# nLab string^c 2-group

under construction

cohomology

# Contents

## Idea

In analogy to how the Lie group spin^c is obtained by twisting the lift through the second stage of the Whitehead tower of $BO$ by the first Chern class

$\begin{array}{ccc}B{\mathrm{Spin}}^{c}& \to & B\left(\mathrm{SO}×U\left(1\right)\right)\\ & & {↓}^{{w}_{1}-{c}_{1}}\\ & & {B}^{2}ℤ\end{array}$\array{ \mathbf{B}Spin^c &\to& \mathbf{B}(SO \times U(1)) \\ && \downarrow^{\mathrlap{\mathbf{w}_1 - \mathbf{c}_1}} \\ && \mathbf{B}^2 \mathbb{Z} }

there is a similar twist by the second Chern class of the lift through the next stage of the Whitehead tower

$\begin{array}{ccc}B{\mathrm{String}}^{{c}_{2}}& \to & B\left(\mathrm{Spin}×\mathrm{SU}\left(n\right)\right)\\ & & {↓}^{\frac{1}{2}{p}_{1}-{c}_{2}}\\ & & {B}^{3}U\left(1\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ \mathbf{B}String^{\mathbf{c}_2} &\to& \mathbf{B}(Spin \times SU(n)) \\ && \downarrow^{\mathrlap{\tfrac{1}{2}\mathbf{p}_1 - \mathbf{c}_2}} \\ && \mathbf{B}^3 U(1) } \,.

Accordingly a lift of the structure group to ${\mathrm{String}}^{c}$ is a ${\mathrm{String}}^{c}$-structure.

For the moment see at twisted smooth cohomology in string theory for more.

## References

Topological ${\mathrm{string}}^{c}$-structures were introduced in

• Qingtao Chen, Fei Han, Weiping Zhang, Generalized Witten Genus and Vanishing Theorems (arXiv:1003.2325)

and discussed in the context of string theory in

Their smooth refinement and their smooth moduli 2-stacks were introduced in

A general discussion is in section 5.2 of

Revised on November 6, 2012 01:20:52 by Urs Schreiber (89.204.130.22)