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\begin{document}

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\section*{string 2-group}

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

\hypertarget{higher_lie_theory}{}\paragraph*{{Higher Lie theory}}\label{higher_lie_theory}

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

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{higher_spin_geometry}{}\paragraph*{{Higher spin geometry}}\label{higher_spin_geometry}

[[!include higher spin geometry - contents]]

\hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory}

[[!include string 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{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{presentations}{Presentations}\dotfill \pageref*{presentations} \linebreak
\noindent\hyperlink{ByLieIntegrationOfStringLie2Algebra}{By Lie integration of the string Lie 2-algebra}\dotfill \pageref*{ByLieIntegrationOfStringLie2Algebra} \linebreak
\noindent\hyperlink{PresentationByStrictTwoGroups}{By strict Lie $2$-group}\dotfill \pageref*{PresentationByStrictTwoGroups} \linebreak
\noindent\hyperlink{as_a_finitedimensional_weak_lie_2group}{As a finite-dimensional weak Lie 2-group}\dotfill \pageref*{as_a_finitedimensional_weak_lie_2group} \linebreak
\noindent\hyperlink{as_an_automorphism_2group_of_fermionic_cft}{As an automorphism 2-group of fermionic CFT}\dotfill \pageref*{as_an_automorphism_2group_of_fermionic_cft} \linebreak
\noindent\hyperlink{as_the_automorphisms_of_the_wesszuminowitten_gerbe_2connection}{As the automorphisms of the Wess-Zumino-Witten gerbe 2-connection}\dotfill \pageref*{as_the_automorphisms_of_the_wesszuminowitten_gerbe_2connection} \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 2-group} is a [[smooth ∞-groupoid|smooth 2-group]]-refinement of the [[topological group]] called the [[string group]]. It is the [[∞-group extension]] induced by the smooth/stacky version of the [[first fractional Pontryagin class]]/[[second Chern class]].

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

A string 2-group extension $String(G)$ is defined for every [[simple Lie group|simple]] [[simply connected topological space|simply connected]] [[compact Lie group]] $G$, such as the [[spin group]] $G = Spin(n)$ or the [[special unitary group]] $G = SU(n)$ (for non-low $n$).

Since [[string structures]] arise predominantly as higher analogs of [[spin structures]], the default choice is $G = Spin$ and in that case one usually just writes $String = String(Spin)$, for short.

Recall first that the [[string group]] in [[Top]] is one step in the [[Whitehead tower]] of the [[orthogonal group]].

\begin{defn}
\label{AbstractDefinitionDiscrete}\hypertarget{AbstractDefinitionDiscrete}{}
For $n \in \mathbb{N}$ let $Spin(n)$ denote the [[spin group]], regarded as a [[topological group]]. Write $B Spin(n) \in$ [[Top]] for its [[classifying space]] and

\begin{displaymath}
\frac{1}{2}p_1 : B Spin(n) \to B^4 \mathbb{Z}
\end{displaymath}
for a representative of the [[characteristic class]] called the first fractional [[Pontryagin class]]. Its [[homotopy fiber]] in the [[(∞,1)-topos]] [[Top]] $\simeq$ [[∞Grpd]] is denoted $B String(n) := B O(n)\langle 7 \rangle$

\begin{displaymath}
\itexarray{
     B String(n) &\to& * 
     \\
     \downarrow && \downarrow
     \\
     B Spin(n) &\stackrel{\frac{1}{2} p_1}{\to}& B^4 \mathbb{Z}
  }
  \,.
\end{displaymath}
The [[loop space]]

\begin{displaymath}
String(n) := \Omega B String(n)
\end{displaymath}
is the [[∞-group]]-object in [[Top]] called the \textbf{[[string group]]}.

