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First of all he says that a vortex is something around which a wavefunction picks up a phase of 2##\pi##. He then says that a single electron in the a laughlin state with angular momentum = m can be seen as an electron with (m-1) vortices attached to it. This interpretation is based on the fact that in the Laughlin wavefunction the terms are of the form ##(z_i-z_j)^m## for which he says that the first ##(z_i-z_j)## is needed for that fermi statistics, whereas the remaining m-1 terms are just vortices.

Why this distinction? Something of the form ##(z_i-z_j)^m## should have a wavefunction pick up a phase of 2##\pi##m regardless of what you call it. i.e. why isnt what he is calling an electron a vortex as well?

And this matters, in equation 3.34 when he is computing the berry phase he has the Aharanov Bohm term and then an additional 2##\pi##(m-1) for each each electron as though only the m-1 vortices contributed a phase and not the "electron".

The other thing I am struggling with is the ##\nu## = 1/2 Landau level. He derives (Eqn. 3.35) that the effective magnetic field is B* = B - (m-1)n##\Phi_0## where n is the density. The density is given by n = ##\nu##B/##Phi_0##. Therefore B* = B(1-##\nu##(m-1)). A Laughlin state with angular momentum = m has filling fraction ##\nu## = 1/m so we get that B* = B(1-(m-1)/m) = B/m. How then does the ##\nu## = 1/2 filled landau level give you B* = 0 in equation 3.40?