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\section*{String Lie 2-algebra}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{lie_theory}{}\paragraph*{{$\infty$-Lie theory}}\label{lie_theory}

[[!include infinity-Lie theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{InComponents}{In components}\dotfill \pageref*{InComponents} \linebreak
\noindent\hyperlink{skeletal_model}{Skeletal model}\dotfill \pageref*{skeletal_model} \linebreak
\noindent\hyperlink{strict_lie_2algebra_model}{Strict Lie 2-algebra model}\dotfill \pageref*{strict_lie_2algebra_model} \linebreak
\noindent\hyperlink{AsHomotopyFiber}{As a homotopy fiber}\dotfill \pageref*{AsHomotopyFiber} \linebreak
\noindent\hyperlink{as_the_prequantum_line_2bundle_of_a_courant_algebroid}{As the prequantum line 2-bundle of a Courant algebroid}\dotfill \pageref*{as_the_prequantum_line_2bundle_of_a_courant_algebroid} \linebreak
\noindent\hyperlink{as_a_heisenberg_lie_2algebra}{As a Heisenberg Lie 2-algebra}\dotfill \pageref*{as_a_heisenberg_lie_2algebra} \linebreak
\noindent\hyperlink{generalizations}{Generalizations}\dotfill \pageref*{generalizations} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{string Lie 2-algebra} is the [[infinitesimal object|infinitesimal approximation]] to the [[Lie 2-group]] that is called the [[string 2-group]].

It is a shifted [[L-infinity-algebra|∞-Lie algebra]] central extension

\begin{displaymath}
0 \to \mathbf{b} \mathfrak{u}(1) \to \mathfrak{string}(n)
  \to \mathfrak{so}(n) \to 0
\end{displaymath}
of the [[Lie algebra]] $\mathfrak{so}(n)$ by the [[Lie 2-algebra]] $\mathbf{b} \mathfrak{u}(1)$ which is classified by the canonical (up to normalization) [[Lie algebra cohomology|Lie algebra 3-cocycle]] $\mu$ on $\mathfrak{so}(n)$, which may itself be understood as a morphism

\begin{displaymath}
\mu : \mathfrak{so}(n) \to b^2 \mathfrak{u}(1)
  \,.
\end{displaymath}
When $\mu$ is normalized such that it represents the image in [[deRham cohomology]] of the generator of the [[integral cohomology]] $H^3(X,Spin(n))$ of the [[Spin group]], then the [[Lie integration]] of the String Lie 2-algebra is the [[String Lie 2-group]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

We spell out first an explicit algebraic realization of the string Lie 2-algebra and then give its abstract definition as a [[homotopy fiber]] or [[principal ∞-bundle]].

\hypertarget{InComponents}{}\subsubsection*{{In components}}\label{InComponents}

As with any [[L-∞ algebra]], we may define the String Lie 2-algebra $\mathfrak{string}(n)$ equivalently in terms of its [[Chevalley-Eilenberg algebra]].

There are various equivalent models we discuss a small one with a trinary bracket, and an infinite dimensional model which is however strict in that it comes from a [[differential crossed module]].

\hypertarget{skeletal_model}{}\paragraph*{{Skeletal model}}\label{skeletal_model}

Write $\mathfrak{g} := \mathfrak{so}(n)$ in the following. The [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$ of $\mathfrak{g}$ has a degree 3 element

\begin{displaymath}
\mu = \langle -, [-,-]\rangle 
  \,,
\end{displaymath}
well defined up to normalization ($\langle - ,- \rangle$ is the canonical bilinear symmetric [[invariant polynomial]] on $\mathfrak{g}$ and $[-,-]$ the Lie bracket), which is closed in $CE(\mathfrak{g})$

\begin{displaymath}
d_{\mathfrak{g}} \mu = 0
  \,.
\end{displaymath}
Hence this is the canonical (up to normalization) 3-cocycle in the [[Lie algebra cohomology]] of $\mathfrak{g}$.

The Chevalley--Eilenberg algebra of $\mathfrak{string}(n)$ is

\begin{displaymath}
CE(\mathfrak{string}(n)) = (\wedge^\bullet (\mathfrak{g}^* \oplus \langle b\rangle), d_{\mathfrak{string}})
  \,,
\end{displaymath}
where

\begin{itemize}%
\item $b$ is a single new generator in degree 2;


\item the differental $d_{\mathfrak{string}}$ coincides with $d_{\mathfrak{g}}$ on $\mathfrak{g}^*$:

\begin{displaymath}
d_{\mathfrak{string}} |_{\mathfrak{g}^*} = d_{\mathfrak{g}}
\end{displaymath}

\item on the new generator it is defined by

\begin{displaymath}
d_{\mathfrak{string}} : b \mapsto \mu
  \,.
\end{displaymath}


\end{itemize}
That the differential defined this way is indeed of degree +1 and squares to 0 is precisely the fact that $\mu$ is a degree 3-cocycle of $\mathfrak{g}$.

