2d Index and Surface operators (1305.0266v2)

Abstract: In this paper we compute the superconformal index of 2d (2,2) supersymmetric gauge theories. The 2d superconformal index, a.k.a. flavored elliptic genus, is computed by a unitary matrix integral much like the matrix integral that computes 4d superconformal index. We compute the 2d index explicitly for a number of examples. In the case of abelian gauge theories we see that the index is invariant under flop transition and CY-LG correspondence. The index also provides a powerful check of the Seiberg-type duality for non-abelian gauge theories discovered by Hori and Tong. In the later half of the paper, we study half-BPS surface operators in N=2 superconformal gauge theories. They are engineered by coupling the 2d (2,2) supersymmetric gauge theory living on the support of the surface operator to the 4d N=2 theory, so that different realizations of the same surface operator with a given Levi type are related by a 2d analogue of the Seiberg duality. The index of this coupled system is computed by using the tools developed in the first half of the paper. The superconformal index in the presence of surface defect is expected to be invariant under generalized S-duality. We demonstrate that it is indeed the case. In doing so the Seiberg-type duality of the 2d theory plays an important role.

• The paper computes the 2D superconformal index using a novel matrix integral method and confirms duality invariances in both abelian and non-abelian setups.
• The paper introduces half-BPS surface operators from 2D-4D gauge couplings that maintain S-duality and enhance computational precision.
• The study reveals modular properties of the elliptic genus and provides a framework for exploring new symmetries in quantum field theories.

Introduction to the Study of 2D Superconformal Indices and Surface Operators

The paper by Abhijit Gadde and Sergei Gukov explores the computation of the superconformal index for $2d$ $(2,2)$ supersymmetric gauge theories. This work analyzes the so-called flavored elliptic genus through a unitary matrix integral akin to the process used for $4d$ superconformal indices. A notable feature is the extension of the paper to include half-BPS surface operators in $=2$ superconformal theories, linking $2d$ and $4d$ systems via gauge couplings, exploring generalizations such as the invariant Seiberg-type duality under S-duality transformations.

Main Contributions

1. Superconformal Index for $2d$ Theories:
   • The paper identifies the $2d$ superconformal index as a tool for counting states in supersymmetric theories, preserving invariance under continuous deformations of the theory.
   • The methodology involves explicit computations for specific gauge theories, showcasing the invariant properties under flop transitions and the CY-LG correspondence in abelian cases, and verifying duality conjectures for non-abelian scenarios.
2. Surface Operators:
   • The research constructs half-BPS surface operators in $=2$ theories as 2d gauge theories coupled to 4d superconformal ones. It exhibits the system's invariance under the general S-duality transformations.
   • The paper introduces techniques for calculating the index of these coupled systems, employing insights from singularity theory in the field configurations and state indices, ensuring precise calculability.

Detailed Insights

• Methodology: The authors craft a detailed matrix integral approach, akin to techniques in $4d$ systems. By dissecting component multiplet indices, they achieve a formulation for the superconformal index that's applicable across various flavors and symmetries within $2d$ theories.
• Dualities and Symmetries: A significant aspect of the research lies in affirming hypothesis' regarding Hori-Tong dualities for different matter configurations within non-abelian theories. They argue that their derived indices exhibit invariances consistent with mathematical symmetries of the theoretical framework.
• Modular Properties and Degeneracies: The paper accentuates the modular properties of the framework through elliptic genus theories and characterizes the resulting degeneracies directly tied to the modular transformations, providing computational tools for interpreting the central charges of these fixed points.

Implications and Future Work

• Practical Implications: Immediate implications are observed in the field of theoretical high-energy physics, particularly within the field of Gaiotto theories and dualities. The refined computational techniques provide a robust foundation for exploring more complex multi-dimensional and multi-symmetry systems.
• Theoretical Insights: The identification and computation of the $2d$ superconformal index provide a template for exploring equivalent and potentially new symmetries in various dimensions. The paper's verification of dualities set the stage for future theoretical and geometric explorations.
• Potential Developments in Field Theory: By establishing a more profound connection between lower-dimensional surface operators and higher-dimensional theories via these indices, there is potential for advancements in understanding quantum field theories' geometric and algebraic aspects.

This paper opens paths to more comprehensive dissertations on the interplay between gauge theories, dualities, and computational indices, laying groundwork towards uncovering novel properties within quantum field theories and their associated symmetries.