Comments on a state-operator correspondence for the torus

TL;DR

This paper investigates whether a state-operator correspondence can be extended from the sphere to states on a spatial torus in a conformal field theory.It argues that any putative torus SOC must arise from a compact Euclidean filling, but explicit checks in holographic CFTs and a free boson show the vacuum cannot be prepared by such fillings, and excited torus states may be tied to line operators.The authors relate this nonexistence to a CFT version of the Horowitz–Myers conjecture for the AdS soliton, and they prove necessary conditions tied to conformal Killing vectors which cannot be satisfied by compact caps with torus boundaries.The results place strong constraints on how modular properties of torus partition functions relate to operator data and point to a line-operator-based framework for torus states.

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.

Figures (8)

• Figure 1: The state operator correspondence
• Figure 2: A picture of the potential line operator/state correspondence. The state is prepared on the boundary of the donut: the two-torus. We can path integrate inwards until we reach the center of the donut where the operator is inserted (the blue line).
• Figure 3: The manifold over which we path integrate, with the two circles forming the two-torus on which the state is prepared.
• Figure 4: Ratios $R$ plotted as a function of the ratio of the two circle lengths $\alpha = L_1/L_2$ for (a) a Holographic CFT, \ref{['Rholography']}, and (b) for a free boson where we summed \ref{['Eboson']} up to $l = 200$. The fact that $R$ shoots up near $\alpha=0$ in (b) is a consequence of this finite sum.
• Figure 5: a) Manifold $M = \mathbf{R}_+ \times \mathbf{T}^2$, which prepares the ground state. This manifold has a Killing vector $\xi$ along the $\tau$ direction. The stress tensor $T_{\tau\tau}$ is integrated over the torus (red shaded region). To consider the effect of an operator insertion, we have inserted one at $\tau = \tau_*$. b) We can deform the region of integration of $T_{\tau\tau}$ from $\Sigma_1$ to $\Sigma_2$. We pick up a bulk contribution from integration over the blue region $B$ between $\Sigma_1$ and $\Sigma_2$. This contribution vanishes when there is a conformal Killing vector.