cohomology

# Contents

## Definition

### Topological

For $n\in ℕ$ the Lie group ${\mathrm{Spin}}^{c}\left(n\right)$ 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 .

By the definition at twisted cohomology, for a given class $\left[c\right]\in {H}^{3}\left(X,ℤ\right)$, a $c$-twisted ${\mathrm{spin}}^{c}$-structure is a choice of homotopy

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

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

$\begin{array}{ccccc}{W}_{3}{\mathrm{Struc}}_{\mathrm{tw}}\left(TX\right)& \to & {W}_{3}{\mathrm{Struc}}_{\mathrm{tw}}\left(X\right)& \stackrel{\mathrm{tw}}{\to }& {H}^{3}\left(X,ℤ\right)\\ ↓& & ↓& & ↓\\ *& \stackrel{TX}{\to }& \mathrm{Top}\left(X,B\mathrm{SO}\left(n\right)\right)& \stackrel{{W}_{3}}{\to }& \mathrm{Top}\left(X,{B}^{2}U\left(1\right)\right)\end{array}\phantom{\rule{thinmathspace}{0ex}},$\array{ W_3 Struc_{tw}(T X) &\to& W_3 Struc_{tw}(X) &\stackrel{tw}{\to}& H^3(X, \mathbb{Z}) \\ \downarrow && \downarrow && \downarrow \\ * &\stackrel{T X}{\to}& Top(X, B SO(n)) &\stackrel{W_3}{\to}& Top(X, B^2 U(1)) } \,,

where the right vertical morphism is the canonical effective epimorphism that picks one point in each connected component.

### 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 Top $\simeq$ ∞Grpd to Smooth∞Grpd.

More precisely, 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.

By the general definition at twisted differential c-structure , the 2-groupoid of smooth twisted ${\mathrm{spin}}^{c}$-structures ${W}_{3}{\mathrm{Struc}}_{\mathrm{tw}}\left(X\right)$ is the joint (∞,1)-pullback

$\begin{array}{ccccc}{W}_{3}{\mathrm{Struc}}_{\mathrm{tw}}\left(TX\right)& \to & {W}_{3}{\mathrm{Struc}}_{\mathrm{tw}}\left(X\right)& \stackrel{\mathrm{tw}}{\to }& {H}_{\mathrm{smooth}}^{2}\left(X,U\left(1\right)\right)\\ ↓& & ↓& & ↓\\ *& \stackrel{TX}{\to }& \mathrm{Smooth}\infty \mathrm{Grpd}\left(X,B\mathrm{SO}\left(n\right)\right)& \stackrel{{W}_{3}}{\to }& \mathrm{Smooth}\infty \mathrm{Grpd}\left(X,{B}^{2}U\left(1\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ \mathbf{W}_3 Struc_{tw}(T X) &\to& \mathbf{W}_3 Struc_{tw}(X) &\stackrel{tw}{\to}& H_{smooth}^2(X, U(1)) \\ \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) } \,.

## Applications

### Anomaly cancellation in physics

The existence of an ordinary spin structure on a space $X$ is, as discussed there, the condition for $X$ to serve as the target space for the spinning particle sigma-model, in that the existence of this structure is precisely the condition that the corresponding fermionic quantum anomaly on the worldline vanishes.

Twisted ${\mathrm{spin}}^{c}$-structures appear similarly as the conditions for the analogous quantum anomaly cancellation, but now of the open type II superstring ending on a D-brane. This is also called the Freed-Witten anomaly cancellation.

More precisely, in these applications the class of ${W}_{3}\left(\mathrm{TX}\right)-H$ need not vanish, it only needs to be $n$-torsion if there is moreover a twisted bundle of rank $n$ on the $D$-brane.

See the references below for details.

## References

### General

The notion of twisted ${\mathrm{Spin}}^{c}$-structures as such were apparently first discussed in section 5 of

More discussion appears in section 3 of

The refinement to smooth twisted structures is discussed in section 4.1 of

### In physics

The need for twisted ${\mathrm{Spin}}^{c}$-structures as Freed-Witten anomaly cancellation condition on the worldvolume of D-branes in string theory was first discussed in

More details are in

A clean formulation and review is provided in

• Loriano Bonora, Fabio Ferrari Ruffino, Raffaele Savelli, Classifying A-field and B-field configurations in the presence of D-branes (arXiv:0810.4291)

• Fabio Ferrari Ruffino, Classifying A-field and B-field configurations in the presence of D-branes - Part II: Stacks of D-branes (arXiv:1104.2798)

• Fabio Ferrari Ruffino, Topics on topology and superstring theory (arXiv:0910.4524)

and

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

In (Laine) the discussion of FW-anomaly cancellation with finite-rank gauge bundles is towards the very end, culminating in equation (3.41).

Revised on August 12, 2012 22:11:26 by Urs Schreiber (82.113.121.69)