Three Dimensional Topological Field Theories and Nahm Sum Formulas

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.

31 pages, v2: some clarifications, refs added, v3: minor changes

References in corpus (25)

• Down the rabbit hole with theories of class S
• M-theoretic Genesis of Topological Phases
• Non-unitary TQFTs from 3D $\mathcal{N}=4$ rank 0 SCFTs
• $\hat{Z}$ at large $N$: from curve counts to quantum modularity
• Classification of Unitary RCFTs with Two Primaries and Central Charge Less Than 25
• Classification of RCFT from Holomorphic Modular Bootstrap: A Status Report
• Expanding 3d $\mathcal{N}=2$ Theories around the Round Sphere
• Boundary vertex algebras for 3d $\mathcal{N}=4$ rank-0 SCFTs
• On Classification of Fermionic Rational Conformal Field Theories
• BPS Indices, Modularity and Perturbations in Quantum K-theory
• A non-unitary bulk-boundary correspondence: Non-unitary Haagerup RCFTs from S-fold SCFTs
• 3d SUSY enhancement and non-semisimple TQFTs from four dimensions
• Fermionic Rational Conformal Field Theories and Modular Linear Differential Equations
• Representation theory of $L_k\left(\mathfrak{osp}(1|2)\right)$ from vertex tensor categories and Jacobi forms
• Three-Dimensional Topological Field Theories and Non-Unitary Minimal Models
• Affine $\mathcal{W}$-algebras and Miura maps from 3d $\mathcal{N}=4$ non-Abelian quiver gauge theories
• On the 4d/3d/2d view of the SCFT/VOA correspondence
• Non-hyperbolic 3-manifolds and 3D field theories for 2D Virasoro minimal models
• Bridging 4D QFTs and 2D VOAs via 3D high-temperature EFTs
• Algebraic K-Theory and Partition Functions in Conformal Field Theory
• 3d-3d Correspondence and 2d $\mathcal{N}=(0,2)$ Boundary Conditions
• Boundary Chiral Algebras and Holomorphic Twists
• 3D bulk field theories for 2D non-unitary N=1 supersymmetric minimal models
• 3D TFTs from 4d $\mathcal{N}=2$ BPS Particles
• Mirror Symmetry and Level-rank Duality for 3d $\mathcal{N}=4$ Rank 0 SCFTs

Cited by in corpus (2)

• The 3d $A$-model and generalised symmetries, Part I: bosonic Chern-Simons theories
• A Family of Vertex Algebras from Argyres-Douglas Theory