A complete axiomatization of infinitary first-order intuitionistic logic over ${\mathcal{L}}_{{\kappa }^{+},\kappa }$

Given a weakly compact cardinal $\kappa$, we give an axiomatization of intuitionistic first-order logic over ${\mathcal{L}}_{{\kappa }^{+},\kappa }$ and prove it is sound and complete with respect to Kripke models. As a consequence we get the disjunction and existence properties for that logic. This generalizes the work of Nadel for intuitionistic logic over ${\mathcal{L}}_{{\omega }_1,\omega }$. When $\kappa$ is a regular cardinal such that ${\kappa }^{<\kappa }=\kappa$, we deduce, by an easy modification of the proof, a complete axiomatization of intuitionistic first-order logic over ${\mathcal{L}}_{{\kappa }^{+},\kappa ,\kappa }$, the language with disjunctions of at most $\kappa$ formulas, conjunctions of less than $\kappa$ formulas and quantification on less than $\kappa$ many variables. In particular, this applies to any regular cardinal under the Generalized Continuum Hypothesis.