) \otimes \mathsf{D}\mathbb{G},\qquad \mathsf{D}\mathscr{L}^0 \cong (C^\infty(Y)\otimes G)\oplus (\Omega^1(Y)\otimes \h).
\end{equation*}

Due to the 2-group structure inherited from $\mathbb{G}$, this perspective allows us to use the whiskering operation (see <ref>) to write

\[(g,g\rhd\Theta)= % https://q.uiver.app/#q=WzAsMixbMCwwLCJcXGFzdCJdLFszLDAsIlxcYXN0Il0sWzAsMSwiZyIsMCx7ImN1cnZlIjotNX1dLFswLDEsImdcXGJhclxcbXVfMVxcbGVmdChcXGV4cFxcaW50X1xcZ2FtbWFcXFRoZXRhX1xccGVycFxccmlnaHQpIiwyLHsiY3VydmUiOjV9XSxbMiwzLCJcXGV4cFxcaW50X1xcZ2FtbWFcXFRoZXRhX1xccGVycCIsMCx7InNob3J0ZW4iOnsic291cmNlIjoyMCwidGFyZ2V0IjoyMH19XV0=
\begin{tikzcd}
\ast &&& \ast
\arrow[""{name=0, anchor=center, inner sep=0}, "g", curve={height=-30pt}, from=1-1, to=1-4]
\arrow[""{name=1, anchor=center, inner sep=0}, "{\bar\mu_1\left(\exp\int g\rhd\Theta\right)g}"', curve={height=30pt}, from=1-1, to=1-4]
\arrow["{\exp\int_\gamma g\rhd \Theta}"{description}, shorten <=8pt, shorten >=8pt, Rightarrow, from=0, to=1]
\end{tikzcd}\]

\[
= \begin{tikzcd}
\ast && \ast &&& \ast
\arrow[""{name=0, anchor=center, inner sep=0}, "1", curve={height=-30pt}, from=1-3, to=1-6]
\arrow[""{name=1, anchor=center, inner sep=0}, "{\bar\mu_1\left(\exp\int \Theta\right)}"', curve={height=30pt}, from=1-3, to=1-6]
\arrow["g", from=1-1, to=1-3]
\arrow["{\exp\int_\gamma \Theta}"{description}, shorten <=8pt, shorten >=8pt, Rightarrow, from=0, to=1]
\end{tikzcd} = g\rhd (1,\Theta).
% \begin{tikzcd}
%   \ast \arrow[r, bend left, "g", ""{name=U,inner sep=1pt,below}]
%   \arrow[r, bend right, "g\bar\mu_1\left(\exp\int_\gamma\Theta_\perp\right)"{below}, ""{name=D,inner sep=1pt}]
%   & \ast
%   \arrow[Rightarrow, from=U, to=D, "g\rhd\Theta_\perp"]
% \end{tikzcd} = \begin{tikzcd}
%   \ast \arrow[r,"g"] & \ast \arrow[r, bend left, "1", ""{name=U,inner sep=1pt,below}]
%   \arrow[r, bend right, "\bar\mu_1\left(\exp_\gamma\Theta_\perp\right)"{below}, ""{name=D,inner sep=1pt}]
%   & \ast
%   \arrow[Rightarrow, from=U, to=D, "\Theta_\perp"]
% \end{tikzcd}.
\]

This explains why $\Theta^g=g\rhd\Theta$ keeps showing up in our theory (<ref>) — it was encoding whiskerings by $g$! The subspace $\Omega^1\otimes\h\subset \mathsf{D}\mathscr{L}^0$, within which the field $\Theta$ itself lives, then naturally acquires the interpretation of face decorations in the derived 2-group.

§.§.§ The differential in D

This graded algebra $\mathsf{D}\mathscr{L}^\bullet$ is equipped with the differential $\hat \rd:\mathsf{D}\mathscr{L}^\bullet\rightarrow \mathsf{D}\mathscr{L}^{\bullet+1}$ that satisfies the usual graded Leibniz rule, but also a compatibility with whiskering $\rhd$. Since $\G$ only has two terms, so does each degree of $\mathsf{D}\mathscr{L}^\bullet$. The differential can then be defined in terms of a tensor product $\hat \rd = \rd_1 \otimes \rd_2$. On elements $(g,g\rhd\Theta)$ of degree-0, we can explicitly write

\begin{equation}
\rd_1(g,\Theta^g) = \overrightarrow{\Delta} g - \mu_1\Theta^g, \qquad \rd_2\Theta^g = \rd\Theta^g - \Theta^g\wedge\Theta^g,\label{differential}
\end{equation}

where $\overrightarrow{\Delta}g= -\rd gg^{-1}$ is the right Maurer-Cartan form and $\rd_2$ is the covariant derivative associated to a connection $\Theta^g$ on a $\mathsf{H}$-bundle over $X$ (recall $\mathsf{H}$ is a $G$-module).

For $g,h\in C^\infty(Y)\otimes G,$ and $ \Theta\in\Omega^1(Y)\otimes\h$, the compatibility with a whiskering by $g$ can then be written as

\begin{equation*}
g\rhd \hat{\rd} (h,h\rhd \Theta) = \hat{D} (gh,(gh)\rhd \Theta)\,,\qquad \hat D = D_1\otimes D_2\,,
\end{equation*}

where $ D_1 =  \rd_1 - [\partial gg^{-1},\cdot]$ and $D_2 = \rd_2 - \mu_2(\partial gg^{-1},\cdot)$. The nilpotency of the differential, $\hat \rd^2=0$, is equivalent to the fake- and 2-flatness of $(L,H)$. This then allows us to identify the (off-shell) currents
$$(L,H) = g\rhd \hat{\rd}(1,\Theta) = \hat D(g,\Theta^g) \in \mathsf{D}\mathscr{L}^1$$
as a degree-1 element in this derived complex.

§.§.§ Characterizing the currents

To characterize the on-shell 2-currents, we slice up the dg algebra $\mathsf{D}\mathscr{L}^\bullet$ into two according to the projection $\operatorname{proj}_\ell$ and its complement. This induces a fibration

\begin{equation*}
\mathsf{D}\mathscr{L}_\perp^\bullet \cong (\mathsf{D}\mathscr{L}^\bullet/\mathsf{D}\mathscr{L}_\ell^\bullet)\hookrightarrow \mathsf{D}\mathscr{L}^\bullet \rightarrow\mathsf{D}\mathscr{L}^\bullet_\ell\rightarrow 0,
\end{equation*}

where $\mathsf{D}\mathscr{L}_\ell^\bullet = \Omega_\ell^\bullet(Y)\otimes \mathsf{D}\mathbb{G}$. This splits the differential $\hat \rd = \hat \rd_\perp + \hat \rd_\ell$, which are individually nilpotent $\hat \rd_\perp^2,\hat \rd_\ell^2 =0$ on the fields $(g,\Theta^g)\in \mathsf{D}\mathscr{L}^0$, as a consequence of the 2-Lax equations (<ref>) and (<ref>). We can then characterize the current algebras as the following subspaces:

\begin{align}
\D &= \{J \in g\rhd \hat{\rd}_\perp(\Omega^1\otimes\h)\mid \hat \rd_\ell J =0\} \label{homotopycurrents}\\
\tilde\D &= \{\tilde J\in g\rhd \hat{\rd}_\ell(\Omega^1\otimes\h)\mid \hat \rd_\perp \tilde J  =0\}.\label{homotopycurrents1}
\end{align}

Note $\mu_1$ appears in both $\hat \rd_\ell,\hat\rd_\perp $ due to the definition (<ref>). More explicit characterizations are possible given the data of a THF $\Phi$ on $Y$; we will use $\Phi$ more explicitly later.

