We describe a web of well-known dualities connecting quantum field theories in $d=1+1$ dimensions. The web is constructed by gauging ${bf Z}_2$ global symmetries and includes a number of perennial favourites such as the Jordan-Wigner transformation,
Kramers-Wannier duality, bosonization of a Dirac fermion, and T-duality. There are also less-loved examples, such as non-modular invariant $c=1$ CFTs that depend on a background spin structure.
Quantum Hall matrix models are simple, solvable quantum mechanical systems which capture the physics of certain fractional quantum Hall states. Recently, it was shown that the Hall viscosity can be extracted from the matrix model for Laughlin states.
Here we extend this calculation to the matrix models for a class of non-Abelian quantum Hall states. These states, which were previously introduced by Blok and Wen, arise from the conformal blocks of Wess-Zumino-Witten conformal field theory models. We show that the Hall viscosity computed from the matrix model coincides with a result of Read, in which the Hall viscosity is determined in terms of the weights of primary operators of an associated conformal field theory.
We study a supersymmetry breaking deformation of the 2d N=(2,2) cigar=Liouville mirror pair, first introduced by Hori and Kapustin. We show that mirror symmetry flows in the infra-red to 2d bosonization, with the theories reducing to massive Thirring
and Sine-Gordon respectively. The exact bosonization map emerges at one-loop. We further compactify non-supersymmetric 3d bosonization dualities on a circle and argue that these too flow to 2d bosonization at long distances.
