Apologies if this is too basic, but I haven't been able to find any info about the question. Is there anything known about the moduli space of flat metrics on an $n$-torus (i.e., $(S^1)^n$)? Information about the moduli space of lattices in families of Lie groups other than $\mathbb{R}^n$ would be of interest as well. (But I want to stress I am curious about families of Lie groups with the dimension going to infinity rather than a fixed-dimensional example.)

Ultimately, I'm interested in any information about the systoles of flat tori (i.e., the length of the shortest non-contractible loop). I do know the Gromov systolic inequality but was wondering what more is known (esp. since the Gromov inequality is straightforward in the flat case.)

I am asking (and the question has the `computational-complexity`

tag) because I've been reading about the "shortest vector problem"*, and it struck me that it is equivalent to finding the systole of a flat torus, and so I was curious about any understanding of the problem from this perspective. (I guess this equivalence should predict that there can't be **too** much understanding of flat metrics on tori since the shortest vector problem is NP hard.)

*The shortest vector problem is: Given the generators of a Euclidean lattice, find the shortest (WRT the Euclidean norm say) element in the lattice.