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\section*{twisted spin^c structure}

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

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

[[!include cohomology - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{topological}{Topological}\dotfill \pageref*{topological} \linebreak
\noindent\hyperlink{smooth}{Smooth}\dotfill \pageref*{smooth} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{anomaly_cancellation_in_physics}{Anomaly cancellation in physics}\dotfill \pageref*{anomaly_cancellation_in_physics} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{ReferencesAnomalyCancellation}{In physics}\dotfill \pageref*{ReferencesAnomalyCancellation} \linebreak


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

\hypertarget{topological}{}\subsubsection*{{Topological}}\label{topological}

For $n \in \mathbb{N}$ the [[Lie group]] $Spin^c(n)$ is a [[central extension]]

\begin{displaymath}
U(1) \to Spin^c(n) \to SO(n)
\end{displaymath}
of the [[special orthogonal group]] by the [[circle group]]. This comes with a long [[fiber sequence]]

\begin{displaymath}
\cdots \to B U(1) \to B Spin^c(n)  \to B SO(n) \stackrel{W_3}{\to} B^2 U(1)
  \,,
\end{displaymath}
where $W_3$ is the \emph{third integral [[Stiefel-Whitney class]]} .

By the definition at [[twisted cohomology]], for a given class $[c] \in H^3(X, \mathbb{Z})$, a \textbf{$c$-twisted $spin^c$-structure} is a choice of [[homotopy]]

\begin{displaymath}
\eta : c \stackrel{\simeq}{\to} W_3(T X)
  \,.
\end{displaymath}
The [[space]]/[[∞-groupoid]] of all twisted $Spin^c$-structures on $X$ is the [[homotopy fiber]] $W_3 Struc_{tw}(T X)$ in the [[pasting diagram]] of [[homotopy pullback]]s

\begin{displaymath}
\itexarray{
    W_3 Struc_{tw}(T X) &\to& W_3 Struc_{tw}(X) &\stackrel{tw}{\to}& H^3(X, \mathbb{Z})
    \\
    \downarrow && \downarrow && \downarrow 
    \\
    * &\stackrel{T X}{\to}& Top(X, B SO(n)) 
    &\stackrel{W_3}{\to}&
    Top(X, B^2 U(1))
  }
  \,,
\end{displaymath}
where the right vertical morphism is the canonical [[effective epimorphism]] that picks one point in each connected component.

\hypertarget{smooth}{}\subsubsection*{{Smooth}}\label{smooth}

Since $U(1) \to Spin^c \to SO$ is a sequence of [[Lie group]]s, the above may be lifted from the [[(∞,1)-topos]] [[Top]] $\simeq$ [[∞Grpd]] to [[Smooth∞Grpd]].

More precisely, by the discussion at [[Lie group cohomology]] (and [[smooth ∞-groupoid -- structures]]) the characteristic map $W_3 : B SO \to B^2 U(1)$ in $\infty Grpd$ has, up to equivalence, a unique lift

\begin{displaymath}
\mathbf{W}_3 : \mathbf{B} SO \to \mathbf{B}^2 U(1)
\end{displaymath}
to [[Smooth∞Grpd]], where on the right we have the [[delooping]] of the smooth [[circle n-group|circle 2-group]].

By the general definition at \emph{[[twisted differential c-structure]]} , the [[2-groupoid]] of \emph{smooth twisted $spin^c$-structures} $\mathbf{W}_3 Struc_{tw}(X)$ is the joint [[(∞,1)-pullback]]

\begin{displaymath}
\itexarray{
    \mathbf{W}_3 Struc_{tw}(T X) &\to& \mathbf{W}_3 Struc_{tw}(X) &\stackrel{tw}{\to}& H_{smooth}^2(X, U(1))
    \\
    \downarrow && \downarrow && \downarrow 
    \\
    * &\stackrel{T X}{\to}& Smooth \infty Grpd(X, \mathbf{B} SO(n)) 
    &\stackrel{\mathbf{W}_3}{\to}&
    Smooth \infty Grpd(X, \mathbf{B}^2 U(1)
  }
  \,.
\end{displaymath}
\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\hypertarget{anomaly_cancellation_in_physics}{}\subsubsection*{{Anomaly cancellation in physics}}\label{anomaly_cancellation_in_physics}

The existence of an ordinary [[spin structure]] on a space $X$ is, as discussed there, the condition for $X$ to serve as the [[target space]] for the [[spinning particle]] [[sigma-model]], in that the existence of this structure is precisely the condition that the corresponding fermionic [[quantum anomaly]] on the [[worldline]] vanishes.

