What's the Matter with 3D Gravity?

Abstract

We revisit the problem of minimally coupling matter to Einstein gravity in three dimensions with negative cosmological constant. By working in the worldline formalism, we construct a classical phase space on an initial time surface $Σ$, which we quantize using geometric quantization. States in the Hilbert space correspond to Virasoro conformal blocks with operators of conformal weight $h<c/24$. As an application of our formalism, we compute the partition function on thermal $\text{AdS}_3$ through equivariant localization. Our answer reproduces the AdS$_3$ Wilson spool and agrees with the known one-loop result. It further serves as a conjecture for the value of the path integral of gravity minimally coupled to a massive scalar field in thermal $\text{AdS}_3$ to all orders in $G_N$.