§.§ Graded Lie algebra structure of the currents

In the following, we will extract the graded Lie algebra structure of the currents. We shall do this from the "dual" perspective of the conserved Noether charges associated to the semilocal symmetries of the theory studied in <ref>, and analyzing their transformation properties.

Taking inspiration from covariant field space approach <cit.>, we first smear the currents with gauge fields $\alpha \in C^\infty(Y)\otimes \g$ and $\Gamma\in\Omega^1(Y)\otimes \h$, which gives us the local Noether charges (also called "charge aspects"),

\begin{equation*}
Q_{(\alpha,\Gamma)} = \langle (\alpha,\Gamma),J\rangle = \langle \alpha, H_\ell\rangle + \langle \Gamma,L_\perp\rangle \,.
\end{equation*}

For the currents $J$ of the theory (<ref>), the above quantities are 2-forms that live in the image of the projection $\operatorname{proj}_\ell$ — that is, they have no legs along $\mathrm{d}\ell$. On-shell of the equations of motion (<ref>), a simple computation gives

\begin{equation*}
\rd Q_{(\alpha,\Gamma)} =\langle \partial_\ell\alpha,H_\ell\rangle + \langle \partial_\ell\Gamma,L_\perp\rangle
\end{equation*}

which vanishes provided we impose the boundary conditions

\begin{equation}\label{bc1}
\partial_\ell\alpha=0\,,\qquad \partial_\ell\Gamma =0\,.
\end{equation}

In other words, the currents $Q_{(\alpha,\Gamma)}$ generate the infinitesimal left-acting symmetries $\L^L_{3d}(\ell)$ of the theory (<ref>). On the other hand, for the 2-current $\tilde J$, the local charge aspects

\begin{equation*}
\tilde Q_{(\alpha,\Gamma)} = \langle(\alpha,\Gamma),\tilde J\rangle = \langle \alpha,\tilde  H_\perp\rangle + \langle \Gamma,\tilde L_\ell\rangle
\end{equation*}

are 2-forms living in the kernel of $\operatorname{proj}_\ell$ — meaning they have only legs including $\mathrm{d}\ell$. On-shell of (<ref>), we find

\begin{align*}
\rd\tilde Q &= \langle \partial_\perp \alpha, \tilde H_{\perp}\rangle + \langle \partial_\perp \cdot\star_2\Gamma,\tilde L_\ell\rangle - \langle \Gamma,\partial_\perp \tilde L_\ell\rangle  \\
&= \langle \partial_\perp \alpha + \mu_1\Gamma,\tilde H_\perp\rangle + \langle \operatorname{tor}\Gamma, \tilde L_\ell\rangle\,,
\end{align*}

which vanishes provided we impose the boundary conditions[Recall we have without loss of generality (cf. <ref>) assumed that $\Gamma$ has no component along the chiral direction $\mathrm{d}\ell$, and hence $\Gamma_\perp=\Gamma$.]

\begin{equation}\label{bc2}
\partial_\perp \alpha + \mu_1\Gamma=0\,,\qquad \operatorname{tor} \Gamma=0\,.
\end{equation}

In other words, the currents $\tilde Q_{(\alpha,\Gamma)}$ generate the (infinitesimal) right-acting symmetries $\mathring{\L}_{3d}^R(\ell)$ of the theory (<ref>).

We now prove the following.

The above conserved Noether charge aspects $Q,\tilde Q$ satisfy the following graded algebra structure:

\begin{align}
[Q_{(\alpha,\Gamma)},Q_{(\alpha',\Gamma')}] &= -Q_{([\alpha',\alpha],\mu_2(\alpha',\Gamma)-\mu_2(\alpha,\Gamma'))} \nonumber\\
&\qquad -\langle \Gamma,\partial \alpha' + \mu_1\Gamma'\rangle + \langle \alpha,\operatorname{tor}\Gamma'\rangle \label{bracket} \\
[\tilde Q_{(\alpha',\Gamma')},\tilde Q_{(\alpha,\Gamma)}] &= -\tilde Q_{([\alpha,\alpha'],\mu_2(\alpha,\Gamma')-\mu_2(\alpha',\Gamma))}  \nonumber\\
&\qquad - \langle \Gamma',\partial_\ell\alpha\rangle +\langle \alpha',\partial_\ell \Gamma\rangle,.\label{bracket1}
\end{align}

Moreover, $[Q,\tilde Q]=0.$

Following <cit.>, we proceed with the computation of the (graded) Lie algebra bracket between the charges using the general formula <cit.>

\begin{equation}
[Q_{(\alpha,\Gamma)}, Q_{(\alpha',\Gamma')}] = -\delta_{(\alpha',\Gamma')}Q_{(\alpha,\Gamma)}\,,
\end{equation}

where $\delta_{(\alpha,\Gamma)}$ denotes a gauge variation. Let us consider first the gauge variations of $Q\in\hat\D$, which is the Noether charges constructed from the 2-current $J = (L_\perp,H_\ell)$. Recall these charges generate the infinitesimal left variation,

\begin{equation}
\label{ec:infleftaction}
\delta_{\alpha'} g = -{\alpha'}\cdot g,\qquad \delta_{(\alpha',\Gamma')} \Theta = -g^{-1}\rhd \Gamma',
\end{equation}

we find that

\begin{align*}
\delta_{(\alpha',\Gamma')} L_\perp &= - [\alpha', L_\perp]+\partial_\perp \alpha' + \mu_1 \Gamma',\\
\delta_{(\alpha',\Gamma')}  H_\ell &= - \mu_2(\alpha', H_\ell) - \mu_1( L_\perp, \Gamma') - \operatorname{tor}\Gamma',
\end{align*}

in which $(\alpha',\Gamma')$ parameterize the symmetries satisfying the boundary conditions (<ref>). This then leads to precisely (<ref>).

