The torus operator in holography

We consider the non-local operator ${\mathcal T}$ defined in 2-dimensional CFTs by the path integral over a torus with two punctures. Using the AdS/CFT correspondence, we study the spectrum and ground state of this operator in holographic such CFTs in the limit of large central charge $c$. In one region of moduli space, we argue that the operator retains a finite gap and has a ground state that differs from the CFT vacuum only by order one corrections. In this region the torus operator is much like the cylinder operator. But in another region of moduli space we find a puzzle. Although our ${\mathcal T}$ is of the manifestly positive form $A^\dagger A$, studying the most tractable phases of $\text{Tr}( {\mathcal T}^n)$ suggests that ${\mathcal T}$ has negative eigenvalues. It seems clear that additional phases must become relevant at large $n$, perhaps leading to novel behavior associated with a radically different ground state or a much higher density of states. By studying the action of two such torus operators on the CFT ground state, we also provide evidence that, even at large $n$, the relevant bulk saddles have $t=0$ surfaces with small genus.