| Résume | The fractional quantum Hall effect arises in certain two-dimensional electron systems under strong magnetic fields, it is defined by the Hall conductance taking only rational values rather than varying
continuously. This apparent quantization suggests that particle behavior is governed by an underlying topological and geometric structure, and provides the physical motivation for the geometric constructions considered in this talk. Fix a compact Riemann surface $C$ and a number of particles $N$. Trial wavefunctions are originally defined as explicit functions on the complex plane to encode a given observed value of the Hall conductance. To define them over an abstract surface $C$, we retain only their features: holomorphicity, (anti)symmetry and prescribed vanishing. They can thus be modeled as sections of certain holomorphic line bundles over the symmetric product $S^NC$ that depend on a parameter on the Jacobian $J(C)$. Varying this parameter produces a sheaf $V$ over $J(C)$. The Hall conductance is then recovered by the first Chern class of $V$ divided by its rank. I will explain how to properly build such sheaves, with emphasis on the one for multilayer Laughlin states, and what tools one can use to compute their Chern character. |
