In theories with supersymmetry, this definition can be generalized to include supergroups and Lie superalgebras.
A central charge is any operator which commutes with all the other supersymmetry generators.
Theories with extended supersymmetry typically have many operators of this kind.
[2] As a result, conformal field theory is characterized by a representation of Virasoro algebra with central charge c. For conformal field theories that are described by modular category, the central charge can be extracted from the Gauss sum.
This definition allows extending the definition to a higher central charge,[4][5] using the higher Gauss sums:[6] The vanishing higher central charge is a necessary condition for the topological quantum field theory to admit topological (gapped) boundary conditions.