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\section*{spin^c structure}

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

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

[[!include higher spin geometry - contents]]

\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{Higher}{Higher $spin^c$-structures}\dotfill \pageref*{Higher} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{PropertiesOfSpinC}{Of $Spin^c$}\dotfill \pageref*{PropertiesOfSpinC} \linebreak
\noindent\hyperlink{as_orientation}{As $KU$-orientation}\dotfill \pageref*{as_orientation} \linebreak
\noindent\hyperlink{RelationToMetaplecticStructures}{Relation to metaplectic structures}\dotfill \pageref*{RelationToMetaplecticStructures} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{FromAlmostComplexStructures}{From almost complex structures}\dotfill \pageref*{FromAlmostComplexStructures} \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{as_orientationanomaly_cancellation_in_type_ii_string_theory}{As $KU$-orientation/anomaly cancellation in type II string theory}\dotfill \pageref*{as_orientationanomaly_cancellation_in_type_ii_string_theory} \linebreak


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

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

For $n \in \mathbb{N}$ the [[Lie group]] [[spin{\tt \symbol{94}}c|spin]] 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]]} .

An [[oriented]] manifold $X$ \textbf{has $Spin^c$-structure} if the [[characteristic class]] $[W_3(X)] \in H^3(X, \mathbb{Z})$

\begin{displaymath}
W_3(X) \coloneqq W_3(T X) 
  \;\colon\;
  X \stackrel{T X}{\to} B SO(n) \stackrel{W_3}{\to} B^2 U(1) \simeq K(\mathbb{Z},3)
\end{displaymath}
is trivial. This is the [[Dixmier-Douady class]] of the [[circle n-bundle|circle 2-bundle]]/[[bundle gerbe]] that [[obstruction|obstructs]] the existence of a $Spin^c$-[[principal bundle]] [[lift of structure group|lifting]] the given [[tangent bundle]].

A manifold $X$ is \textbf{equipped with $Spin^c$-structure} $\eta$ if it is equipped with a choice of trivializaton

\begin{displaymath}
\eta : 1 \stackrel{\simeq}{\to} W_3(T X)
  \,.
\end{displaymath}
The [[homotopy type]]/[[∞-groupoid]] of $Spin^c$-structures on $X$ is the [[homotopy fiber]] $W_3 Struc(T X)$ in the [[pasting diagram]] of [[homotopy pullback]]s

\begin{displaymath}
\itexarray{
    W_3 Struc(T X) &\to& W_3 Struc(X) &\to& * 
    \\
    \downarrow && \downarrow && \downarrow 
    \\
    * &\stackrel{T X}{\to}& Top(X, B SO(n)) 
    &\stackrel{W_3}{\to}&
    Top(X,B^2 U(1))
  }
  \,.
\end{displaymath}
If the class does not vanish and if hence there is no $Spin^c$-structure, it still makes sense to discuss the structure that remains as \emph{[[twisted spin{\tt \symbol{94}}c structure|twisted spin structure]]} .

\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]] $L_{whe}$ [[Top]] $\simeq$ [[∞Grpd]] of [[discrete ∞-groupoids]] to that of [[smooth ∞-groupoids]], [[Smooth∞Grpd]].

More in detail, 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]].

Accordingly, the [[2-groupoid]] of \emph{smooth $spin^c$-structures} $\mathbf{W}_3 Struc(X)$ is the joint [[(∞,1)-pullback]]

\begin{displaymath}
\itexarray{
    \mathbf{W}_3 Struc(T X) &\to& \mathbf{W}_3 Struc(X) &\to& * 
    \\
    \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{Higher}{}\subsubsection*{{Higher $spin^c$-structures}}\label{Higher}

In parallel to the existence of \href{spin+structure#Higher}{higher spin structures} there are higher analogs of $Spin^c$-structures, related to [[quantum anomaly]] cancellation of theories of higher dimensional [[branes]].

