theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...

, a central charge is an operator ''Z'' that commutes with all the other symmetry operators. The adjective "central" refers to the center of the

symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the amb ...

—the

subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ...

of elements that commute with all other elements of the original group—often embedded within a

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...

. In some cases, such as

two-dimensional conformal field theory A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invariant under local conformal transformations. In contrast to other types of conformal field theories, two-dimensional conformal fi ...

, a central charge may also commute with all of the other operators, including operators that are not symmetry generators.

Overview

More precisely, the central charge is the charge that corresponds, by

Noether's theorem Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law. This is the first of two theorems (see Noether's second theorem) published by the mat ...

, to the center of the central extension of the symmetry group. In theories with

supersymmetry Supersymmetry is a Theory, theoretical framework in physics that suggests the existence of a symmetry between Particle physics, particles with integer Spin (physics), spin (''bosons'') and particles with half-integer spin (''fermions''). It propo ...

, 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 physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...

, in the first quantized formalism, these operators also have the interpretation of

winding number In mathematics, the winding number or winding index of a closed curve in the plane (mathematics), plane around a given point (mathematics), point is an integer representing the total number of times that the curve travels counterclockwise aroun ...

s ( topological quantum numbers) of various strings and branes. In

conformal field theory A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometime ...

, 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 The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress ...

. As a result, conformal field theory is characterized by a representation of

Virasoro algebra In mathematics, the Virasoro algebra is a complex Lie algebra and the unique nontrivial central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string theory. It is named after Miguel Ángel ...

with central charge .

Gauss sums and higher central charge

For conformal field theories that are described by modular category, the central charge can be extracted from the Gauss sum. In terms of

anyon In physics, an anyon is a type of quasiparticle so far observed only in two-dimensional physical system, systems. In three-dimensional systems, only two kinds of elementary particles are seen: fermions and bosons. Anyons have statistical proper ...

quantum dimension and topological spin of anyon , the Gauss sum is given by :

\zeta_1 = \frac,

and equals

e^

, where

c_-

is central charge. This definition allows extending the definition to a higher central charge, using the higher Gauss sums: :

\zeta_n = \frac.

The vanishing higher central charge is a necessary condition for the

topological quantum field theory In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory that computes topological invariants. While TQFTs were invented by physicists, they are also of mathemati ...

to admit topological (gapped) boundary conditions.

References

Quantum field theory Anomalies (physics) Conformal field theory {{Quantum-stub

Overview

Gauss sums and higher central charge

See also

References