Now analogously, we compute the bracket on the dual Noether charges $\tilde Q\in \hat{\tilde \D}$ constructed from $\tilde J = (\tilde L_\ell,\tilde H_\perp)$. Recalling these generate the infinitesimal right-variation

\begin{equation}
\label{ec:infrightaction}
\tilde\delta_{\alpha} g = g\cdot\alpha,\qquad \tilde\delta_{(\alpha,\Gamma)} \Theta = -\mu_2(\alpha,\Theta) + \Gamma,
\end{equation}

we can then compute

\begin{align*}
\tilde\delta_\alpha \tilde L_\ell &= -[\alpha,\tilde L_\ell] + \partial_\ell \alpha, \\
\tilde\delta_{(\alpha,\Gamma)} \tilde H_\perp &= -\mu_2(\alpha,\tilde H_\perp) -\mu_2(\tilde L_\ell,\Gamma) + \partial_\ell\Gamma\,,
\end{align*}

where $(\alpha,\Gamma)$ satisfy the boundary conditions (<ref>). These then give rise to the brackets (<ref>). Finally, to show that $[Q,\tilde Q]=0$ we compute

\begin{equation}
\begin{aligned}
\tilde \delta_{\alpha'} L_\perp &=-\mathrm{Ad}_g(\partial_\perp \alpha'+\mu_1(\Gamma')) \\
\tilde \delta_{(\alpha',\Gamma')}H_\ell &= g\rhd \mathrm{tor} \, \Gamma'
\end{aligned}
\end{equation}

where in the variation of $H_\ell$ we have used the graded Jacobi identity and the first boundary condition in (<ref>). In particular, we note that both of the above vanish due to the boundary conditions (<ref>). An entirely analogous computation shows that $\delta_\alpha \tilde L_\ell =0$ and $\delta_{(\alpha,\Gamma)}\tilde H_\perp=0$.

Notice these two brackets (<ref>), (<ref>) take identical forms. In particular, the central terms in both of these expressions can be written collectively as

\begin{equation}
\langle (\alpha,\Gamma),\hat \rd(\alpha',\Gamma')\rangle\,, \label{central}
\end{equation}

depending on the boundary conditions (<ref>), (<ref>) that are satisfied by $(\alpha,\Gamma)$ (recall $\Gamma_\ell=0$). Note $\hat \rd$ is the differential on $\D,\tilde\D$ defined in (<ref>), which includes the map $\mu_1$.

These structures run in complete parallel to the story in the 2d WZW model <cit.>, where the dual charges satisfy identical algebras and they do not have mutual brackets. Moreover, we will show in the following that (<ref>) can in fact be identified as a Lie 2-algebra 2-cocycle characterizing a central extension sequence.

We emphasize here that $\mathrm{d}\ell$ is the direction of the chirality, so our computations so far are a priori independent of the direction of a THF $\Phi$ of $Y$. However, we will use the data of $\Phi$ in the following to construct a differential for the charge algebra $\hat\D$ and $\hat{\tilde\D}$.

§.§ Central extensions in the affine Lie 2-algebra

In this section, we prove that the central term (<ref>) in fact defines a Lie 2-algebra 2-cocycle. Given an arbitrary Lie 2-algebra $\G$ and an Abelian $\G$-module $\mathfrak{V}$ — namely a 2-term complex $\mathfrak{V} = W\xrightarrow{\phi}V$  of vector spaces <cit.> with an action $\rho: \G\rightarrow \mathfrak{gl}(\mathfrak{V})$ by $\G$ — the (twisted) cohomology classes $H^2_\rho(\G,\mathfrak{V})$ of such 2-cocycles have been shown <cit.> to classify central extension sequences of Lie 2-algebras of the form

\begin{equation*}
\mathfrak{V}\rightarrow \hat\G\rightarrow \G
\end{equation*}

over the ground field $\mathbb{K}$ of characteristic 0.

For our purposes, it suffices to consider $\mathfrak{V}$ as a trivial $\G$-module, where $\rho=0$. We first recall the data of a representative 2-cocycle $(s_1,s_2,s_3)$ of an extension class.

Suppose $\mathfrak{V}$ is a trivial $\G$-module, where $\G = \h\xrightarrow{\mu_1}\g$ is a Lie 2-algebra over the ground field $\mathbb{K}$. A Lie 2-algebra 2-cocycle $s=(s_1,s_2,s_3): \G\otimes\G\rightarrow\mathfrak{V}$ with coefficients in $\mathfrak{V}$ is the data of a triple

\begin{equation*}
s_1: \g\otimes\g\rightarrow V,\qquad s_2: \g\otimes\h\rightarrow W,\qquad s_3: \h\rightarrow V,
\end{equation*}

such that

\begin{align}
0 &= ~\circlearrowright ~ s_1([\sfx_1,\sfx_2],\sfx_3),\label{jac2} \\
0 &= s_2(\mu_1(\sfy_1),\sfy_2) + s_2(\mu_1(\sfy_2),\sfy_1) \label{cond1}\\
0&=~ \circlearrowright~ s_2(\mu_1([\sfy_1,\sfy_2]),\sfy_3),\label{jac3} \\
0&= s_1(\sfx,\mu_1(\sfy)) - \phi s_2(\sfx,\sfy) + s_3(\sfx\rhd \sfy),\label{cond2} \\
0&= s_2([\sfx_1,\sfx_2],\sfy) - s_2(\sfx_1,\sfx_2\rhd \sfy) + s_2(\sfx_2,\sfx_1\rhd \sfy),\label{jac}\\
0&= s_2(\sfx,[\sfy_1,\sfy_2]) - s_2(\mu_1(\sfx\rhd \sfy_1),\sfy_2) + s_2(\mu_1(\sfx\rhd \sfy_2),\sfy_1)\label{jac1}
\end{align}

for all $\sfx,\sfx_1,\sfx_2,\sfx_3\in \g$ and $\sfy,\sfy_1,\sfy_2,\sfy_3\in\h$, where $\circlearrowright$ denotes a sum over cyclic permutations.

The full list of conditions for when $\mathfrak{V}$ is not a trivial $\G$-module can be found in <cit.>.

