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It is well known that in two spatial dimensions the fractional quantum Halleffect (FQHE) deals with point-like anyons that carry fractional electriccharge and statistics. Moreover, in presence of a SO(3) order parameter,point-like skyrmions emerge and play a central role in the correspondingquantum Hall ferromagnetic phase. In this work, we show that in six spatialdimensions, the FQHE for extended objects shares very similar features with itstwo-dimensional counterpart. In the higher-dimensional case, theelectromagnetic and hydrodynamical one-form gauge fields are replaced bythree-form gauge fields and the usual point-like anyons are replaced bymembranes, namely two-dimensional extended objects that can carry fractionalcharge and statistics. We focus on skyrmionic membranes, which are associatedto a SO(5) order parameter and give rise to an higher-dimensional generalizatonof the quantum Hall ferromagnetism. We show that skyrmionic membranes naturallycouple to the curved background through a generalized Wen-Zee term and can giveus some insights about the chiral conformal field theory on the boundary. Wethen present a generalization of the Witten effect in six spatial dimensions byshowing that one-dimensional extended monopoles (magnetic strings) in the bulkof the FQH states can acquire electric charge through an axion field bybecoming dyonic strings.

Fractional Quantum Hall Effect for Extended Objects: from Skyrmionic Membranes to Dyonic Strings