TITLE: Occurrence of semi-spin groups
QUESTION [9 upvotes]: In the classification of simple Lie algebras one has the familiar picture of 4 families, $A_n$, $B_n$, $C_n$ and $D_n$, and 5 exceptional groups, $F_4,$ $G_2,$ $E_6$, $E_7$ and $E_8$.  The $D_n$ family has the unique feature that it contains, among all the corresponding Lie groups, groups whose center is non-cyclic: for $Spin(4n+2)$ the center is $\mathbb{Z}_4$, but one has $$Z(Spin(4n))=\mathbb{Z}_2\times \mathbb{Z}_2.$$  One can take the quotient of $Spin(4n)$ by any of the 3 subgroups of its center isomorphic to $\mathbb{Z}_2$ - one  of these quotients is $SO(4n)$, the other are two are isomorphic to each other and are sometimes referred to as the half-spin or semi-spin groups, denoted by $SSpin(4n)$ (the notation may not be completely standard).  They are a little bit the forgotten Lie groups - simple groups that are neither exceptional nor a quotient or covering of a classical group.
My question is: other than occuring in the classification, are there any places where these half-spin groups show up (naturally so to speak)?

REPLY [7 votes]: $SSpin(32)$ appears in two superficially unrelated places that I know of.

There are two heterotic string theories: the one based on the root lattice of $E_8 \oplus E_8$ and the one based on the weight lattice of $SSpin(32)$.  This is often (but incorrectly!) called the $SO(32)$ heterotic string theory.
Milnor gave the first example of isospectral manifolds which are not isometric: the flat tori obtained by quotienting $\mathbb{R}^{16}$ by the above two lattices.

Another factoid is the following.  Although they are isomorphic, it is convenient for this to distinguish between, say, $SSpin(4n)$ and $S'Spin(4n)$, which are the quotients of $Spin(4n)$ by different $\mathbb{Z}_2$ subgroups of the centre, which are not $SO(4n)$.  Then $SSpin(8n)$ and $S'Spin(8n)$ are self-dual (under Langlands duality), whereas $SSpin(8n+4)$ is the Langlands (a.k.a. magnetic) dual of $S'Spin(8n+4)$.