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Many trial wavefunctions for fractional quantum Hall states in a singleLandau level are given by functions called conformal blocks, taken from someconformal field theory. Also, wavefunctions for certain paired states offermions in two dimensions, such as p_x+ip_y states, reduce to such a form atlong distances. Here we investigate the adiabatic transport of suchmany-particle trial wavefunctions using methods from two-dimensional fieldtheory. One context for this is to calculate the statistics of widely-separatedquasiholes, which has been predicted to be non-Abelian in a variety of cases.The Berry phase or matrix (holonomy) resulting from adiabatic transport arounda closed loop in parameter space is the same as the effect of analyticcontinuation around the same loop with the particle coordinates held fixed(monodromy), provided the trial functions are orthonormal and holomorphic inthe parameters so that the Berry vector potential (or connection) vanishes. Weshow that this is the case (up to a simple area term) for paired states(including the Moore-Read quantum Hall state), and present general conditionsfor it to hold for other trial states (such as the Read-Rezayi series). Weargue that trial states based on a non-unitary conformal field theory do notdescribe a gapped topological phase, at least in many cases. By consideringadiabatic variation of the aspect ratio of the torus, we calculate the Hallviscosity, a non-dissipative viscosity coefficient analogous to Hallconductivity, for paired states, Laughlin states, and more general quantum Hallstates. Hall viscosity is an invariant within a topological phase, and isgenerally proportional to the "conformal spin density" in the ground state.

Non-Abelian adiabatic statistics and Hall viscosity in quantum Hall states and p_x+ip_y paired superfluids