Three Dimensional Topological Field Theories and Nahm Sum Formulas

TL;DR

The work develops a physical realization of Nahm-sum modular functions through 3d ${\cal N}=2$ abelian Chern-Simons-matter theories, aiming to realize IR endpoints as either unitary TFTs or ${\cal N}=4$ rank-zero SCFTs with rational boundary VOAs. By leveraging half-indices, superconformal indices and F-maximization, the authors classify admissible CS-level matrices $K$ (up to rank 3) and identify 27 distinct examples plus infinite families that flow to the targeted IR phases, organizing them into eight duality classes. They show that many of the resulting boundary algebras reproduce known RCFT characters (e.g., minimal models and $\text{osp}$-type theories) and, in several cases, align with Zagier’s modular Nahm-sum list, while also revealing generalized Nahm-conjecture structures and new modular data arising from their modified half-sum form. The results illuminate a rich interplay between bulk 3d theories, boundary VOAs, and modular objects, providing a physically motivated extension and refinement of Nahm’s conjecture. The approach offers a framework to systematically explore modular invariants via IR dualities in topological boundary theories with potential broader applications in RCFT classification and holographic contexts.

Abstract

It is known that a large class of characters of 2d conformal field theories (CFTs) can be written in the form of a Nahm sum. In \cite{Zagier:2007knq}, D. Zagier identified a list of Nahm sum expressions that are modular functions under a congruence subgroup of $SL(2,\mathbb{Z})$ and can be thought of as candidates for characters of rational CFTs. Motivated by the observation that the same formulas appear as the half-indices of certain 3d $\mathcal{N}=2$ supersymmetric gauge theories, we perform a general search over low-rank 3d $\mathcal{N}=2$ abelian Chern-Simons matter theories which either flow to unitary TFTs or $\mathcal{N}=4$ rank-zero SCFTs in the infrared. These are exceptional classes of 3d theories, which are expected to support rational and $C_2$-cofinite chiral algebras on their boundary. We compare and contrast our results with Zagier's and comment on a possible generalization of Nahm's conjecture.