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\begin{document}

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\section*{differential string structure}

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

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

[[!include string theory - contents]]

\hypertarget{differential_cohomology}{}\paragraph*{{Differential cohomology}}\label{differential_cohomology}

[[!include differential cohomology - contents]]

\hypertarget{chernweil_theory}{}\paragraph*{{$\infty$-Chern-Weil theory}}\label{chernweil_theory}

[[!include infinity-Chern-Weil 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{GeneralProperties}{General}\dotfill \pageref*{GeneralProperties} \linebreak
\noindent\hyperlink{ChernWeilTheory}{Construction in terms of $L_\infty$-Cech cocycles}\dotfill \pageref*{ChernWeilTheory} \linebreak
\noindent\hyperlink{PresentationOfClassByFibration}{Presentation of the differential class by a fibration}\dotfill \pageref*{PresentationOfClassByFibration} \linebreak
\noindent\hyperlink{ExplicitCocycles}{Explicit Cech cocycles}\dotfill \pageref*{ExplicitCocycles} \linebreak
\noindent\hyperlink{differential_string_structures_and_fermionic_string_quantum_anomalies}{Differential string structures and fermionic string quantum anomalies}\dotfill \pageref*{differential_string_structures_and_fermionic_string_quantum_anomalies} \linebreak
\noindent\hyperlink{InheteroticSugra}{The Green-Schwarz mechanism in heterotic supergravity}\dotfill \pageref*{InheteroticSugra} \linebreak
\noindent\hyperlink{relation_to_string_2connections}{Relation to string 2-connections}\dotfill \pageref*{relation_to_string_2connections} \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}

Where a [[string structure]] is a trivialization of a class in [[integral cohomology]], a \emph{differential string structure} or \emph{geometric string structure} is the trivialization of this class refined to [[ordinary differential cohomology]]:

the first fractional [[Pontryagin class]]

\begin{displaymath}
\frac{1}{2} p_1 : B Spin \to B^4 \mathbb{Z}
\end{displaymath}
in the [[(∞,1)-topos]] [[∞Grpd]] $\simeq$ [[Top]] has a refinement to $\mathbf{H} =$ [[Smooth∞Grpd]] of the form

\begin{displaymath}
\frac{1}{2}\mathbf{p}_1 : \mathbf{B}Spin \to \mathbf{B}^3 U(1)
\end{displaymath}
-- the .

The induced morphism on [[cocycle]] [[∞-groupoid]]s

\begin{displaymath}
\frac{1}{2}\mathbf{p}_1
  :
  \mathbf{H}(X,\mathbf{B} Spin)
  \stackrel{}{\to}
  \mathbf{H}(X,\mathbf{B}^3 U(1))
\end{displaymath}
sends a [[spin group]]-[[principal bundle]] $P$ to its corresponding [[Chern-Simons circle 3-bundle]] $\frac{1}{2}\mathbf{p}_1(P)$.

A choice of trivialization of $\frac{1}{2}p_1(P)$ is a [[string structure]]. The [[2-groupoid]] of smooth string structures is the [[homotopy fiber]] of $\frac{1}{2}\mathbf{p}_1$ over the trivial [[circle n-bundle with connection|circle 3-bundle]].

By [[Chern-Weil theory in Smooth∞Grpd]] this morphism may be further refined to a [[differential characteristic class]] $\frac{1}{2}\hat \mathbf{p}_1$ that lands in the [[ordinary differential cohomology]] $\mathbf{H}_{diff}(X, \mathbf{B}^3 U(1))$, classifying [[circle n-bundle with connection|circle 3-bundles with connection]]

\begin{displaymath}
\frac{1}{2}\hat \mathbf{p}_1
  :
  \mathbf{H}_{conn}(X,\mathbf{B} Spin)
  \stackrel{}{\to}
  \mathbf{H}_{diff}(X,\mathbf{B}^3 U(1))
\end{displaymath}
The [[2-groupoid]] of \textbf{differential string structures} is the [[homotopy fiber]] of this refinement $\frac{1}{2}\hat \mathbf{p}_1$ over the [[circle n-bundle with connection|trivial circle 3-bundle with trivial connection]] or more generally over the trivial circle 3-bundles with possibly non-trivial connection

Such a differential string structure over a [[smooth manifold]] $X$ is characterized by a tuple consisting of

\begin{enumerate}%
\item a [[connection on a bundle|connection]] $\nabla$ on a [[Spin group|Spin]]-[[principal bundle]] on $X$;


\item a choice of trivial [[circle n-bundle with connection|circle 3-bundle]] with connection $(0, H_3)$, hence a differential 3-form $H_3 \in \Omega^3(X)$;


\item a choice of [[equivalence in an (∞,1)-category|equivalence]] $\lambda$ of the [[Chern-Simons circle 3-bundle]] with connection $\frac{1}{2}\hat\mathbf{p}_1(\nabla)$ of $\nabla$ with this chosen 3-bundle



\end{enumerate}
\begin{displaymath}
\lambda : \frac{1}{2}\hat \mathbf{p}_1(\nabla) \stackrel{\simeq}{\to}
    (0,H_3)
    \,.
\end{displaymath}
More generally, one can consider the [[homotopy fiber]]s of $\frac{1}{2}\hat \mathbf{p}_1$ over arbitrary circle 3-bundles with connection $\hat \mathcal{G}_4 \in \mathbf{H}_{diff}^4(X, \mathbf{B}^3 U(1))$ and hence replace $(0,H_3)$ in the above with $\hat \mathcal{G}_4$. According to the general notion of [[twisted cohomology]], these may be thought of as \textbf{twisted differential string structures}, where the class $[\mathcal{G}_4] \in H^4_{diff}(X)$ is the twist.

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

We will assume that the reader is familiar with basics of the discussion at [[Smooth∞Grpd]]. We often write $\mathbf{H} := Smooth \infty Grpd$ for short.

Let $Spin(n) \in$ [[SmoothMfd]] $\hookrightarrow$ [[Smooth∞Grpd]] be the [[Spin group]], for some $n \in \mathbb{N}$, regarded as a [[Lie group]] and thus canonically as an [[∞-group]] object in [[Smooth∞Grpd]]. We shall notationally suppress the $n$ in the following. Write $\mathbf{B}Spin$ for the [[delooping]] of $Spin$ in [[Smooth∞Grpd]]. (See the discussion ). Let moreover $\mathbf{B}^2 U(1) \in Smooth \infty Grpd$ be the  and $\mathbf{B}^3 U(1)$ its [[delooping]].