\begin{itemize}%
\item [[string{\tt \symbol{94}}c structure|string structure]]

\end{itemize}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{PropertiesOfSpinC}{}\subsubsection*{{Of $Spin^c$}}\label{PropertiesOfSpinC}

\begin{defn}
\label{}\hypertarget{}{}
The group $Spin^c$ is the [[fiber product]]

\begin{displaymath}
\begin{aligned}
    Spin^c & := Spin \times_{\mathbb{Z}_2} U(1)
    \\
    & = (Spin \times U(1))/{\mathbb{Z}_2}
    \,,
  \end{aligned}
\end{displaymath}
where in the second line the [[action]] is the diagonal action induced from the two canonical embeddings of [[subgroups]] $\mathbb{Z}_2 \hookrightarrow \mathbb{Z}$ and $\mathbb{Z}_2 \hookrightarrow U(1)$.

\end{defn}
\begin{prop}
\label{SpinCAsHomotopyFiberProductW2C1}\hypertarget{SpinCAsHomotopyFiberProductW2C1}{}
We have a [[homotopy pullback]] diagram

\begin{displaymath}
\itexarray{
     \mathbf{B} Spin^c &\to& \mathbf{B}U(1)
     \\
     \downarrow && \downarrow^{\mathrlap{\mathbf{c}_1 mod 2}}
     \\
     \mathbf{B} SO &\stackrel{w_2}{\to}&
     \mathbf{B}^2 \mathbb{Z}_2 
  }
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
We present this as usual by [[simplicial presheaves]] and [[∞-anafunctors]].

The first [[Chern class]] is given by the [[∞-anafunctor]]

\begin{displaymath}
\itexarray{
     \mathbf{B}(\mathbb{Z} \to \mathbb{R})
     &\stackrel{c_1}{\to}&
     \mathbf{B}(\mathbb{Z} \to 1)
     = 
     \mathbf{B}^2 \mathbb{Z}
     \\
     \downarrow^{\mathrlap{\simeq}}
     \\
     \mathbf{B} U(1)     
  }
  \,.
\end{displaymath}
The second [[Stiefel-Whitney class]] is given by

\begin{displaymath}
\itexarray{
    \mathbf{B}(\mathbb{Z}_2 \to Spin)
    &\stackrel{w_2}{\to}&
    \mathbf{B}(\mathbb{Z}_2 \to 1)
    = 
    \mathbf{B}^2 \mathbb{Z}_2
    \\
    \downarrow^{\mathrlap{\simeq}}
    \\
    \mathbf{B} SO
  }
  \,.
\end{displaymath}
Notice that the top horizontal morphism here is a [[fibration]].

Therefore the [[homotopy pullback]] in question is given by the ordinary pullback

\begin{displaymath}
\itexarray{
    Q
    &\to& 
    \mathbf{B}(\mathbb{Z} \to \mathbb{R})
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}(\mathbb{Z}_2 \to Spin)
    &\to&
    \mathbf{B}^2 \mathbb{Z}_2
  }
  \,.
\end{displaymath}
This pullback is $\mathbf{B}(\mathbb{Z} \stackrel{\partial}{\to} Spin \times \mathbb{R})$, where

\begin{displaymath}
\partial : n \mapsto ( n mod 2 , n)
  \,.
\end{displaymath}
This is equivalent to

\begin{displaymath}
\mathbf{B}(\mathbb{Z}_2 \stackrel{\partial'}{\to} Spin \times U(1))
\end{displaymath}
where now

\begin{displaymath}
\partial' : \sigma \mapsto (\sigma, \sigma)
  \,.
\end{displaymath}
This in turn is equivalent to

\begin{displaymath}
\mathbf{B} (Spin \times_{\mathbb{Z}_2} U(1))
  \,,
\end{displaymath}
which is the original definition.

