spin geometry, string geometry, fivebrane geometry …
rotation groups in low dimensions:
see also
By definition of the spin group $Spin(n)$ there is a canonical inclusion
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 $\mathbb{Z}/2\mathbb{Z}$.
For $n \in \mathbb{N}$, the Lie group $Spin^c(n)$ is the quotient group
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
following the notation Sp(n).Sp(1) (see there).
For $n=3$, the exceptional isomorphism between Spin(3) and SU(2) extends to an isomorphism between $Spin^c(3)$ and the unitary group $U(2)$:
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
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
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
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.
We have a short exact sequence
where $U(1) \to Spin^c$ is the canonical inclusion into the defining product $U(1) \to Spin \times U(1) \to Spin \times_{\mathbb{Z}_2} U(1)$.
We discuss in the following that
the universal third integral Stiefel-Whitney class $W_3$ has an essentially unique lift from ∞Grpd $\simeq$ Top to Smooth∞Grpd;
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 $\mathbf{W}_3$.
We have a homotopy pullback diagram
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.
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
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
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
This pullback is $\mathbf{B}(\mathbb{Z} \stackrel{\partial}{\to} Spin \times \mathbb{R})$, where
This is equivalent to
(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
This in turn is equivalent to
Compare this with the similar but different homotopy pullback that defines the spin group
Under the identificaton $Spin^c \simeq Spin \underset{\mathbb{Z}_2}{\times} U(1)$ the “universal determinant line bundle map”
is given in components by
(where on the right we write the group structure additively).
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.
The third integral Stiefel-Whitney class
has an essentially unique lift through geometric realization ${\vert-\vert}\colon$ Smooth∞Grpd $\stackrel{\Pi}{\to}$ ∞Grpd $\stackrel{\simeq}{\to}$ Top
given by
where $\mathbf{\beta}_2$ 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 $\beta_2$ is presented by the ∞-anafunctor
Accordingly we need to lift the canonical presentation of $\mathbf{\beta}_2$ to a comparable $\infty$-anafunctor. This is accomplished by
Here the top horizontal morphism is induced from the morphism of crossed modules that is given by the commuting diagram
Since $\mathbb{R}$ is contractible, we have indeed under geometric realization an equivalence
The sequence
where $\mathbf{\beta}_2$ is the smoothly refined Bockstein homomorphism from prop. , is a fiber sequence.
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 $\mathbf{c}_1 mod 2$.
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 .
Then consider the pasting diagram of homotopy pullbacks
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.
There is a direct analogy between Spin, Spin^c and the metaplectic groups Mp and Mp^c (see there for more).
spin group, $spin^c$-group, spin^h group
For more see the references at spin^c structure.
The exceptional isomorphisms in low dimension are described in
Last revised on September 17, 2023 at 11:16:57. See the history of this page for a list of all contributions to it.