Fusion rules

Definition

Fusion rules are mathematical descriptions that determine how objects can combine or 'fuse' in the context of modular tensor categories and subfactor theory. They provide a systematic way to understand the relationships and interactions between different representations, ultimately helping to classify them. These rules play a crucial role in constructing principal graphs, analyzing subfactor lattices, and understanding the structure of various types of subfactors, along with conformal nets.

5 Must Know Facts For Your Next Test

1. Fusion rules can often be represented by a fusion matrix, which helps visualize how different simple objects combine and interact with each other.
2. In many contexts, fusion rules are subject to certain axioms that must be satisfied, ensuring consistency across different mathematical frameworks.
3. The values of fusion rules are often related to dimensions of representations or to the number of ways certain types of particles can interact in quantum physics.
4. Fusion rules also help in defining invariants of subfactors that classify them, providing deeper insights into their structure and properties.
5. Different types of subfactors may exhibit distinct fusion rules, which leads to unique principal graphs and lattice structures.

Review Questions

• How do fusion rules relate to principal graphs and what information do they convey about the structure of a subfactor?
  • Fusion rules directly inform the construction of principal graphs by dictating how different simple objects can combine within a subfactor. Each node in a principal graph represents an object, while edges reflect possible fusions based on these rules. This relationship allows us to visualize the interconnections between objects in a categorical framework and understand the underlying algebraic structure.
• Discuss how fusion rules contribute to understanding Bisch-Haagerup subfactors and their representation theory.
  • Fusion rules play a significant role in characterizing Bisch-Haagerup subfactors by providing a detailed account of how their simple objects can interact. These rules establish a clear framework for exploring representation theory in this context, as they determine dimensions and relationships among representations. This understanding is vital for uncovering deeper properties such as duality and modularity within the Bisch-Haagerup framework.
• Evaluate the significance of fusion rules within conformal nets and how they enhance our understanding of quantum field theory.
  • In conformal nets, fusion rules are essential for revealing how local observables can combine or 'fuse' under various transformations. This concept extends into quantum field theory by demonstrating how symmetries and interactions manifest at different scales. Understanding these rules enhances our grasp of physical phenomena, such as particle interactions and phase transitions, illustrating the interplay between mathematical structures and physical reality.

Related terms

Modular Tensor Category: A type of category that arises in the study of quantum groups and conformal field theory, where the fusion rules provide a way to describe how different objects can interact.

Principal Graphs: Graphs that encode information about the fusion rules and the structure of the subfactors, representing how simple objects can combine within a given category.

Bisch-Haagerup Subfactors: A specific type of subfactor that has well-defined fusion rules, illustrating interesting properties about the representation theory of these subfactors.