5D N=1 super QFT: symplectic quivers

We develop a method to build new 5D $\mathcal{N}=1$ gauge models based on Sasaki-Einstein manifolds $Y^{p,q}\mathrm{.}$ These models extend the standard 5D ones having a unitary SU${\left(p\right)}_q$ gauge symmetry based on $$. Particular focus is put on the building of a gauge family with symplectic SP$(2r,\mathbb{R})$ symmetry. These super QFTs are embedded in M-theory compactified on folded toric Calabi-Yau threefolds $% \hat{X}(Y^{2r,0})$constructedfromconical$Y^{2r,0}$. By using outer-automorphism symmetries of 5D $\mathcal{N}=1$$$BPS quivers with unitary SU$\left(2r\right)$ gauge invariance, we also construct BPS quivers with symplectic SP$(2r,\mathbb{R})$ gauge symmetry. Other related aspects are discussed. Keywords: SCFT${}_5$, 5D $\mathcal{N}=1$ super QFT on a finite circle, Sasaki-Einstein manifolds, BPS quivers, outer-automorphisms.