  Many trial wavefunctions for fractional quantum Hall states in a single
Landau level are given by functions called conformal blocks, taken from some
conformal field theory. Also, wavefunctions for certain paired states of
fermions in two dimensions, such as p_x+ip_y states, reduce to such a form at
long distances. Here we investigate the adiabatic transport of such
many-particle trial wavefunctions using methods from two-dimensional field
theory. One context for this is to calculate the statistics of widely-separated
quasiholes, which has been predicted to be non-Abelian in a variety of cases.
The Berry phase or matrix (holonomy) resulting from adiabatic transport around
a closed loop in parameter space is the same as the effect of analytic
continuation around the same loop with the particle coordinates held fixed
(monodromy), provided the trial functions are orthonormal and holomorphic in
the parameters so that the Berry vector potential (or connection) vanishes. We
show that this is the case (up to a simple area term) for paired states
(including the Moore-Read quantum Hall state), and present general conditions
for it to hold for other trial states (such as the Read-Rezayi series). We
argue that trial states based on a non-unitary conformal field theory do not
describe a gapped topological phase, at least in many cases. By considering
adiabatic variation of the aspect ratio of the torus, we calculate the Hall
viscosity, a non-dissipative viscosity coefficient analogous to Hall
conductivity, for paired states, Laughlin states, and more general quantum Hall
states. Hall viscosity is an invariant within a topological phase, and is
generally proportional to the "conformal spin density" in the ground state.