At [[Chern-Weil theory in Smooth∞Grpd]] the following statement is proven (\hyperlink{FSS}{FSS}).

\begin{prop}
\label{LieIntegrationToPOne}\hypertarget{LieIntegrationToPOne}{}
The image under [[Lie integration]] of the canonical [[Lie algebra cohomology|Lie algebra 3-cocycle]]

\begin{displaymath}
\mu = \langle -,[-,-]\rangle : \mathfrak{so} \to b^2 \mathbb{R}
\end{displaymath}
on the [[semisimple Lie algebra]] $\mathfrak{so}$ of the [[Spin group]] -- the [[special orthogonal Lie algebra]] -- is a morphism in [[Smooth∞Grpd]] of the form

\begin{displaymath}
\frac{1}{2} \mathbf{p}_1 : 
  \mathbf{B}Spin
    \to
  \mathbf{B}^3 U(1)
\end{displaymath}
whose image under the  $\Pi : Smooth \infty Grpd \to$ [[∞Grpd]] is the ordinary fractional [[Pontryagin class]]

\begin{displaymath}
\frac{1}{2}p_1 : B Spin \to B^4 \mathbb{Z}
\end{displaymath}
in [[Top]]. Moreover, the corresponding 

\begin{displaymath}
\frac{1}{2}\hat \mathbf{p}_1 : 
  \mathbf{H}_{conn}(-,\mathbf{B}Spin)
  \to
  \mathbf{H}_{diff}(-, \mathbf{B}^3 U(1))
\end{displaymath}
is in [[cohomology]] the corresponding refined [[Chern-Weil homomorphism]]

\begin{displaymath}
[\frac{1}{2}\hat \mathbf{p}_1] : 
  H^1_{Smooth}(X,Spin) \to H_{diff}^4(X)
\end{displaymath}
with values in [[ordinary differential cohomology]] that corresponds to the [[Killing form]] [[invariant polynomial]] $\langle - , - \rangle$ on $\mathfrak{so}$.

\end{prop}
\begin{defn}
\label{DifferentialStringStructure}\hypertarget{DifferentialStringStructure}{}
For any $X \in$ [[Smooth∞Grpd]], the [[2-groupoid]] of \textbf{differential string-structures} on $X$ -- $String_{diff}(X)$ -- is the [[homotopy fiber]] of $\frac{1}{2}\hat \mathbf{p}_1(X)$ over the trivial differential cocycle.

More generally (see [[twisted cohomology]]) the 2-groupoid of \textbf{twisted differential string structures} is the [[(∞,1)-pullback]] $String_{diff,tw}(X)$ in

\begin{displaymath}
\itexarray{
    String_{diff,tw}(X) &\to& H_{diff}^4(X)
    \\
    \downarrow && \downarrow
    \\
    \mathbf{H}_{conn}(X,\mathbf{B}Spin)
      &\stackrel{\frac{1}{2}\hat \mathbf{p}_1}{\to}&
    \mathbf{H}_{diff}(X,\mathbf{B}^3 U(1))
  }
  \,,
\end{displaymath}
where the right vertical morphism is a choice of (any) one point in each [[connected component]] (differential cohomology class) of the [[cocycle]] [[∞-groupoid]] $\mathbf{H}_{diff}(X,\mathbf{B}^3 U(1))$ (the [[homotopy type]] of the [[(∞,1)-pullback]] is independent of this choice).

More specifically, a \textbf{geometric string structure} is a twisted differential string structure whose differential twist has underlying trivial class.

\end{defn}
\begin{note}
\label{}\hypertarget{}{}
In terms of local [[∞-Lie algebra valued differential forms]] data this has been considered in (\hyperlink{SSSIII}{SSSIII}), as we shall discuss \hyperlink{ChernWeilTheory}{below}.

For the case where the the underlying integral class of the twist is trivial -- geometric string structures -- something close to this definition, explicitly modeled on [[bundle 2-gerbe]]s, has been given in (\hyperlink{Waldorf}{Waldorf}). See the discussion \hyperlink{Properties}{below}.

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

\hypertarget{GeneralProperties}{}\subsubsection*{{General}}\label{GeneralProperties}

\begin{prop}
\label{GroupoidOfStringStructures}\hypertarget{GroupoidOfStringStructures}{}
The [[∞-groupoid]] $String_{tw,diff}(X)$ of twisted differential string structures is [[2-truncated]], hence is a [[2-groupoid]].

\end{prop}
\begin{proof}
This follows from the [[long exact sequence of homotopy groups]] associated to the defining [[(∞,1)-pullback]], using that

\begin{itemize}%
\item $\mathbf{H}_{diff}(X, \mathbf{B}^3 U(1))$ is a [[3-groupoid]];


\item $\mathbf{H}(X, \mathbf{B}Spin)$ is a [[1-groupoid]];


\item $H^4_{diff}(X)$ is a [[0-groupoid]].



\end{itemize}
\end{proof}
See also (\hyperlink{Waldorf}{Waldorf, cor. 1.1.5}).

\begin{prop}
\label{PropertiesOfTrivialIntegralTwistClass}\hypertarget{PropertiesOfTrivialIntegralTwistClass}{}
If the underlying [[integral cohomology]] class of the twist is trivial, $c(tw) = 0 \in H^3(X, \mathbb{Z})$, then a $tw$-twisted differential string structures on a $Spin$-[[connection on a bundle|connection]] $\nabla$ are characterized by a globally defined 3-form on $X$.

\end{prop}
This 3-form is the globally defined connection 3-form of an appropriate [[circle n-bundle with connection|circle 3-bundle with connection]] equivalent to the [[Chern-Simons circle 3-bundle]] $CS(\nabla)$ whose underlying 3-bundle is by assumption trivial: on a trivial circle $n$-bundle every connection may be represented by a globally defined $n$-form.

This statement appears as (\hyperlink{Waldorf}{Waldorf, theorem 1.3.3}), where circle 3-bundles are modeled as [[bundle 2-gerbe]]s. The explicit construction of the globally defined 3-form in this model is spelled out in lemma 3.2.4 there.

\begin{proof}
Elements in the defining [[homotopy pullback]] from def. \ref{DifferentialStringStructure} over a given [[connection on a bundle|connection]] $\nabla \in \mathbf{H}(X,\mathbf{B} Spin)_{conn}$ are chracterized by an element $[\alpha] \in H^4_{diff}(X)$ and an [[equivalence in an (infinity,1)-category|equivalence]]

\begin{displaymath}
\Omega : CS(\nabla) \stackrel{\simeq}{\to} \alpha
\end{displaymath}
between the corresponding [[Chern-Simons circle 3-bundle]] and the given [[circle n-bundle with connection|circle 3-bundle]] $\alpha$. In the case at hand, both have underlying trivial class $c(CS(\nabla)) = c(\alpha) = 0$.

By the \href{http://nlab.mathforge.org/nlab/show/ordinary%20differential%20cohomology#AbstractProperties}{characteristic class exact sequence}

\begin{displaymath}
0 \to \Omega^3(X)/\Omega_{int}^{3}(X)
  \to 
   H^4_{diff}(X) \stackrel{c}{\to} H^4(X, \mathbb{Z})
\end{displaymath}
any two classes in $\pi_0 \mathbf{H}_{diff}(X, \mathbf{B}^3 U(1)) \simeq H^4_{diff}(X)$ that have trivial underlying class in $\pi_0 \mathbf{H}(X, \mathbf{B}^3 U(1)) \simeq H^4(X, \mathbb{Z})$ differ by a 3-form modulo a closed 3-form with integral periods.

Therefore both $[\alpha]$ as well as $[CS(\nabla)] \in H^4_{diff}(X)$ are given by a globally defined 3-form modulo an integral form: the global connection 3-form on these trivial [[circle n-bundle with connection|circle 3-bundles with connection]].

