Nondegenerate multidimensional matrices and instanton bundles

Abstract

In this paper we prove that the moduli space of rank $2n$ symplectic instanton bundles on ${\mathbb {P}^{2n+1}}$, defined from the well-known monad condition, is affine. This result was not known even in the case $n=1$, where by Atiyah, Drinfeld, Hitchin, and Manin in 1978 the real instanton bundles correspond to self-dual Yang Mills $Sp(1)$-connections over the $4$-dimensional sphere. The result is proved as a consequence of the existence of an invariant of the multidimensional matrices representing the instanton bundles.