Gauge theories labeled by three-manifolds and 3d N=2 dualities

[atyy]

http://arxiv.org/abs/1108.4389
Gauge Theories Labelled by Three-Manifolds
Tudor Dimofte, Davide Gaiotto, Sergei Gukov
(Submitted on 22 Aug 2011)
We propose a dictionary between geometry of triangulated 3-manifolds and physics of three-dimensional N=2 gauge theories. Under this duality, standard operations on triangulated 3-manifolds and various invariants thereof (classical as well as quantum) find a natural interpretation in field theory. For example, independence of the SL(2) Chern-Simons partition function on the choice of triangulation translates to a statement that S^3_b partition functions of two mirror 3d N=2 gauge theories are equal. Three-dimensional N=2 field theories associated to 3-manifolds can be thought of as theories that describe boundary conditions and duality walls in four-dimensional N=2 SCFTs, thus making the whole construction functorial with respect to cobordisms and gluing.

 

[mitchell porter]

These field theories in 2+1 dimensions result from compactifying one of those nonlagrangian 5+1 dimensional theories on a 3-manifold. This is the 3-manifold "label" mentioned in the title of the paper - the shape of the three compact space dimensions that are being neglected. Different triangulations of the compact 3-manifold, and also mirror-symmetry relations connecting different quantum 3-manifolds, correspond to different formulations of the same 2+1 dimensional theory.

To my mind what it most resembles is deconstruction, because the combinatorial structure of the compact 3-manifold translates to properties of the 2+1 dimensional field theory, just as the "moose diagrams" and other combinatorial structures appearing in deconstruction describe discretized extra dimensions.