This question is relevant to Mirror symmetry and moduli space.
The linear sigma model in $U(k)$ with $Nk$ chiral fields vacua equation can be reduced as \begin{align} \sum_{i,j=1}^k \left(\sum_s^N \bar{\phi}_{is}\phi^{js}-\delta_i^{\phantom{i}j}r\right)=0 \end{align} The solution of $\phi$ is know as Grassmanian $Gr(k, N)$ There is some comment on this in witten's "The Verlinde Algebra and the cohomology of the Grassmannian" paper, which seems plausible, but i don't get it matematically.
Thus first i want to know why this vacua space is Grassmannian $Gr(k, N)$ in mathematically.
Next, I want to know the terminology of " Holomorphic vector bundle over Grassmanian."
For the linear sigma model in $U(k)$ $N_f k$ fundamental chrial fields with $N_a k$ anti-fundamental chiral fields the solution of vaccua equation Holomorphic vector bundle $N_a$ over Grassmanian $Gr(k, N_f)$.
I want to know what this means.