\end{proof}
\hypertarget{ChernWeilTheory}{}\subsubsection*{{Construction in terms of $L_\infty$-Cech cocycles}}\label{ChernWeilTheory}

We use the [[presentable (∞,1)-category|presentation]] of the [[(∞,1)-topos]] [[Smooth∞Grpd]] (as described there) by the local [[model structure on simplicial presheaves]] $[CartSp_{smooth}^{op}, sSet]_{proj,loc}$ to give an explicit construction of twisted differential string structures in terms of [[Cech cohomology|Cech]]-cocycles with coefficients in [[∞-Lie algebra valued differential forms]].

Proofs not displayed here can be found at \emph{[[differential string structure -- proofs]]} .

Recall the following fact from [[Chern-Weil theory in Smooth∞Grpd]] (\hyperlink{FSS}{FSS}).

\begin{prop}
\label{LieIntegrationOfDifferentialPOne}\hypertarget{LieIntegrationOfDifferentialPOne}{}
The differential fractional Pontryagin class $\frac{1}{2} \hat \mathbf{p}_1$ is presented in $[CartSp_{smooth}^{op}, sSet]_{proj}$ by the top morphism of simplicial presheaves in

\begin{displaymath}
\itexarray{
    \mathbf{cosk}_3\exp(\mathfrak{so})_{ChW,smp}
     &\stackrel{\exp(\mu, cs)}{\to}&
    \mathbf{B}^3 \mathbb{R}/\mathbb{Z}_{ChW,smp}     
    \\
    \downarrow && \downarrow
    \\
    \mathbf{cosk}_3\exp(\mathfrak{so})_{diff,smp}
     &\stackrel{\exp(\mu, cs)}{\to}&
    \mathbf{B}^3 \mathbb{R}/\mathbb{Z}_{smp} 
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    \mathbf{B}Spin_{c}
  }
  \,.
\end{displaymath}
\end{prop}
Here the middle morphism is the direct [[Lie integration]] of the [[infinity-Lie algebra cohomology|L-∞ algebra cocycle]] while the top morphisms is its restriction to coefficients for [[connection on an ∞-bundle|∞-connections]].

In order to compute the [[homotopy fiber]]s of $\frac{1}{2}\hat \mathbf{p}_1$ we now find a [[resolution]] of this morphism $\exp(\mu,cs)$ by a fibration in $[CartSp_{smooth}^{op}, sSet]_{proj}$. By the fact that this is a [[simplicial model category]] then also the hom of any cofibrant object into this morphism, computing the cocycle $\infty$-groupoids, is a fibration, and therefore, by the general discussion at [[homotopy pullback]], we obtain the [[homotopy fiber]]s as the ordinary [[fiber]]s of this fibration.

\hypertarget{PresentationOfClassByFibration}{}\paragraph*{{Presentation of the differential class by a fibration}}\label{PresentationOfClassByFibration}

In order to factor $\exp(\mu,cs)$ into a weak equivalence followed by a fibration, we start by considering such a factorization before differential refinement, on the underlying characteristic class $\exp(\mu)$.

To that end, we replace the [[Lie algebra]] $\mathfrak{g} = \mathfrak{so}$ by an equivalent but bigger [[∞-Lie algebra|Lie 3-algebra]] (following \hyperlink{SSSIII}{SSSIII}). We need the following notation:

\begin{itemize}%
\item $\mathfrak{g} = \mathfrak{so}$, the [[special orthogonal Lie algebra]] (the Lie algebra of the [[spin group]]);


\item $b^2 \mathbb{R}$ the [[line Lie n-algebra|line Lie 3-algebra]], the single generator in degee 3 of its [[Chevalley-Eilenberg algebra]] we denote $c \in CE(b^2 \mathbb{R})$, $d c = 0$.


\item $\langle -,-\rangle \in W(\mathfrak{g})$ is the [[Killing form]] [[invariant polynomial]], regarded as an element of the [[Weil algebra]] of $\mathfrak{so}$;


\item $\mu := \langle -,[-,-]\rangle \in CE(\mathfrak{g})$ the degree 3 [[Lie algebra cohomology|Lie algebra cocycle]], identified with a morphism

\begin{displaymath}
CE(\mathfrak{g}) \leftarrow CE(b^2 \mathbb{R}) : \mu
\end{displaymath}
of [[Chevalley-Eilenberg algebra]]s; and normalized such that its continuation to a 3-form on $Spin$ is the image in [[de Rham cohomology]] of $Spin$ of a generator of $H^3(Spin,\mathbb{Z}) \simeq \mathbb{Z}$;


\item $cs \in W(\mathfrak{g})$ is a [[Chern-Simons element]] interpolating between the two; characterized by the fact that it fits into the commuting diagram

\begin{displaymath}
\itexarray{
     CE(\mathfrak{g}) &\stackrel{\mu}{\leftarrow}&
     CE(b^2 \mathbb{R})
     \\
     \uparrow && \uparrow
     \\
     W(\mathfrak{g})
     &\stackrel{cs}{\leftarrow}&
     W(b^2 \mathbb{R})
     \\
     \uparrow && \uparrow
     \\
     inv(\mathfrak{g}) &\stackrel{\langle-,-\rangle}{\leftarrow}&
     inv(b^2 \mathbb{R})
     & = CE(b^3 \mathbb{R})
  }
\end{displaymath}

\item $\mathfrak{g}_\mu := \mathfrak{string}$ the [[string Lie 2-algebra]].



\end{itemize}
\begin{defn}
\label{ResolutionOfLieAlgebra}\hypertarget{ResolutionOfLieAlgebra}{}
Let $(b\mathbb{R} \to \mathfrak{g}_\mu)$ denote the [[L-∞ algebra]] whose [[Chevalley-Eilenberg algebra]] is

\begin{displaymath}
CE(b\mathbb{R} \to \mathfrak{g}_\mu) = 
  (\wedge^\bullet(  \mathfrak{g}^* \oplus \langle b\rangle \oplus \langle c \rangle ), d)
  \,,
\end{displaymath}
with $b$ a generator in degree 2, and $c$ a generator in degree 3, and with differential defined on generators by

\begin{displaymath}
\begin{aligned}
    d|_{\mathfrak{g}^*} & = [-,-]^*
    \\
    d  b & = - \mu + c
    \\
    d c & =  0
  \end{aligned}
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{FactorizationOfTheCocycle}\hypertarget{FactorizationOfTheCocycle}{}
The 3-cocycle $CE(\mathfrak{g}) \stackrel{\mu}{\leftarrow} CE(b^2 \mathbb{R})$ factors as

\begin{displaymath}
CE(\mathfrak{g})
   \stackrel{(c \mapsto \mu, b \mapsto 0)}{\leftarrow}
  CE(b\mathbb{R} \to \mathfrak{g}_\mu)
   \stackrel{(c \mapsto c)}{\leftarrow}
  CE(b^2 \mathbb{R})
  :
  \mu
  \,,
\end{displaymath}
where the morphism on the left (which is the identity when restricted to $\mathfrak{g}^*$ and acts on the new generators as indicated) is a [[quasi-isomorphism]].

