cohomology

# Contents

## Definition

### Topological

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

$U\left(1\right)\to {\mathrm{Spin}}^{c}\left(n\right)\to \mathrm{SO}\left(n\right)$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

$\cdots \to BU\left(1\right)\to B{\mathrm{Spin}}^{c}\left(n\right)\to B\mathrm{SO}\left(n\right)\stackrel{{W}_{3}}{\to }{B}^{2}U\left(1\right)\phantom{\rule{thinmathspace}{0ex}},$\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 ${\mathrm{Spin}}^{c}$-structure if the class $\left[{W}_{3}\left(X\right)\right]\in {H}^{3}\left(X,ℤ\right)$

${W}_{3}\left(X\right):={W}_{3}\left(TX\right):X\stackrel{TX}{\to }B\mathrm{SO}\left(n\right)\stackrel{{W}_{3}}{\to }{B}^{2}U\left(1\right)\simeq K\left(ℤ,3\right)$W_3(X) := W_3(T X) : 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 class of the circle 2-bundle/bundle gerbe that obstructs the existence of a ${\mathrm{Spin}}^{c}$-principal bundle. A given equipped with ${\mathrm{Spin}}^{c}$-structure $\eta$ if it is equipped with a choice of trivializaton

$\eta :1\stackrel{\simeq }{\to }{W}_{3}\left(TX\right)\phantom{\rule{thinmathspace}{0ex}}.$\eta : 1 \stackrel{\simeq}{\to} W_3(T X) \,.

The space/∞-groupoid of ${\mathrm{Spin}}^{c}$-structures on $X$ is the homotopy fiber ${W}_{3}\mathrm{Struc}\left(TX\right)$ in the pasting diagram of homotopy pullbacks

$\begin{array}{ccccc}{W}_{3}\mathrm{Struc}\left(TX\right)& \to & {W}_{3}\mathrm{Struc}\left(X\right)& \to & *\\ ↓& & ↓& & ↓\\ *& \stackrel{TX}{\to }& \mathrm{Top}\left(X,B\mathrm{SO}\left(n\right)\right)& \stackrel{{W}_{3}}{\to }& \mathrm{Top}\left(X,{B}^{2}U\left(1\right)\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 ${\mathrm{Spin}}^{c}$-structure, it still makes sense to discuss the structure that remains as twisted spin^c structure .

### Smooth

Since $U\left(1\right)\to {\mathrm{Spin}}^{c}\to \mathrm{SO}$ is a sequence of Lie groups, the above may be lifted from the (∞,1)-topos ${L}_{\mathrm{whe}}$ 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 ${W}_{3}:B\mathrm{SO}\to {B}^{2}U\left(1\right)$ in $\infty \mathrm{Grpd}$ has, up to equivalence, a unique lift

${W}_{3}:B\mathrm{SO}\to {B}^{2}U\left(1\right)$\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 ${\mathrm{spin}}^{c}$-structures ${W}_{3}\mathrm{Struc}\left(X\right)$ is the joint (∞,1)-pullback

$\begin{array}{ccccc}{W}_{3}\mathrm{Struc}\left(TX\right)& \to & {W}_{3}\mathrm{Struc}\left(X\right)& \to & *\\ ↓& & ↓& & ↓\\ *& \stackrel{TX}{\to }& \mathrm{Smooth}\infty \mathrm{Grpd}\left(X,B\mathrm{SO}\left(n\right)\right)& \stackrel{{W}_{3}}{\to }& \mathrm{Smooth}\infty \mathrm{Grpd}\left(X,{B}^{2}U\left(1\right)\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 ${\mathrm{spin}}^{c}$-structures

In parallel to the existence of higher spin structures there are higher analogs of ${\mathrm{Spin}}^{c}$-structures, related to quantum anomaly cancellation of theories of higher dimensional branes.

## Properties

### Of ${\mathrm{Spin}}^{c}$

###### Definition

The group ${\mathrm{Spin}}^{c}$ is

$\begin{array}{rl}{\mathrm{Spin}}^{c}& :=\mathrm{Spin}{×}_{{ℤ}_{2}}U\left(1\right)\\ & =\left(\mathrm{Spin}×U\left(1\right)\right)/{ℤ}_{2}\phantom{\rule{thinmathspace}{0ex}},\end{array}$\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 ${ℤ}_{2}↪ℤ$ and ${ℤ}_{2}↪U\left(1\right)$.

