Central charge
Commuting Lie algebra operator
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Through Operator (mathematics), Commutative property and Charge (physics)
Overview
More precisely, the central charge is the charge that corresponds, by Noether's theorem, to the center of the central extension of the symmetry group.
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. In string theory, in the first quantized formalism, these operators also have the interpretation of winding numbers (topological quantum 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. As a result, conformal field theory is characterized by a representation of Virasoro algebra with central charge c.
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Gauss sums and higher central charge
For conformal field theories that are described by modular tensor category, the central charge can be extracted from the Gauss sum. In terms of anyon quantum dimension da and topological spin θa of anyon a, the Gauss sum is given by
and equals , where is central charge.
This definition allows extending the definition to a higher central charge, using the higher Gauss sums:
The vanishing higher central charge is a necessary condition for the topological quantum field theory to admit topological (gapped) boundary conditions.