Lecturer: Dr. Semyon Klevtsov
Affiliation: University of Cologne
Abstract:
We consider the Laughlin states for the integer
and fractional Quantum Hall effect on Riemann
surfaces with an arbitrary Riemannian metric.
We ask how does the logarithm of the norm
of the state, or the free energy, depend on the
metric, for the large magnetic flux $k$.
Turns out, some recent methods of Kahler
geometry (which we review) are particularly
useful when solvingthis problem. In the integer
case the problem is fully controlled by the
asymptotic expansion of the so-called Bergman
kernel. We derive the first few termsin the $1/k$
expansion of the free energy and explain the
structure of the expansion to all orders in $1/k$.
In the fractional case we derive the expansion
on the sphere, by a different method, based
on the free field representation.
Physical properties of quantum Hall, such as
Hall conductance, anomalous viscosity etc, are
manifest in the structure of the expansion.
Time permitting, we will discuss some interesting
math applications of the problem, in particular
to the Yau-Tian-Donaldson program in Kahler
geometry and to random metrics.
Based on 1309.7333 and 1410.xxxx.
Additional details of the upcoming Joint Seminars
in Theoretical High Energy Physics can be found
on the following link