\end{defn}
Write now

\begin{displaymath}
(\Pi \dashv Disc \dashv \Gamma \dashv coDisc)
   :
  Smooth\infty Grpd
   \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{coDisc}{\leftarrow}}}}
  \infty Grpd
    \simeq 
  Top
\end{displaymath}
for the [[(∞,1)-topos]] [[Smooth∞Grpd]] of [[smooth ∞-groupoid]]s, regarded as a [[cohesive (∞,1)-topos]] over [[∞Grpd]].

\begin{prop}
\label{SmoothFractionalPontryaginClass}\hypertarget{SmoothFractionalPontryaginClass}{}
There is a lift through $\Pi$ of $\frac{1}{2} p_1$ to the 

\begin{displaymath}
\frac{1}{2}\mathbf{p}_1 : \mathbf{B}Spin(n) \to \mathbf{B}^3 U(1)
\end{displaymath}
in [[Smooth∞Grpd]], where

\begin{itemize}%
\item $\mathbf{B}Spin$ is the [[delooping]] [[Lie groupoid]] of $Spin(n)$ regarded as a [[Lie group]] (see );


\item $\mathbf{B}^3 U(1)$ is the three-fold delooping of the [[circle group]], regarded as a [[Lie group]] (see );


\item $\frac{1}{2}\mathbf{p}_1$ is the image under [[Lie integration]] of the canonical [[Lie algebra cohomology|cocycle]]

\begin{displaymath}
\mu = \langle -,[-,-]\rangle : \mathfrak{so}(n) \to b^2 \mathbb{R}
  \,.
\end{displaymath}
on the [[orthogonal Lie algebra]].



\end{itemize}
\end{prop}
This is shown in (\hyperlink{FSS}{FSS}).

\begin{defn}
\label{AbstractDefinitionSmooth}\hypertarget{AbstractDefinitionSmooth}{}
Write $\mathbf{B}String(n)$ for the [[homotopy fiber]] of the smooth first fractional Pontryagin class

\begin{displaymath}
\itexarray{
    \mathbf{B}String &\to& * 
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}Spin &\stackrel{\frac{1}{2}\mathbf{p}_1}{\to}&
    \mathbf{B}^3 U(1)
  }
\end{displaymath}
in [[Smooth∞Grpd]]. Its [[loop space object]]

\begin{displaymath}
String(n) := \Omega \mathbf{B}String(n)
\end{displaymath}
is the [[∞-Lie group|smooth ∞-group]] called [[generalized the|the]] \textbf{smooth string 2-group}.

\end{defn}
\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

Write

\begin{displaymath}
\vert - \vert := \vert\Pi(-)\vert :
  Smooth \infty Grpd
    \stackrel{\Pi}{\to}
  \infty Grpd
    \stackrel{\vert - \vert}{\to}
  Top
\end{displaymath}
for the  in [[Smooth∞Grpd]].

\begin{prop}
\label{}\hypertarget{}{}
The smooth string 2-group, def. \ref{AbstractDefinitionSmooth}, indeed maps under $\vert-\vert$ to the topological string group:

\begin{displaymath}
\vert \mathbf{B}String(n) \vert
  \simeq
  B String(n)
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
Since $\mathbf{B}^3 U(1)$ is presented by a simplicial presheaf that is degreewise presented by a paracompact smooth manifold (a finite product of the [[circle group]] with itself), it follows from the general properties of $\Pi$ discussed at [[Smooth∞Grpd]] that $\Pi$ preserves the [[homotopy fiber]] of $\frac{1}{2}\mathbf{p}_1$.

\end{proof}
\hypertarget{presentations}{}\subsection*{{Presentations}}\label{presentations}

Several explicit presentations of the string Lie 2-group are known.

\hypertarget{ByLieIntegrationOfStringLie2Algebra}{}\subsubsection*{{By Lie integration of the string Lie 2-algebra}}\label{ByLieIntegrationOfStringLie2Algebra}

We discuss a presentation of the smooth string 2-group by [[Lie integration]] of the skeletal version of the [[string Lie 2-algebra]].

Recall the identification of [[L-∞ algebra]]s $\mathfrak{g}$ with their dual [[Chevalley-Eilenberg algebra]]s $CE(\mathfrak{g})$.

\begin{defn}
\label{}\hypertarget{}{}
Write

\begin{displaymath}
\mu := \langle - ,[-,-]\rangle
  : 
  \mathfrak{so}(n) \to b^2 \mathbb{R}
\end{displaymath}
for the canonical degree-3 [[cocycle]] in the [[Lie algebra cohomology]] of the [[special orthogonal group]], normalized such that the 3-form

\begin{displaymath}
\Omega^\bullet(Spin(n)) \hookleftarrow
  CE(\mathfrak{so}(n))
  \stackrel{\mu}{\leftarrow}
  CE(b^2 \mathbb{R})
\end{displaymath}
represents the image in [[de Rham cohomology]] of a generators of the [[integral cohomology]] group $H^3(G,\mathbb{Z}) \simeq \mathbb{Z}$.

