# Troost-Bourget identity $N \sum_{d|N} 1 = \sum_{d| N} \sum_{l=1}^d \mathrm{gcd}(d,l)$

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In the process of evaluating a "supersymmetric index", Bourget and Troost establish a rather elementary identity:

$$\frac{N}{m} \sum_{d| N} \sum_{l=1}^{\mathrm{gcd}(d,m)} \mathrm{gcd}\left[ \mathrm{gcd}(d,m), n + \frac{ld}{\mathrm{gcd}(d,m)} \right]$$

but later they compute this same supersymmetric index to be another formula:

$$N \sum_{d|N} \mathrm{gcd}\left[ d, m, \frac{N}{d}, \frac{N}{m}, n \right]$$

and the finally they count it come other way and get yet another formula:

$$\sum_{d|N} \sum_{t = 0}^{d-1} \mathrm{gcd}\left[ N \frac{d}{m}, N \frac{m}{d}, N\left( \frac{t}{m} + \frac{n}{d} \right) \right]$$

These formulae should be equivalent for any $N, m, n$ with $m$ dividing $n$... (and possibly other hypotheses missing) Is there a conceptual proof this result?

As a special case they show:

$$N \sum_{d|N} 1 = \sum_{d| N} \sum_{l=1}^d \mathrm{gcd}(d,l)$$

The supersymmetrc index counts just about everthing in hep-th - what could it be counting here?

I can venture a guess these have something to do with the Lie groups they mention:

$$(SU(N)/\mathbb{Z}_m)_n$$

where the meaning of the $n$ is unclear ( the paper says "dionic tilt"). In another section the Smith normal form is mentioned:

$$\frac{\mathbb{Z}}{L \mathbb{Z}} \simeq \bigoplus_{i=1}^n \frac{\mathbb{Z}}{e_i \mathbb{Z}}$$

This looks quite like the chinese remainder theorem

This post imported from StackExchange MathOverflow at 2016-06-23 20:58 (UTC), posted by SE-user john mangual
asked Jun 19, 2016
retagged Jun 23, 2016
You have called your first display an identity, but identities have equals signs.

This post imported from StackExchange MathOverflow at 2016-06-23 20:58 (UTC), posted by SE-user Gerry Myerson
Also, if there is truly no restriction on $N,m,n$, then those formulas involve the gcds of numbers that aren't integers. How is that meant to be understood?

This post imported from StackExchange MathOverflow at 2016-06-23 20:58 (UTC), posted by SE-user Gerry Myerson
@GerryMyerson having re-read the paper I missed some hypotheses; the authors clearly have something in mind but it doesn't quite come through in his paper

This post imported from StackExchange MathOverflow at 2016-06-23 20:58 (UTC), posted by SE-user john mangual
OK, but it's kinda hard to come up with a conceptual proof of a theorem that is missing some hypotheses.

This post imported from StackExchange MathOverflow at 2016-06-23 20:58 (UTC), posted by SE-user Gerry Myerson
@GerryMyerson the original paper itself is not terribly clear -- take a look 3.7, 3.15, 3.24 - arxiv.org/abs/1606.01022 - don't mind the physics jargon it's pretty irrelevant

This post imported from StackExchange MathOverflow at 2016-06-23 20:58 (UTC), posted by SE-user john mangual
The special case from the heading has a straightforward proof.

This post imported from StackExchange MathOverflow at 2016-06-23 20:58 (UTC), posted by SE-user Peter Mueller

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