###### Proposition

We have a homotopy pullback diagram

$\begin{array}{ccc}B{\mathrm{Spin}}^{c}& \to & BU\left(1\right)\\ ↓& & {↓}^{{c}_{1}\mathrm{mod}2}\\ B\mathrm{SO}& \stackrel{{w}_{2}}{\to }& {B}^{2}{ℤ}_{2}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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

$\begin{array}{ccc}B\left(ℤ\to ℝ\right)& \stackrel{{c}_{1}}{\to }& B\left(ℤ\to 1\right)={B}^{2}ℤ\\ {↓}^{\simeq }\\ BU\left(1\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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

$\begin{array}{ccc}B\left({ℤ}_{2}\to \mathrm{Spin}\right)& \stackrel{{w}_{2}}{\to }& B\left({ℤ}_{2}\to 1\right)={B}^{2}{ℤ}_{2}\\ {↓}^{\simeq }\\ B\mathrm{SO}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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

$\begin{array}{ccc}Q& \to & B\left(ℤ\to ℝ\right)\\ ↓& & ↓\\ B\left({ℤ}_{2}\to \mathrm{Spin}\right)& \to & {B}^{2}{ℤ}_{2}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $B\left(ℤ\stackrel{\partial }{\to }\mathrm{Spin}×ℝ\right)$, where

$\partial :n↦\left(n\mathrm{mod}2,n\right)\phantom{\rule{thinmathspace}{0ex}}.$\partial : n \mapsto ( n mod 2 , n) \,.

This is equivalent to

$B\left({ℤ}_{2}\stackrel{\partial \prime }{\to }\mathrm{Spin}×U\left(1\right)\right)$\mathbf{B}(\mathbb{Z}_2 \stackrel{\partial'}{\to} Spin \times U(1))

where now

$\partial \prime :\sigma ↦\left(\sigma ,\sigma \right)\phantom{\rule{thinmathspace}{0ex}}.$\partial' : \sigma \mapsto (\sigma, \sigma) \,.

This in turn is equivalent to

$B\left(\mathrm{Spin}{×}_{{ℤ}_{2}}U\left(1\right)\right)\phantom{\rule{thinmathspace}{0ex}},$\mathbf{B} (Spin \times_{\mathbb{Z}_2} U(1)) \,,

which is the original definition.

## Examples

### From almost complex structures

An almost complex structure canonically induces a ${\mathrm{Spin}}^{c}$-structure:

###### Proposition

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

$\begin{array}{ccc}BU\left(n\right)& \to & BU\left(1\right)\\ ↓& & {↓}^{{c}_{1}\mathrm{mod}2}\\ B\mathrm{SO}\left(2n\right)& \stackrel{{w}_{2}}{\to }& {B}^{2}{ℤ}_{2}\end{array}\phantom{\rule{thinmathspace}{0ex}},$\array{ \mathbf{B}U(n) &\to& \mathbf{B}U(1) \\ \downarrow && \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 $ℂ\to {ℝ}^{2}$, and where the top morphism is the canonical projection $BU\left(n\right)\to BU\left(1\right)$ (induced from $U\left(n\right)$ being the semidirect product group $U\left(n\right)\simeq \mathrm{SU}\left(n\right)⋊U\left(1\right)$).

###### 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.

By prop. 1 this induces a canonical morphism

$k:BU\left(n\right)\to B{\mathrm{Spin}}^{c}$k \colon \mathbf{B}U(n) \to \mathbf{B}Spin^c

and this is the universal morphism from almost complex structures:

For $c:X\to BU\left(n\right)$ modulating an almost complex structure/complex vector bundle over $X$, the composite

$kc:X\stackrel{c}{\to }BU\left(n\right)\stackrel{k}{\to }B{\mathrm{Spin}}^{c}$k c \colon X \stackrel{c}{\to} \mathbf{B}U(n) \stackrel{k}{\to} \mathbf{B}Spin^c

is the corresponding ${\mathrm{Spin}}^{c}$-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, ${\mathrm{Spin}}^{c}$-manifolds (pdf)

• Stable complex and ${\mathrm{Spin}}^{c}$-structures (pdf)

### As anomaly cancellation in type II string theory

That the $U\left(1\right)$-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 ${\mathrm{spin}}^{c}$-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 $\mathrm{Spin}$ and ${\mathrm{Spin}}^{c}$ structures in the M-theory partition function (arXiv:1005.1700)