One can equivalently describe the $L_\infty$-algebra structure of $\mathfrak{string}(n)$ in terms of lots of brackets

\begin{displaymath}
[-,-,\dots,-]_k:\wedge^k \mathfrak{string}(n)\to \mathfrak{string}(n),
\end{displaymath}
of degree $2-k$. In addition to the Lie bracket of $\mathfrak{g}$, there is only a further nontrivial bracket: it is the 3-bracket

\begin{displaymath}
[-,-,-]_3:\wedge^3 \mathfrak{g}\to \langle b\rangle^*
\end{displaymath}
given by

\begin{displaymath}
[x,y,z]_3=\mu(x,y,z)\cdot \beta,
\end{displaymath}
where $\beta:\langle b\rangle\to\mathbb{R}$ is the dual of $b$.

\hypertarget{strict_lie_2algebra_model}{}\paragraph*{{Strict Lie 2-algebra model}}\label{strict_lie_2algebra_model}

\textbf{Proposition} The string Lie 2-algebra given above is equivalent to the infinite-dimensional Lie 2-algebra coming from the [[differential crossed module]]

\begin{displaymath}
\hat \Omega \mathfrak{g} \to P \mathfrak{g}
\end{displaymath}
of the [[universal central extension]] of the [[loop Lie algebra]] mapping into the path Lie algebra, which acts on the former in the evident way.

This is proven in \hyperlink{BCSS}{BCSS}.

\hypertarget{AsHomotopyFiber}{}\subsubsection*{{As a homotopy fiber}}\label{AsHomotopyFiber}

Up to equivalence, the string Lie 2-algebra is the [[homotopy fiber]] of the cocycle $\mu : \mathfrak{so}(n) \to \mathbf{b}^2 \mathfrak{u}(1)$, hence is the canonical $\mathbf{b} \mathfrak{u}(1)$-[[principal ∞-bundle]] over $\mathfrak{so}(n)$.

Here we take by definition the [[(∞,1)-category]] of [[∞-Lie algebroid]]s to be that [[presentable (∞,1)-category|presented]] by the opposite (after passing to [[Chevalley-Eilenberg algebras]]) of the [[model structure on dg-algebras]]. For a detailed discussion of the recognition of this [[homotopy fiber]] see section 3.1 and specifically example 3.5.4 of (\hyperlink{FiorenzaRogersSchreiber13}{Fiorenza-Rogers-Schreiber 13}).

In terms of dg-algebras, the cocycle is dually a morphism

\begin{displaymath}
CE(\mathfrak{so}(n)) \leftarrow CE(\mathbf{b}^2 \mathfrak{u}(1)) : \mu
\end{displaymath}
and the [[homotopy fiber]] in question is dually modeled by the [[homotopy pushout]]

\begin{displaymath}
\itexarray{
    && 0
    \\
    && \uparrow
    \\
    CE(\mathfrak{so}(n)) &\stackrel{\mu}{\leftarrow}&
    CE(\mathbf{b}^2 \mathfrak{u}(1))
  }
  \,.
\end{displaymath}
By the general rules for computing [[homotopy pushout]]s, this may be computed by an ordinary [[pushout]] if we choose a [[resolution]] of $CE(\mathbf{b}^2 \mathfrak{u}(1)) \to 0$ by a cofibration and ensure that all three objects in the pushout diagram are cofibrations.

For the resolution we take the standard one by the CE-algebra of the $\mathbf{b}^2 \mathfrak{u}(1)$-[[universal principal ∞-bundle]] $\mathbf{e b} \mathfrak{u}(1)$, which is the dg-algebra

\begin{displaymath}
CE(\mathbf{e b} \mathfrak{u}(1)) = (\wedge^\bullet( \langle b\rangle \oplus \langle c\rangle ), d)
\end{displaymath}
where $b$ is a generator of degree 2, $c$ one of degree 3 and the differential is given by

\begin{displaymath}
d b = c
\end{displaymath}
and

\begin{displaymath}
d c = 0
  \,.
\end{displaymath}
The morphism $CE(\mathbf{b}^2 \mathfrak{u}(1)) \to CE(e b \mathfrak{u}(1))$ is the one that identifies the two degree-3 generators.

