Three dimensional topological field theories and Nahm sum formulas

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 [D. Zagier, in Frontiers in number theory, physics, and geometry II, Springer, Berlin (2007)], D. Zagier identified a list of Nahm sum expressions that are modular functions under a congruence subgroup of SL(2,Z) S L ( 2 , ℤ ) , which 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 N=2 𝒩 = 2 supersymmetric gauge theories, we perform a general search over low-rank 3d N=2 𝒩 = 2 Abelian Chern-Simons matter theories which either flow to unitary TFTs or N=4 𝒩 = 4 rank-zero SCFTs in the infrared. These are exceptional classes of 3d theories, which are expected to support rational and C_2 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.