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twisted spin^c structure

Definition

Topological

For $n\in \mathbb{N}$ the Lie group ${\mathrm{Spin}}^c(n)$ is a central extension

$$U(1)\to {\mathrm{Spin}}^c(n)\to \mathrm{SO}(n)$$U(1) \to Spin^c(n) \to SO(n)

of the special orthogonal group by the circle group. This comes with a long fiber sequence

$$\cdots \to BU(1)\to B{\mathrm{Spin}}^c(n)\to B\mathrm{SO}(n)\stackrel{W_3}{\to }{B}^{2}U(1),$$\cdots \to B U(1) \to B Spin^c(n) \to B SO(n) \stackrel{W_3}{\to} B^2 U(1) \,,

where $W_3$ is the third integral Stiefel-Whitney class .

By the definition at twisted cohomology, for a given class $[c]\in H^3(X,\mathbb{Z})$, a $c$-twisted ${\mathrm{spin}}^c$-structure is a choice of homotopy

$$\eta :c\stackrel{\simeq }{\to }{W}_3(TX).$$\eta : c \stackrel{\simeq}{\to} W_3(T X) \,.

The space/∞-groupoid of all twisted ${\mathrm{Spin}}^c$-structures on $X$ is the homotopy fiber $W_3{\mathrm{Struc}}_{\mathrm{tw}}(TX)$ in the pasting diagram of homotopy pullbacks

$$\begin{array}{ccccc}{W}_3{\mathrm{Struc}}_{\mathrm{tw}}(TX)& \to & W_3{\mathrm{Struc}}_{\mathrm{tw}}(X)& \stackrel{\mathrm{tw}}{\to }& H^3(X,\mathbb{Z})\\ \downarrow & & \downarrow & & \downarrow \\ *& \stackrel{TX}{\to }& \mathrm{Top}(X,B\mathrm{SO}(n))& \stackrel{W_3}{\to }& \mathrm{Top}(X,B^{2}U(1))\end{array},$$\array{ W_3 Struc_{tw}(T X) &\to& W_3 Struc_{tw}(X) &\stackrel{tw}{\to}& H^3(X, \mathbb{Z}) \\ \downarrow && \downarrow && \downarrow \\ * &\stackrel{T X}{\to}& Top(X, B SO(n)) &\stackrel{W_3}{\to}& Top(X, B^2 U(1)) } \,,

where the right vertical morphism is the canonical effective epimorphism that picks one point in each connected component.

Smooth

Since $U(1)\to {\mathrm{Spin}}^c\to \mathrm{SO}$ is a sequence of Lie groups, the above may be lifted from the (∞,1)-topos Top $\simeq$ ∞Grpd to Smooth∞Grpd.

More precisely, by the discussion at Lie group cohomology (and smooth ∞-groupoid -- structures) the characteristic map $W_3:B\mathrm{SO}\to B^{2}U(1)$ in $\mathrm{\infty }\mathrm{Grpd}$ has, up to equivalence, a unique lift

$$W_3:B\mathrm{SO}\to B^{2}U(1)$$\mathbf{W}_3 : \mathbf{B} SO \to \mathbf{B}^2 U(1)

to Smooth∞Grpd, where on the right we have the delooping of the smooth circle 2-group.

By the general definition at twisted differential c-structure , the 2-groupoid of smooth twisted ${\mathrm{spin}}^c$-structures $W_3{\mathrm{Struc}}_{\mathrm{tw}}(X)$ is the joint (∞,1)-pullback

$$\begin{array}{ccccc}{W}_3{\mathrm{Struc}}_{\mathrm{tw}}(TX)& \to & W_3{\mathrm{Struc}}_{\mathrm{tw}}(X)& \stackrel{\mathrm{tw}}{\to }& H_{\mathrm{smooth}}^2(X,U(1))\\ \downarrow & & \downarrow & & \downarrow \\ *& \stackrel{TX}{\to }& \mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}(X,B\mathrm{SO}(n))& \stackrel{W_3}{\to }& \mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}(X,B^{2}U(1)\end{array}.$$\array{ \mathbf{W}_3 Struc_{tw}(T X) &\to& \mathbf{W}_3 Struc_{tw}(X) &\stackrel{tw}{\to}& H_{smooth}^2(X, U(1)) \\ \downarrow && \downarrow && \downarrow \\ * &\stackrel{T X}{\to}& Smooth \infty Grpd(X, \mathbf{B} SO(n)) &\stackrel{\mathbf{W}_3}{\to}& Smooth \infty Grpd(X, \mathbf{B}^2 U(1) } \,.

Applications

Anomaly cancellation in physics

The existence of an ordinary spin structure on a space $X$ is, as discussed there, the condition for $X$ to serve as the target space for the spinning particle sigma-model, in that the existence of this structure is precisely the condition that the corresponding fermionic quantum anomaly on the worldline vanishes.

Twisted ${\mathrm{spin}}^c$-structures appear similarly as the conditions for the analogous quantum anomaly cancellation, but now of the open type II superstring ending on a D-brane. This is also called the Freed-Witten anomaly cancellation.

More precisely, in these applications the class of $W_3(\mathrm{TX})-H$ need not vanish, it only needs to be $n$-torsion if there is moreover a twisted bundle of rank $n$ on the $D$-brane.

See the references below for details.

Related concepts

• (twisted differential) c-structures

  • orientation

  • spin structure, twisted spin structure

    spin^c structure, twisted ${\mathrm{spin}}^c$ structure

  • string structure, differential string structure

  • fivebrane structure, differential fivebrane structure

References

General

The notion of twisted ${\mathrm{Spin}}^c$-structures as such were apparently first discussed in section 5 of

• Christopher Douglas, On the Twisted K-Homology of Simple Lie Groups Topology, Volume 45, Issue 6, November 2006, Pages 955-988 (arXiv:0402082)

More discussion appears in section 3 of

• Bai-Ling Wang, Geometric cycles, index theory and twisted K-homology J. Noncommut. Geom. 2 (2008), no. 4, 497–552 (arXiv:0710.1625)

The refinement to smooth twisted structures is discussed in section 4.1 of

• Urs Schreiber, differential cohomology in a cohesive topos

In physics

The need for twisted ${\mathrm{Spin}}^c$-structures as Freed-Witten anomaly cancellation condition on the worldvolume of D-branes in string theory was first discussed in

• Daniel Freed, Edward Witten, Anomalies in String Theory with D-Branes (arXiv:hep-th/9907189)

More details are in

• Anton Kapustin, D-branes in a topologically nontrivial B-field , Adv. Theor. Math. Phys. 4, no. 1, pp. 127–154 (2000), (arXiv:hep-th/9909089)

A clean formulation and review is provided in

• Loriano Bonora, Fabio Ferrari Ruffino, Raffaele Savelli, Classifying A-field and B-field configurations in the presence of D-branes (arXiv:0810.4291)

• Fabio Ferrari Ruffino, Classifying A-field and B-field configurations in the presence of D-branes - Part II: Stacks of D-branes (arXiv:1104.2798)

• Fabio Ferrari Ruffino, Topics on topology and superstring theory (arXiv:0910.4524)

and

• Kim Laine, Geometric and topological aspects of Type IIB D-branes (arXiv:0912.0460)

In (Laine) the discussion of FW-anomaly cancellation with finite-rank gauge bundles is towards the very end, culminating in equation (3.41).