Higher spin geometry

Group Theory




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

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

of the group of order 2. For Spin(n)GL 1(Cl( n))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}\{+1,-1\}.

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


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

Spin c Spin× 2U(1) =(Spin×U(1))/ 2, \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 2\mathbb{Z}_2 \hookrightarrow \mathbb{Z} and 2U(1)\mathbb{Z}_2 \hookrightarrow U(1).


Group extension


We have a short exact sequence

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

where U(1)Spin cU(1) \to Spin^c is the canonical inclusion into the defining product U(1)Spin×U(1)Spin× 2U(1)U(1) \to Spin \times U(1) \to Spin \times_{\mathbb{Z}_2} U(1).


As the homotopy fiber of the smooth W 3\mathbf{W}_3

We discuss in the following that

  1. the universal third integral Stiefel-Whitney class W 3W_3 has an essentially unique lift from ∞Grpd \simeq Top to Smooth∞Grpd;

  2. the smooth delooping BSpin cSmoothGrpd\mathbf{B}Spin^c \in Smooth\infty Grpd is the homotopy fiber of W 3\mathbf{W}_3, hence is the circle 2-bundle over BSO\mathbf{B} SO classified by W 3\mathbf{W}_3.


We have a homotopy pullback diagram

BSpin c Bdet BU(1) c 1mod2 BSO w 2 B 2 2 \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


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

B() c 1 B(1)=B 2 BU(1), \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 1G 0)(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

B( 2Spin) w 2 B( 21)=B 2 2 BSO. \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 QQ in

Q B() B( 2Spin) B 2 2. \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(Spin×)\mathbf{B}(\mathbb{Z} \stackrel{\partial}{\to} Spin \times \mathbb{R}), where

:n(nmod2,n). \partial\colon n \mapsto ( n \,mod\, 2 , n) \,.

This is equivalent to

(Spin×) ( 2Spin×(/2)) ( 2Spin×U(1)), \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)/(2)U(1) \simeq \mathbb{R}/(2\mathbb{Z}) here, which is important below in prop. 3 for the identification of detdet) where now \partial' is the diagonal embedding of the subgroup

:σ(σ,σ). \partial'\colon \sigma \mapsto (\sigma, \sigma) \,.

This in turn is equivalent to

B(Spin× 2U(1)), \mathbf{B} (Spin \times_{\mathbb{Z}_2} U(1)) \,,

which is def. 1.


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

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

Under the identificaton Spin cSpin× 2U(1)Spin^c \simeq Spin \underset{\mathbb{Z}_2}{\times} U(1) the “universal determinant line bundle map”

det:Spin cU(1) det \colon Spin^c \to U(1)

is given in components by

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

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


By the proof of prop. 2 the U(1)U(1)-factor in Spin cSpin× 2U(1)Spin^c \simeq Spin \underset{\mathbb{Z}_2}{\times}U(1) arises from the identification U(1)/2U(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.


The third integral Stiefel-Whitney class

W 3β 2w 2:BSOw 2B 2 2β 2B 3 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 Π\stackrel{\Pi}{\to} ∞Grpd \stackrel{\simeq}{\to} Top

given by

W 3=β 2w 2:BSO(n)w 2B 2 2β 2B 2U(1), \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 β 2\mathbf{\beta}_2 is simply given by the canonical subgroup embedding.


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 β 2\beta_2 is presented by the ∞-anafunctor

B 2(2) B 2(1)=B 3 B 2 2. \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 β 2\mathbf{\beta}_2 to a comparable \infty-anafunctor. This is accomplished by

B 2(2) β^ 2 B 2(2) B 2 2 β 2 B 2U(1). \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

id 2 2 . \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

|B 2(2)| |β^ 2| |B 2(2)| |B 2(2)| |B 2(1)| |B 2 2| β 2 |B 3|. \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 } \,.

The sequence

BU(1)c 1mod2B 2 2β 2B 2U(1), \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 β 2\mathbf{\beta}_2 is the smoothly refined Bockstein homomorphism from prop. 4, is a fiber sequence.


The homotopy fiber of B 2BU(1)\mathbf{B} \mathbb{Z}_2 \to \mathbf{B}U(1) is U(1)/ 2U(1)U(1)/\mathbb{Z}_2 \simeq U(1). Thinking of this is (1/2)(\mathbb{Z} \stackrel{\cdot 1/2}{\to} \mathbb{R}) one sees that the inclusion of this fiber is indeed c 1mod2\mathbf{c}_1 mod 2.


The delooping BSpin c\mathbf{B}Spin^c of the Lie group Spin cSpin^c in Smooth∞Grpd is the homotopy fiber of the universal third smooth integral Stiefel-Whitney class from 4.

BSpin cBSOW 3B 2U(1), \mathbf{B}Spin^c \to \mathbf{B} SO \stackrel{\mathbf{W}_3}{\to} \mathbf{B}^2 U(1) \,,

Then consider the pasting diagram of homotopy pullbacks

BSpin c BU(1) * c 1mod2 BSO w 2 B 2 2 β 2 B 2U(1). \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. 5. The left square is a homotopy pullback by prop. 2. The bottom composite is the smooth W 3\mathbf{W}_3 by prop 4.

This implies by claim by the pasting law.

Relation to metaplectic group Mp cMp^c

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

Revised on January 2, 2015 23:46:07 by Urs Schreiber (