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\begin{document}

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\section*{Chan-Paton bundle}

\begin{quote}%
under construction


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

\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{the_field_as_a_prequantum_2bundle}{The $B$-field as a prequantum 2-bundle}\dotfill \pageref*{the_field_as_a_prequantum_2bundle} \linebreak
\noindent\hyperlink{the_chanpaton_gauge_field}{The Chan-Paton gauge field}\dotfill \pageref*{the_chanpaton_gauge_field} \linebreak
\noindent\hyperlink{the_open_string_sigmamodel}{The open string sigma-model}\dotfill \pageref*{the_open_string_sigmamodel} \linebreak
\noindent\hyperlink{the_anomalyfree_open_string_coupling_to_the_chanpaton_gauge_field}{The anomaly-free open string coupling to the Chan-Paton gauge field}\dotfill \pageref*{the_anomalyfree_open_string_coupling_to_the_chanpaton_gauge_field} \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}

In the context of [[string theory]], the [[background gauge field]] for the [[open string]] [[sigma-model]] over a [[D-brane]] in [[bosonic string theory]] or [[type II string theory]] is a unitary [[principal bundle]] [[connection on a bundle|with connection]], or rather, by the Kapustin-part of the [[Freed-Witten-Kapustin anomaly cancellation]] mechanism, a [[twisted bundle|twisted unitary bundle]], whose twist is the restriction of the ambient [[B-field]] to the [[D-brane]].

The first hint for the existence of such [[background gauge fields]] for the [[open string]] 2d-[[sigma-model]] comes from the fact that the open string's endpoint can naturally be taken to carry labels $i \in \{1, \cdots n\}$. Further analysis then shows that the lowest excitations of these $(i,j)$-strings behave as the quanta of a $U(n)$-[[gauge field]], the $(i,j)$-excitation being the given [[matrix]] element of a $U(n)$-valued connection 1-form $A$.

This original argument goes back work by Chan and Paton. Accordingly one speaks of \emph{Chan-Paton factors} and \emph{Chan-Paton bundles} .

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

We discuss the Chan-Paton gauge field and its [[quantum anomaly cancellation]] in [[extended prequantum field theory]].

Throughout we write $\mathbf{H} =$ [[Smooth∞Grpd]] for the [[cohesive (∞,1)-topos]] of [[smooth ∞-groupoids]].

\hypertarget{the_field_as_a_prequantum_2bundle}{}\subsubsection*{{The $B$-field as a prequantum 2-bundle}}\label{the_field_as_a_prequantum_2bundle}

For $X$ a [[type II supergravity]] [[spacetime]], the [[B-field]] is a map

\begin{displaymath}
\nabla_B \;\colon\; X \to \mathbf{B}^2 U(1)
  \,.
\end{displaymath}
If $X = G$ is a [[Lie group]], this is the [[prequantum 2-bundle]] of $G$-[[Chern-Simons theory]]. Viewed as such we are to find a canonical [[∞-action]] of the [[circle 2-group]] $\mathbf{B}U(1)$ on some $V \in \mathbf{H}$, form the corresponding [[associated ∞-bundle]] and regard the [[sections]] of that as the [[prequantum 2-states]] of the theory.

The Chan-Paton gauge field is such a prequantum 2-state.

\hypertarget{the_chanpaton_gauge_field}{}\subsubsection*{{The Chan-Paton gauge field}}\label{the_chanpaton_gauge_field}

We discuss the [[Chan-Paton gauge fields]] over [[D-branes]] in [[bosonic string theory]] and over $Spin^c$-D-branes in [[type II string theory]].

We fix throughout a natural number $n \in \mathbb{N}$, the \emph{[[rank]]} of the Chan-Paton gauge field.

\begin{prop}
\label{TheLongSequenceOfTheProjectiveUnitaryExtension}\hypertarget{TheLongSequenceOfTheProjectiveUnitaryExtension}{}
The [[extension of groups|extension]] of [[Lie groups]]

\begin{displaymath}
U(1) \to U(n) \to PU(n)
\end{displaymath}
exhibiting the [[unitary group]] as a [[circle group]]-extension of the [[projective unitary group]] sits in a long [[homotopy fiber sequence]] of [[smooth ∞-groupoids]] of the form

\begin{displaymath}
U(1) \to U(n) \to PU(n) \to \mathbf{B}U(1)
  \to \mathbf{B}U(n) \to \mathbf{B}PU(n) \stackrel{\mathbf{dd}_n}{\to}
  \mathbf{B}^2 U(1)
  \,,
\end{displaymath}
where for $G$ a [[Lie group]] $\mathbf{B}G$ is its [[delooping]] [[Lie groupoid]], hence the [[moduli stack]] of $G$-[[principal bundles]], and where similarly $\mathbf{B}^2 U(1)$ is the [[moduli ∞-stack|moduli 2-stack]] of [[circle 2-group]] [[principal 2-bundles]] ([[bundle gerbes]]).

