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.

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    $\begingroup$ The "systolic geometry of flat tori" is exactly the same a the "geometry of numbers" and you will find the info you want more readily if you look under that heading. I like the book by Gruber and Lekkerkerker. Try that as starting point. $\endgroup$ – alvarezpaiva Aug 21 '17 at 7:47

The moduli space of $n$-tori (of volume 1) is the locally symmetric space (orbifold) $SO(n)\backslash SL(n,\mathbb{R})/SL(n,\mathbb{Z})$. The symmetric space is also the space of positive-definite symmetric matrices of determinant 1, and $SL(n,\mathbb{Z})$ acts on these as the symmetric square representation (i.e. by reparameterizing quadratic forms). Various things are known about its spine (the well-rounded retract), and the systole is known to be a topological Morse function.

But really when you are asking about systoles of tori, you are asking about lattice sphere packings, of which much is known. Given a systole of length $l$, there exists a ball at any point in the torus whose radius is $l/2$, and whose interior is embedded, and conversely. Hence lattices with maximal sphere packings achieve the maximal systole for volume 1. In a given dimension, this is equivalent to determining Hermite's constant. The optimal lattice packings are known in dimensions $\leq 8$ and $24$. Check out the encyclopedic

Conway, J.H.; Sloane, N.J.A., Sphere packings, lattices and groups. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov., Grundlehren der Mathematischen Wissenschaften. 290. New York, NY: Springer. lxxiv, 703 p. (1999). ZBL0915.52003.

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Gruber and Lekkerkerker is out of date. Some of the best work on the subject is by

Banaszczyk, W. Inequalities for convex bodies and polar reciprocal lattices in R^n. II. Application of K -convexity. Discrete Comput. Geom. 16 (1996), no. 3, 305-311.

or for a more elementary treatment

Bangert, Victor; Katz, Mikhail. Stable systolic inequalities and cohomology products. Dedicated to the memory of Jürgen K. Moser. Comm. Pure Appl. Math. 56 (2003), no. 7, 979-997.

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