\end{prop}
The point of introducing the resolution $(b \mathbb{R} \to \mathfrak{g}_\mu)$ in the above way is that it naturally supports the obstruction theory of lifts from $\mathfrak{g}$-[[connection on a bundle|connections]] to [[string Lie 2-algebra]] [[connection on an infinity-bundle|2-connection]]

\begin{prop}
\label{LongFiberSequenceOnLieAlgebras}\hypertarget{LongFiberSequenceOnLieAlgebras}{}
The defining projection $\mathfrak{g}_\mu \to \mathfrak{g}$ factors through the above quasi-isomorphism $(b \mathbb{R} \to \mathfrak{g}_\mu) \to \mathfrak{g}$ by the canonical inclusion

\begin{displaymath}
\mathfrak{g}_\mu \to (b \mathbb{R} \to \mathfrak{g}_\mu)
  \,,
\end{displaymath}
which dually on $CE$-algebras is given by

\begin{displaymath}
t^a \mapsto t^a
\end{displaymath}
\begin{displaymath}
b \mapsto - b
\end{displaymath}
\begin{displaymath}
c \mapsto 0
  \,.
\end{displaymath}
In total we are looking at a convenient presentation of the long [[fiber sequence]] of the [[string Lie 2-algebra]] extension:

\begin{displaymath}
\itexarray{   
    && && (b \mathbb{R} \to \mathfrak{g}_\mu) &\to& b^2 \mathbb{R}
    \\
    && & \nearrow & \downarrow^{\mathrlap{\simeq}}
    \\
    b \mathbb{R} &\to& \mathfrak{g}_\mu &\to& \mathfrak{g}
  }
  \,.
\end{displaymath}
\end{prop}
(The signs appearing here are just unimportant convention made in order for some of the formulas below to come out nice.)

\begin{prop}
\label{BareFibration}\hypertarget{BareFibration}{}
The image under [[Lie integration]] of the above factorization is

\begin{displaymath}
\exp(\mu) 
  :
   \mathbf{cosk}_3\exp(\mathfrak{g})
   \to 
   \mathbf{cosk}_3\exp(b \mathbb{R} \to \mathfrak{g}_\mu)
   \to 
   \mathbf{B}^3 \mathbb{R}/\mathbb{Z}_c
\end{displaymath}
where the first morphism is a weak equivalence followed by a fibration in the [[model structure on simplicial presheaves]] $[CartSp_{smooth}^{op}, sSet]_{proj}$.

\end{prop}
\begin{proof}
To see that the left morphism is objectwise a [[weak homotopy equivalence]], notice that a $[k]$-cell of $\exp(b \mathbb{R} \to \mathfrak{g}_\mu)$ consists of a triple $(A,B,C)$, where $A$ is a vertical flat $\mathfrak{g}$-valued 1-form on $U\times\Delta^k$, $B$ is a vertical 2-form and $C$ a 3-form on $U\times\Delta^k$, such that $d B=C-\mu(A,A,A)$ and $d C=0$, since $A$ is flat. Therefore $C$ is uniquely determined by $A$ and $B$, and there are no conditions on $B$. This means that a $[k]$-cell of $\exp(b \mathbb{R} \to \mathfrak{g}_\mu)$ is identified with a pair consisting of a based [[smooth function]] $f : \Delta^k \to Spin$ and a [[vertical differential form|vertical 2-form]] $B \in \Omega^2_{si,vert}(U \times \Delta^k)$, (both suitably with sitting instants perpendicular to the boundary of the simplex). Since there is no further condition on the 2-form, it can always be extended from the boundary of the $k$-simplex to the interior (for instance simply by radially rescaling it smoothly to 0). Accordingly the [[simplicial homotopy group]]s of $\exp(b \mathbb{R} \to \mathfrak{g}_\mu)(U)$ are the same as those of $\exp(\mathfrak{g})(U)$. The morphism between them is the identity in $f$ and picks $B = 0$ and is hence clearly an isomorphism on homotopy groups.

We turn now to discussing that the second morphism is a fibration. The nontrivial degrees of the lifting problem

\begin{displaymath}
\itexarray{
    \Lambda[k]_i 
     &\to&
     \mathbf{cosk}_3\exp(b\mathbb{R} \to \mathfrak{g}_\mu)(U)
    \\
    \downarrow && \downarrow
    \\
    \Delta[k] &\to& \mathbf{B}^3 \mathbb{R}/\mathbb{Z}_c(U)
  }
\end{displaymath}
are $k = 3$ and $k = 4$.

Notice that a $3$-cell of $\mathbf{B}^3 \mathbb{R}/ \mathbb{Z}_c(U)$ is a [[smooth function]] $U \to \mathbb{R}/\mathbb{Z}$ and that the morphism $\exp(b\mathbb{R} \to \mathfrak{g}_\mu) \to \mathbf{B}^3 \mathbb{R}/\mathbb{Z}_c$ sends the pair $(f,B)$ to the [[fiber integration]] $\int_{\Delta^3}(f^* \langle \theta \wedge [\theta \wedge \theta]\rangle + d B) mod \mathbb{Z}$.

Our lifting problem in degree 3, has given a function $c : U \times \Delta^3 \to \mathbb{R}/\mathbb{Z}$ and a smooth function (with sitting instants at the subfaces) $f : U \times \Lambda^3_i \to Spin$ together with a 2-form $B$ on the [[horn]] $U \times \Lambda^3_i$.

By pullback along the standard [[continuous function|continuous]] [[retract]] $\Delta^3 \to \Lambda^3_i$ which is non-smooth only where $f$ has sitting instants, we can always extend $f$ to a smooth function $f' : U \times \Delta^3 \to Spin$ with the property that $\int_{\Delta^3} (f')^* \langle \theta \wedge [\theta \wedge \theta]\rangle = 0$. (Following the general discussion at [[Lie integration]].)

In order to find a horn filler for the 2-form component, consider any smooth 2-form with sitting instants and non-vanishing integeral on $\Delta^2$, regarded as the missing face of the [[horn]]. By multiplying it with a suitable smooth function on $U$ we can obtain an extension $\tilde B \in \Omega^3_{si,vert}(U \times \partial \Delta^3)$ of $B$ to all of $U \times \partial \Delta^3$ with the property that its integral over $\partial \Delta^3$ is the given $c$. By the [[Stokes theorem]] it remains to extend $\tilde B$ to the interior of $\Delta^3$ in any way, as long as it is smooth and has sitting instants.

To that end, we can find in a similar fashion a smooth $U$-parameterized family of closed 3-forms $H$ with sitting instants on $\Delta^3$, whose integral over $\Delta^3$ equals $c$. Since by sitting instants this 3-form vanishes in a neighbourhood of the boundary, the standard formula for the [[Poincare lemma]] applied to it produces a 2-form $B' \in \Omega^2_{si, vert}(U \times \Delta^3)$ with $d B' = C$ that itself is radially constant at the boundary. By construction the difference $\tilde B - B'|_{\partial \Delta^3}$ has vanishing surface integral. By the discussion at [[Lie integration]] it follows that the difference extends smoothly and with sitting instants to a closed 2-form $\hat B \in \Omega^2_{si,vert}(U \times \Delta^3)$. Therefore the sum

\begin{displaymath}
B' + \hat B \in \Omega^2_{si,vert}(U \times \Delta^3)
\end{displaymath}
equals $B$ when restricted to $\Lambda^k_i$ and has the property that its integral over $\Delta^3$ equals $c$. Together with our extension $f'$, this constitutes a pair that solves the lifting problem.

