Higher spin geometry
Special and general types
For 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 is the third integral Stiefel-Whitney class .
An oriented manifold has -structure if the characteristic class
is trivial. This is the Dixmier-Douady class of the circle 2-bundle/bundle gerbe that obstructs the existence of a -principal bundle lifting the given tangent bundle.
A manifold is equipped with -structure if it is equipped with a choice of trivializaton
The homotopy type/∞-groupoid of -structures on is the homotopy fiber in the pasting diagram of homotopy pullbacks
If the class does not vanish and if hence there is no -structure, it still makes sense to discuss the structure that remains as twisted spin^c structure .
Since is a sequence of Lie groups, the above may be lifted from the (∞,1)-topos Top ∞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 in 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 -structures is the joint (∞,1)-pullback
In parallel to the existence of higher spin structures there are higher analogs of -structures, related to quantum anomaly cancellation of theories of higher dimensional branes.
The group is the fiber product
where in the second line the action is the diagonal action induced from the two canonical embeddings of subgroups and .
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 , where
This is equivalent to
This in turn is equivalent to
which is the original definition.
This factors the above characterization of as the homotopy fiber of :
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 an oriented manifold, the map is generalized oriented in periodic complex K-theory precisely if has a -structure.
Let be a compact symplectic manifold equipped with a Kähler polarization hence a Kähler manifold structure . A metaplectic structure of this data is a choice of square root of the canonical line bundle. This is equivalently a spin structure on (see the discussion at Theta characteristic).
Now given a prequantum line bundle , in this case the Dolbault quantization of coincides with the spin^c quantization of the spin^c structure induced by and .
This appears as (Paradan 09, prop. 2.2).
From almost complex structures
An almost complex structure canonically induces a -structure:
For all we have a homotopy-commuting diagram
where the vertical morphism is the canonical morphism induced from the identification of real vector spaces , and where the top morphism is the canonical projection (induced from being the semidirect product group ).
and this is the universal morphism from almost complex structures:
For modulating an almost complex structure/complex vector bundle over , the composite
is the corresponding -structure.
A canonical textbook reference is
- H.B. Lawson and M.-L. Michelson, Spin Geometry , Princeton University Press, Princeton, NJ, (1989)
Other accounts include
Blake Mellor, -manifolds (pdf)
Stable complex and -structures (pdf)
Peter Teichner, Elmar Vogt, All 4-manifolds have -structures (pdf)
As -orientation/anomaly cancellation in type II string theory
That the -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 -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
- Kim Laine, Geometric and topological aspects of Type IIB D-branes (arXiv:0912.0460)
The relation to metaplectic corrections is discussed in
- O. Hijazi, S. Montiel, F. Urbano, -geometry of Kähler manifolds and the Hodge Laplacian on minimal Lagrangian submanifolds (pdf)