3d N=2 Theories from Cluster Algebras

TL;DR

This paper introduces a novel way to describe certain 3d N=2 theories using cluster algebras, linking their partition functions to 3-manifold invariants and extending to theories from 6d compactifications.

Contribution

It defines a new formalism for 3d N=2 theories via quivers and mutation sequences, connecting them to cluster partition functions and 3-manifold invariants.

Findings

Partition functions match cluster partition functions.

Includes theories from 6d (2,0) compactifications.

Relates 3d theories to Chern-Simons invariants.

Abstract

We propose a new description of 3d $\mathcal{N}=2$ theories which do not admit conventional Lagrangians. Given a quiver $Q$ and a mutation sequence $m$ on it, we define a 3d $\mathcal{N}=2$ theory $\mathcal{T}[(Q,m)]$ in such a way that the $S^3_b$ partition function of the theory coincides with the cluster partition function defined from the pair $(Q, m)$. Our formalism includes the case where 3d $\mathcal{N}=2$ theories arise from the compactification of the 6d $(2,0)$ $A_{N-1}$ theory on a large class of 3-manifolds $M$, including complements of arbitrary links in $S^3$. In this case the quiver is defined from a 2d ideal triangulation, the mutation sequence represents an element of the mapping class group, and the 3-manifold is equipped with a canonical ideal triangulation. Our partition function then coincides with that of the holomorphic part of the $SL(N)$ Chern-Simons partition function on $M$.