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\begin{document}

%-------------------------------------------------------------------


\section*{differential string structure -- proofs}

This page contains technical details to be used at the main page \emph{[[differential string structure]]} . See there for context.


\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{FactorizationOfTheLInfintiyCocycle}{Factorization of the $L_\infty$-cocycle}\dotfill \pageref*{FactorizationOfTheLInfintiyCocycle} \linebreak
\noindent\hyperlink{FactorizationOfTheLInfintiyCocycle}{Factorization of the differential $L_\infty$-cocycle}\dotfill \pageref*{FactorizationOfTheLInfintiyCocycle} \linebreak
\noindent\hyperlink{ModifiedWeilAlgebra}{The modified Weil algebra}\dotfill \pageref*{ModifiedWeilAlgebra} \linebreak
\noindent\hyperlink{TheLInftyDifferentialLift}{The differential lift}\dotfill \pageref*{TheLInftyDifferentialLift} \linebreak


\hypertarget{FactorizationOfTheLInfintiyCocycle}{}\subsection*{{Factorization of the $L_\infty$-cocycle}}\label{FactorizationOfTheLInfintiyCocycle}

\begin{prop}
\label{FactorizationOfTheCocycle}\hypertarget{FactorizationOfTheCocycle}{}
The $L_\infty$-algebra cocycle

\begin{displaymath}
\mu : \mathfrak{so} \to b^2 \mathbb{R}
\end{displaymath}
factors as

\begin{displaymath}
\mathfrak{so} \stackrel{}{\to} (b \mathbb{R} \to \mathfrak{string})
  \stackrel{}{\to}
  b^2 \mathbb{R}
\end{displaymath}
given dually on CE-algebras by

\begin{displaymath}
CE(\mathfrak{so}) 
     \stackrel{
       \left(
       \itexarray{
        t^a \mapsto t^a
        \\  
        b \mapsto 0
        \\   
        c \mapsto \mu
     }
     \right)
     }{\leftarrow} 
  CE(b \mathbb{R} \to \mathfrak{string})
   \stackrel{
     \left(
     \itexarray{
        c \mapsto c
     }
     \right)
   }{\leftarrow}
  CE(b^2 \mathbb{R})
  \,.
\end{displaymath}
The left morphism is a [[quasi-isomorphism]].

\end{prop}
\begin{proof}
To see that we have a quasi-isomorphism, notice that the dg-algebra $CE(b \mathbb{R} \to \mathfrak{string})$ is [[isomorphic]] to the one with generators $\{t^a, b, c'\}$ and differentials

\begin{displaymath}
\begin{aligned}
    d|_{\mathfrak{g}^*} & = [-,-]^*
    \\
    d  b & = c'
    \\
    d c' & =  0
  \end{aligned}
  \,,
\end{displaymath}
where the isomorphism is given by the identity on the $t^a$s and on $b$ and by

\begin{displaymath}
c \mapsto c' + \mu
  \,.
\end{displaymath}
The primed dg-algebra is the [[tensor product]] $CE(\mathfrak{g}) \otimes CE( inn(b \mathbb{R}))$, where the second factor is manifestly cohomologically trivial.

\end{proof}
\hypertarget{FactorizationOfTheLInfintiyCocycle}{}\subsection*{{Factorization of the differential $L_\infty$-cocycle}}\label{FactorizationOfTheLInfintiyCocycle}

We now give a concrete construction showing

\begin{prop}
\label{DifferentialLInfinityLift}\hypertarget{DifferentialLInfinityLift}{}
The factorization

\begin{displaymath}
CE(\mathfrak{so}) 
     \stackrel{
     \left(
     \itexarray{
        t^a \mapsto t^a
        \\  
        b \mapsto 0
        \\   
        c \mapsto \mu
     }
     \right)
     }{\leftarrow} 
  CE(b \mathbb{R} \to \mathfrak{string})
   \stackrel{
     \left(
     \itexarray{
        c \mapsto c
     }
     \right)
   }{\leftarrow}
  CE(b^2 \mathbb{R})
\end{displaymath}
from \hyperlink{FactorizationOfTheLInfintiyCocycle}{above} lifts to a factorization of differential $L_\infty$-algebraic cocycles

