Conical spaces, modular invariance and $c_{p,1}$ holography

We propose a non-unitary example of holography for the family of two-dimensional logarithmic conformal field theories with negative central charge $c= c_{p,1} = - 6p +13 - 6 p^{-1}$. We argue that at large $p$, these models have a semiclassical gravity-like description which contains, besides the global AdS$_3$ spacetime, a tower of solitonic solutions describing conical excess angles. Evidence comes from the fact that the central charge and the natural modular invariant partition function of such a theory coincide with those of the $c_{p,1}$ model. These theories have an extended chiral W-algebra whose currents have large spin of order $|c|$, and which in the bulk are realized as spinning conical solutions. As a by-product we also find a direct link between geometric actions for exceptional Virasoro coadjoint orbits, which describe fluctuations around the conical spaces, and Felder's free field construction of degenerate representations.