Gauge Theories and Macdonald Polynomials

TL;DR

The paper develops a two-parameter Macdonald-based TQFT framework for the N=2 superconformal index of A-type class S theories, enabling explicit formulas across Lagrangian and non-Lagrangian quivers. By focusing on limits with enhanced supersymmetry (Hall-Littlewood, Schur, Macdonald, and Coulomb-branch), it shows that the index reduces to well-studied symmetric polynomials and 2d Yang–Mills-like structures, with Macdonald polynomials diagonally organizing the structure constants. The authors conjecture comprehensive HL and Macdonald expressions for general SU(k) quivers with arbitrary punctures, test them against dualities and Argyres-Seiberg-type relations, and explore large-k and higher-rank E-type SCFTs (E6–E8), uncovering deep connections to Higgs-branch Hilbert series, refined Chern–Simons, and holographic interpretations. The work suggests a unified, highly structured picture linking 4d protected spectra to 2d topological theories, with potential extensions to full elliptic Macdonald indices and a microscopic 6d origin via elliptic integrable systems. Overall, it provides a powerful toolkit for computing and understanding the N=2 index in broad families of class S theories.

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.

Figures (6)

• Figure 1: Association of flavor fugacities for the vertex corresponding to the $331$ of the $SU(3)$ quivers. Here $a_1a_2a_3=1$ and $b_1b_2b_3=1$.
• Figure 2: Association of the flavor fugacities for a generic puncture. Punctures are classified by embeddings of $SU(2)$ in $SU(k)$, so they are specified by the decomposition of the fundamental representation of $SU(k)$ into irreps of $SU(2)$, that is, by a partition of $k$. Graphically we represent the partition by an auxiliary Young diagram $\Lambda$ with $k$ boxes, read from left to right. In the figure we have the fundamental of $SU(26)$ decomposed as $\mathbf{5} + \mathbf{5} + \mathbf{4} + \mathbf{4} + \mathbf{4} + \mathbf{2} + \mathbf{1} + \mathbf{1}$. The commutant of the embedding gives the residual flavor symmetry, in this case $S(U(3)\times U(2)\times U(2)\times U(1))$, where the $S(\dots)$ constraint amounts to removing the overall $U(1)$. The $\tau$ variable is viewed here as an $SU(2)$ fugacity, while the Latin variables are fugacities of the residual flavor symmetry. The $S(\dots)$ constraint implies that the flavor fugacities satisfy $(ab)^{5}(cde)^4f^2gh=1$.
• Figure 3: The factors ${\frak a}^i_k$ associated to a generic Young diagram. The upper index is the row index and the lower is the column index. In $\bar{\frak a}^i_k$ one takes the inverse of flavor fugacities while $\tau$ is treated as real number. As before, the flavor fugacities in this example satisfy $(ab)^{5}(cde)^4f^2gh=1$.
• Figure 4: Association of the flavor fugacities for the $E_7$ vertex. Here $\prod_{i=1}^4b_i=\prod_{i=1}^4a_i=1$.
• Figure 5: Association of the flavor fugacities for the $E_8$ vertex. Here $\prod_{i=1}^3b_i=\prod_{i=1}^6a_i=1$.