\begin{displaymath}
\itexarray{
    CE(\mathfrak{so}) 
     &
       \underoverset{\simeq}{
       \left(
       \itexarray{   
          t^a \mapsto t^a
          \\  
          b \mapsto 0
          \\   
          c \mapsto \mu
       }
       \right)
       }{\leftarrow} 
     &
    CE(b \mathbb{R} \to \mathfrak{string}) 
     &
     \stackrel{
       \left(
       \itexarray{
          c \mapsto c
       }
       \right)
     }{\leftarrow}
     &
    CE(b^2 \mathbb{R})
    \\
    \uparrow^\mathrlap{} 
     && 
    \uparrow^\mathrlap{}  
     && 
    \uparrow^\mathrlap{}  
    \\    
    W(\mathfrak{so}) 
      &
       \underoverset{\simeq}{
       }{\leftarrow} 
      &
    W(b \mathbb{R} \to \mathfrak{string}) 
      &
     \stackrel{
     }{\leftarrow}
     &
    W(b^2 \mathbb{R})
    \\
    \uparrow^\mathrlap{} 
     && 
    \uparrow^\mathrlap{}  
     && 
    \uparrow^\mathrlap{}  
    \\    
    inv(\mathfrak{so}) 
      &
       \underoverset{\simeq}{
       }{\leftarrow} 
      &
     \tilde inv(b \mathbb{R} \to \mathfrak{string})
      &
     \stackrel{
     }{\leftarrow}
     &
    inv(b^2 \mathbb{R})
  }
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
This is at its heart trivial, but potentially a bit tedious. We proceed in two steps:

\begin{enumerate}%
\item consider a ``modified Weil algebra'' of the twisted string Lie 3-algebra $(b \mathbb{R} \to \mathfrak{string})$

in \hyperlink{ModifiedWeilAlgebra}{The modified Weil algebra};


\item construct the desired factorization by factoring itself through two fairly evident morphisms into and out of the modified Weil algebra,

in \href{TheLInftyDifferentialLift}{The differential lift}.



\end{enumerate}
\end{proof}
\hypertarget{ModifiedWeilAlgebra}{}\subsubsection*{{The modified Weil algebra}}\label{ModifiedWeilAlgebra}

Our factorization \href{}{below} makes use of an isomorphic copy of the Weil algebra $W(b\mathbb{R} \to \mathfrak{g}_\mu)$.

\begin{prop}
\label{ShiftedWeilAlgebra}\hypertarget{ShiftedWeilAlgebra}{}
The [[Weil algebra]] $W(b\mathbb{R} \to \mathfrak{g}_\mu)$ of $(b^2 \mathbb{R} \to \mathfrak{g})$ is given on the extra shifted generators $\{r^a := \sigma t^a, h := \sigma b, g := \sigma c\}$ (where $\sigma$ is the shift operator extended as a graded derivation, see [[Weil algebra]]) by

\begin{itemize}%
\item $d t^a = -\frac{1}{2} C^a{}_{b c} t^b \wedge t^c + r^a$;


\item $d b = - \mu + c + h$;


\item $d c = g$,



\end{itemize}
with [[Bianchi identities]]

\begin{itemize}%
\item $d r^a = - C^a{}_{b c} t^b \wedge r^c$


\item $d h = \sigma \mu - g$;


\item $d g = 0$.



\end{itemize}
Let $\tilde W(b\mathbb{R} \to \mathfrak{g}_\mu)$ be the dg-algebra with the same underlying graded algebra as $W(b\mathbb{R} \to \mathfrak{g}_\mu)$ but with the differential modified as follows

\begin{itemize}%
\item $d t^a = -\frac{1}{2} C^a{}_{b c} t^b \wedge t^c + r^a$;


\item $d r^a  = - C^a{}_{b c} t^b \wedge r^a$;


\item $d b = - cs + c + \tilde h$;


\item $d \tilde h = \langle -,-\rangle - g$;


\item $d c = g$ .


\item $d g = 0$,



\end{itemize}
where ``$\tilde h$'' is the new name for the generator that used to be called ``$h$''

There is an [[isomorphism]]

\begin{displaymath}
W(b\mathbb{R} \to \mathfrak{g}_\mu)
  \to 
  \tilde W(b\mathbb{R} \to \mathfrak{g}_\mu)
\end{displaymath}
in [[dgAlg]] that is the identity on all generators except on $h$, where it is given by

\begin{displaymath}
h \mapsto \tilde h + (\mu - cs)
  \,.
\end{displaymath}
\end{prop}
\begin{note}
\label{RemarkOnShiftedWeilAlgebra}\hypertarget{RemarkOnShiftedWeilAlgebra}{}
Where the formula for the differential of $W(b\mathbb{R}\to \mathfrak{g}_\mu)$ has the 3-cocycle $\mu$ that for $\tilde W(b\mathbb{R}\to \mathfrak{g}_\mu)$ has the [[Chern-Simons element]] $cs$. The shift by $cs-\mu$ is precisely what shifts the curvature characteristic $d_{W(\mathfrak{g})}\mu$ into the shifted copy of $\mathfrak{g}^*$ in the Weil algebra, thus exhibiting the modified $h$ as an [[invariant polynomial]].