The extension problem in degree 4 amounts to a similar construction: by coskeletalness the condition is that for a given $c : U \to \mathbb{R}/\mathbb{Z}$ and a given vertical 2-form on $U \times \partial \Delta^3$ such that its integral equals $c$, as well as a function $f : U \times \partial \Delta^3 \to Spin$, we can extend the 2-form and the function along $U \times \partial \Delta^3 \to U \times \Delta^3$. The latter follows from the fact that $\pi_2 Spin = 0$ which guarantees a continuous filler (with sitting instants), and using the [[Steenrod-Wockel approximation theorem]] to make this smooth. We are left with the problem of extending the 2-form, which is the same problem we discussed above after the choice of $\tilde B$.

\end{proof}
We now proceed to extend this factorization to the exponentiated differential coefficients (see [[connection on an ∞-bundle]]).

\begin{prop}
\label{PresentationByFibration}\hypertarget{PresentationByFibration}{}
\textbf{(presentation of the differential class by a fibration)}

Under [[Lie integration]] the \hyperlink{FactorizationOfTheCocycle}{above factorization} of the Lie algebra cocycle\newline maps to the factorization

\begin{displaymath}
\exp(\mu, cs)
   :
  \mathbf{cosk}_3 \exp(\mathfrak{g})_{ChW} 
   \stackrel{\simeq}{\to} 
  \mathbf{cosk}_3 \exp((b \mathbb{R} \to \mathfrak{g}_\mu))_{ChW}
   \to
  \mathbf{B}^3 U(1)_{ChW,ch}
\end{displaymath}
of $\exp(\mu,cs)$ in $[CartSp^{op}, sSet]_{proj}$, where the first morphism is a weak equivalence and the second a fibration.

\end{prop}
\begin{proof}
The following proof makes use of details discussed at \emph{[[differential string structure -- proofs]]} .

We discuss that the first morphism is an equivalence. Clearly it is injective on homotopy groups: if a sphere of $A$-data cannot be filled, then also adding the $(B,C)$-data does not yield a filler. So we need to check that it is also surjective on homotopy groups: if the $A$-data can be filled, then also the corresponding $(B,C)$-data has a filler. Since the curvature $H$ is horizontal it is already extended. We may extend $B$ in any smooth way to $U \times \Delta^k$ (for instance by rescaling it smoothly to zero at the center of the $k$-simplex) and then take the equation $d B = - CS(A) + C + H$ to define the extension of $C$.

We now check that the second morphism is a fibration. It is itself the composite

\begin{displaymath}
\mathbf{cosk}_{3} \exp(b \mathbb{R} \to \mathfrak{g}_\mu)_{ChW}
  \to 
  \exp(b^2 \mathbb{R})_{ChW}/\mathbb{Z}
  \stackrel{\int_{\Delta^\bullet}}{\to}
  \mathbf{B}^3 \mathbb{R}/\mathbb{Z}_{ChW,ch}
  \,.
\end{displaymath}
Here the second morphism is a degreewise surjection of simplicial abelian groups, hence a degreewise surjection under the [[normalized chain complex]] functor, hence is itself already a projective fibration. Therefore it is sufficient to show that the first morphism here is a fibration.

In degree $k = 0$ to $k = 3$ the lifting problems

\begin{displaymath}
\itexarray{
     \Lambda[k]_i 
       &\to & 
      \exp(b\mathbb{R} \to \mathfrak{g}_{\mu})_{smp,ChW}(U)
      \\
      \downarrow && \downarrow
      \\
      \Delta[k] 
        &\to&
      \exp(b^2 \mathbb{R})_{smp,ChW}/\mathbb{Z}(U)
  }
\end{displaymath}
may all be equivalently reformulated as lifting against a [[cylinder]] $D^k \hookrightarrow D^k \times [0,1]$ by using the sitting instants of all forms.

We have then a 3-form $C \in \Omega^3_{si}(U \times D^{k-1}\times [0,1])$ with horizontal curvature $\mathcal{G} \in \Omega^4(U)$ and differential form data $(A,B)$ on $U \times D^{k-1}$ given. We may always extend $A$ along the cylinder direction $[0,1]$ (its vertical part is equivalently a based smooth function to $Spin$ which we may extend constantly). $H$ has to be horizontal so it is to be constantly extended along the cylinder.

We can then use the kind of formula that proves the [[Poincare lemma]] to extend $B$. Let $\Psi : (D^k \times [0,1]) \times [0,1] \to (D^k  \times [0,1])$ be a smooth contraction. Then while $d(H - CS(A) + C)$ may be non-vanishing, by horizonatlity of their [[curvature characteristic form]]s we still have that $\iota_{\partial_t} \Psi_t^* d(H - CS(A) + C)$ vanishes (since the contraction vanishes).

Therefore the 2-form

\begin{displaymath}
\tilde B := \int_{[0,1]} \iota_{\partial_t} \Psi_t^*(H - CS(A) + C)
\end{displaymath}
satisfies $d \tilde B = (H - CS(A) + C)$. It may however not coincide with our given $B$ at $t = 0$. But the difference $B - \tilde B|_{t = 0}$ is a closed form on the left boundary of the cylinder. We may find some closed 2-form on the other boundary such that the integral around the boundary vanishes. Then the argument from the proof of the [[Lie integration]] of the [[line Lie n-algebra]] applies and we find an extension $\lambda$ to a closed 2-form on the interior. The sum

\begin{displaymath}
\hat B := \tilde B + \lambda
\end{displaymath}
then still satisfies $d \hat B = H - CS(A) - C$ and it coincides with $B$ on the left boundary.

Notice that here $\tilde B$ indeed has sitting instants: since $H$, $CS(A)$ and $C$ have sitting instants they are constant on their value at the boundary in a neighbourhood perpendicular to the boundary, which means for these 3-forms in the degrees $\leq 3$ that they \emph{vanish} in a neighbourhood of the boundary, hence that the above integral is towards the boundary over a vanishing integrand.

In degree 4 the nature of the lifting problem

\begin{displaymath}
\itexarray{
  \Lambda[4]_i &\to& \mathbf{cosk}_3\exp(b\mathbb{R} \to \mathfrak{g}_\mu)(U)
  \\
  \downarrow && \downarrow
  \\
  \Delta[4] &\to& \mathbf{B}^3 \mathbb{R}/\mathbb{Z}_{ChW,ch}
  }
\end{displaymath}
starts out differently, due to the presence of $\mathbf{cosk}_3$, but it then ends up amounting to the same kind of argument:

We have four functions $U \to \mathbb{R}/\mathbb{Z}$ which we may realize as the [[fiber integration]] of a 3-form $C$ on $U \times (\partial \Delta[4] \setminus \delta_i \Delta[3])$, and we have a lift to $(A,B,C, H)$-data on $U \times (\partial \Delta[4]\setminus \delta_i(\Delta[3]))$ (the boundary of the 4-simplex minus one of its 3-simplex faces).

We observe that we can

\begin{itemize}%
\item always extend $C$ smoothly to the remaining 3-face such that its [[fiber integration]] there reproduces the signed difference of the four given functions corresponding to the other faces (choose any smooth 3-form with sitting instants and with non-vanishing integral and rescale smoothly);


\item fill the $A$-data horizonatlly due to the fact that $\pi_2 (Spin) = 0$.