§.§.§ Constructing the differential in D

To begin, we first endow a grading where the $\Gamma$-charges have degree-0 and the $\alpha$-charges have degree-1. The idea is to perform a pushforward at the differential forms factor $\Omega^\bullet$ of $\hat{\tilde{\D}}$ (or $\hat\D$). This is accomplished by viewing the THF $\Phi$ as inducing a fibration <cit.>

\begin{equation*}
\R_{\mathrm{d}x_3} \hookrightarrow Y\rightarrow M
\end{equation*}

over a (compact) complex manifold $M$, where $\mathbb{R}_{\mathrm{d}x_3}$ is the 1-dimensional fibre along the covector direction $\mathrm{d}x_3$ of the THF (which may differ from the chirality $\mathrm{d}\ell$). For simplicity, we compactify the fibre from $\mathbb{R}_{\mathrm{d}x_3}$ to a circle $S^1_{\mathrm{d}x_3}$ such that $Y$ defines a Seifert fibration $p:Y\rightarrow M$ <cit.>, then we can perform a fibrewise integration $\pi_*:\Omega^{\bullet}(Y)\rightarrow \Omega^{\bullet-1}(M)$ such that,[This gives rise to the well-known Gysin homomorphism $\pi^!:H^\bullet(Y)\rightarrow H^{\bullet-\operatorname{rank}E}(M)$ <cit.> on cohomology, where $E$ is the compact fibre of $Y\rightarrow M$.] on 1-forms,

\begin{equation}\label{integrate}
(\pi_*\omega)_{(w,\bar w)} = \int_{S^1_{\mathrm{d}x_3} = p^{-1}(w,\bar w)}\omega,\qquad \omega\in\Omega^1(Y).
\end{equation}

We make use of this map to define the differential $\hat\mu_1:\hat\D_1\rightarrow \hat\D_0$ as a pullback by $\pi_*\otimes\mu_1$,

\begin{equation*}
(\hat\mu_1\tilde Q)_\Gamma = \tilde Q_{(\pi_*\otimes \mu_1)\Gamma} = \langle \mu_1(\pi_*\Gamma),H_\perp\rangle,
\end{equation*}

assigning a $\Gamma$-charge to an $\alpha$-charge. We construct the same differential for the dual charges $\hat\D$.

We are finally to prove the main theorem of this section.

The tuple $(\hat{\tilde{\D}},\hat\mu_1,\hat\mu_2=[-,-])$ with $[-,-]$ given by (<ref>) is a Lie 2-algebra fitting into the central extension sequence

\begin{equation*}
\underline{\mathbb{K}} \rightarrow \hat{\tilde\D}\rightarrow \tilde\D,
\end{equation*}

iff (<ref>) defines a Lie 2-algebra 2-cocycle associated to this extension, where $\underline{\mathbb{K}} = \mathbb{K}\xrightarrow{0}\mathbb{K}$.

Recall $\Gamma_\ell=0$. We will first perform some general computations before we specialize $(\alpha,\Gamma)$ to the space $\mathring{\L}_{3d}^R$ parameterizing the charges $\hat{\tilde{\D}}$. From the central term (<ref>), we shall define

\begin{equation}
s_1(\Gamma,\Gamma') = \langle \Gamma,\mu_1\Gamma'\rangle,\qquad s_2(\Gamma,\alpha) = \langle \operatorname{tor}\Gamma,\alpha\rangle,\qquad s_3=0\label{2cocycle}.
\end{equation}

It is useful to extend $s_2$ to a skew-symmetric map by an integration by parts,

\begin{equation*}
s_2(\alpha,\Gamma) = \langle \partial \alpha,\Gamma\rangle =- \langle \operatorname{tor}\Gamma,\alpha\rangle = -s_2(\Gamma,\alpha).
\end{equation*}

As $\phi=0,s_3=0$, and since $\pi_*$ acts trivially on the $\alpha$-charges, we see that (<ref>), (<ref>) are trivially satisfied.

Let us now check the graded Leibniz rule,

\begin{equation*}
\hat\mu_1\circ[-,-] = [-,-]\circ (\hat\mu_1\otimes 1 +(-1)^\text{deg} 1\otimes \hat\mu_1).
\end{equation*}

From (<ref>), the left-hand side gives

\begin{align*}
\hat\mu_1[Q_{(\alpha,\Gamma)},Q_{(\alpha',\Gamma')}] &= -\hat\mu_1(Q_{([\alpha',\alpha],\mu_2(\alpha',\Gamma) - \mu_2(\alpha,\Gamma'))} + \langle (\alpha,\Gamma),\hat d(\alpha',\Gamma')\rangle )\\
&= - Q_{[\alpha',(\pi_*\otimes \mu_1)\Gamma] - [\alpha,(\pi_*\otimes \mu_1)\Gamma']}\\
&= -Q_{[\alpha',\mu_1(\pi_*\Gamma)]} - Q_{[\alpha,\mu_1(\pi_*\Gamma')]},
\end{align*}

while the right-hand side gives

\begin{align*}
[(\hat\mu_1Q)_\Gamma,Q_{(\alpha',\Gamma')}] - [Q_{(\alpha,\Gamma)},(\hat\mu_1Q)_{\Gamma'}] &= -Q_{([\alpha',(\pi_*\otimes\mu_1)\Gamma],-\mu_2((\pi_*\otimes\mu_1)\Gamma,\Gamma'))} \\
&\qquad + Q_{[(\pi_*\otimes\mu_1)\Gamma',\alpha], \mu_2((\pi_*\otimes\mu_1)\Gamma',\Gamma)} \\
&\qquad +\langle \mu_1(\pi_*\Gamma),\operatorname{tor}\Gamma'\rangle - \langle \Gamma,\partial \mu_1(\pi_*\Gamma')\rangle \\
&= -Q_{[\alpha',\mu_1(\pi_*\Gamma)]} + Q_{[\mu_1(\pi_*\Gamma'),\alpha]}  \\
&\qquad + \langle \mu_1(\pi_*\Gamma),\operatorname{tor}\Gamma'\rangle + \langle \operatorname{tor}\Gamma,\mu_1(\pi_*\Gamma')\rangle\\
&\qquad - (Q_{\mu_2(\mu_1(\pi_*\Gamma'),\Gamma)}+ Q_{\mu_2(\mu_1(\pi_*\Gamma),\Gamma')}).
\end{align*}

Now given the parameters $\Gamma$ in $\mathring{\L}^R_{3d}$ must be valued in the maximal Abelian $\t\subset\h$, the final terms involving $\mu_2(\mu_1(\pi_*\Gamma'),\Gamma) = [\pi_*\Gamma',\Gamma] =0$ vanish. The two sides then coincide iff $s_2$ (<ref>) satisfies (<ref>).

We now check the graded Jacobi identities. Due to the form of the bracket (<ref>), the graded Jacobi identities for the non-central extension terms follow directly from those of $\G$. The remaining terms read

\begin{align}
\langle [\alpha,\alpha'],\operatorname{tor}\Gamma\rangle - \langle \alpha',\operatorname{tor}\mu_2(\alpha,\Gamma)\rangle + \langle \alpha,\operatorname{tor}\mu_2(\alpha',\Gamma)\rangle &=0 \label{proofcond},
\\
\langle \mu_1\Gamma,\mu_2(\alpha,\Gamma')\rangle + \langle \alpha,[\Gamma',\Gamma]\rangle - \langle \mu_1\Gamma',\mu_2(\alpha,\Gamma')\rangle &=0 \nonumber,
\end{align}

which are precisely the conditions (<ref>), (<ref>) for $s$ (<ref>). However, due to the lack of a $\langle\Gamma,\mu_1\Gamma'\rangle$ contribution in the central term of the bracket (<ref>) for $\hat{\tilde{\D}}$, (<ref>) is trivially satisfied for $\hat{\tilde\D}$. And finally, since $\Gamma$ is Abelian, (<ref>) is also trivially satisfied. This completes the proof.

