2 theories from cluster algebras

We propose a new description of 3d theories which do not admit conventional Lagrangians. Given a quiver and a mutation sequence on it, we define a 3d theory in such a way that the partition function of the theory coincides with the cluster partition function defined from the pair . Our formalism includes the case where 3d theories arise from the compactification of the 6d theory on a large class of 3-manifolds , including complements of arbitrary links in . 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 ChernSimons partition function on .