Contents

spin geometry

string geometry

group theory

# Contents

## 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 & \coloneqq Spin \times_{\mathbb{Z}_2} U(1) \\ & = (Spin \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 $\mathbb{Z}_2 \hookrightarrow \mathbb{Z}$ and $\mathbb{Z}_2 \hookrightarrow U(1)$.

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).

## 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 $U(1) \to Spin \times U(1) \to Spin \times_{\mathbb{Z}_2} U(1)$.

### 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

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

in Smooth∞Grpd, where

###### 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).

## References

• Robert Gompf, $Spin^c$ structures and homotopy equivalences, Geom. Topol. 1 (1997) 41-50 (arXiv:math/9705218)