Construction of stable rank $2$ 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 $c_1 = e \in \{-1, 0\}$ and $c_2 = n \geqslant 1$ on the projective space $\mathbb{P}^3$. We construct the 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 whose middle term 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 $n \geqslant 146$). $\Sigma_0$ yields the next example, after the series of instanton components, of an infinite series of components of $\mathcal{B}(0, n)$ satisfying this property.