nLab
string^c 2-group

under construction

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

$$\begin{array}{ccc}B{\mathrm{Spin}}^c& \to & B(\mathrm{SO}\times U(1))\\ & & {\downarrow }^{w_1-c_1}\\ & & B^2\mathbb{Z}\end{array}$$\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

$$\begin{array}{ccc}B{\mathrm{String}}^{c_2}& \to & B(\mathrm{Spin}\times \mathrm{SU}(n))\\ & & {\downarrow }^{\frac{1}{2}{p}_1-c_2}\\ & & B^{3}U(1)\end{array}.$$\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.

Related concepts

• string 2-group

• twisted differential string structure

• (twisted, differential) c-structures

  • orientation

  • spin structure, twisted spin structure, differential spin structure

    spin^c structure, twisted spin^c structure,

  • string structure, twisted differential string structure,

  • ${\mathrm{string}}^c$ structure

  • fivebrane structure, twisted differential fivebrane structure

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

• Hisham Sati, Geometric and topological structures related to M-branes II: Twisted $\mathrm{String}$ and ${\mathrm{String}}^c$ structures, J. Australian Math. Soc. 90 (2011), 93-108 (arXiv:1007.5419)

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

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory

A general discussion is in section 5.2 of

• Urs Schreiber, differential cohomology in a cohesive topos