Wess-Zumino-Witten Model

Definition

The Wess-Zumino-Witten model is a two-dimensional conformal field theory that describes the behavior of certain types of nonlinear sigma models, particularly those that are associated with Lie groups and their representations. This model is significant because it incorporates both topological and geometric aspects, making it an essential tool in understanding infinite-dimensional geometry and integrable systems, especially in the context of string theory and statistical mechanics.

5 Must Know Facts For Your Next Test

1. The Wess-Zumino-Witten model can be defined on a Riemann surface and is crucial for studying string theories in two dimensions.
2. It introduces a topological term known as the Wess-Zumino term, which modifies the usual sigma model action and ensures conformal invariance.
3. The model can be associated with current algebras, where the symmetries correspond to the representations of the underlying Lie group.
4. In terms of integrable systems, the Wess-Zumino-Witten model provides examples of solitons and other stable solutions that arise in two-dimensional field theories.
5. The Wess-Zumino-Witten model has deep connections to statistical mechanics, particularly in the study of phase transitions and critical phenomena in two-dimensional systems.

Review Questions

• How does the Wess-Zumino term contribute to the conformal invariance of the Wess-Zumino-Witten model?
  • The Wess-Zumino term adds a crucial topological component to the action of the Wess-Zumino-Witten model, ensuring that the model remains invariant under conformal transformations. This term captures global properties of the field configurations defined on a Riemann surface and allows for a richer set of symmetries than a typical nonlinear sigma model would have. By including this term, one can analyze more complex interactions and behaviors in the theory while maintaining consistency with two-dimensional conformal field theory principles.
• Discuss how current algebras relate to the representation theory of Lie groups within the context of the Wess-Zumino-Witten model.
  • Current algebras play a key role in understanding symmetries within the Wess-Zumino-Witten model, as they arise from the model's conserved currents associated with its symmetries. These currents are linked to representations of Lie groups, allowing for a powerful description of physical states in terms of these algebraic structures. The representation theory helps classify these states and understand their transformations under the symmetry group, providing insights into both quantum field theories and statistical mechanics.
• Evaluate how the Wess-Zumino-Witten model serves as a bridge between geometry and physics, particularly in integrable systems.
  • The Wess-Zumino-Witten model serves as an important link between geometry and physics by illustrating how geometric structures can manifest in physical theories. It showcases how infinite-dimensional geometry influences integrable systems through its solutions, which often exhibit solitonic behavior. The interplay between these geometric ideas and physical models allows for profound insights into phenomena such as phase transitions and critical behavior in statistical mechanics, demonstrating that deep mathematical concepts can directly inform our understanding of physical systems.

Related terms

Conformal Field Theory: A quantum field theory that is invariant under conformal transformations, focusing on the properties of scale and angles rather than distances.

Lie Group:

A group that is also a smooth manifold, where the group operations are smooth functions, serving as a fundamental structure in geometry and physics.

Integrable Systems:

Dynamical systems that can be solved exactly due to the presence of a sufficient number of conserved quantities, often exhibiting rich mathematical structures.