Define the \textbf{[[string Lie 2-algebra]]}

\begin{displaymath}
\mathfrak{string}(n) := \mathfrak{so}(n)_\mu
\end{displaymath}
to be given by the [[Chevalley-Eilenberg algebra]]

\begin{displaymath}
CE(\mathfrak{string}(n)) := \wedge^\bullet ( \mathfrak{so}(n)^* \oplus \langle b\rangle , d_{\mathfrak{string}})
\end{displaymath}
which is that of $\mathfrak{so}(n)$ with a single generator $b$ in degree 3 adjoined and the differential given by

\begin{displaymath}
d_{\mathfrak{string}}|_{\mathfrak{so}(n)^*} = d_{\mathfrak{so}(n)};
\end{displaymath}
\begin{displaymath}
d_{\mathfrak{string}} : b \mapsto \mu
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{HomotopyFiberOfLInftyAlgebraCocycle}\hypertarget{HomotopyFiberOfLInftyAlgebraCocycle}{}
We have a [[pullback]] square in $L_\infty Alg$

\begin{displaymath}
\itexarray{
    \mathfrak{string}(n) &\to& e b \mathbb{R}
    \\
    \downarrow && \downarrow
    \\
    \mathfrak{so}(n) &\stackrel{\mu}{\to}& b^2 \mathbb{R}
  }
  \,.
\end{displaymath}
\end{prop}
See [[string Lie 2-algebra]] for more discussion.

\begin{prop}
\label{}\hypertarget{}{}
The [[Lie integration]] of $\mathfrak{string}(n)$ yields a presentation of the smooth String 2-group, def. \ref{AbstractDefinitionSmooth}

\begin{displaymath}
\mathbf{cosk}_3 \exp(\mathfrak{string}(n))
  \simeq
  \mathbf{B} String(n)
  \,.
\end{displaymath}
\end{prop}
This is essentially the model considered in (\hyperlink{Henriques}{Henriques}), discussed here in the context of [[Smooth∞Grpd]] as described in (\hyperlink{FSS}{FSS}).

\begin{proof}
We observe the image under [[Lie integration]] of the $L_\infty$-algebra pullback diagram from prop. \ref{HomotopyFiberOfLInftyAlgebraCocycle} is a pullback diagram in $[CartSp_{smooth}^{op}, sSet]_{proj}$ that presents the defining [[homotopy fiber]]. Before applying the [[coskeleton]] operation we have immediately

\begin{displaymath}
\exp(-) 
    \; :\; 
  \left(
   \itexarray{
     \mathfrak{string}(n) &\to& e b \mathbb{R}
     \\
     \downarrow && \downarrow
     \\
     \mathfrak{so}(n) &\stackrel{\mu}{\to}& b^2 \mathbb{R}
   }
  \right)
  \;\mapsto \;
  \left(
   \itexarray{
     \exp(\mathfrak{string}(n)) &\to& \exp(e b \mathbb{R})
     \\
     \downarrow && \downarrow
     \\
     \exp(\mathfrak{so}(n)) &\stackrel{\mu}{\to}& \exp(b^2 \mathbb{R})
   }
  \right)
\end{displaymath}
such that on the right we still have a [[pullback]] diagram.

We discuss the descent o this pullback diagram along the projection $\exp(\mathfrak{so}(n)) \to \mathbf{cosk}_3 \exp(\mathfrak{so}(n))$.

Notice from [[Lie integration]] the weak equivalence

\begin{displaymath}
\int_{\Delta^\bullet} : \exp(b^n \mathbb{R}) \simeq \mathbf{B}^{n+1}\mathbb{R}_c
  \,.
\end{displaymath}
Let $I$ be the set of maps $\partial \Delta[4] \to \exp(b^2 \mathbb{R})$ that fit into a diagram

\begin{displaymath}
\itexarray{
    \partial \Delta[4] &\to& \exp(b^2 \mathbb{R})
    \\
    \downarrow && \downarrow^{\mathrlap{\int_{\Delta^\bullet}}}
    \\
    && \mathbf{B}^3 \mathbb{R}_c
    \\
    \downarrow && \downarrow 
    \\
    \Delta[4] &\to& \mathbf{B}^3 (\mathbb{Z} \to \mathbb{R})_c
  }
\end{displaymath}
(closed 3-forms on 3-balls whose integral is an integer).

