Gauge Theories and Macdonald Polynomials (1110.3740v3)

Abstract: We study the N=2 four-dimensional superconformal index in various interesting limits, such that only states annihilated by more than one supercharge contribute. Extrapolating from the SU(2) generalized quivers, which have a Lagrangian description, we conjecture explicit formulae for all A-type quivers of class S, which in general do not have one. We test our proposals against several expected dualities. The index can always be interpreted as a correlator in a two-dimensional topological theory, which we identify in each limit as a certain deformation of two-dimensional Yang-Mills theory. The structure constants of the topological algebra are diagonal in the basis of Macdonald polynomials of the holonomies.

• The paper presents a novel method for computing the N=2 superconformal index using a trace formula expanded in three superconformal fugacities.
• It interprets the index as a TQFT correlator whose structure constants are diagonalized by Macdonald polynomials, even in non-Lagrangian settings.
• The study reveals simplified limits and combinatorial identities that extend computational techniques and deepen insights into A-type quiver gauge theories.

Overview of "Gauge Theories and Macdonald Polynomials"

The paper "Gauge Theories and Macdonald Polynomials" presents a detailed paper of the superconformal index in four-dimensional $\mathcal{N}=2$ superconformal field theories (SCFTs). The authors explore this index in various interesting limits, specifically focusing on its behavior when contributions are limited to states annihilated by more than one supercharge. The work extrapolates from $SU(2)$ generalized quivers to conjecture explicit formulae for all $A$-type quivers of class $\mathcal{S}$, despite most lacking a Lagrangian description.

Key Contributions

1. Superconformal Index Expansion: The paper explores the computation of the $\mathcal{N}=2$ superconformal index, emphasizing its computation using a trace formula in the space of states on $S^{d-1}$. The index is expanded in terms of three superconformal fugacities, typically $(p, q, t)$, which relate to specific combinations of conformal dimensions and R-charges.
2. Interpretation as a TQFT: A significant insight of this paper is the interpretation of the superconformal index as a correlator in a two-dimensional topological quantum field theory (TQFT). This formalism provides a powerful framework for analyzing the index by considering Riemann surfaces obtained from gluing together three-punctured spheres, each associated with different SCFTs, interconnected by cylinders representing gauge interactions.
3. Structure Constants and Orthogonality: The authors propose that the superconformal index of a three-punctured sphere, or $T_N$ theory, can be expressed in terms of Macdonald polynomials. They illustrate that the orthonormal polynomials diagonalize the structure constants of the TQFT and suggest a corresponding basis that involves a new measure.
4. Limits and Simplifications:

The paper explores limits where the index is simplified due to enhanced supersymmetry: - Hall-Littlewood Index: Simplified to contributions from specific short multiplets related to the Higgs branch. - Schur Index: Related to $q$-deformed Yang-Mills and characterizes theories using single parameter fugacities. - Macdonald Index: Generalized to include two parameters and relies on the Macdonald polynomials.

1. Combinatorial Proofs and Identities: In a striking use of representation theory, the paper provides a "physics proof" of certain Macdonald polynomial identities based on the gauge theory setup, illustrating a deep interplay between combinatorial structures and field theoretic constructs.
2. Practical Implications and Deductions: The authors propose that the insights obtained could provide a blueprint for computations in broader classes of theories within the $A$-series and suggest extensions to $D$ and $E$-series. These implications are significant for understanding dualities and extending index computations across various SCFTs lacking straightforward Lagrangian forms.

Implications and Future Directions

The results have profound implications for theoretical physics, particularly in enhancing the understanding of SCFTs, dualities, and topological field theories. The paper's conjectures about the structure constants and the diagonalization basis in terms of Macdonald polynomials offer a starting point for further exploration within non-Lagrangian field theories and their connections to mathematical frameworks, such as symmetric polynomials and quantum integrable systems.

Future speculations include extending these results and techniques to more complex theories and perhaps establishing connections with emerging areas in mathematical physics, such as refined Chern-Simons theory or elliptic Calogero-Moser systems.

Conclusion

This paper makes a substantial contribution to the field of gauge theories by providing new computational tools and theoretical insights into the structure of $\mathcal{N}=2$ SCFTs and their superconformal indices. By leveraging the power of Macdonald polynomials and topological quantum field theory frameworks, the authors pave the way for further developments in the understanding of these complex quantum field theories. The depth of mathematical and physical intersections explored in this work enriches the landscape of both theoretical physics and modern mathematical analysis.