\item the $H$-form is already horizontal, hence already filled.



\end{itemize}
Moreover, by the fact that the 2-form $B$ already is defined on all of $\partial \Delta[4] \setminus \delta_i(\Delta[3])$ its fiber integral over the boundary $\partial \Delta[3]$ coincides with the fiber integral of $H - CS(A) + C$ over $\partial \Delta[4] \setminus \delta_i (\Delta[3])$). But by the fact that we have lifted $C$ and the fact that $\mu(A_{vert}) = CS(A)|_{\Delta^3}$ is an integral cocycle, it follows that this equals the fiber integral of $C - CS(A)$ over the remaining face.

Use then as above the vertical Poincare lemma-formula to find $\tilde B$ on $U \times \Delta^3$ with sitting instants that satisfies the equation $d B = H - CS(A) + C$ there. Then extend the closed difference $B - \tilde B|_{0}$ to a closed smooth 2-form on $\Delta^3$. As before, the difference

\begin{displaymath}
\hat B := \tilde B  + \lambda
\end{displaymath}
is an extension of $B$ that constitutes a lift.

\end{proof}
\hypertarget{ExplicitCocycles}{}\paragraph*{{Explicit Cech cocycles}}\label{ExplicitCocycles}

\begin{cor}
\label{PresentationBySimplicialPresheaves}\hypertarget{PresentationBySimplicialPresheaves}{}
For any $X \in$ [[SmoothMfd]] $\hookrightarrow$ [[Smooth∞Grpd]], for any choice of differentiaby [[good open cover]] with corresponding cofibrant presentation $\hat X = C(\{C_i\})\in [CartSp_{smooth}^{op}, sSet]_{proj}$ we have that the [[2-groupoid]]s of \hyperlink{DifferentialStringStructure}{twisted different String structures}s are presented by the ordinary [[fiber]]s of the morphism of [[Kan complex]]es

\begin{displaymath}
[CartSp^{op}, sSet](\hat X,\exp(\mu,cs))
  :
  [CartSp^{op}, sSet](\hat X, \mathbf{cosk}_3 \exp(b \mathbb{R} 
   \to \mathfrak{g}_\mu)_{ChW})
  \to 
  [CartSp^{op}, sSet](\hat X, \mathbf{B}^3 U(1)_{ChW})
  \,.
\end{displaymath}
over any basepoints in the connected components of the [[Kan complex]] on the right, which correspond to the elements\newline $[\hat \mathbf{C}_3] \in H_{diff}^4(X)$ in the [[ordinary differential cohomology]] of $X$.

\end{cor}
\begin{proof}
Since $[CartSp_{smooth}^{op}, sSet]_{proj}$ is a [[simplicial model category]] the morphism $[CartSp^{op}, sSet](\hat X,\exp(\mu,cs))$ is a fibration because $\exp(\mu,cs)$ is and $\hat X$ is cofibrant.

It follows from the discussion at [[homotopy pullback]] that the ordinary [[pullback]] of [[simplicial presheaves]]

\begin{displaymath}
\itexarray{
    String_{diff,tw}(X) &\to& H_{diff}^4(X)
    \\
    \downarrow && \downarrow
    \\
    [CartSp^{op}, sSet](\hat X, \mathbf{cosk}_3 \exp(b \mathbb{R} 
     \to \mathfrak{g}_\mu)_{ChW})
    &\to& 
  [CartSp^{op}, sSet](\hat X, \mathbf{B}^3 U(1)_{ChW})
  }
\end{displaymath}
is a presentation for the defining [[(∞,1)-pullback]] for $String_{diff,tw}(X)$, as defined \hyperlink{DifferentialStringStructure}{above}.

\end{proof}
We unwind the explicit expression for a twisted differential string structure under this equivalence.

Any twisting cocycle is in the above presentation given by a [[Cech cohomology|Cech]] [[Deligne cohomology|Deligne]]-cocycle (as discussed at [[circle n-bundle with connection]])

\begin{displaymath}
\hat \mathbf{H}_3 = ((H_3)_i, \cdots)
\end{displaymath}
with local connection 3-form $(H_3)_i \in \Omega^3(U_i)$ and globally defined [[curvature]] 4-form $\mathcal{G}_4 \in \Omega^4(X)$.

\begin{note}
\label{UnwindingTheLocalData}\hypertarget{UnwindingTheLocalData}{}
A differential string structure on $X$ twisted by this cocycles is on patches $U_i$ a morphism

\begin{displaymath}
\Omega^\bullet(U_i)
  \leftarrow
  \tilde W(b\mathbb{R}\to \mathfrak{g}_\mu)
\end{displaymath}
in [[dgAlg]], subject to some horizontality constraints. The components of this are over each $U_i$ a collection of differential forms of the following structure

\begin{displaymath}
\left(
    \itexarray{ 
      F_\omega =& d \omega + \frac{1}{2}[\omega \wedge \omega]
      \\
      H_3 =& \nabla B := d B + CS(\omega) - C_3 
      \\
      \mathcal{G}_4  =& d C_3
      \\
      d F_\omega =& - [\omega \wedge F_\omega]
      \\
      d H_3 =&  \langle F_\omega \wedge F_\omega\rangle - \mathcal{G}_4
      \\
      d \mathcal{G}_4 =& 0
    }
  \right)_i
  \;\;\;\;
  \stackrel{
    \itexarray{
       t^a & \mapsto \omega^a
       \\
       r^a & \mapsto F^a_\omega
       \\
       b & \mapsto B
       \\
       c & \mapsto C_3
       \\
       h & \mapsto H_3
       \\
       g & \mapsto \mathcal{G}_4
    }
  }{\leftarrow}|
  \;\;\;\;
  \left(
    \itexarray{  
       r^a  =& d t^a + \frac{1}{2}C^a{}_{b c} t^b \wedge t^c 
       \\
       h = & d b + cs - c     
       \\
       g  =& d c
       \\
       d r^a  =&  - C^a{}_{b c} t^b \wedge r^a
       \\
       d h =& \langle -,-\rangle - g 
       \\
       d g =&  0 
    }
  \right)
  \,.
\end{displaymath}
\end{note}
Here we are indicating on the right the generators and their relation in $\tilde W(b\mathbb{R} \to \mathfrak{g}_\mu)$ and on the left their images and the images of the relations in $\Omega^\bullet(U_i)$. This are first the definitions of the [[curvature]]s themselves and then the [[Bianchi identities]] satisfied by these.

\hypertarget{differential_string_structures_and_fermionic_string_quantum_anomalies}{}\subsubsection*{{Differential string structures and fermionic string quantum anomalies}}\label{differential_string_structures_and_fermionic_string_quantum_anomalies}

The [[Pfaffian line bundle]] controlling the [[fermionic path integral]] of the [[heterotic superstring]] propagating on target $X$ trivializes precisely if the target has a (geometric) string structure.

One shows that the [[Pfaffian line bundle]] on the worldsheet is [[isomorphic]] as a bundle with connection with the [[transgression]] of the differential string structure on the target space to the mapping space $[\Sigma,X]$. So the target space having a (differential) string structure is a sufficient condition for the cancellation of the [[quantum anomaly]].

(First argued in \hyperlink{Killingback}{Killingback}, later made precise in (\hyperlink{Bunke}{Bunke})).

