nLab spinᶜ

Definition

Remark

By definition of the spin group $Spin(n)$ there is a canonical inclusion

$$\mathbb{Z}/2\mathbb{Z}\hookrightarrow Spin(n)$$

of the group of order 2. For $Spin(n)\hookrightarrow GL_1(Cl(\mathbb{R}^n))$ canonically realized by even Clifford algebra elements of unit norm, this is given by the inclusion of $\{+1,-1\}$.

We frequently write $\mathbb{Z}_2$ as shorthand for $\mathbb{Z}/2\mathbb{Z}$.

Definition

For $n \in \mathbb{N}$, the Lie group $Spin^c(n)$ is the quotient group

$$\begin{aligned} Spin^c(n) & \coloneqq Spin(n) \times_{\mathbb{Z}_2} U(1) \\ & = (Spin(n) \times U(1))/{\mathbb{Z}_2} \,, \end{aligned}$$

of the product of the spin group with the circle group by the common sub-group of order 2 $\mathbb{Z}_2 \hookrightarrow Spin$ and $\mathbb{Z}_2 \hookrightarrow U(1)$ (i.e.: the central product group).

Usually the only the case $n \geq 3$ is considered.

Some authors (e.g. Gompf 97, p. 2) denote this as

$$\begin{aligned} Spin^c(n) & \coloneqq Spin(n)\cdot Spin(2) \\ & \simeq Spin(n) \cdot U(1) \end{aligned}$$

following the notation Sp(n).Sp(1) (see there).

Examples

• For $n=3$, the exceptional isomorphism between Spin(3) and SU(2) extends to an isomorphism between $Spin^c(3)$ and the unitary group $U(2)$:

  $$Spin^c(3) \;\simeq\; U(2)$$

  over the exceptional isomorphism $SO(3) \simeq PU(2)$, as both of these quotient groups are quotients by the respective centers, both identifiable with the circle group $U(1)$) (Ozbagci–Stipsicz 2004 Section 6.2). This isomorphism follows from considering the surjective homomorphism $SU(2) \times U(1) \to U(2)$ given by $(A,z) \mapsto z A$, and noticing its kernel is precisely $\{\pm(I,1)\}$, hence $U(2)$ satisfies the universal property defining $Spin^c(3)$ as a quotient.

• For $n=4$, we have

  $$Spin^c(4) = \big(SU(2)\times SU(2)\times U(1)\big) / \{\pm(I,I,1)\} \simeq U(2)\times_{U(1)} U(2).$$

  This latter group is the fibre product of groups over $U(1)$, namely $\{(A,B)\in U(2)\times U(2)\mid \det(A) = \det(B)\}$ (Ozbagci–Stipsicz 2004 Section 6.3). The construction can be seen by considering the surjective homomorphism

  $$SU(2) \times SU(2) \times U(1) \to SU(2) \times SU(2) \times U(1) \times U(2) \to U(2) \times U(2)$$

  defined by $(A,B,z) \mapsto (z A,z B)$, which has image $U(2)\times_{U(1)} U(2)$. Similarly to the case of $Spin^c(3)$, the kernel consists of triples $(A,B,z)$ such that $z A = I = z B$, hence that $A = B = (1/z) I$. Since $\det(A) = 1/z^2 = 1$, we must have $z=\pm 1$, and hence $A=B=\pm I$ with the same sign as $z$. Thus the kernel is precisely $\{\pm(I,I,1)\}$, and so again by the universal property we get the isomorphism as stated.

  The $Spin(4)$ subgroup can be seen as the subgroup of pairs of unitary matrices with both of them having determinant 1.

• Using the exceptional isomorphism $Spin(6) \simeq SU(4)$, and the multiplication map $SU(4)\times U(1) \to U(4)$ analogous to the above, it can be seen that the group $Spin^c(6)$ is the connected double cover of $U(4)$ corresponding to the (unique) index-2 subgroup $2\mathbb{Z} \hookrightarrow \mathbb{Z} \simeq \pi_1(U(4))$. This is because the multiplication map is surjective and has kernel canonically isomorphic to $\mu_4$, the fourth roots of unity, via the map $\mu_4\to SU(4)\times U(1)$ sending $\zeta\mapsto (\zeta^{-1}I,\zeta)$. Hence there are a pair of 2:1 surjective homomorphisms

  $$SU(4) \times U(1) \to Spin^c(6) \to U(4),$$

  and hence the result.

The last two examples can also be considered as exceptional isomorphisms, even if not to one of the more classical Lie groups.

Properties

Group extension

Proposition

We have a short exact sequence

$$U(1) \to Spin^c \to SO \,,$$

where $U(1) \to Spin^c$ is the canonical inclusion into the defining product $U(1) \to Spin \times U(1) \to Spin \times_{\mathbb{Z}_2} U(1)$.

