Is the number of string vacua finite?
In plain words
String theory offers many ways to curl up extra dimensions. Whether the number of distinct low-energy universes is finite is unknown.
Precise statement
Determine whether the set of string compactifications to 4D with a bounded number of light fields and effective theories valid below a fixed cutoff is finite, including whether the number of topologically distinct Calabi-Yau threefolds is finite.
What would settle it
A finiteness theorem for all compactifications below a fixed cutoff, or an infinite family of distinct consistent vacua.
Status in the literature
Finiteness of tadpole-bounded flux vacua on a fixed Calabi-Yau orientifold was proven with tame geometry (Bakker, Grimm, Schnell, Tsimerman 2021); finiteness of Calabi-Yau threefold topologies remains open.
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