Construction of stable rank 2 vector bundles on $\mathbb{P}^3$ via symplectic bundles

In this article we study the Gieseker-Maruyama moduli spaces $\mathcal{B}(e,n)$ of stable rank 2 algebraic vector bundles with Chern classes $c_1=e\in\{-1,0\},\ c_2=n\ge1$ on the projective space $\mathbb{P}^3$. We construct two new infinite series $\Sigma_0$ and $\Sigma_1$ of irreducible components of the spaces $\mathcal{B}(e,n)$, for $e=0$ and $e=-1$, respectively. General bundles of these components are obtained as cohomology sheaves of monads, the middle term of which is a rank 4 symplectic instanton bundle in case $e=0$, respectively, twisted symplectic bundle in case $e=-1$. We show that the series $\Sigma_0$ contains components for all big enough values of $n$ (more precisely, at least for $n\ge146$). $\Sigma_0$ yields the next example, after the series of instanton components, of an infinite series of components of $\mathcal{B}(0,n)$ satisfying this property.