General

As the homotopy fiber of the smooth $\mathbf{W}_3$

We discuss in the following that

1. the universal third integral Stiefel-Whitney class $W_3$ has an essentially unique lift from ∞Grpd $\simeq$ Top to Smooth∞Grpd;

2. the smooth delooping $\mathbf{B}Spin^c \in Smooth\infty Grpd$ is the homotopy fiber of $\mathbf{W}_3$, hence is the circle 2-bundle over $\mathbf{B} SO$ classified by $\mathbf{W}_3$.

Proposition

We have a homotopy pullback diagram

$$\array{ \mathbf{B} Spin^c &\stackrel{\mathbf{B}det}{\longrightarrow}& \mathbf{B}U(1) \\ \big\downarrow && \big\downarrow{}^{\mathrlap{\mathbf{c}_1 mod 2}} \\ \mathbf{B} SO &\stackrel{\mathbf{w}_2}{\longrightarrow}& \mathbf{B}^2 \mathbb{Z}_2 }$$

in Smooth∞Grpd, where

• the right morphism is the universal first Chern class modulo 2;

• the bottom morphism is the universal second Stiefel-Whitney class.

Proof

We present the sitation as usual in the projective model structure on simplicial presheaves over CartSp by ∞-anafunctors.

The first Chern class is given by the ∞-anafunctor

$$\array{ \mathbf{B}(\mathbb{Z} \to \mathbb{R}) &\stackrel{\mathbf{c}_1}{\to}& \mathbf{B}(\mathbb{Z} \to 1) = \mathbf{B}^2 \mathbb{Z} \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B} U(1) } \,,$$

where $(G_1 \to G_0)$ denotes a presentation of a strict 2-group by a crossed module.

The second Stiefel-Whitney class is given by

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

Notice that the top horizontal morphism here is a fibration.

Therefore the homotopy pullback in question is (as discussed there) given by the ordinary pullback $Q$ in

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

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

$$\partial\colon n \mapsto ( n \,mod\, 2 , n) \,.$$

This is equivalent to

$$\begin{aligned} (\mathbb{Z} \stackrel{\partial}{\to} Spin \times \mathbb{R}) & \simeq (\mathbb{Z}_2 \stackrel{\partial'}{\to} Spin \times (\mathbb{R}/2\mathbb{Z})) \\ & \simeq (\mathbb{Z}_2 \stackrel{\partial'}{\to} Spin \times U(1)) \end{aligned} \,,$$

(notice the non-standard identification $U(1) \simeq \mathbb{R}/(2\mathbb{Z})$ here, which is important below in prop. for the identification of $det$) where now $\partial'$ is the diagonal embedding of the subgroup

$$\partial'\colon \sigma \mapsto (\sigma, \sigma) \,.$$

This in turn is equivalent to

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

which is def. .

Remark

Compare this with the similar but different homotopy pullback that defines the spin group

$$\array{ \mathbf{B}Spin &\to& * \\ \downarrow && \downarrow \\ \mathbf{B}SO &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 }$$

Proposition

Under the identificaton $Spin^c \simeq Spin \underset{\mathbb{Z}_2}{\times} U(1)$ the “universal determinant line bundle map”

$$det \colon Spin^c \to U(1)$$

is given in components by

$$(g,c) \mapsto 2 c$$

(where on the right we write the group structure additively).

Proof

By the proof of prop. the $U(1)$-factor in $Spin^c \simeq Spin \underset{\mathbb{Z}_2}{\times}U(1)$ arises from the identification $U(1) \simeq \mathbb{R}/2\mathbb{Z}$. But under the horizontal map as it appears in the homotopy pullback in that proof this corresponds to multiplication by 2.

Proposition

The third integral Stiefel-Whitney class

$$W_3 \coloneqq \beta_2 \circ w_2 \colon B SO \stackrel{w_2}{\to} B^2 \mathbb{Z}_2 \stackrel{\beta_2}{\to} B^3 \mathbb{Z}$$

has an essentially unique lift through geometric realization ${\vert-\vert}\colon$ Smooth∞Grpd $\stackrel{\Pi}{\to}$ ∞Grpd $\stackrel{\simeq}{\to}$ Top

given by

$$\mathbf{W}_3 = \mathbf{\beta}_2 \circ \mathbf{w}_2 \colon \mathbf{B} SO(n) \stackrel{w_2}{\to} \mathbf{B}^2 \mathbb{Z}_2 \stackrel{\mathbf{\beta}_2}{\to} \mathbf{B}^2 U(1) \,,$$

where $\mathbf{\beta}_2$ is simply given by the canonical subgroup embedding.