Hence $\check\mu_1$ indeed preserves the bracket.

From (<ref>), we see that the above general computations also applies to the charges $\hat\D$. Whence, under mild assumptions,[For $\hat\D$, we must keep quadratic terms $\sim [\Gamma,\Gamma'],\langle\Gamma,\mu_1\Gamma'\rangle$ in our computations. Assuming the quantity $[\pi_*\Gamma,\Gamma'] = \llbracket \Gamma,\Gamma'\rrbracket$ defines a Lie bracket $Q_{\llbracket \Gamma,\Gamma'\rrbracket} = [Q_{\Gamma'},Q_\Gamma]$ on the charges, (<ref>) follows from (<ref>) given $\hat\mu_1$ preserves this bracket. One can check that this is indeed the case:

\begin{equation*}
(\pi_*\otimes\mu_1)(\llbracket\Gamma,\Gamma'\rrbracket) = \int_{S^1_{\mathrm{d}x_3}} \mu_1[(\pi_*\Gamma)_{(w,\bar w)},\Gamma'_{(x_3,w',\bar w')}] = [\mu_1(\pi_*\Gamma_{(w,\bar w)}),\mu_1(\pi_*\Gamma_{(w',\bar w')})],
\end{equation*}

where $(w,\bar w),(w',\bar w')\in M$ are local coordinates on the base space $M$ and $x_3\in S^1_\Phi$ is along the fibre.] one can prove that it also forms a centrally extended Lie 2-algebra. We call $\hat\D,\hat{\tilde{\D}}$ the homotopy affine Lie algebras of planar currents.

Recall that $\Gamma_\ell=0$. In the affine Lie 2-algebras $\hat\D,\hat{\tilde\D}$, the pushforward $\pi_*=0$ (<ref>) is trivial if the THF $\Phi$ aligns with $\mathrm{d}\ell$. This renders the charge algebra $\hat\D$ both strict and skeletal, and hence lacks a degree-3 Postnikov class $\kappa=0$ <cit.>; see also <ref>. As one can construct gauge-invariant local perturbative observables from such a class (strictly speaking one does this from the cohomology $H(\hat\D)$ <cit.>, which contains the Postnikov class as a $k$-invariant), this suggests that $S_{3d}$ hosts no perturbative anomalies on 3-manifolds. This is consistent with our result found in <ref> that, in the aligned case, the surface holonomies are bordism invariants and thus only hosts non-perturbative data.

All in all, the conserved Noether charges in our theory $S_{3d}$ form centrally-extended Lie 2-algebras $\hat\D,\hat{\tilde\D}$. These algebras are parameterized by the semilocal residual symmetries $\L_\text{sym}\subset C^\infty(Y)\oplus \Omega^1(Y)$ of $S_{3d}$, which are function spaces of infinite-dimension over $\mathbb{K}$. This is of course in direct analogy with the affine Lie algebra underlying the charges in the 2d WZW model <cit.>, which is also infinite-dimensional.

§ OUTLOOK

In this paper, we have constructed a 3d integrable field theory from a higher-gauge theory by following through with the Costello-Yamazaki localization procedure <cit.>. We also studied in detail the properties of this 3d field theory, and proved several very interesting facts about its properties. We believe that there are many more open questions to be answered in regards to this theory, and the connection between higher-gauge theories and higher-dimensional integrable field theories in general. Here, we list a few of them in this section.

Categorified integrability. Recall that the conserved 2-holonomies (<ref>) in our 3d field theory come labelled by the categorical characters $\mathcal{X}_k$ of the Lie 2-group $\mathbb{G}$ and homotopy classes $[\Sigma]$ of surfaces in $Y$. We are lead to two very interesting problems to tackle when considering the inteegrabiliy of these charges.

* The label $[\Sigma]$: differential gerbes and 2-tangle invariants. In 3d TQFTs, it is known that the algebra of the Wilson lines define invariants of framed links <cit.>. The usual singular homology is not enough to capture such data: indeed, framed link invariants satisfy certain knot and satellite relations that are seen only by differential — possibly Dolbeault, when the chirality and foliation are misaligned — cohomology <cit.> (eg. the Deligne-Beilinson cohomology <cit.> in Abelian Chern-Simons theory). As such, the label $[\Sigma]$ should be treated as differential non-Abelian gerbes <cit.> which classify invariants of framed 2-tangles <cit.> in $Y$ (or its thickening $Y\times [0,1]$).

* The label $k$: orthogonality of categorical characters. In regular representation theory, one can prove that there are countably many orthogonal irreducible characters of a compact Lie group that span the representation ring <cit.>. Correspondingly, the label $k$ should run over all indecomposable $\mathbb{G}$-module categories, but categorical character theory is not yet powerful enough to prove this (see eg. 9.6 in <cit.> and Proposition 1.2.19 in <cit.>). Moreover, while an inner product is known to exist for categorical characters <cit.>, there is no satisfactory notion of "orthogonality" for them at the moment. One of the authors (HC) of this paper is currently investigating this problem.

Resolving these two issues would pin down precisely the integrable structure of the 3d model $S_{3d}$, but they are also of major interest individually. For instance, higher-tangle invariants would help define 4d non-semisimple TQFTs (which have a chance of definition exotic 4-manifold invariants <cit.>), and categorical character theory would have very explicit applications in the construction of 4d gapped phases in condensed matter theory <cit.>.

Quantization of $S_{3d}$: the 2-Kac-Moody algebra. An obvious next step to take would be to try and quantize the theory $S_{3d}$. Towards this, we wish to integrate the affine Lie 2-algebra $\hat\D$ (or $\hat{\tilde\D}$) through a normal mode expansion on $\mathbb{C}P^1\subset Y$, each mutually orthogonal with respect to an appropriate integration pairing
$$\langle-.-\rangle: \hat\D\odot\hat\D \rightarrow\Bbbk,$$
in order to describe a higher homotopy version "$\widehat{\Sigma_s\mathbb{G}}$" (where $s$ is the extension 2-cocycle (<ref>)) of the Kac-Moody algebra that controls the monodromy matrices (<ref>). As $S_{3d}$ is a topological-holomorphic field theory, it would be interesting to understand the structure of this object $\widehat{\Sigma_s\mathbb{G}}$ and its relation to the raviolo vertex operator algebra <cit.>.

