Irreducibility of the moduli space of orthogonal instanton bundles on ${\mathbb{P}}^n$

In order to obtain existence criteria for orthogonal instanton bundles on ${\mathbb{P}}^n$, we provide a bijection between equivalence classes of orthogonal instanton bundles with no global sections and symmetric forms. Using such correspondence we are able to provide explicit examples of orthogonal instanton bundles with no global sections on ${\mathbb{P}}^n$ and prove that every orthogonal instanton bundle with no global sections on ${\mathbb{P}}^n$ and charge $c\ge 3$ has rank $r\le (n-1)c$. We also prove that when the rank $r$ of the bundles reaches the upper bound, ${\mathcal{M}}_{{\mathbb{P}}^n}^{\mathcal{O}}(c,r)$, the coarse moduli space of orthogonal instanton bundles with no global sections on ${\mathbb{P}}^n$, with charge $c\ge 3$ and rank $r$, is affine, reduced and irreducible. Last, we construct Kronecker modules to determine the splitting type of the bundles in ${\mathcal{M}}_{{\mathbb{P}}^n}^{\mathcal{O}}(c,r)$, whenever is non-empty.