N = 2 theories from cluster algebras

We propose a new description of 3d N=2 theories which do not admit conventional Lagrangians. Given a quiver Q and a mutation sequence m on it, we define a 3d N=2 theory T[(Q,m)] in such a way that the S_b³ partition function of the theory coincides with the cluster partition function defined from the pair (Q,m). Our formalism includes the case where 3d 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 S³. 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