\end{note}
\begin{cor}
\label{TheInvariantPolynomials}\hypertarget{TheInvariantPolynomials}{}
The [[invariant polynomial]]s on $(b \mathbb{R} \to \mathfrak{g}_\mu)$ are generated from those of $\mathfrak{g}_\mu$ together with $\tilde h$ and $g$:

\begin{displaymath}
\tilde inv(b \mathbb{R} \to \mathfrak{string})
  =
  (inv(\mathfrak{so})\otimes \langle \tilde h, g\rangle)/(d \tilde h = \langle -,-\rangle - g)
  \,.
\end{displaymath}
\end{cor}
\hypertarget{TheLInftyDifferentialLift}{}\subsubsection*{{The differential lift}}\label{TheLInftyDifferentialLift}

We now use the isomorphism

\begin{displaymath}
W(b \mathbb{R} \to \mathfrak{string}) \stackrel{\simeq}{\to}
  \tilde W(b \mathbb{R} \to \mathfrak{string})
\end{displaymath}
from prop. \ref{ShiftedWeilAlgebra} and obtain the desired factorization, as the composite

\begin{displaymath}
\itexarray{
    CE(\mathfrak{so}) 
     &
       \underoverset{\simeq}{
       \left(
       \itexarray{   
          t^a \mapsto t^a
          \\  
          b \mapsto 0
          \\   
          c \mapsto \mu
       }
       \right)
       }{\leftarrow} 
     &
    CE(b \mathbb{R} \to \mathfrak{string}) 
     &
     =
     &
    CE(b \mathbb{R} \to \mathfrak{string}) 
     &
     \stackrel{
       \left(
       \itexarray{
          c \mapsto c
       }
       \right)
     }{\leftarrow}
     &
    CE(b^2 \mathbb{R})
    \\
    \uparrow^\mathrlap{i^*_{\mathfrak{so}}} 
     && 
    \uparrow^\mathrlap{i^*_{(b\mathbb{R} \to \mathfrak{string})}}  
     && 
    \uparrow^\mathrlap{}  
     && 
    \uparrow^\mathrlap{i^*_{b^2 \mathbb{R}}}  
    \\    
    W(\mathfrak{so}) 
      &
       \underoverset{\simeq}{
       \left(
       \itexarray{   
         t^a \mapsto t^a
         \\
         b \mapsto 0
         \\
         c \mapsto cs
         \\
         r^a \mapsto r^a
         \\
         \tilde h \mapsto 0  
         \\
         g \mapsto \langle-,-\rangle
       }
       \right)
       }{\leftarrow} 
      &
    W(b \mathbb{R} \to \mathfrak{string}) 
     &
     \underoverset{\simeq}{
      \left(
        \itexarray{
          \tilde h \mapsto h + (cs - \mu)
        }
      \right)
     }
     {\leftarrow}
     &
    \tilde W(b \mathbb{R} \to \mathfrak{string}) 
      &
     \stackrel{
       \left(
       \itexarray{
         c \mapsto c
         \\
         g \mapsto g
       }
       \right)
     }{\leftarrow}
     &
    W(b^2 \mathbb{R})
    \\
    \uparrow^\mathrlap{p^*_{\mathfrak{so}}} 
     && 
    \uparrow^\mathrlap{}  
     && 
    \uparrow^\mathrlap{p^*_{(b \mathbb{R} \to \mathfrak{string})}}  
     && 
    \uparrow^\mathrlap{p^*_{b^2 \mathbb{R}}}  
    \\    
    inv(\mathfrak{so}) 
      &
       \underoverset{\simeq}{
       \left(
       \itexarray{
         \tilde h \mapsto 0
         \\   
         g \mapsto \langle -,-\rangle
         \\   
         \langle \cdots \rangle
         \mapsto 
         \langle \cdots \rangle
       }
       \right)
       }{\leftarrow} 
      &
    \tilde inv(b \mathbb{R} \to \mathfrak{string}) 
      &
       =
      &
    \tilde inv(b \mathbb{R} \to \mathfrak{string}) 
      &
     \stackrel{
       \left(
       \itexarray{
         g \mapsto g
       }
       \right)
     }{\leftarrow}
     &
    inv(b^2 \mathbb{R})
  }
  \,.
\end{displaymath}
Here

\begin{itemize}%
\item the unlabelled vertical morphisms are defined as the unique ones that make the respective square commute;


\item the notation $\langle \cdots \rangle$ stands for all the [[invariant polynomial]]s of $\mathfrak{so}$ and $\langle-,-\rangle$ specifically for the [[Killing form]].



\end{itemize}


\end{document}