\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
Here

\begin{displaymath}
\mathbf{dd}_n \;\colon\; \mathbf{B} PU(n) \to \mathbf{B}^2 U(1)
\end{displaymath}
is a smooth refinement of the universal [[Dixmier-Douady class]]

\begin{displaymath}
dd_n \;\colon\; B PU(n) \to K(\mathbb{Z}, 3)
\end{displaymath}
in that under [[geometric realization of cohesive ∞-groupoids]] ${\vert- \vert} \colon$ [[Smooth∞Grpd]] $\to$ [[∞Grpd]] we have

\begin{displaymath}
{\vert \mathbf{dd}_n \vert}
  \simeq
  dd_n
  \,.
\end{displaymath}
\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
By the discussion at \emph{[[∞-action]]} the [[homotopy fiber sequence]] in prop. \ref{TheLongSequenceOfTheProjectiveUnitaryExtension}

\begin{displaymath}
\itexarray{
    \mathbf{B} U(n)
    &\to&
    \mathbf{B} PU(n)
    \\
    && \downarrow
    \\
    && \mathbf{B}^2 U(1)
  }
\end{displaymath}
in $\mathbf{H}$ exhibits a smooth[[∞-action]] of the [[circle 2-group]] on the [[moduli stack]] $\mathbf{B}U(n)$ and it exhibits an equivalence

\begin{displaymath}
\mathbf{B} PU(n) \simeq (\mathbf{B}U(n))//(\mathbf{B} U(1))
\end{displaymath}
of the moduli stack of projective unitary bundles with the [[∞-quotient]] of this [[∞-action]].

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
For $X \in \mathbf{H}$ a [[smooth manifold]] and $\mathbf{c} \;\colon\; X \to \mathbf{B}^2 U(1)$ modulating a [[circle 2-group]]-[[principal 2-bundle]], maps

\begin{displaymath}
\mathbf{c} \to \mathbf{dd}_n
\end{displaymath}
in the [[slice (∞,1)-topos]] $\mathbf{H}_{/\mathbf{B}^2 U(1)}$, hence [[diagrams]] of the form

\begin{displaymath}
\itexarray{
    X &&\stackrel{}{\to}&& \mathbf{B} PU(n)
    \\
    & {}_{\mathllap{\mathbf{c}}}\searrow 
    &\swArrow& 
    \swarrow_{\mathrlap{\mathbf{dd}_n}}
    \\
    && \mathbf{B}^2 U(1)
  }
\end{displaymath}
in $\mathbf{H}$ are equivalently rank-$n$ unitary [[twisted bundles]] on $X$, with the twist being the class $[\mathbf{c}] \in H^3(X, \mathbb{Z})$.

\end{prop}
\begin{prop}
\label{DifferentialRefinementOfSMoothDDClass}\hypertarget{DifferentialRefinementOfSMoothDDClass}{}
There is a further differential refinement

\begin{displaymath}
\itexarray{
     (\mathbf{B}U(n))//(\mathbf{B}U(1))_{conn}
     &\stackrel{\widehat \mathbf{dd}_n}{\to}&
     \mathbf{B}^2 U(1)_{conn}
     \\
     \downarrow && \downarrow
     \\
     (\mathbf{B}U(n))//(\mathbf{B}U(1))
     &\stackrel{\widehat \mathbf{dd}_n}{\to}&
     \mathbf{B}^2 U(1)
  }
  \,,
\end{displaymath}
where $\mathbf{B}^2 U(1)_{conn}$ is the universal moduli 2-stack of [[circle n-bundle with connection|circle 2-bundles with connection]] ([[bundle gerbes]] with connection).

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

\begin{displaymath}
\left(
  \left(\mathbf{B}U\left(n\right)//\mathbf{B}U\left(1\right)\right)_{conn}
   \stackrel{\mathbf{Fields}}{\to} 
  \mathbf{B}^2 U\left(1\right)_{conn}
  \right)
  \;\;
  \in \mathbf{H}_{/\mathbf{B}^2 U(1)_{conn}}
\end{displaymath}
for the differential smooth universal Dixmier-Douady class of prop. \ref{DifferentialRefinementOfSMoothDDClass}, regarded as an object in the [[slice (∞,1)-topos]] over $\mathbf{B}^2 U(1)_{conn}$.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
Let

\begin{displaymath}
\iota_X   
    \;\colon\;
  Q \hookrightarrow X
\end{displaymath}
be an inclusion of [[smooth manifolds]] or of [[orbifolds]], to be thought of as a [[D-brane]] [[worldvolume]] $Q$ inside an ambient [[spacetime]] $X$.

