The group $O(n,n,\mathbb{Z})$ and its properties.

Question

While reading "String Theory" by Becker$^2$, Schwarz I encountered a group called $O(n,m,\mathbb{Z})$, which is a group of symmetries of the bosonic/heterotic strings compactified on a torus. I would like to learn some more details about this group, like its relations to other groups and isomorphisms. Where can I do that?

Answer 1

In general, there is a large literature on integral orthogonal groups, which are a special case of arithmetic groups; a good start is perhaps this MO-question, which discusses algorithms to compute such groups, but also gives more background, e.g., to quadratic forms over the integers. Concerning isomorphism theory the following article might be helpful: Isomorphism theory for orthogonal groups over arbitrary integral domains.