# Recent advances and open questions on the susy structure of the chiral de Rham Complex

## TL;DR

This paper reviews the supersymmetry structures within the chiral de Rham complex on special holonomy manifolds, discussing different constructions, topological twists, and algebraic reductions.

## Contribution

It provides a comprehensive comparison of supersymmetry subalgebra constructions and introduces new perspectives on their algebraic origins and topological twists.

## Key findings

- Different constructions of supersymmetry subalgebras are compared.
- Topological twists for $G_2$ and $Spin_7$ manifolds are described.
- Algebras are constructed via quantum Hamiltonian reduction of Lie superalgebras.

## 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.