Then a \textbf{field configuration} of a \emph{[[B-field]]} on $X$ together with a compatible rank-$n$ \textbf{Chan-Paton gauge field} on the [[D-brane]] is a map

\begin{displaymath}
\phi 
   \;\colon\;
  \iota_X \to \mathbf{Fields}
\end{displaymath}
in the [[arrow (∞,1)-topos]] $\mathbf{H}^{(\Delta^1)}$, hence a [[diagram]] in $\mathbf{H}$ of the form

\begin{displaymath}
\itexarray{
    Q &\stackrel{\nabla_{gauge}}{\to}& (\mathbf{B}U(n)//\mathbf{B}U(1))
    \\
    {}^{\iota_X}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\hat \mathbf{dd}_n}}
    \\
    X &\stackrel{\nabla_B}{\to}& \mathbf{B}^2 U(1)_{conn}
  }
\end{displaymath}
\end{defn}
This identifies a [[twisted bundle]] with connection on the D-brane whose twist is the class in $H^3(X, \mathbb{Z})$ of the bulk [[B-field]].

This relation is the Kapustin-part of the [[Freed-Witten-Kapustin anomaly]] cancellation for the [[bosonic string]] or else for the [[type II string]] on $Spin^c$ D-branes. (\hyperlink{FSS}{FSS})

\begin{remark}
\label{}\hypertarget{}{}
If we regard the [[B-field]] as a [[background field]] for the [[Chan-Paton gauge field]], then remark \ref{PullbackAlongGeneralizedLocalDiffeomorphisms} determines along which maps of the B-field the Chan-Paton gauge field may be transformed.

\begin{displaymath}
\itexarray{
    Y &\stackrel{}{\to}& X &\stackrel{}{\to}& (\mathbf{B}U(n)//\mathbf{B}U(1))_{conn}
    \\
    & \searrow & \downarrow & \swarrow
    \\
    &&\mathbf{B}^2 U(1)_{conn}
  }
  \,.
\end{displaymath}
On the local connection forms this acts as

\begin{displaymath}
A \mapsto A + \alpha
  \,.
\end{displaymath}
\begin{displaymath}
B \mapsto B + d \alpha
\end{displaymath}
This is the famous gauge transformation law known from the string theory literature.

\end{remark}
\hypertarget{the_open_string_sigmamodel}{}\subsubsection*{{The open string sigma-model}}\label{the_open_string_sigmamodel}

\begin{remark}
\label{}\hypertarget{}{}
The [[D-brane]] inclusion $Q \stackrel{\iota_X}{\to} X$ is the [[target space]] for an [[open string]] with [[worldsheet]] $\partial \Sigma \stackrel{\iota_\Sigma}{\hookrightarrow} \Sigma$: a [[field (physics)|field]] configuration of the open string [[sigma-model]] is a map

\begin{displaymath}
\phi \;\colon\; \iota_\Sigma \to \iota_X
\end{displaymath}
in $\mathbf{H}^{\Delta^1}$, hence a [[diagram]] of the form

\begin{displaymath}
\itexarray{
    \partial \Sigma &\stackrel{\phi_{bdr}}{\to}& Q
    \\
    \downarrow^{\mathrlap{\iota_\Sigma}} 
     &\swArrow& 
    \downarrow^{\mathrlap{\iota_X}}
    \\
    \Sigma &\stackrel{\phi_{bulk}}{\to}& X
  }
  \,.
\end{displaymath}
For $X$ and $Q$ ordinary [[manifolds]] just says that a field configuration is a map $\phi_{bulk} \;\colon\; \Sigma \to X$ subject to the constraint that it takes the [[boundary]] of $\Sigma$ to $Q$. This means that this is a [[trajectory]] of an [[open string]] in $X$ whose endpoints are constrained to sit on the [[D-brane]] $Q \hookrightarrow X$.

If however $X$ is more generally an [[orbifold]], then the [[homotopy]] filling the above diagram imposes this constraint only up to orbifold transformations, hence exhibits what in the physics literature are called ``orbifold twisted sectors'' of open string configurations.

