The 4d Superconformal Index from q-deformed 2d Yang-Mills

Abstract: We identify the 2d topological theory underlying the N=2 4d superconformal index with an explicit model: q-deformed 2d Yang-Mills. By this route we are able to evaluate the index of some strongly-coupled 4d SCFTs, such as Gaiotto's T_N theories.

• The paper proposes a novel duality connecting the 4d superconformal index with 2d q-deformed Yang-Mills theory for N=2 SCFTs.
• It simplifies computation by reducing the index to a single fugacity, validated through SU(2) and SU(3) generalized quiver tests and S-duality checks.
• The work paves the way for generalizing full indices and exploring elliptic quantum group structures in higher-dimensional field theories.

The $4d$ Superconformal Index from $q$-deformed $2d$ Yang-Mills

This paper presents a novel duality connecting four-dimensional ($4d$) superconformal field theories (SCFTs) and two-dimensional ($2d$) $q$-deformed Yang-Mills (qYM) theories. Specifically, the authors establish a correspondence between the superconformal index of $4d$ ${\cal N}=2$ SCFTs and qYM theory in two dimensions, providing insight into the operator spectrum of strongly-coupled $4d$ gauge theories. This significant insight into the physics of SCFTs relies on the framework of $q$-deformed two-dimensional Yang-Mills theory's representation theory.

Overview of Approach

• Duality and Topology: The authors propose a duality where the $4d$ index, a twisted supersymmetric partition function on $S^3 \times S^1$, corresponds to the zero-area limit of $2d$ qYM theory. This duality simplifies calculations by transforming a strongly-coupled $4d$ problem into a more tractable $2d$ setting.
• Reduced Index: By focusing on a reduced index dependent on a single fugacity $q$, they manage a significant simplification in calculations. The $4d$ SCFTs considered lack weak coupling descriptions, complicating direct evaluations of traditional indices.
• Associative Algebra: The work shows how these indices form an associative algebra using $2d$ topological quantum field theory (TQFT), setting the stage for comparing generalized quiver gauge theories via S-dualities.

Key Results and Validation

• $SU(2)$ and $SU(3)$ Quivers: Initial tests were performed on $SU(2)$ and $SU(3)$ generalized quivers, showing that the reduced superconformal index can be computed using qYM structures. For $SU(2)$, the tri-fundamental hypermultiplet’s structure constants collapse into those of qYM, and a similar structure is proposed for $SU(3)$ based on the $E_6$ SCFT.
• Conjecture for General Rank SU(N): The proposal is that indices for three maximal puncture spheres (e.g., Gaiotto’s $T_N$ theories) align with correlators in $q$-deformed Yang-Mills, supporting the correctness of the outlined duality.
• Consistency Tests: Their conjecture has undergone consistency checks with known dualities and matchings of operator spectrums, such as the Argyres-Seiberg duality.

Theoretical Implications and Future Directions

The paper's proposed duality holds practical implications for evaluating indices of strongly-coupled SCFTs, which is otherwise challenging due to the absence of Lagrangian descriptions. Understanding the qYM theory's exact contribution to these indices offers a robust tool for probing non-perturbative dynamics in four dimensions.

Future developments might encompass:

• Full Index Generalization: Extending their findings beyond the reduced index to fully involve all parameters, thereby potentially uncovering deeper algebraic structures akin to higher quantum groups.
• Elliptic Extensions: Given the symmetric roles of fugacities $q$ and $p$, the work hints at the potential of elliptically-deformed quantum group structures, drawing a tantalizing parallel to the broader mathematical frameworks of quantum integrable systems.
• Deeper Conceptual Insights: Expanding the scope of this correspondence from a higher-dimensional manifold perspective, such as those inspired by six-dimensional insights related to Riemann surfaces, may yield further theoretical enlightenment. Leveraging these for practical computations within higher-dimensional theories presents an intriguing field for exploration.

The research's detailed proposal of bridging $4d$ SCFTs through these $2d$ frameworks not only reveals hidden structures within these field theories but also lays down comprehensive connections which can be fundamental in future theoretical physics advancements.