# Electro-Magnetic Duality, Magnetic monopoles and Topological Insulators

Thesis

The Maxwell equations of electrodynamics acquire an additional symmetry if one assumes the existence of hypothetical particles-magnetic monopoles, carrying a magnetic charge. The additional internal symmetry is the electromagnetic duality generated by the rotations in the space of electric and magnetic charges. In this project we revise the electromagnetic duality in his global aspect starting with the celebrated Dirac monopole, a singular solution in a slightly modified Maxwell theory. We then take account of the new insight on the duality in the broken SO(3) gauge theory where the magnetic monopoles arose as finite-energy smooth solution (found by ’t Hooft and Polyakov). The stability of these monopoles is guaranteed by the conservation of topological invariants, i.e., these are topologically protected states. The spectrum of the gauge theory states enjoys a symmetry between the electrically charged gauge boson and the magnetic monopole, manifesting a quantum electro-magnetic duality which turns out to be a part of larger SL(2, Z)-group symmetry acting on the 2-dimensional charge lattice. Recently the idea of magnetic monopoles and dyons was revived by the discovery of new kind of materials known as topological insulators. The theoretical considerations in the modified axion electrodynamics show that the electric charges on the boundary of a topological insulator induce mirror images carrying magnetic charges. We consider carefully the mirror images in the case of topological insulator with planar and spherical boundary. We then provide a description of the induced mirror images in a manifestly SL(2, Z)-covariant form.