Central Charge

Definition

The central charge is a key parameter in conformal field theory that describes the scaling dimensions of primary fields and plays a vital role in defining the algebra of conserved currents. It quantifies how the conformal transformations act on a system and is essential for understanding the symmetry properties and physical implications of conformal field theories. The central charge is often denoted by 'c' and is linked to the degree of freedom of the system, influencing the conformal anomaly.

5 Must Know Facts For Your Next Test

1. The central charge 'c' can take on different values depending on the specific conformal field theory being studied, which can classify the theory's universality class.
2. In two-dimensional conformal field theories, the central charge determines the number of degrees of freedom and is crucial in identifying critical points in statistical mechanics.
3. The value of the central charge can affect various physical properties, such as correlation functions and scaling behaviors of operators within the theory.
4. When calculating the partition function on a torus, the central charge appears as a multiplicative factor that influences modular invariance.
5. The central charge is directly related to the conformal anomaly, where theories with non-zero central charges exhibit unexpected behavior under conformal transformations.

Review Questions

• How does the central charge influence the scaling dimensions of primary fields in a conformal field theory?
  • The central charge plays a crucial role in determining the scaling dimensions of primary fields within a conformal field theory. It acts as a parameter that quantifies how these fields transform under scale transformations, affecting their behavior at critical points. Specifically, different values of the central charge can indicate different universality classes for phase transitions, which can lead to significant variations in physical phenomena like critical exponents.
• Discuss the relationship between central charge and conformal anomaly in quantum field theories.
  • The central charge is intimately connected to the concept of conformal anomaly in quantum field theories. When a classical theory exhibits conformal symmetry, this symmetry may be broken at the quantum level due to divergences arising from loop corrections. The presence of a non-zero central charge indicates such an anomaly, reflecting how quantum effects alter the behavior and symmetries of the original classical theory, leading to observable consequences like modified correlation functions.
• Evaluate how varying central charges affect the classification of different conformal field theories and their physical implications.
  • Varying central charges provide a framework for classifying different conformal field theories into distinct universality classes. The value of the central charge determines properties such as critical behavior, spectrum of operators, and correlation functions. For instance, theories with higher central charges typically exhibit richer structures with more degrees of freedom. Understanding these classifications helps physicists connect mathematical frameworks with physical phenomena like phase transitions and critical points in statistical mechanics.

Related terms

Conformal Symmetry: A type of symmetry that preserves angles but not necessarily distances, crucial for the analysis of two-dimensional quantum field theories.

Primary Fields: The basic building blocks of conformal field theory that transform in a specific way under conformal transformations and are characterized by their scaling dimensions.

Conformal Anomaly: A phenomenon that occurs when a theory that is classically conformally invariant exhibits a breakdown of this symmetry at the quantum level, often characterized by a non-zero central charge.