  In this work, we extend a 2d topological gravity model of Marolf and Maxfield
to have as its bulk action any open/closed TQFT obeying Atiyah's axioms. The
holographic duals of these topological gravity models are ensembles of 1d
topological theories with random dimension. Specifically, we find that the TQFT
Hilbert space splits into sectors, between which correlators of boundary
observables factorize, and that the corresponding sectors of the boundary
theory have dimensions independently chosen from different Poisson
distributions. As a special case, we study in detail the gravity model built
from the bulk action of 2d Dijkgraaf-Witten theory, with or without
end-of-the-world branes, and for arbitrary finite group $G$. The dual of this
Dijkgraaf-Witten gravity model can be interpreted as a 1d topological theory
whose Hilbert space is a random representation of $G$ and whose aforementioned
sectors are labeled by the irreducible representations of $G$. These
holographic interpretations of our gravity models require projecting out
negative-norm states from the baby universe Hilbert space, which Marolf and
Maxfield achieved by the (only seemingly) ad hoc solution of adding a nonlocal
boundary term to the bulk action. In order to place their solution in the
completely local framework of a TQFT with defects, we couple the boundaries of
the gravity model to an auxiliary 2d TQFT in a non-gravitational (i.e. fixed
topology) region. In this framework, the difficulty of negative-norm states can
be remedied in a local way by the introduction of a defect line between the
gravitational and non-gravitational regions. The gravity model is then
holographically dual to an ensemble of boundary conditions in an open/closed
TQFT without gravity.
