spin geometry

string geometry

cohomology

# Contents

## Definition

### Topological

For $n \in \mathbb{N}$ the Lie group spin^c is a central extension

$U(1) \to Spin^c(n) \to SO(n)$

of the special orthogonal group by the circle group. This comes with a long fiber sequence

$\cdots \to B U(1) \to B Spin^c(n) \to B SO(n) \stackrel{W_3}{\to} B^2 U(1) \,,$

where $W_3$ is the third integral Stiefel-Whitney class .

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

$W_3(X) \coloneqq W_3(T X) \;\colon\; X \stackrel{T X}{\to} B SO(n) \stackrel{W_3}{\to} B^2 U(1) \simeq K(\mathbb{Z},3)$

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

$\eta : 1 \stackrel{\simeq}{\to} W_3(T X) \,.$

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

$\array{ W_3 Struc(T X) &\to& W_3 Struc(X) &\to& * \\ \downarrow && \downarrow && \downarrow \\ * &\stackrel{T X}{\to}& Top(X, B SO(n)) &\stackrel{W_3}{\to}& Top(X,B^2 U(1)) } \,.$

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 .

### Smooth

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

$\mathbf{W}_3 : \mathbf{B} SO \to \mathbf{B}^2 U(1)$

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

$\array{ \mathbf{W}_3 Struc(T X) &\to& \mathbf{W}_3 Struc(X) &\to& * \\ \downarrow && \downarrow && \downarrow \\ * &\stackrel{T X}{\to}& Smooth \infty Grpd(X, \mathbf{B} SO(n)) &\stackrel{\mathbf{W}_3}{\to}& Smooth \infty Grpd(X,\mathbf{B}^2 U(1)) } \,.$

### Higher $spin^c$-structures

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.

## Properties

### Of $Spin^c$

###### Definition

The group $Spin^c$ is the fiber product

\begin{aligned} Spin^c & := Spin \times_{\mathbb{Z}_2} U(1) \\ & = (Spin \times U(1))/{\mathbb{Z}_2} \,, \end{aligned}

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 $\mathbb{Z}_2 \hookrightarrow U(1)$.

###### Proposition

We have a homotopy pullback diagram

$\array{ \mathbf{B} Spin^c &\to& \mathbf{B}U(1) \\ \downarrow && \downarrow^{\mathrlap{\mathbf{c}_1 mod 2}} \\ \mathbf{B} SO &\stackrel{w_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 } \,.$
###### Proof

We present this as usual by simplicial presheaves and ∞-anafunctors.

The first Chern class is given by the ∞-anafunctor

$\array{ \mathbf{B}(\mathbb{Z} \to \mathbb{R}) &\stackrel{c_1}{\to}& \mathbf{B}(\mathbb{Z} \to 1) = \mathbf{B}^2 \mathbb{Z} \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B} U(1) } \,.$

The second Stiefel-Whitney class is given by

$\array{ \mathbf{B}(\mathbb{Z}_2 \to Spin) &\stackrel{w_2}{\to}& \mathbf{B}(\mathbb{Z}_2 \to 1) = \mathbf{B}^2 \mathbb{Z}_2 \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B} SO } \,.$

Notice that the top horizontal morphism here is a fibration.

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

$\array{ Q &\to& \mathbf{B}(\mathbb{Z} \to \mathbb{R}) \\ \downarrow && \downarrow \\ \mathbf{B}(\mathbb{Z}_2 \to Spin) &\to& \mathbf{B}^2 \mathbb{Z}_2 } \,.$

This pullback is $\mathbf{B}(\mathbb{Z} \stackrel{\partial}{\to} Spin \times \mathbb{R})$, where

$\partial : n \mapsto ( n mod 2 , n) \,.$

This is equivalent to

$\mathbf{B}(\mathbb{Z}_2 \stackrel{\partial'}{\to} Spin \times U(1))$

where now

$\partial' : \sigma \mapsto (\sigma, \sigma) \,.$

This in turn is equivalent to

$\mathbf{B} (Spin \times_{\mathbb{Z}_2} U(1)) \,,$

which is the original definition.

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

###### Proposition

We have a pasting diagram of homotopy pullbacks of smooth infinity-groupoids of the form

$\array{ \mathbf{B} Spin^c &\to& \mathbf{B}U(1) &\to& \ast \\ \downarrow && \downarrow^{\mathrlap{c_1 \, mod\, 2}} && \downarrow \\ \mathbf{B}SO &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 \stackrel{\mathbf{\beta}_2}{\to} \mathbf{B}^2 U(1) } \,.$

This is discussed at Spin^c – Properties – As the homotopy fiber of smooth w3.

### As $KU$-orientation

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.

### Relation to metaplectic structures

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).

## Examples

### From almost complex structures

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

###### Proposition

For all $n \in \mathbb{N}$ we have a homotopy-commuting diagram

$\array{ \mathbf{B}U(n) &\to& \mathbf{B}U(1) \\ \downarrow &\swArrow& \downarrow^{\mathrlap{\mathbf{c_1} mod 2}} \\ \mathbf{B}SO(2n) &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 } \,,$

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 $U(n) \simeq SU(n) \rtimes U(1)$).

###### Proof

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.

###### Remark

By prop. 1 and the universal property of the homotopy pullback this induces a canonical morphism

$k \colon \mathbf{B}U(n) \to \mathbf{B}Spin^c \,.$

and this is the universal morphism from almost complex structures:

###### Definition

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

$k c \colon X \stackrel{c}{\to} \mathbf{B}U(n) \stackrel{k}{\to} \mathbf{B}Spin^c$

is the corresponding $Spin^c$-structure.

## References

### General

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, $Spin^c$-manifolds (pdf)

• Stable complex and $Spin^c$-structures (pdf)

• Peter Teichner, Elmar Vogt, All 4-manifolds have $Spin^c$-structures (pdf)

### As $KU$-orientation/anomaly 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

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

A more recent review is provided in

• Kim Laine, Geometric and topological aspects of Type IIB D-branes (arXiv:0912.0460)

• Hisham Sati, Geometry of $Spin$ and $Spin^c$ structures in the M-theory partition function (arXiv:1005.1700)
• O. Hijazi, S. Montiel, F. Urbano, $Spin^c$-geometry of Kähler manifolds and the Hodge Laplacian on minimal Lagrangian submanifolds (pdf)