The quantum Hilbert space associated to $S_{3d}$ are then given by unitary (categorical) representations of this "2-Kac-Moody algebra",

\begin{equation*}
\operatorname{UMod}_{\mathsf{2Vect}}(\widehat{\Sigma_s\mathbb{G}}).
\end{equation*}

In analogy with the modules of the Kac-Mody algebra $\widehat{\Omega_kG}$ underlying the 2d WZW model <cit.>, this 2-category should have equipped higher modular and braided tensor structures, which are of major interest in both pure categorical algebra <cit.> as well as the construction of higher-skein "lasagna modules" in 4d TQFTs <cit.>.

Other choices of disorder operators In this article we have considered a unique choice of disorder operator $\omega = z^{-1}\rd z$, which led to a rich family of three-dimensional theories satisfying properties which are higher homotopical analogues of those satisfied by the WZW model in two dimensions. Notably, in <cit.> they considered a variety of disorder operators, except for the one hereby considered. Their construction led to different $3$d theories which have a Lax connection with spectral parameter. It would be very interesting to perform a deep analysis of the holonomies, focusing on the invariance under homotopies relative boundary for the Lax connections constructed there. Morever, it would be appealing to analyze the symmetries of the theories obtained in <cit.>, and construct the corresponding current algebras.

§ CLASSIFICATION OF LIE 2-GROUPS AND LIE 2-ALGEBRAS

Let us begin with a brief overview of the notion of 2-groups, Lie 2-algebras and 2-gauge theory. Recall a 2-group $\mathbb{G}$ is, by definition, a monoidal groupoid $\Gamma\rightrightarrows G$ in which all objects are invertible. There are many different but equivalent characterizations of a 2-group, namely as a pointed 2-groupoid, as a group object internal to the category of groups, or as a 2-term crossed-complex of groups as in Definition <ref>.

The fundamental theorem of Lie 2-groups state that there is a one-to-one correspondence between connected, simply-connected Lie 2-groups and Lie 2-algebras. Definition <ref> gives Lie 2-algebras as a differential crossed-complex <cit.>, which is equivalent to the $L_2$-algebra definition by simply identifying $\mu_1=t$ and $\mu_2 = (\rhd,[-,-])$. The homotopy Jacobi identities are equivalent to (<ref>) and the 2-Jacobi identities.

The goal of this section is to briefly recall some classification results for Lie 2-groups and Lie 2-algebras. If $\mathbb{G},\mathbb{G}'$ are two Lie 2-groups, a Lie 2-group homomorphism $(\Phi,\Psi): \mathbb{G}\rightarrow\mathbb{G}'$ is a tuple of Lie group maps $\Phi:G\rightarrow G',\Psi:\mathsf{H}\rightarrow\mathsf{H}'$ such that

* $\bar\mu_1'\circ \Phi = \Psi\circ\bar\mu_1$, and

* $\Phi(x\rhd y) = (\Psi x)\rhd' (\Phi y)$ for all $x\in G,y\in\mathsf{H}$.

Similarly, If $\mathfrak{G},\mathfrak{G}'$ are two Lie 2-algebras, then a Lie 2-algebra homomorphism $(\phi,\psi):\mathfrak{G}\rightarrow\mathfrak{G}'$ is a tuple of Lie algebra maps $\phi: \h\rightarrow \h',\psi: \g\rightarrow \g'$ such that

* $\mu_1'\circ \phi = \psi \circ \mu_1$, and

* $\phi(\sfx\rhd \sfy) = (\psi \sfx)\rhd'(\phi \sfy)$ for each $\sfx\in \g,\sfy\in \h$.

As in the Lie 1-algebra case, this notion transports to maps between principal 2-bundles $\mathcal{P}\rightarrow\mathcal{P}$ <cit.>.

In contrast to Lie groups, the notion of "isomorphism" for Lie groupoids is categorical equivalence <cit.>. A bit more explicitly, we say $\mathbb{G}\simeq\mathbb{G}'$ are equivalent iff there exist Lie 2-group maps $(\Phi,\Psi):\mathbb{G}\rightarrow\mathbb{G}'$ and $(\Phi,\Psi)^{-1}:\mathbb{G}'\rightarrow\mathbb{G}$ such that there are invertible natural transformations

\begin{equation*}
\mathcal{H}:(\Phi,\Psi)^{-1}\circ (\Phi,\Psi)\Rightarrow\id_{\mathbb{G}}, \qquad \mathcal{E}: (\Phi,\Psi)\circ (\Phi,\Psi)^{-1}\Rightarrow \id_{\mathbb{G}'}.
\end{equation*}

The following is a classic result first proven by Hoàng Xuân Sính <cit.>, who is a Vietnamese mathematician taught by Alexander Grothendieck.

2-groups $\mathbb{G}$ are classified by the following data: a group $\Pi_1=\operatorname{coker}\bar\mu_1$, a $\Pi_1$-module $\Pi_2=\operatorname{ker}\bar\mu_1$, and a degree-3 group cohomology class $\tau\in H^3(\Pi_1,\Pi_0)$ called the Postnikov class.

Note $\Pi_2\subset\mathsf{H}$ must be an Abelian group by the Peiffer identity (<ref>). The tuple $(\Pi_1,\Pi_2,\tau)$ is often called the Hoàng data of $\mathbb{G}$.

An analogous result holds for Lie 2-algebras. Here, one can state equivalence in the context of chain complexes. We say $\G\simeq\G'$ are equivalent iff there exist Lie 2-algebra maps $(\phi,\psi):\G\rightarrow \G',(\phi,\psi)^{-1}:\G'\rightarrow \G$ and invertible chain homotopies

\begin{equation*}
\eta: (\phi,\psi)^{-1}\circ(\phi,\psi)\Rightarrow \id_\G,\qquad \varepsilon:(\phi,\psi)\circ (\phi,\psi)^{-1}\Rightarrow\id_{\G'}.
\end{equation*}

The following is also a classic result, which was claimed to be first proven by Gerstenhaber by Wagemann in <cit.>.

Lie 2-algebras are classified up to equivalence by the following data: a Lie algebra $\mathfrak{n}=\operatorname{coker}\mu_1$, a $\mathfrak{n}$-module $V=\operatorname{ker}\mu_1$ and a degree-3 Lie algebra cohomology class $\kappa\in H^3(\mathfrak{n},V)$, called the Postnikov class.

One can show that, if two principal 2-bundles $\mathcal{P},\mathcal{P}'\rightarrow X$ have structure 2-groups $\mathbb{G},\mathbb{G}'$ that have distinct Postnikov classes, then they cannot be isomorphic 2-bundles over $X$.

