Central charge

Definition

The central charge is a crucial quantity in conformal field theory that characterizes the scaling dimensions of primary fields and their behavior under the conformal group. It plays a significant role in the classification of conformal field theories and is linked to the anomaly in the energy-momentum tensor, influencing correlation functions and operator product expansions.

5 Must Know Facts For Your Next Test

1. The central charge is often denoted by 'c' and is a key feature in the Virasoro algebra, where it appears in the commutation relations of the Virasoro generators.
2. In two-dimensional conformal field theories, the value of the central charge can determine critical behaviors such as phase transitions and universality classes.
3. A central charge of zero usually corresponds to theories without anomalies, while non-zero values indicate the presence of anomalies and impact correlation functions.
4. The central charge can influence the number of degrees of freedom in a given conformal field theory, affecting its overall statistical properties.
5. In string theory, the central charge plays a role in defining consistent models by imposing constraints on the spectrum of physical states.

Review Questions

• How does the central charge relate to the classification of conformal field theories and their critical phenomena?
  • The central charge serves as an important parameter in classifying conformal field theories by influencing their scaling dimensions and symmetry properties. It determines key aspects of critical phenomena, such as universality classes, where different systems exhibit similar behavior near critical points despite having different microscopic details. Understanding how the central charge manifests in correlation functions allows for deeper insights into phase transitions and critical behavior in statistical mechanics.
• Discuss the implications of a non-zero central charge in terms of anomalies within conformal field theories.
  • A non-zero central charge indicates that a conformal field theory has anomalies, which are deviations from classical symmetries that arise when quantizing fields. These anomalies affect the conservation laws associated with the energy-momentum tensor and impact physical observables such as correlation functions. The presence of anomalies can also signify constraints on how operators are defined within the theory, thus affecting predictions related to critical phenomena.
• Evaluate how understanding central charge enhances our knowledge of operator product expansions in conformal field theory.
  • Understanding central charge is fundamental to evaluating operator product expansions (OPE) in conformal field theories because it influences how operators combine at short distances. The central charge modifies the coefficients appearing in OPEs, which dictate how primary operators interact with one another under scaling transformations. By analyzing these interactions through OPEs, one can derive important physical properties, including correlation functions and scaling dimensions, thereby enriching our understanding of critical phenomena.

Related terms

conformal symmetry: A type of symmetry that includes transformations such as scaling, translation, rotation, and special conformal transformations, preserving angles but not necessarily distances.

primary operator: An operator in a conformal field theory that transforms simply under the action of the conformal group and has a well-defined scaling dimension.

Virasoro algebra: An algebraic structure that describes the symmetries of conformal field theories and is generated by the Virasoro operators, which include contributions from the central charge.