  We introduce a new kind of foliated quantum field theory (FQFT) of gapped
fracton orders in the continuum. FQFT is defined on a manifold with a layered
structure given by one or more foliations, which each decompose spacetime into
a stack of layers. FQFT involves a new kind of gauge field, a foliated gauge
field, which behaves similar to a collection of independent gauge fields on
this stack of layers. Gauge invariant operators (and their analogous particle
mobilities) are constrained to the intersection of one or more layers from
different foliations. The level coefficients are quantized and exhibit a
duality that spatially transforms the coefficients. This duality occurs because
the FQFT is a foliated fracton order. That is, the duality can decouple 2+1D
gauge theories from the FQFT through a process we dub exfoliation.
