BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters

Abstract: We study symmetries of quantum field theories involving topologically distinct sectors of the field space. To exhibit these symmetries we define special gauge invariant observables, which we call the $qq$-characters. In the context of the BPS/CFT correspondence, using these observables, we derive an infinite set of Dyson-Schwinger-type relations. These relations imply that the supersymmetric partition functions in the presence of $\Omega$-deformation and defects obey the Ward identities of two dimensional conformal field theory and its $q$-deformations. The details will be discussed in the companion papers.

• The paper presents a non-perturbative framework leveraging qq-characters to extend Dyson-Schwinger equations in supersymmetric gauge theories.
• It integrates the BPS/CFT correspondence with quiver varieties to analyze partition functions and instanton contributions in detail.
• The work reveals deep algebraic and geometric insights that advance our understanding of gauge symmetries and quantum integrable systems.

BPS/CFT Correspondence and Non-Perturbative Dyson-Schwinger Equations in Quantum Field Theory

The paper under examination discusses a sophisticated construction within the framework of the BPS/CFT correspondence, specifically addressing non-perturbative Dyson-Schwinger equations and the concept of $qq$-characters in quantum field theory. Authored by Nikita Nekrasov, the research intricately weaves together advanced concepts from supersymmetric quantum field theories, conformal field theories (CFT), and complex mathematical structures such as quiver varieties.

Overview of Key Concepts

1. BPS/CFT Correspondence: This duality relates supersymmetric field theories in higher dimensions, notably those with eight supercharges, to two-dimensional conformal field theories. The correspondence is often realized through the partition functions of supersymmetric gauge theories, interpreted as special functions that generalize several well-known mathematical structures.
2. Quantum Field Theory Partitions: The paper provides a detailed analysis of the partition functions of ${\CalN}=2$ supersymmetric gauge theories, defined over a quiver, which is a directed graph representing both the symmetry and the matter content of the theory. Within this framework, both tree-level and one-loop contributions to the partition function are considered, alongside an intricate account of the moduli space of quiver-graded torsion-free sheaves utilized for calculating the instanton partition function via localization techniques.
3. Non-Perturbative Dyson-Schwinger Equations: The paper extends classical Dyson-Schwinger equations, traditionally encoding symmetries of quantum field theories, to the non-perturbative regime. By introducing the $qq$-characters, these equations can encapsulate transformations between homology classes in gauge theories, further elucidating symmetries and relationships between different topological sectors of field configurations.
4. $qq$-characters: These gauge invariant observables generalize the concept of characters from representation theory to the setting of quantum groups, providing a powerful tool to explore the algebraic structure underpinning the BPS/CFT correspondence. They offer insights into the spectrum of the quantum integrable systems associated with these field theories.
5. Dyson-Schwinger Propositions: The paper proposes a framework where $qq$-character equations are satisfied for all vectors and weights in the chosen quiver. This reflects the mathematical structure of the quantum field theory and its symmetries, aligning with known expectations about Yangians and integrating these into a coherent mathematical entity applicable to both finite and affine quivers.

Implications and Future Directions

The work presented elucidates profound implications in both theoretical physics and mathematics:

• Understanding Gauge Theory: By leveraging $qq$-characters, this research contributes to a more unified understanding of gauge theories' underlying algebraic structures, offering a pathway to decoding the complex symmetries and dualities that define quantum field theories.
• Mathematical Structures: The introduction of $qq$-characters and the utilization of quiver varieties mark significant advancements in understanding the connections between physical theories and geometrical spaces, contributing to both fields of algebraic geometry and representation theory.
• Further Exploration of the BPS/CFT Correspondence: This research lays a solid foundation for further exploration of the BPS/CFT correspondence, particularly in its application to diverse physical phenomena and more generalized algebraic frameworks.
• Potential for New Physical Insights: The rigorous mathematical formulation provided could lead to novel physical predictions and interpretations, especially when extending these concepts to non-supersymmetric or more complex interacting field theories.

In conclusion, the paper presents a structured and thorough treatment of the interplay between quantum field theory, algebra, and geometry, pushing the boundaries of our understanding of the BPS/CFT correspondence and offering a robust framework to potentially unlock further secrets of supersymmetric gauge theories and their associated mathematical structures.