  Monolayers of transition metal dichalcogenides (TMDC) became one of the most
studied nanostructures in the last decade. Combining two different TMDC
monolayers results in a heterostructure whose properties can be individually
tuned by the twist angle between the lattices of the two van-der-Waals layers
and the relative placement of the layers, leading to Moir\'e cells. For small
twist angles, lattice reconstruction leads to strong strain fields in the
Moir\'e cells. In this paper, we combine an existing theory for lattice
reconstruction with a quantum dynamic theory for interlayer excitons and their
dynamics due to exciton phonon scattering using a polaron transformation. The
exciton theory is formulated in real space instead of the commonly used
quasi-momentum space to account for imperfections in the heterolayer breaking
lattice translational symmetry. We can analyze the structure of the localized
and delocalized exciton states and their exciton-phonon scattering rates for
single phonon processes using Born-Markov approximation and multi-phonon
processes using a polaron transformation. Furthermore, linear optical spectra
and exciton relaxation Green functions are calculated and discussed. A
P-stacked MoSe$_2$/WSe$_2$ heterolayer is used as an illustrative example. It
shows excitons localized in the potential generated through the Moir\'e-pattern
and strain and a delocalized continuum. The exciton-phonon relaxation times
vary depending on the strain and range from sub-pico seconds up to nanoseconds.
