[[!redirects string^c 2-group]] [[!redirects string^c 2-group]] > under construction +-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Cohomology +--{: .hide} [[!include cohomology - contents]] =-- #### Higher spin geometry +-- {: .hide} [[!include higher spin geometry - contents]] =-- #### String theory +-- {: .hide} [[!include string theory - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## 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 [[reduction and lift of structure groups|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|(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 {#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 * {#ChenHanZhang10} [[Qingtao Chen]], [[Fei Han]], [[Weiping Zhang]], _Generalized Witten Genus and Vanishing Theorems_, Journal of Differential Geometry 88.1 (2011): 1-39. ([arXiv:1003.2325](http://arxiv.org/abs/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](http://arxiv.org/abs/1206.5955)) * Haibao Duan, [[Fei Han]], Ruizhi Huang, _$String^c$ Structures and Modular Invariants_, Trans. AMS 2020 ([arXiv:1905.02093](https://arxiv.org/abs/1905.02093)) The [[push-forward in generalized cohomology|push-forward]] in [[twisted cohomology|twisted]] [[tmf]] induced by a $string^c$-structure is discussed in * {#ABG} [[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](http://www.ams.org/bookstore-getitem/item=PSPUM-81), American Mathematical Society ([arXiv:1002.3004](http://arxiv.org/abs/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](http://arxiv.org/abs/1007.5419)) Their smooth refinement and their [[smooth infinity-groupoid|smooth moduli 2-stacks]] were introduced in * [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:7d Chern-Simons theory and the 5-brane|Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory]]_ see also * [[Urs Schreiber]], section 5.2 of _[[schreiber:differential cohomology in a cohesive topos]]_ [[!redirects stringᶜ 2-groups]] [[!redirects stringᶜ structure]] [[!redirects stringᶜ structures]] [[!redirects string^c 2-group]] [[!redirects string^c 2-groups]] [[!redirects String^c 2-group]] [[!redirects String^c 2-groups]] [[!redirects String^c-structure]] [[!redirects String^c-structures]] [[!redirects String^c structure]] [[!redirects String^c structures]] [[!redirects string^c structure]] [[!redirects string^c structures]]