Comments on a state-operator correspondence for the torus

TL;DR

This paper examines the possibility of a state-operator correspondence on the torus in conformal field theories, arguing that such a correspondence likely does not exist and providing evidence from field theory.

Contribution

It presents a novel argument against the existence of a torus state-operator correspondence, extending the Horowitz-Myers conjecture to CFTs.

Findings

No compact Euclidean path integral can produce the vacuum on the torus.

States on the torus are more naturally associated with line operators.

The paper formulates a CFT version of the Horowitz-Myers conjecture for the AdS soliton.

Abstract

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.