spin geometry, string geometry, fivebrane geometry …
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
For $n \in \mathbb{N}$ the Lie group spin^c is a central extension
of the special orthogonal group by the circle group. This comes with a long fiber sequence
where $W_3$ is the third integral Stiefel-Whitney class .
An oriented manifold $X$ has $Spin^c$-structure if the characteristic class $[W_3(X)] \in H^3(X, \mathbb{Z})$
is trivial. This is the Dixmier-Douady class of the circle 2-bundle/bundle gerbe that obstructs the existence of a $Spin^c$-principal bundle lifting the given tangent bundle.
A manifold $X$ is equipped with $Spin^c$-structure $\eta$ if it is equipped with a choice of trivializaton
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 pullbacks
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 twisted spin^c structure .
Since $U(1) \to Spin^c \to SO$ is a sequence of Lie groups, 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
to Smooth∞Grpd, where on the right we have the delooping of the smooth circle 2-group.
Accordingly, the 2-groupoid of smooth $spin^c$-structures $\mathbf{W}_3 Struc(X)$ is the joint (∞,1)-pullback
In parallel to the existence of higher spin structures there are higher analogs of $Spin^c$-structures, related to quantum anomaly cancellation of theories of higher dimensional branes.
The group $Spin^c$ is the fiber product
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 $\mathbb{Z}_2 \hookrightarrow U(1)$.
We have a homotopy pullback diagram
We present this as usual by simplicial presheaves and ∞-anafunctors.
The first Chern class is given by the ∞-anafunctor
The second Stiefel-Whitney class is given by
Notice that the top horizontal morphism here is a fibration.
Therefore the homotopy pullback in question is given by the ordinary pullback
This pullback is $\mathbf{B}(\mathbb{Z} \stackrel{\partial}{\to} Spin \times \mathbb{R})$, where
This is equivalent to
where now
This in turn is equivalent to
which is the original definition.
This factors the above characterization of $\mathbf{B}Spin^c$ as the homotopy fiber of $\mathbf{W}_3$:
We have a pasting diagram of homotopy pullbacks of smooth infinity-groupoids of the form
This is discussed at Spin^c – Properties – As the homotopy fiber of smooth w3.
For $X$ an oriented manifold, the map $X \to \ast$ is generalized oriented in periodic complex K-theory precisely if $X$ has a $Spin^c$-structure.
Let $(X,\omega)$ be a 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 Theta characteristic).
Now given a prequantum line bundle $L_\omega$, in this case the Dolbault quantization of $L_\omega$ coincides with the spin^c quantization of the spin^c structure induced by $J$ and $L_\omega \otimes \sqrt{\Omega^{0,n}}$.
This appears as (Paradan 09, prop. 2.2).
An almost complex structure canonically induces a $Spin^c$-structure:
For all $n \in \mathbb{N}$ we have a homotopy-commuting diagram
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 $U(n) \simeq SU(n) \rtimes U(1)$).
By the general relation between $c_1$ of an almost complex structure and $w_2$ of the underlying orthogonal structure, discussed at Stiefel-Whitney class – Relation to Chern classes.
By prop. 1 and the universal property of the homotopy pullback this induces a canonical morphism
and this is the universal morphism from almost complex structures:
For $c \colon X \to \mathbf{B}U(n)$ modulating an almost complex structure/complex vector bundle over $X$, the composite
is the corresponding $Spin^c$-structure.
A canonical textbook reference is
Other accounts include
Blake Mellor, $Spin^c$-manifolds (pdf)
Stable complex and $Spin^c$-structures (pdf)
Peter Teichner, Elmar Vogt, All 4-manifolds have $Spin^c$-structures (pdf)
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
The twisted spin^c structure (see there for more details) on the worldvolume of D-branes in the presence of a nontrivial B-field was discussed in
See at Freed-Witten-Kapustin anomaly cancellation.
A more recent review is provided in
See also
The relation to metaplectic corrections is discussed in
See also