Orthogonal and Symplectic Parabolic Connections and Stack of Roots

Let $D$ be an effective divisor on a smooth projective variety $X$ over an algebraically closed field $k$ of characteristic $0$. We show that there is a one-to-one correspondence between the class of orthogonal (respectively, symplectic) parabolic vector bundles on $X$ with parabolic structure along $D$ and having rational weights and the class of orthogonal (respectively, symplectic) vector bundles on certain root stacks associated to this data. Using this, we describe the orthogonal (respectively, symplectic) vector bundles on the root stack as reductions of the structure group to orthogonal (respectively, symplectic) groups. When $D$ is a divisor with strict normal crossings, we prove a one-to-one correspondence between the class of orthogonal (respectively, symplectic) parabolic connections on $X$ with rational weights, and the class of orthogonal (respectively, symplectic) logarithmic connections on certain fiber product of root stacks with poles along a divisor with strict normal crossings.