Now $CE(\mathbf{b}^2 \mathfrak{u}(1))$ and $CE(\mathbf{e b} \mathfrak{u}(1))$ are [[Sullivan algebra]]s, hence are cofibrant objects in the [[model structure on dg-algebra]]s. The dg-algebra $CE(\mathfrak{g})$ is not quite a Sullivan algebra, but almost: it is a [[semifree dga]] and only fails to have the filtering property on the differential. This is sufficient for computing the desired [[homotopy fiber]], as discussed at \href{infinity-Lie+algebra+cohomology#Extensions}{∞-Lie algebra cohomology -- Extensions}.

One observes now that

\begin{displaymath}
\itexarray{
    CE(\mathfrak{string}) &\leftarrow& CE(\mathbf{e b} \mathfrak{u}(1))
    \\
    \uparrow && \uparrow
    \\
    CE(\mathfrak{so}(n)) &\leftarrow& CE(\mathbf{b}^2 \mathfrak{u}(1))
  }
\end{displaymath}
is a [[pushout]] diagram. Dually, this exhibits $\mathfrak{string}$ as the [[(∞,1)-pullback]]

\begin{displaymath}
\itexarray{
    \mathfrak{string}(n) &\to& *
    \\
    \downarrow && \downarrow
    \\
    \mathfrak{so}(n) &\stackrel{\mu}{\to}& \mathbf{b}^2 \mathfrak{u}(1)
  }
  \,.
\end{displaymath}
And this may be taken to be the abstract definition of the string Lie 2-algebra.

By the general logic of [[fiber sequence|fiber sequences]] this implies that also

\begin{displaymath}
\mathbf{b} \mathfrak{u}(1) \to \mathfrak{string} \to \mathfrak{so}(n)
\end{displaymath}
is a fiber sequence. By analogous reasoning as before, we see that this is modeled by the ordinary pushout

\begin{displaymath}
\itexarray{
    CE(\mathbf{b} \mathfrak{u}(1)) &\leftarrow& *
    \\
    \uparrow && \uparrow
    \\
    CE(\mathfrak{string}) &\leftarrow& CE(\mathfrak{g})
  }
  \,.
\end{displaymath}
This is indeed a homotopy pushout even without resolving the point, because $CE(\mathfrak{string}) \leftarrow CE(\mathfrak{g})$ is already a cofibration, being the pushout of a cofibration by the above.

\hypertarget{as_the_prequantum_line_2bundle_of_a_courant_algebroid}{}\subsubsection*{{As the prequantum line 2-bundle of a Courant algebroid}}\label{as_the_prequantum_line_2bundle_of_a_courant_algebroid}

The [[delooping]] of a [[semisimple Lie algebra]] $\mathfrak{g}$ to a 1-object [[L-infinity algebroid]] $b \mathfrak{g}$ carries the [[Killing form]] as a quadratic bilinear [[invariant polynomial]] and is as such a [[symplectic Lie n-algebroid]] over the point for $n = 2$, hence a [[Courant Lie 2-algebroid]] over the point.

As described at [[symplectic infinity-groupoid]] one can consider the higher analog of [[geometric quantization]] of these objects. This is again the homotopy fiber as \href{AsHomotopyFiber}{above}.

\hypertarget{as_a_heisenberg_lie_2algebra}{}\subsubsection*{{As a Heisenberg Lie 2-algebra}}\label{as_a_heisenberg_lie_2algebra}

The String Lie 2-algebra identifies also with the [[Heisenberg Lie 2-algebra]] of the [[string]] [[sigma-model]] for the specialization to the [[WZW model]] (\hyperlink{BaezRogers10}{Baez-Rogers 10}). See at \emph{[[2-plectic geometry]]} for more.