§ HOMOTOPY TRANSFER

The BV-BRST formalism is a way to organize the algebraic and geometric data in a field theory with gauge symmetry into cochain complexes, which are designed such that their cohomologies describes precisely the gauge-invariant on-shell quantities. The mathematical structure that arises out of this construction is an $L_\infty$-algebra, which can be described as a $\mathbb{Z}$-graded differential complex

$$\mathfrak{L}\cong \dots \rightarrow \mathfrak{l}_{-2}\xrightarrow{\ell_1}\l_{-1}\xrightarrow{\mu_1}\l_0\xrightarrow{\ell_1}\l_1\xrightarrow{\ell_1}\l_2\rightarrow \dots$$
equipped with higher-brackets $\ell_n\in \operatorname{Hom}^{2-n}(\L^{n\otimes},\L)$ for $n\geq 2$ satisfying the Koszul-Jacobi identities. In gauge field theories with structure (2-)group $\mathbb{G}$, such as the homotopy Maurer-Cartan theories <cit.>, for instance, this complex is given by $\L = \Omega^\bullet(X)\otimes \G$ where $\G=\operatorname{Lie}\mathbb{G}$.

Now from a purely mathematical perspective, it is known that $L_\infty$-algebras retracts onto its cohomology <cit.>.[$L_\infty$-algebras are objects in the derived category of Lie algebras. More generally in any derived Abelian category, equivalences are precisely quasi-isomorphisms, ie. derived chain maps that induce isomorphisms in cohomologies.] In other words, given a $L_\infty$-algebra $\L$, the inclusion $\iota:H(\L)\hookrightarrow \L$ treating $H^\bullet(\L)\cong\operatorname{ker}\ell_1/\operatorname{im}\ell_1$ as a subcomplex of $\L$ admits a homotopy inverse (ie. an adjunction) given by the projection $p:\L\rightarrow H(\L)$ — there exists chain homotopies $\eta: \iota\circ p\Rightarrow \id_{\G}$, $\varepsilon: p\circ \iota\Rightarrow \id_{H(\G)}$ that "witness" this retraction. To be a chain homotopy, the map $\eta=(\eta_n)$ — where $\eta_n: \L_n\rightarrow \L_{n-1}$ — for instance must satisfy

\begin{equation*}
\iota \circ p - 1 = \eta \circ \ell_1 + \ell_1\circ \eta
\end{equation*}

as linear maps on $\L$.

In this context, it can be shown that a $L_\infty$-algebra structure, ie. the higher brackets $\tilde\ell=(\tilde\ell_n)_{n\geq 2}$, on $H(\L)$ can be induced from that $\ell=(\ell_n)_{n\geq 2}$ on $\L$ by using the projector $p$; notice, by construction, the cohomology $H(\L)$ has trivial differential. This is given by the formula

\begin{equation}\label{homotopytransfer}
\tilde\ell = p\circ (\ell + \ell\circ \eta\circ \ell + \ell \circ \eta\circ \ell\circ \eta\circ \ell + \dots ) \circ \iota,
\end{equation}

and it can be proven that $\tilde\ell$ indeed satisfies the Koszul-Jacobi identities on $H(\L)$ <cit.>. This is the celebrated homotopy transfer theorem for $L_\infty$-algebras: the $L_\infty$-algebra structure on $\L$ is transferred to a $L_\infty$-algebra structure on its cohomology $H(\L)$ through the projection quasi-isomorphism $p$.

An interesting case occurs when $\L$ is only a dgla, such that $\ell_n=0$ for all $n>2$. The formula (<ref>) implies that homotopy transfer in fact induces higher homotopy brackets on $H(\L)$. Indeed, at degree-3, we see that we have a generally non-trivial contribution given for each $\sfx_1,\sfx_2,\sfx_3\in\L$ by

\begin{equation*}
\tilde\ell_3([\sfx_1],[\sfx_2],[\sfx_3]) = ~\circlearrowright  [\ell_2(\eta(\ell_2([\sfx_1],[\sfx_2])),\eta([\sfx_3]))],
\end{equation*}

where $\circlearrowright$ denotes a cyclic sum and we have denoted the cohomology class of an element $\sfx\in\L$ by $p(\sfx)=[\sfx]\in H(\L)$. This map $\tilde\ell_3$ is usually called the homotopy map in the literature <cit.>, and can be seen to measure the failure of the chain homotopy $\eta$ to respect the graded Jacobi identity. This is relevant for us, as given a strict Lie 2-algebra $\G$, the associated BV-BRST $L_\infty$-algebra $\L = \Omega^\bullet(X)\otimes \G$ is indeed a dgla with $\ell_1=d - \mu_1$. However, upon performing a homotopy transfer, we see that a degree-3 homotopy map is induced on the cohomology $H(\L)$.

This idea can be used to immediately to prove Gerstenhaber's theorem in <ref>. The cohomology $H(\G)$ of a Lie 2-algebra $\G$ consist of $M\xrightarrow{0}\mathfrak{n}$, and the homotopy map $\tilde\ell_3$ is given by the Postnikov class $\kappa\in H^3(\mathfrak{n},M)$.

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journaltitleTheory and Applications of Categories
titleRepresentation and character theory of finite categorical groups

journaltitleJournal of Algebra
titleOn Lagrangian algebras in group-theoretical braided fusion categories

journaltitleprivate communiations
titleUniversal traces and 2-characters for 2-representations of groups

journaltitlearXiv e-prints
titleThe 2-character theory for finite 2-groups

Mathematics - Representation Theory,Mathematics - Category Theory,Mathematics - Quantum Algebra

titleA cohomology theory for Lie 2-algebras and Lie 2-groups
Differential Geometry (math.DG),Category Theory (math.CT),FOS: Mathematics,FOS: Mathematics

abstractWe investigate quantum field theory in two dimensions invariant with respect to conformal (Virasoro) and non-abelian current (Kac-Moody) algebras. The Wess-Zumino model is related to the special case of the representations of these algebras, the conformal generators being quadratically expressed in terms of currents. The anomalous dimensions of the Wess-Zumino fields are found exactly, and the multipoint correlation functions are shown to satisfy linear differential equations. In particular, Witten's non-abelean bosonisation rules are proven.
journaltitleNuclear Physics B
titleCurrent algebra and Wess-Zumino model in two dimensions

title5d 2-Chern-Simons theory and 3d integrable field theories

journaltitleTheor. Appl. Categor.
titleOn Yetter's invariant and an extension of the Dijkgraaf-Witten invariant to categorical groups

journaltitlearXiv: Category Theory
titleStrict 2-Groups are Crossed Modules

Mount Allison University, Department of Mathematics
Computer Science, Sackville

journaltitleTheory and Applications of Categories [electronic only]
titleHigher-dimensional algebra. V: 2-Groups.
ern-Simons theory; group cohomology

