# nLab string^c 2-group

under construction

cohomology

spin geometry

string geometry

# 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 $\mathbf{B}O$ by the first Chern class

$\array{ \mathbf{B}Spin^c &\to& \mathbf{B}(SO \times U(1)) \\ && \downarrow^{\mathrlap{\mathbf{w}_1 - \mathbf{c}_1}} \\ && \mathbf{B}^2 \mathbb{Z} }$

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

$\array{ \mathbf{B}String^{\mathbf{c}_2} &\to& \mathbf{B}(Spin \times SU(n)) \\ && \downarrow^{\mathrlap{\tfrac{1}{2}\mathbf{p}_1 - \mathbf{c}_2}} \\ && \mathbf{B}^3 U(1) } \,.$

Accordingly a lift of the structure group to $String^c$ is a $String^c$-structure.

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

## References

Topological $string^c$-structures were introduced

• Bai-Ling Wang, Geometric cycles, index theory and twisted K-homology. J. Noncommut. Geom., 2(4):497–552, 2008.

and shown to induce a twisted Witten genus in

• Qingtao Chen, Fei Han, Weiping Zhang, Generalized Witten Genus and Vanishing Theorems, Journal of Differential Geometry 88.1 (2011): 1-39. (arXiv:1003.2325)

• Jianqing Yu, Bo Liu, On the Witten Rigidity Theorem for $String^c$ Manifolds, Pacific Journal of Mathematics 266.2 (2013): 477-508. (arXiv:1206.5955)

The push-forward in twisted tmf induced by a $string^c$-structure is discussed in section 11 of

A discussion explicitly in the context of string theory is in

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

A general discussion is in section 5.2 of

Revised on March 22, 2014 05:26:29 by Urs Schreiber (88.128.80.11)