Write

\begin{displaymath}
\exp(b^2 \mathbb{R}/\mathbb{Z}) 
    :=
  \mathbf{cosk}_3
   \left(
     (I \times \Delta[4])\coprod_{I \times \partial \Delta[4]} \mathbf{cosk_3} \exp(b^2 \mathbb{R})
   \right)
\end{displaymath}
for the result of filling all these by 4-cells. Similarly define $\exp(e b \mathbb{R}/\mathbb{Z})$.

Then applying the [[coskeleton]] functor to the above pullback diagram and using the projection (\hyperlink{FSS}{FSS})

\begin{displaymath}
\itexarray{
    \exp(\mathfrak{so}(n)) &\stackrel{\exp(\mu)}{\to}& \exp(b^2 \mathbb{R})
    \\
    \downarrow && \downarrow
    \\
    \mathbf{cosk}_3\exp(\mathfrak{so}(n)) 
      &\stackrel{\frac{1}{2}\mathbf{p}_1}{\to}&
    \exp(b^2 \mathbb{R}/\mathbb{Z})
  }
\end{displaymath}
we get the diagram

\begin{displaymath}
\itexarray{
    \mathbf{cosk}_3 (\mathfrak{string}(n)) 
      &\to&
    \exp(e b \mathbb{R}/\mathbb{Z})
    \\
    \downarrow && \downarrow
    \\
    \mathbf{cosk}_3 \exp(\mathfrak{so})
    &\stackrel{\frac{1}{2}\mathbf{p}_1}{\to}&
    \exp(b^2 \mathbb{R}/\mathbb{Z})
  }
  \,.
\end{displaymath}
This is again a pullback diagram of a fibration resolution of the point inclusion, hence presents the homotopy fiber in question.

\end{proof}
\hypertarget{PresentationByStrictTwoGroups}{}\subsubsection*{{By strict Lie $2$-group}}\label{PresentationByStrictTwoGroups}

A realization of the string 2-group as a [[strict 2-group]] [[internalization|internal]] to [[diffeological space]]s was given in (\hyperlink{BCSS}{BCSS}).

This is one of three different (there should be more), weakly equivalent such [[strict 2-group]] [[internalization|internal]] to [[diffeological space]] models that are discussed in the (to date unpublished)

\begin{itemize}%
\item \href{http://www.math.uni-hamburg.de/home/schreiber/nactwist.pdf#page=91}{nactwist, section 5.2.3}

\end{itemize}
(This particular section, and its results, are joint work of [[Urs Schreiber]] and [[Danny Stevenson]]).

We have the following pattern of routes through [[Lie integration]]:

\begin{displaymath}
\itexarray{
    StrLie \omega Grpd
    &&&&
    StrLie \omega Grpd
    &\stackrel{\simeq}{\leftarrow}&
    LieCrsdCmplx
    \\
    \uparrow^{\Pi_n S CE}
    &&&&
    \uparrow
    &&
    \uparrow^{\exp(-)}
    \\
    L_\infty Algebras
    &&
    \leftarrow&&
    Str L_\infty Algebras
    &\to&
    DiffCrsdCmplx
  }
\end{displaymath}
Here $StrLie \omega Grpd$ is [[strict omega-groupoid]]s internal to [[diffeological space]]s, $LieCrsCmplx$ is accordingly smooth [[crossed complex]]es , $L_\infty Algebra$ is all [[L-infinity algebra]]s and $Str L_\infty Algebra$ is \emph{strict} $L_\infty$-algebras. The vertical morphism on the right is term-wise ordinary [[Lie integration]]. The other vertical morphisms take an [[L-infinity algebra]], form the [[sheaf]] on [[Diff]] of flat [[schreiber:∞-Lie algebroid valued differential forms|∞-Lie algebroid differential form]]s, and then take [[path n-groupoid]] $\Pi_n(-)$ of that.

