Level N Teichm\"uller TQFT and Complex Chern-Simons Theory

In this manuscript we review the construction of the Teichm"{u}ller TQFT in [AK1], upgrading it to a theory dependent on an extra odd integer $N$ using results developed in [AK3]. We also describe how this theory is related with quantum Chern–Simons Theory at level $N$ with gauge group $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{C})$.