The Superconformal Index of Class S Theories of Type D

TL;DR

This work extends the 4d/2d dictionary of class ${ m S}$ theories to type ${ m D}$ by incorporating $ twist lines and deriving a Macdonald-limit ${2d}$ TQFT that computes the ${ m S^3\times S^1}$ superconformal index. The authors propose a diagonal basis formed from Macdonald polynomials for ${ m SO}$ and ${ m USp}$ sectors, enabling closed-form, duality-consistent expressions for basic building blocks and their gluing in the presence of twist lines. They provide explicit results for low-rank theories ${ m D_2}$ and ${ m D_3}$ in Hall-Littlewood, Schur, and Macdonald limits, and conjecture a general Macdonald-limit structure for ${ m D_n}$ theories, including a genus-$g$ partition function interpretation linked to refined Chern-Simons theory. The paper further generalizes the index to partially closed punctures via Young diagrams, detailing how to assign fugacities and ${ m K}$-factors, and illustrating nontrivial relationships between ${ m SO}$ and ${ m USp}$ data that preserve diagonality and dualities. Overall, the results deepen the 4d/2d correspondence for D-type theories, support generalized S-duality in twisted settings, and connect to known A-type reductions through ${ m D_2}$ and ${ m A_1}$ decompositions, with potential implications for surface operators and refined quantum field theories.

Abstract

We consider the superconformal index of class S theories of type D, which arise by compactification of the (2,0) D_n theories on a punctured Riemann surface C. We also allow for the presence of twist lines on C associated to the Z_2 outer automorphism of D_n. For a two-parameter slice (p=0, q, t) in the space of superconformal fugacities, we determine the 2d TQFT that computes the index.

Figures (10)

• Figure 1: Quiver and corresponding curve for the $\mathrm{SO}(2n)$ gauge theory with $N_f = 2n-2$ hypermultiplets in the vector representation of $\mathrm{SO}(2n)$.
• Figure 2: Quiver and corresponding curve for the $\mathrm{USp}(2n-2)$ gauge theory with $N_f = 2n$ hypermultiplets in the fundamental representation of $\mathrm{USp}(2n-2)$.
• Figure 3: Construction of the $T_{\mathrm{SO}(2n)}$ theory. The $\cdots$ in the middle of the top quiver stand for a sequence of alternating $\mathrm{SO}(2n)$ and $\mathrm{USp}(2n-2)$ gauge groups. The other $\cdots$ on the left/right stand for a sequence of alternating odd $\mathrm{SO}$ and $\mathrm{USp}$ gauge groups of increasing/decreasing rank. The symbol $\subset$ means that we gauge a $\mathrm{SO}(2n-1)$ subgroup of $\mathrm{SO}(2n)$.
• Figure 4: Construction of the $\tilde{T}_{SO(2n)}$ theory. The $\cdots$ in the middle of the top quiver stand for a sequence of alternating $\mathrm{SO}(2n)$ and $\mathrm{USp}(2n-2)$ gauge groups. The other $\cdots$ on the left/right stand for a sequence of alternating odd $\mathrm{SO}$ and $\mathrm{USp}$ gauge groups of increasing/decreasing rank. The symbol $\subset$ means that we gauge a $\mathrm{SO}(2n-1)$ subgroup of $\mathrm{SO}(2n)$.
• Figure 5: The three-punctured spheres with maximal $\mathrm{SO}$ punctures and maximal or empty $\mathrm{USp}$ punctures. We indicate the flavor fugacity assignments and our notation for the corresponding index.