  Electromagnetic field (EMF) is the most fundamental field in condensed-matter
physics. Interaction between electrons, electron-ion interaction, and ion-ion
interaction are all of the electromagnetic origin, while the other 3
fundamental forces, i.e., gravitational force, weak and strong interactions are
irrelevant in the energy/length scales of condensed-matter physics. Also the
physical properties of condensed-matters such as transport, optical, magnetic
and dielectric properties, are almost described as their electromagnetic
responses. In addition to this EMF in the low energy sector, it often happens
that the gauge fields appear as the emergent phenomenon due to the projection
of the electronic wavefunctions onto the curved manifold of the Hilbert
sub-space. These emergent electromagnetic fields (EEMF's) play important roles
in many places in condensed-matter physics including the quantum Hall effect,
strongly correlated electrons, and also in non-interacting electron systems. In
this article, we describe its fundamental idea and some of the applications
recently studied.
