Central Charge

Definition

Central charge is a term used in the context of the Virasoro algebra, referring to an additional parameter that quantifies the extent to which a conformal field theory deviates from being a purely classical theory. It acts as an additional scalar quantity that modifies the algebra's structure, playing a crucial role in defining the properties of the conformal field theory, such as its primary fields and their scaling dimensions.

5 Must Know Facts For Your Next Test

1. The central charge is denoted as 'c' and arises in the context of two-dimensional conformal field theories, often associated with the number of degrees of freedom in the theory.
2. In the Virasoro algebra, the central charge appears in the commutation relations, where it contributes to the anomaly present in the transformation properties of conformal fields.
3. Different values of the central charge can lead to different types of critical behavior in statistical mechanics and are closely related to various phases of matter.
4. In string theory, the central charge helps in determining consistency conditions for various string models, impacting their physical properties and interactions.
5. The central charge can also be connected to modular invariance, which is essential for understanding partition functions and vacuum states in conformal field theories.

Review Questions

• How does the central charge influence the structure of the Virasoro algebra and its role in conformal field theories?
  • The central charge introduces an extra scalar parameter 'c' into the Virasoro algebra's commutation relations, affecting how operators within the algebra interact. This parameter directly influences the classification of primary fields based on their scaling dimensions and allows for distinguishing between different conformal field theories. Consequently, it plays a vital role in understanding anomalies and physical properties within these theories.
• Discuss the implications of varying central charge values in relation to critical phenomena observed in two-dimensional statistical mechanics.
  • Different values of the central charge correspond to various universality classes in critical phenomena. As 'c' varies, it indicates changes in symmetry properties and physical behavior near phase transitions. This relationship helps categorize systems based on their scaling behavior and enables predictions about critical exponents, facilitating a deeper understanding of how systems behave at critical points.
• Evaluate the significance of central charge in string theory and its role in ensuring consistency among different string models.
  • In string theory, central charge is crucial for maintaining modular invariance across different string models. It determines consistency conditions that must be satisfied for a viable string theory, impacting aspects such as vacuum states and interactions. The connection between central charge and physical characteristics allows physicists to derive meaningful results regarding string propagation and compactification, thereby enhancing our understanding of fundamental particles and forces.

Related terms

Virasoro Algebra: A central extension of the Witt algebra, which plays a fundamental role in conformal field theories by providing a framework for studying symmetries and operator product expansions.

Conformal Field Theory: A quantum field theory that is invariant under conformal transformations, allowing for the study of critical phenomena and string theory through its rich symmetry properties.

Primary Field: A field in a conformal field theory that transforms in a simple manner under conformal transformations, characterized by its scaling dimension and associated with the central charge.