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string^c 2-group

Contents

under construction

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Higher spin geometry

String theory

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\mathbf{B}O by the first Chern class

BSpin c B(SO×U(1)) w 1c 1 B 2 \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

BString c 2 B(Spin×SU(n)) 12p 1c 2 B 3U(1). \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 cString^c is a String cString^c-structure.

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

References

Topological string cstring^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 cString^c Manifolds, Pacific Journal of Mathematics 266.2 (2013): 477-508. (arXiv:1206.5955)

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

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

A discussion explicitly in the context of string theory is in

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

see also

Last revised on November 17, 2020 at 14:59:29. See the history of this page for a list of all contributions to it.