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string^c 2-group

under construction

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cohomology

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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)

The push-forward in twisted tmf induced by a string cstring^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)