Noncommutative Geometry, Quantum Hall Effect and Berry Phase
Abstract
Taking resort to Haldane's spherical geometry we can visualize fractional quantum Hall effect on the noncommutative manifold
M
4
×
Z
N
with
N>2
and odd. The discrete space leads to the deformation of symplectic structure of the continuous manifold such that the symplectic area is given by
△p.△q=2πmℏ
with
m
an odd integer which is related to the Berry phase and the filling factor is given by
1
m
. We here argue that this is equivalent to the noncommutative field theory as prescribed by Susskind and Polychronakos which is characterized by area preserving diffeomorphism. The filling factor
1
m
is determined from the change in chiral anomaly and hence the Berry phase as envisaged by the star product.