# Torus wormhole In [[0002 3D gravity|3d gravity]], the path integral of the torus wormhole (aka Cotler-Jensen wormhole) is computed by Cotler and Jensen to be$Z_{\mathbb{T}^2 \times I}\left(\tau_1, \tau_2\right)=\frac{1}{2 \pi^2} Z_0\left(\tau_1\right) Z_0\left(\tau_2\right) \sum_{\gamma \in P S L(2 ; \mathbf{Z})} \frac{\operatorname{Im}\left(\tau_1\right) \operatorname{Im}\left(\gamma \tau_2\right)}{\left|\tau_1+\gamma \tau_2\right|^2},$where$\quad Z_0(\tau)=\frac{1}{\sqrt{\operatorname{Im}(\tau)}|\eta(\tau)|^2}$is the partition function of a non-compact boson. Here $\eta(\tau)$ is the Dedekind delta function. This is only one contribution to the boundary quantity we are computing: other topologies contribute too. At low temperature and fixed large spin, this is the dominant contribution, following from the work of [[2019#Ghosh, Maxfield, Turiaci]] that 3d gravity has an affective [[0050 JT gravity|JT gravity]] description in this limit (so a genus expansion makes sense). ## Derivation - constrain-first approach: not guaranteed to work unless constraining commutes with quantising, which turns out to be the case here ## Comments The Cotler-Jensen computation gives a finite answer whereas [[0596 Virasoro TQFT|Virasoro TQFT]] gives a divergent answer for the path integral on this wormhole. The difference arises from the choice of the measure: Cotler-Jensen uses $\int d\Delta\sim \int PdP$ while VTQFT has $\int dP$. ## Refs - [[2020#Cotler, Jensen (Jun)]]: original computation - [[2020#Cotler, Jensen (Jul)]]: Virasoro symmetry and modular invariance fix the answer up to an overall factor, which is then fixed from the JT limit