On the essential dimension of symplectic vector bundles over curves

Let $X$ be a smooth geometrically connected projective curve of genus at least 2 over a field of characteristic zero. We compute the essential dimension of the moduli stack of symplectic bundles over $X$. Unlike the case of vector bundles, we are able to precisely compute the essential dimension as the generic gerbe of the moduli stack has period 2 over it’s moduli space.