Title: Recent advances and open questions on the susy structure of the chiral de Rham Complex,
Abstract: We review different constructions of the supersymmetry subalgebras of the
chiral de Rham complex on special holonomy manifolds. We describe the
difference between the holomorphic-anti-holomorphic sectors based on a local
free ghost system vs the decomposition in left-right sectors from a local
Boson-Fermion system. We describe the topological twist in the case of $G_2$
and $Spin_7$ manifolds. We describe the construction of these algebras as
quantum Hamiltonian reduction of Lie superalgebras at the minimal or
superprincipal nilpotent.