\hypertarget{InheteroticSugra}{}\subsubsection*{{The Green-Schwarz mechanism in heterotic supergravity}}\label{InheteroticSugra}

We discuss the application of twisted differential string structures in [[supergravity]] and [[string theory]].

Local differential form data as in note \ref{UnwindingTheLocalData} above is known in [[higher category theory and physics|theoretical physics]] in the context of the [[Green-Schwarz mechanism]] for 10-dimensional [[supergravity]].

In this context

\begin{itemize}%
\item $\omega$ is called the [[spin connection]];


\item the components $((H_3)_i, \cdots)$ of the above cocycle are known as the $\hat \mathcal{G}_4$-twisted [[Kalb-Ramond field]].



\end{itemize}
In this application the twisting cocycle $\hat \mathcal{G}_4 \in H^4_{diff}(X)$ is itself the [[Chern-Simons circle 3-bundle]] of a [[unitary group]]-[[principal bundle]] with local connection form $A \in \Omega^1(U, \mathfrak{u})$. Therefore in this case $C_3 = CS(A)$ and the above local form data becomes

\begin{displaymath}
H_3 = d B + CS(\omega) - CS(A)
\end{displaymath}
\begin{displaymath}
d H_3 = \langle F_\omega \wedge F_\omega \rangle
   - 
   \langle F_A \wedge F_A \rangle
  \,.
\end{displaymath}
Since $H_3$ is the would-be [[curvature]] of a [[circle n-bundle with connection|circle 2-bundle with connection]], this is the first higher [[Maxwell equation]] that exhibits

\begin{displaymath}
j_{mag}
  :=
  \langle F_\omega \wedge F_\omega \rangle
   - 
   \langle F_A \wedge F_A \rangle
\end{displaymath}
as the [[magnetic charge]] distribution that twists this 2-bundle. This may be interpreted as the magnetic charge density of a classical background density of magnetic [[fundamental brane|fivebranes]]. For more details on this see [[Green-Schwarz mechanism]].

More precisely, the twisted differential string structure of the [[Green-Schwarz mechanism]] in heterotic [[supergravity]] for fixed gauge bundles are therefore given by the [[(∞,1)-pullback]]

\begin{displaymath}
\itexarray{
    GSBackground_{fixed gauge bundle}(X) &\to& \pi_0 \mathbf{H}_{conn}(X, \mathbf{B}U)
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\hat \mathbf{c}_2}}
    \\
    \mathbf{H}_{conn}(X,\mathbf{B}Spin)
     &\stackrel{\frac{1}{2}\hat \mathbf{p}_1}{\to}&
    \mathbf{H}_{dR}(X, \mathbf{B}^3 U(1))
  }
  \,.
\end{displaymath}
Clearly, if we take into account also gauge transformations of the gauge bundle, we should replace this by the full

\begin{displaymath}
\itexarray{
    GSBackground(X) &\to& \mathbf{H}_{conn}(X, \mathbf{B}U)
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\hat \mathbf{c}_2}}
    \\
    \mathbf{H}_{conn}(X,\mathbf{B}Spin)
     &\stackrel{\frac{1}{2}\hat \mathbf{p}_1}{\to}&
    \mathbf{H}_{dR}(X, \mathbf{B}^3 U(1))
  }
  \,.
\end{displaymath}
The look of this diagram makes manifest how in this situation we are looking at the structures that homotopically cancel the differential classes $\frac{1}{2}\hat \mathbf{p}$ and $\hat \mathbf{c}_2$ against each other.

More discussion of this is in (\hyperlink{SSSIII}{SSSIII}).

Since $\mathbf{H}_{dR}(X, \mathbf{B}^3 U(1))$ is abelian, we may consider the corresponding  sequence by realizing $GSBackground(X)$ equivalently as the [[homotopy fiber]] of the difference of differential cocycles $\frac{1}{2}\hat \mathbf{p}_1 - \hat \mathbf{c}_2$.

\begin{displaymath}
\itexarray{
    GSBackground(X) &\to& *
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow
    \\
    \mathbf{H}_{conn}(X,\mathbf{B}Spin \times \mathbf{B}U)
     &\stackrel{\frac{1}{2}\hat \mathbf{p}_1-\hat \mathbf{c}_2}{\to}&
    \mathbf{H}_{dR}(X, \mathbf{B}^4 U(1))
  }
  \,.
\end{displaymath}
Indeed, the above explicit presentation by simplicial presheaves generalizes immediately to describe this case, realizing $U$-twisted differential string structures equivalently as differential ``untwisted $U$-twisted-string-structures''.

We may usefully formalize this further by defining the $String^{\mathbf{c}_2}$-2-group to be the homotopy fiber

\begin{displaymath}
\itexarray{
    \mathbf{B} String^{\mathbf{c}_2} &\to& *
    \\
    \downarrow && \downarrow
    \\
    \mathbf{G}(Spin \times U)
    &\stackrel{\frac{1}{2}\mathbf{p}_1 - \mathbf{c}_2}{\to}&
    \mathbf{B}^3 U(1)
  }
  \,.
\end{displaymath}
We have then that $GSBackground(X)$ is the 3-groupoid of \emph{untwisted} differential $\mathbf{B}String^{\mathbf{c}_2}$-structures.

\begin{displaymath}
\itexarray{
    GSBackground(X) &\to& *
    \\
    \downarrow && \downarrow^{0}
    \\ 
    \mathbf{H}_{conn}(\mathbf{B} (Spin \times U))
    &\stackrel{\frac{1}{2}\hat \mathbf{p}_1 - \mathbf{c}_2}{\to}&
    \mathbf{H}_{dR}(X, \mathbf{B}^4 U(1))
  }
  \,.
\end{displaymath}
More on this in (\hyperlink{FiSaSc}{FiSaSc}).

This is supposed to be (see section 12 of (\hyperlink{DFM}{DFM})) the restriction to the boundary of the [[supergravity C-field]], which is the $(\infty,1)$-pullback

\begin{displaymath}
\itexarray{
    C Field(Y) &\to& H^4_{dR}(Y)
    \\
    \downarrow && \downarrow^{\mathrlap{d C_3}}
    \\ 
    \mathbf{H}(Y,\mathbf{B} (Spin \times U))
    &\stackrel{\frac{1}{2}\hat \mathbf{p}_1 - \mathbf{c}_2}{\to}&
    \mathbf{H}(Y, \mathbf{B}^4 U(1))
  }
  \,.
\end{displaymath}
where $Y$ is 11-dimensional with $\parital Y = X$. Notice that here in the bottom left we have bundles \emph{without} connection, or equivalently (when computing the [[homotopy pullback]] by an ordinary pullback along a fibration) with [[pseudo-connection]]s.

By the discussion at [[supergravity C-field]] under a shift of connection $\nabla_1 \mapsto \nabla_2$ the $C$-field transforms as

\begin{displaymath}
C_2 = C_1 + CS(\nabla_1, \nabla_2)
  \,,
\end{displaymath}
where on the right we have the relative [[Chern-Simons form]]. This vanishes precisely on the genuine gauge transformations. Hence as we restrict from 11-dimensions to 10, two things happen:

\begin{enumerate}%
\item the supergravity $C$-field vanishes,


\item the gauge bundles develop genuine connections.



