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We investigate the existence of a state-operator correspondence on the torus. This
correspondence would relate states of the CFT Hilbert space living on a spatial torus
to the path integral over compact Euclidean manifolds with operator insertions. Unlike
the states on the sphere that are associated to local operators, we argue that those
on the torus would more naturally be associated to line operators. We find evidence
that such a correspondence cannot exist and in particular, we argue that no compact
Euclidean path integral can produce the vacuum on the torus. Our arguments come solely
from field theory and formulate a CFT version of the Horowitz-Myers conjecture for
the AdS soliton.