  It is a great discovery in physics of the twentieth century that the
elementary particles in Nature are dictated by gauge forces, characterized by a
nonintegrable phase factor that an elementary particle of charge $q$ acquires
from A to B points: $$P \exp\left(i\frac{q}{\hbar c} \int_A^B A_\mu
dx^\mu\right)$$ where $A_\mu$ is the gauge potential and $P$ stands for path
ordering. In a many-body system of strongly correlated electrons, if the
so-called Mott gap is opened up by interaction, the corresponding Hilbert space
will be fundamentally changed. A novel nonintegrable phase factor known as
phase-string will appear and replace the conventional Fermi statistics to
dictate the low-lying physics. Protected by the Mott gap, which is clearly
identified in the high-$T_c$ cuprate with a magnitude > 1.5 eV, such a singular
phase factor can enforce a fractionalization of the electrons, leading to a
dual world of exotic elementary particles with a topological gauge structure. A
non-Fermi-liquid "parent" state will emerge, in which the gapless Landau
quasiparticle is only partially robust around the so-called Fermi arc regions,
while the main dynamics are dominated by two types of gapped spinons.
Antiferromagnetism, superconductivity, and a Fermi liquid with full Fermi
surface can be regarded as the low-temperature instabilities of this new parent
state. Both numerics and experiments provide direct evidence for such an
emergent physics of the Mottness, which lies in the core of a high-$T_c$
superconducting mechanism.