\end{enumerate}
\hypertarget{relation_to_string_2connections}{}\subsubsection*{{Relation to string 2-connections}}\label{relation_to_string_2connections}

By the discussion at \emph{[[connection on an ∞-bundle]]} we have that for $\mathfrak{g}$ an [[L-∞ algebra]] and

\begin{displaymath}
\mathbf{B}G := \mathbf{cosk}_{n+1} \exp(\mathfrak{g})
\end{displaymath}
the [[delooping]] of the [[smooth ∞-groupoid|smooth Lie n-group]] obtained from it by [[Lie integration]], the coefficient for [[connection on an ∞-bundle|∞-connections]] on $G$-[[principal ∞-bundle]]s is

\begin{displaymath}
\mathbf{B}G_{conn} := \mathbf{cosk}_{n+1} \exp(\mathfrak{g})_{conn}
  \,,
\end{displaymath}
where on the very right we have the [[simplicial presheaf]]

\begin{displaymath}
\exp(\mathfrak{g}) : (U,[k]) \mapsto
   \left\{
     A : CE(\mathfrak{g}) \to \Omega^\bullet(U \times \Delta^n)
     |
     A\;is\;vertically\;flat\; and\;F_A\;is \; horizontal
   \right\}
   \,.
\end{displaymath}
(See [[∞-Chern-Weil homomorphism]] for details).

\begin{prop}
\label{StringConnectionsFromDiffStringStructzres}\hypertarget{StringConnectionsFromDiffStringStructzres}{}
The [[2-groupoid]] of entirely untwisted differential string structures on $X$ (the twist being $0 \in H^4_{diff}(X)$) is equivalent to that of [[string 2-group]] [[principal 2-bundle]]s with [[connection on an ∞-bundle|2-connection]]:

\begin{displaymath}
String_{diff, tw = 0}(X) \simeq String 2Bund_{\nabla}(X)
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
By the above discussion of \hyperlink{ExplicitCocycles}{Cech cocycles} we compute $String_{diff, tw = 0}(X)$ as the ordinary fiber of the morphism of simplicial presheaves

\begin{displaymath}
[CartSp^{op}, sSet](
  C(\{U_i\}),  \mathbf{cosk}_3 \exp(b \mathbb{R} \to \mathfrak{g}_\mu))
  \to 
  [CartSp^{op}, sSet](
  C(\{U_i\}), \mathbf{B}^3 U(1)_{diff})
\end{displaymath}
over the identically vanishing cocycle.

In terms of the component formulas spelled out in the \hyperlink{InheteroticSugra}{above discussion} of the GS-mechanism, this amounts to restricting to those cocyles for which n each degree the equations

\begin{displaymath}
C = 0
\end{displaymath}
\begin{displaymath}
G = 0
\end{displaymath}
holds.

Comparing this to the explicit formulas for $\exp(b \mathbb{R} \to \mathfrak{g}_\mu)$ and $\exp(b \mathbb{R} \to \mathfrak{g}_\mu)_{conn}$ in the \hyperlink{PresentationOfClassByFibration}{above} we see that these cocycles are exactly those that factor through the canonical inclusion

\begin{displaymath}
\mathfrak{g}_\mu \to (b \mathbb{R} \to \mathfrak{g}_\mu)
\end{displaymath}
from observation \ref{LongFiberSequenceOnLieAlgebras} of the [[string Lie 2-algebra]] into the [[mapping cone]] Lie 3-algebra of the extension $b \mathbb{R} \to \mathfrak{g}_\mu \to \mathfrak{g}$ that defines it.

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

[[Whitehead tower]] of the [[orthogonal group]]

\begin{itemize}%
\item [[orientation]]


\item [[spin structure]], [[twisted spin structure]], [[differential spin structure]]

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


\item [[string structure]], \textbf{differential string structure}

\begin{itemize}%
\item [[chiral Dolbeault complex]]

\end{itemize}

\item [[fivebrane structure]],


\item [[twisted differential c-structure]]

\begin{itemize}%
\item \textbf{differential string structure}


\item [[supergravity C-field]]


\item [[twisted differential fivebrane structure]]


\item [[differential T-duality]]



\end{itemize}


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

A discussion of differential string structures in terms of [[bundle 2-gerbe]]s is given in

\begin{itemize}%
\item [[Konrad Waldorf]], \emph{String Connections and Chern-Simons Theory}, Trans. Amer. Math. Soc. \textbf{365} (2013), 4393-4432, doi:\href{https://doi.org/10.1090/S0002-9947-2013-05816-3}{10.1090/S0002-9947-2013-05816-3} arXiv:\href{http://arxiv.org/abs/0906.0117}{0906.0117}.

\end{itemize}
The description of the [[gauge transformation]]s of the [[supergravity C-field]] is in section 3 of

E. Diaconescu, [[nLab:Greg Moore]], [[nLab:Dan Freed]], \emph{The $M$-theory 3-form and $E_8$-gauge theory} (\href{http://arxiv.org/abs/hep-th/0312069}{arXiv:hep-th/0312069})

The local data for the [[∞-Lie algebra valued differential forms]] for the description of twisted differential string structures as above was given in

\begin{itemize}%
\item [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], \emph{[[schreiber:Twisted Differential String and Fivebrane Structures]]}

\end{itemize}
The full Cech-Deligne cocycles induced by this (but not yet the homotopy fibers over them) were discussed in

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

\end{itemize}
The 2-group $String^{\mathbf{c}_2}$ and its differential structurs, etc. are discussed in

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{Higher differential $Spin^c$-structures}

\end{itemize}
A comprehensive discussion including all the formal background and the applications is attempted at

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

\end{itemize}
in section 4.2.

The relation to [[quantum anomaly]] cancellation in [[heterotic string theory]] has been first discussed in

\begin{itemize}%
\item Killingback, \emph{World-sheet anomalies and loop geometry} Nuclear Physics B Volume 288, 1987, Pages 578-588

\end{itemize}
and given a rigorous treatment in

\begin{itemize}%
\item [[Ulrich Bunke]], \emph{String structures and trivialisations of a Pfaffian line bundle} (\href{http://arxiv.org/abs/0909.0846}{arXiv})

\end{itemize}
More discussion on the relation to spin structures on smooth loop space is in

\begin{itemize}%
\item Alessandra Capotosti, \emph{[[From String structures to Spin structures on loop spaces]]}, Ph.D. thesis, Universit\`a{} degli Studi Roma Tre, Rome, April 2016

\end{itemize}
[[!redirects differential string structures]]

[[!redirects differential String structure]] [[!redirects differential String structures]] [[!redirects twisted differential String structure]] [[!redirects twisted differential String structures]]

[[!redirects differential string-structure]] [[!redirects differential string-structures]]

[[!redirects twisted differential string structure]] [[!redirects twisted differential string structures]] [[!redirects twisted differential string-structure]] [[!redirects twisted differential string-structures]]

[[!redirects geometric string structure]] [[!redirects geometric string structures]] [[!redirects geometric string-structure]] [[!redirects geometric string-structures]]

[[!redirects twisted string structure]] [[!redirects twisted String structure]] [[!redirects twisted string structures]] [[!redirects twisted String structures]]

[[!redirects smooth first fractional Pontryagin class]]



\end{document}