  Understanding the formation and evolution of low surface brightness galaxies
(LSBGs) is critical for explaining their wide-ranging properties. However,
studies of LSBGs in deep photometric surveys are often hindered by a lack of
distance estimates. In this work, we present a new catalogue of 479 LSBGs,
identified in deep optical imaging data from the Hyper Suprime-Cam Subaru
Strategic Program (HSC-SSP). These galaxies are found across a range of
environments, from the field to groups. Many are likely to be ultra-diffuse
galaxies (UDGs). We see clear evidence for a bimodal population in colour -
S\'ersic index space, and split our sample into red and blue LSBG populations.
We estimate environmental densities for a subsample of 215 sources by
statistically associating them with nearby spectroscopic galaxies from the
overlapping GAMA spectroscopic survey. We find that the blue LSBGs are
statistically consistent with being spatially randomised with respect to local
spectroscopic galaxies, implying they exist predominantly in low-density
environments. However, the red LSBG population is significantly spatially
correlated with local structure. We find that 26+/-5% of isolated, local LSBGs
belong to the red population, which we interpret as quiescent. This indicates
that high environmental density plays a dominant, but not exclusive, role in
producing quiescent LSBGs. Our analysis method may prove to be very useful
given the large samples of LSB galaxies without distance information expected
from e.g. the Vera C. Rubin observatory (aka LSST), especially in combination
with upcoming comprehensive wide field spectroscopic surveys.