For the String-case this yields

\begin{displaymath}
\itexarray{
    \Pi_2(\Omega^\bullet_{fl}(-,\mathfrak{so}_{\mu_3}))
    &\stackrel{\simeq}{\mapsto}&
    \mathbf{B} String_{Mick}
    &\stackrel{\simeq}{\mapsto}&
    \mathbf{B} String_{BCSS}
    &\leftarrow|&
    (\hat \Omega Spin \to P Spin)
    \\
    \uparrow &&&\nearrow& \uparrow  && \uparrow
    \\
    \mathfrak{so}_{\mu_3}
    &&\stackrel{\simeq}{\mapsto}&&
    \mathfrak{string}
    &\mapsto&
    (\hat \Omega \mathfrak{so} \to P \mathfrak{so})    
  }
  \,,
\end{displaymath}
where

\begin{itemize}%
\item $\mathfrak{so}_{\mu_3}$ denotes the weak, skeletal [[String Lie 2-algebra]]


\item $\mathfrak{string}$ its equivalent strict version given by BCSS


\item the diagonal morphism is the construction in BCSS.


\item the strict 2-groupoid $\Pi_2(\Omega^\bullet_{fl}(-,\mathfrak{g}_{\mu_3}))$ has, notice, as morphism smooth paths in $Spin(n)$ that are composed by concatenation


\item the 2-groupoid $\mathbf{B}String_{Mick}$ is a version of the String Lie 2-group that manifestly uses the [[Mickelsson cocycle]] (morphism are paths in $Spin(n)$ that are composed using the group product)


\item the 2-groupoid $\mathbf{B}String_{BCSS}$ is the version given in BCSS (morhisms again are paths in $Spin(n)$ that are composed using the group product).



\end{itemize}
\hypertarget{as_a_finitedimensional_weak_lie_2group}{}\subsubsection*{{As a finite-dimensional weak Lie 2-group}}\label{as_a_finitedimensional_weak_lie_2group}

(\hyperlink{Schommer-Pries}{Schommer-Pries})

\hypertarget{as_an_automorphism_2group_of_fermionic_cft}{}\subsubsection*{{As an automorphism 2-group of fermionic CFT}}\label{as_an_automorphism_2group_of_fermionic_cft}

The string 2-group also appears as a certain [[automorphism 2-group]] inside the [[3-category of fermionic conformal nets]] (\hyperlink{DouglasHenriques}{Douglas-Henriques})

\hypertarget{as_the_automorphisms_of_the_wesszuminowitten_gerbe_2connection}{}\subsubsection*{{As the automorphisms of the Wess-Zumino-Witten gerbe 2-connection}}\label{as_the_automorphisms_of_the_wesszuminowitten_gerbe_2connection}

For $G$ a compact simply connected simple [[Lie group]], there is the ``[[WZW gerbe]]'', hence the [[circle n-bundle with connection|circle 2-bundle with connection]] on $G$ whose [[curvature]] 3-form is the [[left invariant differential form|left invariant]] extension $\langle \theta \wedge [\theta \wedge \theta]\rangle$ of the canonical Lie algebra 3-cocycle to the group

\begin{displaymath}
\mathcal{L}_{WZW} \;\colon\; G \longrightarrow \mathbf{B}^2 
  \,.
\end{displaymath}
\begin{prop}
\label{}\hypertarget{}{}
The string 2-group is the [[smooth infinity-group|smooth 2-group]] of [[automorphism infinity-group|automorphism]] of $\mathcal{L}_{WZW}$ which cover the left [[action]] of $G$ on itself (hence the ``[[Heisenberg 2-group]]'' of $\mathcal{L}_{WZW}$ regarded as a [[prequantum 2-bundle]])

\begin{displaymath}
\mathbf{Aut}(\mathcal{L}_{WZW}) \simeq String(G)
  \,,
\end{displaymath}
\end{prop}
This is due to (\hyperlink{FiorenzaRogersSchreiber13}{Fiorenza-Rogers-Schreiber 13, section 2.6.1}).

