Central charge

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In theoretical physics, a central charge is an operator Z that commutes with all the other symmetry operators. The adjective "central" refers to a center of the symmetry group -- the subgroup of elements that commute with all other elements of the original group -- or the center of a symmetry Lie algebra. In some cases, like 2D conformal field theory, but not all, a central charge also commutes with all the other operators (this includes operators which are not symmetry operators).

See also central extension. The article about central extension analyzes the notion of a central charge mathematically.

In theories with supersymmetry, this definition can be generalized to include supergroups and Lie superalgebras and 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. In string theory, in the first quantized formalism, these operators also have the interpretation of winding numbers of various strings and branes.

In conformal field theory, the central charge is a c-number (commutes with every other operator) term that appears in the commutator of two components of the stress energy tensor.