Springer Science
Business Media LLC

journaltitleCommunications in Mathematical Physics
titleLie 2-Bialgebras

journaltitleTheor. Appl. Categor.
titleHigher-Dimensional Algebra VI: Lie 2-Algebras

journaltitleFortsch. Phys.
title$L_\infty$-Algebras of Classical Field Theories and the Batalin-Vilkovisky Formalism

journaltitleNucl. Phys. B
titleClosed string field theory: Quantum action and the B-V master equation

titleGauging the Gauge and Anomaly Resolution

journaltitleLett. Math. Phys.
titleFour-Dimensional BF theory with cosmological term as a topological quantum field theory
uadratic charges in 4d BF theory and related TQFTs

titleDiscretization of 4d Poincaré BF theory: from groups to 2-groups

journaltitleFortsch. Phys.
titleHomotopy Transfer and Effective Field Theory I: Tree-level

journaltitleJ. Diff. Geom.
titlePoisson 2-groups

journaltitleHigher Structures
title$L_\infty$ and $A_\infty$ structures: then and now

title4-D semistrict higher Chern-Simons theory I

journaltitleCommun. Math. Phys.
titleHomotopical Analysis of 4d Chern-Simons Theory and Integrable Field Theories

titleOn covariant phase space methods

journaltitleAdv. Theor. Math. Phys.
titleLie crossed modules and gauge-invariant actions for 2-BF theories

titleHamiltonian analysis of the BFCG theory for a strict Lie 2-group

journaltitleProceedings of the International Geometry Center
titleOn transversely holomorphic foliations with homogeneous transverse structure

Elsevier BV

journaltitleJournal of Geometry and Physics
titleHigher current algebras, homotopy Manin triples, and a rectilinear adelic complex

Springer Science
Business Media LLC

journaltitleCommunications in Mathematical Physics
titleBundle Gerbes for Chern-Simons and Wess-Zumino-Witten Theories

WORLD SCIENTIFIC

titleNonabelian Multiplicative Integration on Surfaces

Annals of Mathematics

journaltitleAnnals of Mathematics
titleOn Adding Relations to Homotopy Groups

Cambridge University Press

journaltitleLondon Mathematical Society Lecture Note Series
titleComputing Homotopy Types Using Crossed $N$-Cubes of Groups

International Press of Boston

journaltitleHomology, Homotopy and Applications
titleOn the homotopy type and the fundamental crossed complex of the skeletal filtration of a CW-complex
crossed complex,CW-complex,higher homotopy van Kampen Theorem,invariants of homotopy types,skeletal filtration

titleHigher categorical groups and the classification of topological defects and textures

titleTwisted Quantum Double Model of Topological Orders with Boundaries

De Gruyter

titleCrossed Modules

Springer Science
Business Media LLC

journaltitleJournal of High Energy Physics
titleDynamics of a lattice 2-group gauge theory model

titleOn unitary 2-representations of finite groups and topological quantum field theory
Mathematics - Quantum Algebra,Mathematical Physics,Mathematics - Algebraic Topology,Mathematics - Category Theory,Mathematics - Representation Theory

American Mathematical Society (AMS)

journaltitleMemoirs of the American Mathematical Society
titleInfinite-Dimensional Representations of 2-Groups

titleNon-invertible Symmetries and Higher Representation Theory I

Princeton University Press

abstractOne of the great achievements of contemporary mathematics is the new understanding of four dimensions. Michael Freedman and Frank Quinn have been the principals in the geometric and topological development of this subject, proving the Poincar and Annulus conjectures respectively. Recognition for this work includes the award of the Fields Medal of the International Congress of Mathematicians to Freedman in 1986. In Topology of 4-Manifolds these authors have collaborated to give a complete and accessible account of the current state of knowledge in this field. The basic material has been considerably simplified from the original publications, and should be accessible to most graduate students. The advanced material goes well beyond the literature; nearly one-third of the book is new. This work is indispensable for any topologist whose work includes four dimensions. It is a valuable reference for geometers and physicists who need an awareness of the topological side of the field.Originally published in 1990.ThePrinceton Legacy Libraryuses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
titleTopology of 4-Manifolds (PMS-39)

Johns Hopkins University Press

issn00029327, 10806377
journaltitleAmerican Journal of Mathematics
titleOn the Structure of Manifolds

Princeton University Press

titleLectures on the H-Cobordism Theorem

Berlin, Heidelberg

Springer Berlin Heidelberg

booktitleArbeitstagung Bonn 1984
titleLoop groups

booktitleThe Shoucheng Zhang Memorial Workshop
titleAnomaly Inflow and the $\eta$-Invariant

titleMost general theory of 3d gravity: Covariant phase space, dual diffeomorphisms, and more

Princeton University Press

seriesAnnals of Mathematics Studies
titleCharacteristic Classes

titleDirac charge quantization and generalized differential cohomology

journaltitlePacific Journal of Mathematics
titleFOUR EQUIVALENT VERSIONS OF NONABELIAN GERBES

abstractWe give a definition of weakn-categories based on the theory of operads. We work with operads having an arbitrary setSof types, or “S-operads,” and given such an operadO, we denote its set of operations by elt(O). Then for anyS-operadOthere is an elt(O)-operadO+whose algebras areS-operads overO. LettingIbe the initial operad with a one-element set of types, and definingI0+=I,I(i+1)+=(Ii+)+, we call the operations ofI(n−1)+the “n-dimensional opetopes.” Opetopes form a category, and presheaves on this category are called “opetopic sets.” A weakn-category is defined as an opetopic set with certain properties, in a manner reminiscent of Street's simplicial approach to weakω-categories. Similarly, starting from an arbitrary operadOinstead ofI, we define “n-coherentO-algebras,” which arentimes categorified analogs of algebras ofO. Examples include “monoidaln-categories,” “stablen-categories,” “virtualn-functors” and “representablen-prestacks.” We also describe hown-coherentO-algebra objects may be defined in any (n+1)-coherentO-algebra.
journaltitleAdvances in Mathematics
titleHigher-Dimensional Algebra III.n-Categories and the Algebra of Opetopes

journaltitleInventiones mathematicae
titleTannaka-Krein duality for compact matrix pseudogroups. Twisted SU (N) groups.
eorem; matrix pseudogroups

titleExcitations in strict 2-group higher gauge models of topological phases

titleRaviolo vertex algebras

titleRaviolo vertex algebras, cochains and conformal blocks

American Mathematical Society

issn08940347, 10886834
journaltitleJournal of the American Mathematical Society
titleTensor Structures Arising from Affine Lie Algebras. III

American Mathematical Society (AMS)

journaltitleJournal of the American Mathematical Society
titleMinimal nondegenerate extensions

journaltitleGeometry & Topology
titleInvariants of 4–manifolds from Khovanov–Rozansky link homology

journaltitleAdvances in Mathematics
titleSkein lasagna modules and handle decompositions

journaltitleCommunications of The Korean Mathematical Society
titleCROSSED MODULES AND STRICT GR-CATEGORIES

journaltitlearXiv preprint arXiv:2308.05119
titleHoàng Xuân Sính's Thesis: Categorifying Group Theory