  Fuzzy dark matter (FDM) is a proposed modification for the standard cold dark
matter (CDM) model motivated by small-scale discrepancies in low-mass galaxies.
Composed of ultra-light (mass $\sim 10^{-22}$ eV) axions with kpc-scale de
Broglie wavelengths, this is one of a class of candidates that predicts that
the first collapsed objects form in relatively massive dark matter halos. This
implies that the formation history of the first stars and galaxies would be
very different, potentially placing strong constraints on such models. Here we
numerically simulate the formation of the first stars in an FDM cosmology,
following the collapse in a representative volume all the way down to
primordial protostar formation including a primordial non-equilibrium chemical
network and cooling for the first time. We find two novel results: first, the
large-scale collapse results in a very thin and flat gas "pancake"; second,
despite the very different cosmology, this pancake fragments until it forms
protostellar objects indistinguishable from those in CDM. Combined, these
results indicate that the first generation of stars in this model are also
likely to be massive and, because of the sheet morphology, do not
self-regulate, resulting in a massive Pop III starburst. We estimate the total
number of first stars forming in this extended structure to be $10^4$ over 20
Myr using a simple model to account for the ionizing feedback from the stars,
and should be observable with JWST. These predictions provide a potential
smoking gun signature of FDM and similar dark matter candidates.