\end{remark}
\begin{prop}
\label{TheTypeIIOpenStringSigmaModelModuliStackOfFields}\hypertarget{TheTypeIIOpenStringSigmaModelModuliStackOfFields}{}
The [[moduli stack]] $[\iota_\Sigma, \iota_X]$ of such field configurations is the [[homotopy pullback]]

\begin{displaymath}
\itexarray{
    [\iota_{\Sigma}, \iota_X]
    &\to&
    [\Sigma, X]
    \\
    \downarrow &\swArrow& \downarrow
    \\
    [S^1, Q]
    &\to&
    [S^1, X]
  }
  \,.
\end{displaymath}
\end{prop}
\hypertarget{the_anomalyfree_open_string_coupling_to_the_chanpaton_gauge_field}{}\subsubsection*{{The anomaly-free open string coupling to the Chan-Paton gauge field}}\label{the_anomalyfree_open_string_coupling_to_the_chanpaton_gauge_field}

\begin{prop}
\label{}\hypertarget{}{}
For $\Sigma$ a [[smooth manifold]] with [[boundary]] $\partial \Sigma$ of [[dimension]] $n$ and for $\nabla \;\colon \; X \to \mathbf{B}^n U(1)_{conn}$ a [[circle n-bundle with connection]] on some $X \in \mathbf{H}$, then the [[transgression]] of $\nabla$ to the [[mapping space]] $[\Sigma, X]$ yields a [[section]] of the [[complex line bundle]] [[associated bundle|associated]] to the pullback of the ordinary transgression over the mapping space out of the boundary: we have a diagram

\begin{displaymath}
\itexarray{  
    [\Sigma, X]
      &\stackrel{\exp(2 \pi i \int_{\Sigma})}{\to}& 
    \mathbb{C}//U(1)_{conn}
    \\
    \downarrow^{\mathrlap{[\partial \Sigma, X]}} 
    && 
    \downarrow^{\mathrlap{\overline{\rho}}_{conn}}
    \\
    [\partial \Sigma, X]
    &\stackrel{\exp(2 \pi i \int_{\partial \Sigma})}{\to}&
    \mathbf{B} U(1)_{conn}
  }
  \,.
\end{displaymath}
\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
This is the \emph{[[higher parallel transport]]} of the $n$-connection $\nabla$ over maps $\Sigma \to X$.

\end{remark}
\begin{prop}
\label{TheTwistedHolonomyMapOnTwistedUnitaryBundles}\hypertarget{TheTwistedHolonomyMapOnTwistedUnitaryBundles}{}
The operation of forming the [[holonomy]] of a twisted unitary connection around a curve fits into a [[diagram]] in $\mathbf{H}$ of the form

\begin{displaymath}
\itexarray{
    [S^1, (\mathbf{B}U(n))//(\mathbf{B}U(1))_{conn}]
    &\stackrel{hol_{S^1}}{\to}&
    \mathbb{C}//U(1)_{conn}
    \\
    \downarrow^{\mathrlap{[S^1, \widehat\mathbf{dd}_n]}} 
     &\swArrow_{\simeq}& 
    \downarrow^{\mathrlap{\overline{\rho}_{conn}}}
    \\
    [S^1, \mathbf{B}^2 U(1)_{conn}]
    &\stackrel{\exp(2 \pi i \int_{S^1})}{\to}&
    \mathbf{B}U(1)_{conn}
  }
  \,.
\end{displaymath}
\end{prop}
\begin{remark}
\label{}\hypertarget{}{}
By the discussion at \emph{[[∞-action]]} the diagram in prop. \ref{TheTwistedHolonomyMapOnTwistedUnitaryBundles} says in particular that forming traced [[holonomy]] of twisted unitary bundles constitutes a [[section]] of the [[complex line bundle]] on the [[moduli stack]] of twisted unitary connection on the circle which is the [[associated bundle]] to the [[transgression]] $\exp(2 \pi i \int_{S^1} [S^1, \widehat\mathbf{dd}_n])$ of the universal differential [[Dixmier-Douady class]].

