quantum field theory - PDFs expressed through matrix elements of bi-local operators

Extracted from 'At the frontier of ParticlePhysics, handbook of QCD, volume 2',

'...in the physical Bjorken $x$-space formulation, an equivalent definition of PDFs can be given in terms of matrix elements of bi-local operators on the lightcone. The distribution of quark 'a' in a parent 'X' (either hadron or another parton) is defined as $$f^a_X (\zeta, \mu) = \frac{1}{2} \int \frac{\text{d}y^-}{2\pi} e^{-i \zeta p^+ y^-} \langle X | \bar \psi_a(0, y^-, \mathbf 0) \gamma^+ U \psi_a(0) | X \rangle ,'$$ where $$U = \mathcal P \exp \left( -ig \int_0^{y^-} \text{d}z^- A_a^+(0,z^-, \mathbf 0) t_a \right)$$ is the Wilson line.

My questions are:

1) Where does this definition come from? I'd like to particularly understand in detail the content of the rhs (i.e the arguments of the spinors, why an integral over $y^-$ etc)

2) The review also mentions that in the physical gauge $A^+=0$, $U$ becomes the identity operator in which case $f^a_X$ is manifestly the matrix element of the number operator for finding quark 'a' in X with plus momentum fraction $p_a^+ = \zeta p_X^+, p_a^T=0$. Why is $A^+=0$ the physical gauge?