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

\begin{itemize}%
\item [[Platonic 2-group]]

\end{itemize}
[[fivebrane 6-group]] $\to$ \textbf{string 2-group} $\to$ [[spin group]] $\to$ [[special orthogonal group]] $\to$ [[orthogonal group]] $\hookrightarrow$ [[general linear group]]

\begin{itemize}%
\item [[spin group]], [[spin structure]], [[twisted spin structure]]


\item [[spin{\tt \symbol{94}}c]], [[spin{\tt \symbol{94}}c structure]], [[twisted spin{\tt \symbol{94}}c structure]]


\item [[string group]], [[string 2-group]], [[string structure]], [[twisted string structure]]


\item [[T-duality 2-group]]


\item [[string{\tt \symbol{94}}c 2-group]]


\item [[fivebrane group]], [[fivebrane 6-group]]



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

A [[crossed module]] presentation of a topological realization of the string 2-group is implicit in

\begin{itemize}%
\item [[Stephan Stolz]], [[Peter Teichner]], \emph{What is an elliptic object?} (\href{http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.146.5463&rep=rep1&type=pdf}{pdf})

\end{itemize}
A realization of the string 2-group in [[∞-groupoid]]s [[internalization|internal to]] [[Banach space]]s by [[Lie integration]] of the skeletal version of the [[string Lie 2-algebra]] is in

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

\end{itemize}
A realization of the string 2-group in [[strict 2-group]]s internal to [[Frechet manifold]]s by [[Lie integration]] of a [[strict Lie 2-algebra]] incarnation of the [[string Lie 2-algebra]] in in

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

\end{itemize}
A realization of the string 2-group as a [[2-group]] in finite-dimensional [[smooth manifold]]s in in

\begin{itemize}%
\item [[Chris Schommer-Pries]], \emph{Central Extensions of Smooth 2-Groups and a Finite-Dimensional String 2-Group} (\href{http://arxiv.org/abs/0911.2483}{arXiv:0911.2483})

\end{itemize}
A discussion as an [[∞-group]] object in [[Smooth∞Grpd]] and the realization of the smooth first fractional Pontryagin class is in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{Cech cocycles for differential characteristic classes} ()

\end{itemize}
and in section 4.1 of

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

\end{itemize}
A 2-group model which has a smoothening of the \emph{topological} [[string group]] in lowest degree has been given in

\begin{itemize}%
\item [[Thomas Nikolaus]], [[Christoph Sachse]], [[Christoph Wockel]], \emph{A Smooth Model for the String Group} (\href{http://arxiv.org/abs/1104.4288}{arXiv:1104.4288})

\end{itemize}
A construction explicitly in terms of the ``basic'' [[bundle gerbe]] on $G$ is discussed in

\begin{itemize}%
\item [[Konrad Waldorf]], \emph{A Construction of String 2-Group Models using a Transgression-Regression Technique} (\href{http://arxiv.org/abs/1201.5052}{arXiv:1201.5052})

\end{itemize}
Via fermionic nets/[[2-Clifford algebra]]:

\begin{itemize}%
\item [[Chris Douglas]], [[André Henriques]], \emph{Geometric string structures} (\href{http://www.staff.science.uu.nl/~henri105/PDF/TringWP.pdf}{pdf})

\end{itemize}
The realization of the string 2-group as the [[Heisenberg 2-group]] of the [[WZW gerbe]] is due to

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], \emph{[[schreiber:Higher geometric prequantum theory]]}, 2013 (\href{http://arxiv.org/abs/1304.0236}{arXiv:1304.0236})

\end{itemize}
A model of the string 2-group using the smooth [[free loop space]] (instead of the based [[loop space]]) is dicussed in

\begin{itemize}%
\item [[Michael Murray]], [[David Roberts]], [[Christoph Wockel]], \emph{Quasi-periodic paths and a string 2-group model from the free loop group} (\href{https://arxiv.org/abs/1702.01514}{arXiv:1702.01514})

\end{itemize}
Discussion in the context of [[matrix factorizations]] and [[equivariant K-theory]] is in

\begin{itemize}%
\item [[Daniel S. Freed]], [[Constantin Teleman]], \emph{Dirac families for loop groups as matrix factorizations}, \href{http://arxiv.org/abs/1409.6051}{arxiv/1409.6051}

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

[[!redirects smooth string 2-group]]



\end{document}