$\mathbb{A}^1$-connectedness of moduli of vector bundles on a stacky curve
Let $\mathfrak{C}$ be a projective stacky curve over an infinite field $k$ with the coarse space $C$, a geometrically connected smooth projective curve, together with a generating sheaf $\mathcal{E}$ on $\mathfrak{C}$. We consider the moduli stack of $\mathcal{E}$-semistable vector bundles of fixed determinant and multiplicities on $\mathfrak{C}$, and show that it is $\mathbb{A}^1$-connected if and only if it admits a $k$-rational point. We show that this holds whenever genus of $C$ is at least $2$ and $C(k)\neq \emptyset$. As a consequence, we conclude that, for any system of weights $\boldsymbolα$, the moduli stack of parabolic $\boldsymbolα$-semistable vector bundles of fixed determinant and quasi-parabolic data on a geometrically connected smooth projective curve is $\mathbb{A}^1$-connected.