Proof

Once we establish that this is a lift at all, the essential uniqueness follows from the respective theorem at smooth ∞-groupoid – structures.

The ordinary Bockstein homomorphism $\beta_2$ is presented by the ∞-anafunctor

$$\array{ \mathbf{B}^2(\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z}) &\to& \mathbf{B}^2 (\mathbb{Z} \to 1) = \mathbf{B}^3 \mathbb{Z} \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}^2 \mathbb{Z}_2 } \,.$$

Accordingly we need to lift the canonical presentation of $\mathbf{\beta}_2$ to a comparable $\infty$-anafunctor. This is accomplished by

$$\array{ \mathbf{B}^2(\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z}) &\stackrel{\hat \mathbf{\beta}_2}{\to}& \mathbf{B}^2 (\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{R}) \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}^2 \mathbb{Z}_2 &\stackrel{\mathbf{\beta}_2}{\to}& \mathbf{B}^2 U(1) } \,.$$

Here the top horizontal morphism is induced from the morphism of crossed modules that is given by the commuting diagram

$$\array{ \mathbb{Z} &\stackrel{id}{\to}& \mathbb{Z} \\ \downarrow^{\mathrlap{\cdot 2}} && \downarrow^{\mathrlap{\cdot 2}} \\ \mathbb{Z} &\stackrel{}{\hookrightarrow}& \mathbb{R} } \,.$$

Since $\mathbb{R}$ is contractible, we have indeed under geometric realization an equivalence

$$\array{ \vert\mathbf{B}^2(\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z})\vert &\stackrel{\vert \hat {\mathbf{\beta}}_2\vert}{\to}& \vert \mathbf{B}^2 (\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{R}) \vert \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ \vert\mathbf{B}^2(\mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z})\vert &\to& \vert\mathbf{B}^2(\mathbb{Z} \to 1)\vert \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ \vert B^2 \mathbb{Z}_2\vert & \stackrel{\beta_2}{\to}& \vert B^3 \mathbb{Z}\vert } \,.$$

Proposition

The sequence

$$\mathbf{B} U(1) \stackrel{\mathbf{c}_1 mod 2}{\to} \mathbf{B}^2 \mathbb{Z}_2 \stackrel{\mathbf{\beta}_2}{\to} \mathbf{B}^2 U(1) \,,$$

where $\mathbf{\beta}_2$ is the smoothly refined Bockstein homomorphism from prop. , is a fiber sequence.

Proof

The homotopy fiber of $\mathbf{B} \mathbb{Z}_2 \to \mathbf{B}U(1)$ is $U(1)/\mathbb{Z}_2 \simeq U(1)$. Thinking of this is $(\mathbb{Z} \stackrel{\cdot 1/2}{\to} \mathbb{R})$ one sees that the inclusion of this fiber is indeed $\mathbf{c}_1 mod 2$.

Proposition

The delooping $\mathbf{B}Spin^c$ of the Lie group $Spin^c$ in Smooth∞Grpd is the homotopy fiber of the universal third smooth integral Stiefel-Whitney class from .

$$\mathbf{B}Spin^c \to \mathbf{B} SO \stackrel{\mathbf{W}_3}{\to} \mathbf{B}^2 U(1) \,,$$

Proof

Then consider the pasting diagram of homotopy pullbacks

$$\array{ \mathbf{B}Spin^c &\to& \mathbf{B} U(1) &\to& {*} \\ \downarrow && \downarrow^{\mathrlap{\mathbf{c}_1 mod 2}} && \downarrow \\ \mathbf{B} SO &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 &\stackrel{\mathbf{\beta}_2}{\to}& \mathbf{B}^2 U(1) } \,.$$

The right square is a homotopy pullback by prop. . The left square is a homotopy pullback by prop. . The bottom composite is the smooth $\mathbf{W}_3$ by prop .

This implies by claim by the pasting law.

Relation to metaplectic group $Mp^c$

There is a direct analogy between Spin, Spin^c and the metaplectic groups Mp and Mp^c (see there for more).

Related concepts

• spin group, $spin^c$-group, spinʰ group

• spin structure

• MSpinᶜ

• spinᶜ structure, twisted spinᶜ structure

• spinʰ structure

• string 2-group, string structure

• stringᶜ 2-group

References

• H. Blaine Lawson, Marie-Louise Michelsohn, Appendix D in: Spin geometry, Princeton University Press (1989)

For more see the references at spin^c structure.

The exceptional isomorphisms in low dimension are described in

• B. Ozbagci, A. I. Stipsicz, $Spin^c$ Structures on 3- and 4-Manifolds, in: Surgery on Contact 3-Manifolds and Stein Surfaces Bolyai Society Mathematical Studies 13, Springer (2004) [doi:10.1007/978-3-662-10167-4_6]