Three Dimensional Topological Field Theories and Nahm Sum Formulas

TL;DR

This paper explores the connection between 3d supersymmetric gauge theories, Nahm sum formulas, and modular functions, aiming to identify theories that support rational boundary algebras and relate to Zagier's work on Nahm sums.

Contribution

It systematically searches for low-rank 3d $ ext{N}=2$ abelian theories that flow to topological field theories or rank-zero SCFTs, linking them to Nahm sum formulas and modularity.

Findings

Identification of specific 3d theories with rational boundary algebras

Comparison of these theories' indices with Zagier's Nahm sums

Insights into a possible generalization of Nahm's conjecture

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.