\hypertarget{generalizations}{}\subsection*{{Generalizations}}\label{generalizations}

More generally, for $\mathfrak{g}$ an [[L-infinity-algebra|∞-Lie algebra]] and $\mu \in CE(\mathfrak{g})$ an $\infty$-Lie algebra cocycle (a closed element in the [[Chevalley-Eilenberg algebra]] of $\mathfrak{g}$) of degree $k$, there is a corresponding shifted central extension

\begin{displaymath}
0 \to \mathbf{b}^{k-2} \mathfrak{u}(1) \to \mathfrak{g}_\mu  \to \mathfrak{g} \to 0
  \,.
\end{displaymath}
For instance the [[supergravity Lie 3-algebra]] is such an extension of the [[super Poincare Lie algebra]] by a [[super Lie algebra]] 4-cocycle.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[special orthogonal Lie algebra]]


\item \textbf{string Lie 2-algebra}


\item [[fivebrane Lie 6-algebra]]


\item [[type II supergravity Lie 2-algebra]]


\item [[supergravity Lie 3-algebra]], [[supergravity Lie 6-algebra]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

In one incarnation or other the String Lie 2-algebra has been considered in literature of [[dg-algebra]]s, but its [[Lie theory|Lie theoretic]] interpretation as a Lie 2-algebra has been made fully explicit only in

\begin{itemize}%
\item [[John Baez]], [[Alissa Crans]], Higher-dimensional Algebra V: Lie 2-algebras, \emph{Theory and Applications of Categories} \textbf{12} (2004), 492-528. (\href{http://www.tac.mta.ca/tac/volumes/12/15/12-15abs.html}{web}) (\href{http://arxiv.org/abs/math.QA/0307263}{arXiv:math.QA/0307263})

\end{itemize}
In

\begin{itemize}%
\item [[Andre Henriques]], \emph{Integrating $L_\infty$-algebras} (\href{http://arxiv.org/abs/math/0603563}{arXiv:0603563})

\end{itemize}
the string Lie 2-algebra is integrated to the [[string 2-group]] using the general abstract method described at [[Lie integration]].

In

\begin{itemize}%
\item [[John Baez]], [[Alissa Crans]], [[Urs Schreiber]] and [[Danny Stevenson]], From loop groups to 2-groups, \emph{Homotopy, Homology and Applications} \textbf{9} (2007), 101-135. (\href{http://arxiv.org/abs/math.QA/0504123}{arXiv:math.QA/0504123})

\end{itemize}
the equivalent strict model given by a differential crossed module is found, which is then integrated termwise as ordinary Lie algebras to a [[crossed module]] of Frechet-Lie groups, hence to a Lie [[strict 2-group]] model of the String Lie 2-group.

The string Lie 2-algebra as the [[Heisenberg Lie 2-algebra]] on the group $G$ is discussed in

\begin{itemize}%
\item [[John Baez]], [[Chris Rogers]], \emph{Categorified Symplectic Geometry and the String Lie 2-Algebra}, Homology Homotopy Appl. Volume 12, Number 1 (2010), 221-236. (\href{http://arxiv.org/abs/0901.4721}{arXiv:0901.4721},\href{https://projecteuclid.org/euclid.hha/1296223828}{euclid}).

\end{itemize}
The string Lie 2-algebra is also considered in a certain context in

\begin{itemize}%
\item Sati, Schreiber, Stasheff, \emph{$L_\infty$-algebra connections}

\end{itemize}
where also the relation to the [[supergravity Lie 3-algebra]] and other structures is discussed.

The super-$L_\infty$-version of the string $L_\infty$-algebra was considered in

\begin{itemize}%
\item [[John Baez]], [[John Huerta]], \emph{Division Algebras and Supersymmetry II} (\href{http://arxiv.org/abs/1003.3436}{arXiv:1003.3436}).

\end{itemize}
See also [[division algebra and supersymmetry]].

Discussion of the string Lie 2-algebra as the homotopy fiber of the underlying 3-cocycle is around prop. 3.3.96 in

\begin{itemize}%
\item \emph{[[schreiber:differential cohomology in a cohesive topos]]}

\end{itemize}
and example 3.5.4 in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], \emph{[[schreiber:L-∞ algebras of local observables from higher prequantum bundles]]} (\href{http://arxiv.org/abs/1304.6292}{arXiv:1304.6292})

\end{itemize}
More on this in

\begin{itemize}%
\item [[Lennart Schmidt]], \emph{Twisted Weil Algebras for the String Lie 2-Algebra}, in [[Christian Saemann]], [[Urs Schreiber]], [[Martin Wolf]] (eds.) \emph{\href{http://www.maths.dur.ac.uk/lms/109/index.html}{Higher Structures in M-Theory}} \href{http://www.maths.dur.ac.uk/lms/}{Durham Symposium} 2018, Fortschritte der Physik 2019 (\href{https://arxiv.org/abs/1903.02873}{arXiv:1903.02873})

\end{itemize}
[[!redirects string Lie 2-algebra]]



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