  Pair density Waves (PDW) are a inhomogeneous superconducting states whose
Cooper pairs posses a finite momentum resulting in a oscillatory gap in space,
even in the absence of an external magnetic field. There is growing evidence
for the existence of PDW superconducting order in many strongly correlated
materials particularly in the cuprate superconductios and in several other
different types of systems. A feature of the PDW state is that inherently it
has a CDW as a composite order associated with it. Here we study the structure
of the electronic topological defects of the PDW, paying special attention to
the half-vortex and its electronic structure that can be detected in STM
experiments. We discuss tell-tale signatures of the defects in violations of
inversion symmetry, in the excitation spectrum and their spectral functions in
the presence of topological defects. We discuss the "Fermi surface" topology of
Bogoliubov quasi-particle of the PDW phases and we briefly discuss the role of
quasi-particle-interference.
