Special Legendrian submanifolds in toric Sasaki-Einstein manifolds

We show that every toric Sasaki-Einstein manifold $S$ admits a special Legendrian submanifold $L$ which arises as the link $\mathrm{f}\mathrm{i}\mathrm{x}(\tau )\cap S$ of the fixed point set $\mathrm{f}\mathrm{i}\mathrm{x}(\tau )$ of an anti-holomorphic involution $\tau$ on the cone $C(S)$. In particular, an irregular toric Sasaki-Einstein manifold $S^2\times S^3$ has a special Legendrian torus $S^1\times S^1$. Moreover, we also obtain a special Legendrian submanifold in $\sharp m(S^{2}\times S^{3})$foreach$m\ge 1$.