Cluster-like coordinates in supersymmetric quantum field theory

Area of Science:

• Theoretical Physics
• Algebraic Geometry

Background:

• N=2 supersymmetric quantum field theories possess hyperkähler moduli spaces.
• Cluster algebras provide a combinatorial framework with applications in various mathematical fields.

Purpose of the Study:

• To review the connection between N=2 supersymmetric quantum field theories and cluster algebras.
• To explore the extended cluster variety structures found in the moduli spaces of these theories.

Main Methods:

• Examining the hyperkähler moduli spaces associated with N=2 supersymmetric quantum field theories.
• Investigating the emergence of cluster variables and generalized mutations within these moduli spaces.
• Focusing on examples involving moduli spaces of flat connections on surfaces.

Main Results:

• The moduli spaces of N=2 supersymmetric quantum field theories exhibit structures that generalize cluster varieties.
• These structures include standard cluster variables and mutations, as well as novel extra variables and generalized mutations.
• Specific examples highlight the role of moduli spaces of flat connections on surfaces, as studied by Fock and Goncharov.

Conclusions:

• There is a significant and intricate relationship between N=2 supersymmetric quantum field theories and cluster algebras.
• The moduli spaces of these quantum field theories offer a rich ground for exploring extended notions of cluster structures.
• This connection provides new insights into both quantum field theory and algebraic structures.