Bootstrapping the superconformal index with surface defects (1207.3577v1)

Abstract: The analytic properties of the N = 2 superconformal index are given a physical interpretation in terms of certain BPS surface defects, which arise as the IR limit of supersymmetric vortices. The residue of the index at a pole in flavor fugacity is interpreted as the index of a superconformal field theory without this flavor symmetry, but endowed with an additional surface defect. The residue can be efficiently extracted by acting on the index with a difference operator of Ruijsenaars-Schneider type. By imposing the associativity constraints of S-duality, we are then able to evaluate the index of all generalized quiver theories of type A, for generic values of the three superconformal fugacities, with or without surface defects.

• The paper introduces a residue calculus method that connects superconformal indices with surface defects using integrable model techniques.
• It employs difference operators from the elliptic Ruijsenaars-Schneider equation to capture key analytic properties of the index.
• The framework enables bootstrapping TQFT diagonalization in class S theories, enriching insights into dualities in quantum field theory.

Overview of "Bootstrapping the Superconformal Index with Surface Defects"

The paper in the paper "Bootstrapping the Superconformal Index with Surface Defects" by Gaiotto, Rastelli, and Razamat explores the intricate relationship between the superconformal index of 4d $\mathcal{N}=2$ supersymmetric theories and BPS surface defects. The authors provide a framework to compute the superconformal index in the presence of these defects through a residue calculus approach, offering new insights into the structure of these theories by leveraging certain integrable models.

Key Contributions and Methods

The superconformal index is a powerful tool that captures valuable topological information about supersymmetric field theories. By considering the analytic properties of these indices, the authors formulate a connection between the residues at specific poles of the index and the introduction of surface defects characterized by these residues.

The paper proposes interpreting these residues in terms of the insertion of surface defects into the theory, which are realized as solutions to a generalized elliptic Ruijsenaars-Schneider difference equation. This model is inherently related to the underlying geometry and algebraic structures of the field theory, as the difference operators obtained are connected to the Hamiltonians of these integrable systems.

Analytical Framework and Results

The investigation focuses on class $\mathcal{S}$ theories derived from M5 branes compactified on a Riemann surface. These theories span a rich landscape of $\mathcal{N}=2$ theories characterized by different types of punctures, encoding flavor symmetries of the theory.

1. Residue Prescription: The authors detail a method for extracting residues of the index at poles associated with flavor fugacities. These residues are shown to match indices of the related theories with additional surface defects, thereby closing some of the punctures on the Riemann surface.
2. Difference Operators: The analysis reveals that these residues can be captured by operators akin to the differences in Ruijsenaars-Schneider models. The operators exhibit properties such as self-adjointness and commutativity, which are indicative of integrable structure.
3. Diagonalization and Bootstrapping: The paper brings forth a bootstrap approach for diagonalizing the $2d$ topological quantum field theory (TQFT) associated with these indices. The diagonal form elucidates the structure constants and facilitates computations of indices for generalized quiver gauge theories.

Implications and Future Directions

By associating surface defects with known mathematical structures such as difference operators, this paper extends the utility of the superconformal index beyond counting protected operators, embedding it into the field of algebraic geometry and integrable systems. This connection has profound implications for understanding dualities and emergent phenomena in these quantum field theories.

Looking forward, the paper opens several avenues for deeper exploration:

• Non-Lagrangian Theories: The techniques developed hold promise for further exploration into non-Lagrangian examples of class $\mathcal{S}$ where traditional methods might not offer direct computational tools.
• Extension to Other Types of Defects: While the focus here is on surface defects corresponding to specific puncture types, extending this framework to other defects could unveil more about the interplay between geometry and field theory.
• Generalization to $\mathcal{N}=1$ Theories: Transitioning these insights into theories with less symmetry could illuminate how these structures behave under reduced supersymmetry and potentially unveil new dualities.

In summary, this work is a significant step towards a comprehensive understanding of the algebraic and geometric facets underlying the superconformal indices enriched by surface defects. It opens up potential pathways to leverage known mathematical frameworks for describing more complex quantum field theory behavior, aligning with broader efforts to categorize and understand the rich landscape of supersymmetric theories.