\end{proof}
This factors the above characterization of $\mathbf{B}Spin^c$ as the homotopy fiber of $\mathbf{W}_3$:

\begin{prop}
\label{}\hypertarget{}{}
We have a [[pasting diagram]] of [[homotopy pullbacks]] of [[smooth infinity-groupoids]] of the form

\begin{displaymath}
\itexarray{
    \mathbf{B} Spin^c &\to& \mathbf{B}U(1) &\to& \ast
    \\
    \downarrow && \downarrow^{\mathrlap{c_1 \, mod\, 2}}
    && \downarrow
    \\
   \mathbf{B}SO 
    &\stackrel{\mathbf{w}_2}{\to}&
   \mathbf{B}^2 \mathbb{Z}_2
   &\stackrel{\mathbf{\beta}_2}{\to}&
   \mathbf{B}^2 U(1)
  }
  \,.
\end{displaymath}
\end{prop}
This is discussed at \emph{\href{spin%5Ec#AsHomotopyFiberOfSmoothW3}{Spin{\tt \symbol{94}}c -- Properties -- As the homotopy fiber of smooth w3}}.

\hypertarget{as_orientation}{}\subsubsection*{{As $KU$-orientation}}\label{as_orientation}

For $X$ an [[orientation|oriented]] [[manifold]], the map $X \to \ast$ is [[orientation in generalized cohomology|generalized oriented]] in [[periodic complex K-theory]] precisely if $X$ has a $Spin^c$-structure.

\hypertarget{RelationToMetaplecticStructures}{}\subsubsection*{{Relation to metaplectic structures}}\label{RelationToMetaplecticStructures}

Let $(X,\omega)$ be a [[compact topological space|compact]] [[symplectic manifold]] equipped with a [[Kähler polarization]] $\mathcal{P}$ hence a [[Kähler manifold]] structure $J$. A [[metaplectic structure]] of this data is a choice of square root $\sqrt{\Omega^{0,n}}$ of the [[canonical line bundle]]. This is equivalently a [[spin structure]] on $X$ (see the discussion at \emph{[[Theta characteristic]]}).

Now given a [[prequantum line bundle]] $L_\omega$, in this case the \href{geometric+quantization#IndexOfDolbeaultDiracOperator}{Dolbault quantization} of $L_\omega$ coincides with the [[spin{\tt \symbol{94}}c quantization|spin quantization]] of the [[spin{\tt \symbol{94}}c structure|spin structure]] induced by $J$ and $L_\omega \otimes \sqrt{\Omega^{0,n}}$.

This appears as (\hyperlink{Paradan09}{Paradan 09, prop. 2.2}).

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{FromAlmostComplexStructures}{}\subsubsection*{{From almost complex structures}}\label{FromAlmostComplexStructures}

An [[almost complex structure]] canonically induces a $Spin^c$-structure:

\begin{prop}
\label{}\hypertarget{}{}
For all $n \in \mathbb{N}$ we have a homotopy-[[commuting diagram]]

\begin{displaymath}
\itexarray{
    \mathbf{B}U(n) &\to& \mathbf{B}U(1)
    \\
    \downarrow 
     &\swArrow& 
    \downarrow^{\mathrlap{\mathbf{c_1} mod 2}}
    \\
    \mathbf{B}SO(2n) &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2
  }
  \,,
\end{displaymath}
where the vertical morphism is the canonical morphism induced from the identification of real vector spaces $\mathbb{C} \to \mathbb{R}^2$, and where the top morphism is the canonical projection $\mathbf{B}U(n) \to \mathbf{B}U(1)$ (induced from $U(n)$ being the [[semidirect product group]] $U(n) \simeq SU(n) \rtimes U(1)$).