\end{remark}
It follows that on the moduli space of the open string [[sigma-model]] of prop. \ref{TheTypeIIOpenStringSigmaModelModuliStackOfFields} above there are two $\mathbb{C}//U(1)$-valued [[action functionals]] coming from the bulk field and the boundary field

\begin{displaymath}
\itexarray{
    [\iota_{\Sigma}, \iota_X]
    &\to&
    [\Sigma, X]
    &\stackrel{exp(2 \pi i \int_{\Sigma}[\Sigma, \nabla_B]  ) }{\to}&
    \mathbb{C}//U(1)_{conn}
    \\
    \downarrow &\swArrow& \downarrow
    \\
    [S^1, Q]
    &\to&
    [S^1, X]
    \\
    \downarrow^{\mathrlap{hol_{S^1}([S^1, \nabla_{gauge}])}}
    \\
    \mathbb{C}//U(1)_{conn}
  }
  \,.
\end{displaymath}
Neither is a well-defined $\mathbb{C}$-valued function by itself. But by [[pasting]] the above diagrams, we see that both these constitute [[sections]] of the same [[complex line bundle]] on the moduli stack of fields:

\begin{displaymath}
\itexarray{
    [\iota_{\Sigma}, \iota_X]
    &\to&
    [\Sigma, X]
    &\stackrel{[\Sigma, \nabla_B]}{\to}&
    [S^1, \mathbf{B}^2 U(1)_{conn}]
    &\stackrel{\exp(2 \pi i \int_{\Sigma})}{\to}&
    \mathbb{C}//U(1)_{conn}
    \\
    \downarrow &\swArrow& \downarrow
    && && \downarrow
    \\
    [S^1, Q]
    &\to&
    [S^1, X]
    \\
    \downarrow^{\mathrlap{[S^1, \nabla_{gauge}]}} &&  & \searrow^{\mathrlap{[S^1, \nabla_B]}}
    & && \downarrow
    \\
    [S^1, (\mathbf{B}U(n))//(\mathbf{B}U(1))_{conn}] & &\stackrel{[S^1, \widehat \mathbf{dd}_n]}{\to}& & [S^1, \mathbf{B}^2 U(1)_{conn}]
    \\
    \downarrow^{\mathrlap{hol_{S^1}}} 
      && && &  \searrow^{\mathrlap{\exp(2 \pi i \int_{S^1}(-))}}
    \\
    \mathbb{C}//U(1)_{conn} &\to& &\to& &\to&  \mathbf{B}U(1)_{conn}
  }
  \,.
\end{displaymath}
Therefore the product action functional is a well-defined function

\begin{displaymath}
[\iota_\Sigma, \iota_X]
  \stackrel{
    \exp(2 \pi i \int_{\Sigma} [\Sigma, \nabla_b] )
    \cdot
    hol_{S^1}( [S^1, \widehat {\mathbf{dd}}_n] )^{-1}
  }{\to}
  U(1)
  \,.
\end{displaymath}
This is the [[Freed-Witten-Kapustin anomaly|Kapustin anomaly]]-free [[action functional]] of the [[open string]].

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

\begin{itemize}%
\item [[B-field]]


\item [[twisted K-theory]]


\item [[Freed-Witten anomaly cancellation]]


\item [[Dirac-Born-Infeld action]]



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

In the traditional physicist's string theory introductions one finds Chan-Paton bundles discussed for instance

\begin{itemize}%
\item [[Clifford Johnson]], section 2.4 of \emph{D-Brane primer} (\href{http://arxiv.org/abs/hep-th/0007170}{arXiv:hep-th/0007170})


\item David Tong, around p. 66 of \emph{Lectures on string theory} (\href{http://front.math.ucdavis.edu/0908.0333}{arxiv/0908.0333})



\end{itemize}
These lectures tend to ignore most of the global subtleties though. For traditional discussion of the \emph{[[Freed-Witten-Kapustin anomaly]]}, see there. The above account in terms of [[higher geometry]] and [[extended prequantum field theory]] is due to section 5.4 of

\begin{itemize}%
\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:A higher stacky perspective on Chern-Simons theory]]} .

\end{itemize}
Lecture notes along these lines are at

\begin{itemize}%
\item \emph{[[geometry of physics]]}

\end{itemize}
Discussion of the derivation of the [[Yang-Mills theory]] on the D-brane from open [[string scattering amplitudes]]/[[string field theory]] includes

\begin{itemize}%
\item [[David Gross]], [[Edward Witten]], \emph{Superstring modifications of Einstein's equations}, Nuclear Physics B Volume 277, 1986, Pages 1-10

\end{itemize}
and for the non-abelian case:

\begin{itemize}%
\item Semyon Klevtsov, \emph{Yang-Mills theory from String field theory on D-branes} (\href{http://www.mi.uni-koeln.de/~klevtsov/cargese02.pdf}{pdf})

\end{itemize}
[[!redirects Chan-Paton bundles]]

[[!redirects Chan-Paton vector bundle]] [[!redirects Chan-Paton vector bundles]]

[[!redirects Chan-Paton gauge field]] [[!redirects Chan-Paton gauge fields]]

[[!redirects Chan-Paton gauge bundle]] [[!redirects Chan-Paton gauge bundles]]



\end{document}