Modified instanton sum and 4-group structure in 4d $\mathcal{N}=1$ $SU(M)$ SYM from holography

We study the decomposition of the holographic 4d $\mathcal{N}=1$ $SU(M)$ gauge theory with in the Klebanov-Strassler set-up. In particular, we propose a consistent framework for defining a modified instanton sum and a 4-group structure for the SYM theory, derived from its $AdS/CFT$ construction. To achieve this, we analyse symmetry topological operators associated with continuous $(-1)$-form symmetries, derive the corresponding 5-dimensional Symmetry Topological Field Theory (SymTFT), and impose specific discrete gaugings.