\end{prop}
\begin{proof}
By the general relation between $c_1$ of an [[almost complex structure]] and $w_2$ of the underlying orthogonal structure, discussed at \emph{\hyperlink{Stiefel-Whitney+class#RelationToChernClasses}{Stiefel-Whitney class -- Relation to Chern classes}}.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
By prop. \ref{SpinCAsHomotopyFiberProductW2C1} and the [[universal property]] of the [[homotopy pullback]] this induces a canonical morphism

\begin{displaymath}
k \colon \mathbf{B}U(n) \to \mathbf{B}Spin^c
  \,.
\end{displaymath}
\end{remark}
and this is the universal morphism from almost complex structures:

\begin{defn}
\label{}\hypertarget{}{}
For $c \colon X \to \mathbf{B}U(n)$ modulating an [[almost complex structure]]/[[complex vector bundle]] over $X$, the composite

\begin{displaymath}
k c \colon X \stackrel{c}{\to} \mathbf{B}U(n) \stackrel{k}{\to} \mathbf{B}Spin^c
\end{displaymath}
is the corresponding $Spin^c$-structure.

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

\begin{itemize}%
\item [[spin{\tt \symbol{94}}c|spin]]


\item [[twisted differential c-structures|(twisted, differential) c-structures]]

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


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

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

[[K-orientation]]

[[Spin{\tt \symbol{94}}c Dirac operator|Spin Dirac operator]]


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


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


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



\end{itemize}

\item [[spin{\tt \symbol{94}}c quantization|spin quantization]]



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

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

A canonical textbook reference is

\begin{itemize}%
\item H.B. Lawson and M.-L. Michelson, \emph{Spin Geometry} , Princeton University Press, Princeton, NJ, (1989)

\end{itemize}
Other accounts include

\begin{itemize}%
\item Blake Mellor, \emph{$Spin^c$-manifolds} ([[BlakeSpinC.pdf:file]])


\item \emph{Stable complex and $Spin^c$-structures} ([[StableComplexSpinC.pdf:file]])


\item [[Peter Teichner]], Elmar Vogt, \emph{All 4-manifolds have $Spin^c$-structures} (\href{http://people.mpim-bonn.mpg.de/teichner/Papers/spin.pdf}{pdf})



\end{itemize}
\hypertarget{as_orientationanomaly_cancellation_in_type_ii_string_theory}{}\subsubsection*{{As $KU$-orientation/anomaly cancellation in type II string theory}}\label{as_orientationanomaly_cancellation_in_type_ii_string_theory}

That the $U(1)$-[[gauge field]] on a [[D-brane]] in [[type II string theory]] in the absense of a [[B-field]] is rather to be regarded as part of a $spin^c$-structure was maybe first observed in

\begin{itemize}%
\item [[Edward Witten]], \emph{Baryons and Branes In Anti de Sitter Space}, JHEP 9807:006 (1998) (\href{http://arxiv.org/abs/hep-th/9912086}{arXiv:hep-th/9805112}).

\end{itemize}
The [[twisted spin{\tt \symbol{94}}c structure|twisted spin structure]] (see there for more details) on the [[worldvolume]] of [[D-branes]] in the presence of a nontrivial [[B-field]] was 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}
See at \emph{[[Freed-Witten-Kapustin anomaly cancellation]]}.

A more recent review is provided in

\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}
See also

\begin{itemize}%
\item [[Hisham Sati]], \emph{Geometry of $Spin$ and $Spin^c$ structures in the M-theory partition function} (\href{http://arxiv.org/abs/1005.1700}{arXiv:1005.1700})

\end{itemize}
The relation to [[metaplectic corrections]] is discussed in

\begin{itemize}%
\item [[Paul-Emile Paradan]], \emph{Spin-quantization commutes with reduction} (\href{http://arxiv.org/abs/0911.1067}{arXiv:0911.1067})

\end{itemize}
See also

\begin{itemize}%
\item O. Hijazi, S. Montiel, F. Urbano, \emph{$Spin^c$-geometry of K\"a{}hler manifolds and the Hodge Laplacian on minimal Lagrangian submanifolds} (\href{http://hal.archives-ouvertes.fr/docs/00/09/81/71/PDF/lagrangian0412.pdf}{pdf})

\end{itemize}
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\end{document}