  We generalize the results of Chang-Li, Kim-Oh and Chang-Li on the moduli of
$p$-fields to the setting of (quasi-)maps to complete intersections in
arbitrary smooth Deligne-Mumford stacks with projective coarse moduli. In
particular, we show that the virtual cycle of stable (quasi-)maps to a complete
intersection can be recovered by the cosection localized virtual cycle of the
moduli of $p$-fields of the ambient space.
