Restricted shifted Yangians and restricted finite $W$-algebras

We study the truncated shifted Yangian $Y_{n,l}(\sigma )$ over an algebraically closed field $\mathbb{k}$ of characteristic $p>0$, which is known to be isomorphic to the finite $W$-algebra $U(\mathfrak{g},e)$ associated to a corresponding nilpotent element $e\in \mathfrak{g} = \mathfrak{gl}_N($\mathbb{k}$)$. We obtain an explicit description of the centre of $Y_{n,l}(\sigma )$, showing that it is generated by its Harish-Chandra centre and its $p$-centre. We define $Y_{n,l}^{[p]}(\sigma )$ to be the quotient of $Y_{n,l}(\sigma )$ by the ideal generated by the kernel of trivial character of its $p$-centre. Our main theorem states that $Y_{n,l}^{[p]}(\sigma )$ is isomorphic to the restricted finite $W$-algebra $U^{[p]}(\mathfrak{g},e)$. As a consequence we obtain an explicit presentation of this restricted $W$-algebra.