  We extend a 2d topological model of the gravitational path integral to
include sums over spin structure, corresponding to Neveu-Schwarz (NS) or Ramond
(R) boundary conditions for fermions. The Euclidean path integral vanishes when
the number of R boundaries is odd. This path integral corresponds to a
correlator of boundary creation operators on a non-trivial baby universe
Hilbert space. The non-factorization necessitates a dual interpretation of the
bulk path integral in terms of a product of partition functions (associated to
NS boundaries) and Witten indices (associated to R boundaries), averaged over
an ensemble of theories with varying Hilbert space dimension and different
numbers of bosonic and fermionic states. We also consider a model with
End-of-the-World (EOW) branes: the dual ensemble then includes a sum over
randomly chosen fermionic and bosonic states. We propose two modifications of
the bulk path integral which restore an interpretation in a single dual theory:
(i) a geometric prescription where we add extra boundaries with a sum over
their spin structures, and (ii) an algebraic prescription involving "spacetime
D-branes". We extend our ideas to Jackiw-Teitelboim gravity, and propose a dual
description of a single unitary theory with spin structure in a system with
eigenbranes.
