  Moir\'e exciton features tunable Dirac dispersion and spatially dependent
optical selection rules. With the long lifetime due to its interlayer nature,
it is promising to realize a Bose-Einstein condensation of moir\'e excitons.
Here we study the properties of moir\'e exciton condensate within the
mean-field theory, with special focus on exciton-exciton interaction effect on
the nonlinear Bloch band and Bloch waves of exciton condensate in moir\'e
potential. We find the nonlinear dispersion of the moir\'e exciton condensate
exhibits complex loop structure induced by exciton-exciton interaction. A
nonlinear Dirac cone emerges at $\boldsymbol{\Gamma}$ point of the Bloch band,
with a three-fold degenerate nonlinear Dirac point. Each degenerate Bloch wave
at nonlinear Dirac point spontaneously breaks $C3$ rotational symmetry of the
moir\'e potential, and themselves differ by $C3$ rotations. Since they reside
in the light cone, these nontrivial Bloch band structure and broken-symmetry
Bloch waves can be experimentally detected by examining light emission from
those states. Symmetry breaking of Bloch states implies unusual optical
selection rules compared with single exciton case: moir\'e exciton condensate
at $\boldsymbol{\Gamma}$ point emits light with all three components of
polarization instead of only left or right circular polarization. We further
propose that, by applying in-plane electric field on one layer to drive an
initially optically dark exciton condensate towards light cone, the light
polarization of final states as well as their dependence on field direction
serves as the smoking gun for experimental observation.
