IIRC
The elves were using the sword as the core of a barrier that protected their village, or something like that. The problem is, the village isn't on a magic leyline like Eizo's house is, so the magic in the sword slowly ran out, until it shattered from the shield being too much strain on it. So Lidy took the shards out into the world to find a smith that could reforge the blade and imbue it with magic again, and saw the rapier he made with a lot of magic in it, and so set out to find Eizo to fix the sword so they can get the barrier going again.
I might be off, it's been a bit since I read the LN.