nLab stringᶜ 2-group

under construction

Idea

In analogy to how the Lie group spinᶜ 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.

Related concepts

• string 2-group

• twisted differential string structure

• (twisted, differential) c-structures

  • orientation

  • spin structure, twisted spin structure, differential spin structure

    spinᶜ structure, twisted spinᶜ structure,

  • string structure, twisted differential string structure,

  • $string^c$ structure

  • fivebrane structure, twisted differential fivebrane structure

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)

• Haibao Duan, Fei Han, Ruizhi Huang, $String^c$ Structures and Modular Invariants, Trans. AMS 2020 (arXiv:1905.02093)

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

• Matthew Ando, Andrew Blumberg, David Gepner, Section 11 of Twists of K-theory and TMF, in Robert S. Doran, Greg Friedman, Jonathan Rosenberg, Superstrings, Geometry, Topology, and $C^*$-algebras, Proceedings of Symposia in Pure Mathematics vol 81, American Mathematical Society (arXiv:1002.3004)

A discussion explicitly in the context of string theory is in

• Hisham Sati, Geometric and topological structures related to M-branes II: Twisted $String$ and $String^c$ 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

see also

• Urs Schreiber, section 5.2 of differential cohomology in a cohesive topos