  In this paper we explore possible phases arising from doping neutral excitons
into a Mott insulator in the context of moir\'e bilayers. We consider two
moir\'e layers coupled together only through inter-layer repulsion and there is
a U(2)$\times$ U(2) symmetry. The densities of the two layers can be tuned to
be $n_t=x,n_b=1-x$ with $n_t+n_b=1$. $x=0$ limit is a layer polarized Mott
insulator and small $x$ regime can be reached by doping inter-layer excitons at
density $x$. Charge gap can remain finite at small $x$, as is demonstrated
experimentally in the WSe$_2$-hBN-WSe$_2$/WS$_2$ system. To capture the
intertwinement of the spin and exciton degree of freedom, we propose a
four-flavor spin model. In addition to the obvious possibility of single
exciton condensation phase, we also identified more exotic physics with
fractionalization: (I) We define spin gap $\Delta_t, \Delta_b$ for the two
layers respectively. As long as the spin gap at either layer is finite, single
exciton condensation is impossible and we can only have paired exciton
condensation. If both spin gaps are finite, it can be a fractional exciton
superfluid with paired exciton condensation coexisting with $Z_2$ spin liquid.
Numerical evidences for such a phase will be provided. (II) If the layer
polarized Mott insulator at $x=0$ is in a U(1) spin liquid with spinon Fermi
surface, the natural phase at $x>0$ hosts neutral Fermi surface formed by
fermionic excitons. There are metallic counter-flow transport and also Friedel
oscillations in layer polarization in this exotic phase. Our numerical
simulation in one dimension observes an analog of this neutral Fermi surface
phase. (III) There could be a metal-insulator transition (MIT) by tuning $x$ in
the universality class of a bandwidth tuned MIT. We provide one theory for such
a continuous Mott transition and predicted a universal drag resistivity.
