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spin^c structure

Definition

Topological

For $n\in \mathbb{N}$ the Lie group spin^c 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 .

An oriented manifold $X$ has ${\mathrm{Spin}}^c$-structure if the class $[W_3(X)]\in H^3(X,\mathbb{Z})$

$$W_3(X):=W_3(TX):X\stackrel{TX}{\to }B\mathrm{SO}(n)\stackrel{W_3}{\to }{B}^{2}U(1)\simeq K(\mathbb{Z},3)$$W_3(X) := W_3(T X) : X \stackrel{T X}{\to} B SO(n) \stackrel{W_3}{\to} B^2 U(1) \simeq K(\mathbb{Z},3)

is trivial. This is the class of the circle 2-bundle/bundle gerbe that obstructs the existence of a ${\mathrm{Spin}}^c$-principal bundle. A given equipped with ${\mathrm{Spin}}^c$-structure $\eta$ if it is equipped with a choice of trivializaton

$$\eta :1\stackrel{\simeq }{\to }{W}_3(TX).$$\eta : 1 \stackrel{\simeq}{\to} W_3(T X) \,.

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

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

If the class does not vanish and if hence there is no ${\mathrm{Spin}}^c$-structure, it still makes sense to discuss the structure that remains as twisted spin^c structure .

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 $L_{\mathrm{whe}}$ Top $\simeq$ ∞Grpd of discrete ∞-groupoids to that of smooth ∞-groupoids, Smooth∞Grpd.

More in detail, 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.

Accordingly, the 2-groupoid of smooth ${\mathrm{spin}}^c$-structures $W_3\mathrm{Struc}(X)$ is the joint (∞,1)-pullback

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

Higher ${\mathrm{spin}}^c$-structures

In parallel to the existence of higher spin structures there are higher analogs of ${\mathrm{Spin}}^c$-structures, related to quantum anomaly cancellation of theories of higher dimensional branes.

• string^c structure

Properties

Of ${\mathrm{Spin}}^c$

Definition

The group ${\mathrm{Spin}}^c$ is

$$\begin{array}{rl}{\mathrm{Spin}}^c& :=\mathrm{Spin}{\times }_{{\mathbb{Z}}_2}U(1)\\ & =(\mathrm{Spin}\times U(1))/{\mathbb{Z}}_2,\end{array}$$\begin{aligned} Spin^c & := Spin \times_{\mathbb{Z}_2} U(1) \\ & = (Spin \times U(1))/{\mathbb{Z}_2} \,, \end{aligned}

where in the second line the action is the diagonal action induced from the two canonical embeddings of subgroups ${\mathbb{Z}}_2\hookrightarrow \mathbb{Z}$ and ${\mathbb{Z}}_2\hookrightarrow U(1)$.

Proposition

We have a homotopy pullback diagram

$$\begin{array}{ccc}B{\mathrm{Spin}}^c& \to & BU(1)\\ \downarrow & & {\downarrow }^{c_1\mathrm{mod}2}\\ B\mathrm{SO}& \stackrel{w_2}{\to }& B^2{\mathbb{Z}}_2\end{array}.$$\array{ \mathbf{B} Spin^c &\to& \mathbf{B}U(1) \\ \downarrow && \downarrow^{\mathrlap{\mathbf{c}_1 mod 2}} \\ \mathbf{B} SO &\stackrel{w_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 } \,.

Proof

We present this as usual by simplicial presheaves and ∞-anafunctors.

The first Chern class is given by the ∞-anafunctor

$$\begin{array}{ccc}B(\mathbb{Z}\to ℝ)& \stackrel{c_1}{\to }& B(\mathbb{Z}\to 1)=B^2\mathbb{Z}\\ {\downarrow }^{\simeq }\\ BU(1)\end{array}.$$\array{ \mathbf{B}(\mathbb{Z} \to \mathbb{R}) &\stackrel{c_1}{\to}& \mathbf{B}(\mathbb{Z} \to 1) = \mathbf{B}^2 \mathbb{Z} \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B} U(1) } \,.

The second Stiefel-Whitney class is given by

$$\begin{array}{ccc}B({\mathbb{Z}}_2\to \mathrm{Spin})& \stackrel{w_2}{\to }& B({\mathbb{Z}}_2\to 1)=B^2{\mathbb{Z}}_2\\ {\downarrow }^{\simeq }\\ B\mathrm{SO}\end{array}.$$\array{ \mathbf{B}(\mathbb{Z}_2 \to Spin) &\stackrel{w_2}{\to}& \mathbf{B}(\mathbb{Z}_2 \to 1) = \mathbf{B}^2 \mathbb{Z}_2 \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B} SO } \,.

Notice that the top horizontal morphism here is a fibration.

