In the Norse mythology, the Norns writing the fate of the gods, but what are they writing it on?
For example, the Greek Fates weave destiny on a loom.
If so, Does it have a name?
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If you mean some sort of book, there doesn't appear to be anything of the kind. However...
In the poem Voluspa, the Norns do write, or at least carve on wood (verse 20).
From there came three maidens, knowing much
three from the lake that stands under the tree
Destiny they called one, Becoming the second -
they carved on wood tablets - Shall-be the third;
laws they laid down, lives they chose
for the children of mankind, the fates of men.
It varies with the translation, but Larrington has "carved on wooden slips", Hollander has "they scores did cut" and the version at Sacred Texts has "wood they scored". So it's probaby closer to runic inscriptions or some form of divination.
In the later Prose Edda, which follows the earlier version pretty closely, Snorri Sturluson only says that the Norns "shape" or "determine" fate, without stating how. So it may be that there were different ideas about exactly how the Norns laid down fate.
If this is something that really interests you, you can find Karen Bek-Pedersen's book on the Norns at Questia.