Twisted $spin^c$-structures appear similarly as the conditions for the analogous quantum anomaly cancellation, but now of the open [[type II superstring]] ending on a [[D-brane]]. This is also called the \textbf{[[Freed-Witten anomaly cancellation]]}.

More precisely, in these applications the class of $W_3(TX) - H$ need not vanish, it only needs to be $n$-[[torsion]] if there is moreover a [[twisted bundle]] of rank $n$ on the $D$-brane.

See the \href{ReferencesAnomalyCancellation}{references below} for details.

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

\begin{itemize}%
\item [[twisted differential c-structure|(twisted differential) c-structures]]

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


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

[[spin{\tt \symbol{94}}c structure]], \textbf{twisted $spin^c$ structure}


\item [[string structure]], [[differential string structure]]


\item [[fivebrane structure]], [[differential fivebrane structure]]



\end{itemize}


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

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

The notion of twisted $Spin^c$-structures as such were apparently first discussed in section 5 of

\begin{itemize}%
\item [[Christopher Douglas]], \emph{On the Twisted K-Homology of Simple Lie Groups} Topology, Volume 45, Issue 6, November 2006, Pages 955-988 (\href{http://arxiv.org/abs/math/0402082}{arXiv:0402082})

\end{itemize}
More discussion appears in section 3 of

\begin{itemize}%
\item [[Bai-Ling Wang]], \emph{Geometric cycles, index theory and twisted K-homology} J. Noncommut. Geom. 2 (2008), no. 4, 497--552 (\href{http://arxiv.org/abs/0710.1625}{arXiv:0710.1625})

\end{itemize}
The refinement to smooth twisted structures is discussed in section 4.1 of

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

\end{itemize}
\hypertarget{ReferencesAnomalyCancellation}{}\subsubsection*{{In physics}}\label{ReferencesAnomalyCancellation}

The need for twisted $Spin^c$-structures as [[Freed-Witten anomaly cancellation]] condition on the [[worldvolume]] of [[D-branes]] in [[string theory]] was first discussed in

\begin{itemize}%
\item [[Daniel Freed]], [[Edward Witten]], \emph{Anomalies in String Theory with D-Branes} (\href{http://arxiv.org/abs/hep-th/9907189}{arXiv:hep-th/9907189})

\end{itemize}
More details are in

\begin{itemize}%
\item [[Anton Kapustin]], \emph{D-branes in a topologically nontrivial B-field} , Adv. Theor. Math. Phys. 4, no. 1, pp. 127--154 (2000), (\href{http://arxiv.org/abs/hep-th/9909089}{arXiv:hep-th/9909089})

\end{itemize}
A clean formulation and review is provided in

\begin{itemize}%
\item Loriano Bonora, Fabio Ferrari Ruffino, Raffaele Savelli, \emph{Classifying A-field and B-field configurations in the presence of D-branes} (\href{http://arxiv.org/abs/0810.4291}{arXiv:0810.4291})


\item Fabio Ferrari Ruffino, \emph{Classifying A-field and B-field configurations in the presence of D-branes - Part II: Stacks of D-branes} (\href{http://arxiv.org/abs/1104.2798}{arXiv:1104.2798})


\item Fabio Ferrari Ruffino, \emph{Topics on topology and superstring theory} (\href{http://arxiv.org/abs/0910.4524}{arXiv:0910.4524})



\end{itemize}
and

\begin{itemize}%
\item Kim Laine, \emph{Geometric and topological aspects of Type IIB D-branes} (\href{http://arxiv.org/abs/0912.0460}{arXiv:0912.0460})

\end{itemize}
In (\hyperlink{Laine}{Laine}) the discussion of FW-anomaly cancellation with finite-rank gauge bundles is towards the very end, culminating in equation (3.41).

[[!redirects twisted spin{\tt \symbol{94}}c structures]]

[[!redirects twisted spin{\tt \symbol{94}}c-structure]] [[!redirects twisted spin{\tt \symbol{94}}c-structures]]



\end{document}