Therefore the homotopy pullback in question is given by the ordinary pullback

$$\begin{array}{ccc}Q& \to & B(\mathbb{Z}\to ℝ)\\ \downarrow & & \downarrow \\ B({\mathbb{Z}}_2\to \mathrm{Spin})& \to & B^2{\mathbb{Z}}_2\end{array}.$$\array{ Q &\to& \mathbf{B}(\mathbb{Z} \to \mathbb{R}) \\ \downarrow && \downarrow \\ \mathbf{B}(\mathbb{Z}_2 \to Spin) &\to& \mathbf{B}^2 \mathbb{Z}_2 } \,.

This pullback is $B(\mathbb{Z}\stackrel{\partial }{\to }\mathrm{Spin}\times ℝ)$, where

$$\partial :n\mapsto (n\mathrm{mod}2,n).$$\partial : n \mapsto ( n mod 2 , n) \,.

This is equivalent to

$$B({\mathbb{Z}}_2\stackrel{\partial \prime }{\to }\mathrm{Spin}\times U(1))$$\mathbf{B}(\mathbb{Z}_2 \stackrel{\partial'}{\to} Spin \times U(1))

where now

$$\partial \prime :\sigma \mapsto (\sigma ,\sigma ).$$\partial' : \sigma \mapsto (\sigma, \sigma) \,.

This in turn is equivalent to

$$B(\mathrm{Spin}{\times }_{{\mathbb{Z}}_2}U(1)),$$\mathbf{B} (Spin \times_{\mathbb{Z}_2} U(1)) \,,

which is the original definition.

Examples

From almost complex structures

An almost complex structure canonically induces a ${\mathrm{Spin}}^c$-structure:

Proposition

For all $n\in \mathbb{N}$ we have a homotopy-commuting diagram

$$\begin{array}{ccc}BU(n)& \to & BU(1)\\ \downarrow & & {\downarrow }^{c_1\mathrm{mod}2}\\ B\mathrm{SO}(2n)& \stackrel{w_2}{\to }& B^2{\mathbb{Z}}_2\end{array},$$\array{ \mathbf{B}U(n) &\to& \mathbf{B}U(1) \\ \downarrow && \downarrow^{\mathrlap{\mathbf{c_1} mod 2}} \\ \mathbf{B}SO(2n) &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 } \,,

where the vertical morphism is the canonical morphism induced from the identification of real vector spaces $ℂ\to {ℝ}^2$, and where the top morphism is the canonical projection $BU(n)\to BU(1)$ (induced from $U(n)$ being the semidirect product group $U(n)\simeq \mathrm{SU}(n)⋊U(1)$).

Proof

By the general relation between $c_1$ of an almost complex structure and $w_2$ of the underlying orthogonal structure, discussed at Stiefel-Whitney class – Relation to Chern classes.

By prop. 1 this induces a canonical morphism

$$k:BU(n)\to B{\mathrm{Spin}}^c$$k \colon \mathbf{B}U(n) \to \mathbf{B}Spin^c

and this is the universal morphism from almost complex structures:

For $c:X\to BU(n)$ modulating an almost complex structure/complex vector bundle over $X$, the composite

$$kc:X\stackrel{c}{\to }BU(n)\stackrel{k}{\to }B{\mathrm{Spin}}^c$$k c \colon X \stackrel{c}{\to} \mathbf{B}U(n) \stackrel{k}{\to} \mathbf{B}Spin^c

is the corresponding ${\mathrm{Spin}}^c$-structure.

Related concepts

• spin^c

• (twisted, differential) c-structures

  • orientation

  • spin structure, twisted spin structure, differential spin structure

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

    K-orientation

    Spin^c Dirac operator

  • string structure, twisted differential string structure,

  • string^c 2-group, string^c structure

  • fivebrane structure, twisted differential fivebrane structure

• spin^c quantization

References

General

A canonical textbook reference is

• H.B. Lawson and M.-L. Michelson, Spin Geometry , Princeton University Press, Princeton, NJ, (1989)

Other accounts include

• Blake Mellor, ${\mathrm{Spin}}^c$-manifolds (pdf)

• Stable complex and ${\mathrm{Spin}}^c$-structures (pdf)

As anomaly cancellation in type II string theory

That the $U(1)$-gauge field on a D-brane in type II string theory in the absense of a B-field is rather to be regarded as part of a ${\mathrm{spin}}^c$-structure was maybe first observed in

• Edward Witten, Baryons and Branes In Anti de Sitter Space, JHEP 9807:006 (1998) (arXiv:hep-th/9805112).

The twisted spin^c structure (see there for more details) on the worldvolume of D-branes in the presence of a nontrivial B-field was discussed in

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

A more recent review is provided in

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

See also

• Hisham Sati, Geometry of $\mathrm{Spin}$ and ${\mathrm{Spin}}^c$ structures in the M-theory partition function (arXiv:1005.1700)