# nLab string^c 2-group

Contents

under construction

cohomology

## Spin geometry

spin geometry

Dynkin labelsp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
B1SO(3)Spin(3)Pin(3)
D2SO(4)Spin(4)Pin(4)
B2SO(5)Spin(5)Pin(5)
D3SO(6)Spin(6)
B3SO(7)Spin(7)
D4SO(8)Spin(8)SO(8)
B4SO(9)Spin(9)
D5SO(10)Spin(10)
B5SO(11)Spin(11)
D6SO(12)Spin(12)
$\vdots$$\vdots$
D8SO(16)Spin(16)SemiSpin(16)
$\vdots$$\vdots$
D16SO(32)Spin(32)SemiSpin(32)

string geometry

# 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 $\mathbf{B}O$ 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.

## 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

A discussion explicitly in the context of string theory is in

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