Gravitational dual of averaged free CFT's over the Narain lattice

Abstract: It has been recently argued that the averaging of free CFT's over the Narain lattice can be holographically described through a Chern-Simons theory for $U\left(1\right)^{D}\times U\left(1\right)^{D}$ with a precise prescription to sum over three-dimensional handlebodies. We show that a gravitational dual of these averaged CFT's would be provided by Einstein gravity on AdS$_{3}$ with $U\left(1\right)^{D-1}\times U\left(1\right)^{D-1}$ gauge fields, endowed with a precise set of boundary conditions closely related to the "soft hairy" ones. Gravitational excitations then go along diagonal $SL\left(2,\mathbb{R}\right)$ generators, so that the asymptotic symmetries are spanned by $U\left(1\right)^{D}\times U\left(1\right)^{D}$ currents. The stress-energy tensor can then be geometrically seen as composite of these currents through a twisted Sugawara construction. Our boundary conditions are such that for the reduced phase space, there is a one-to-one map between the configurations in the gravitational and the purely abelian theories. The partition function in the bulk could then also be performed either from a non-abelian Chern-Simons theory for two copies of $SL\left(2,\mathbb{R}\right)\times U\left(1\right)^{D-1}$ generators, or formally through a path integral along the family of allowed configurations for the metric. The new boundary conditions naturally accommodate BTZ black holes, and the microscopic number of states then appears to be manifestly positive and suitably accounted for from the partition function in the bulk. The inclusion of higher spin currents through an extended twisted Sugawara construction in the context of higher spin gravity is also briefly addressed.