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In this article, we shall describe the center of the universal affine vertex superalgebra   $V^{\kappa _{c}}(\mathfrak{g})$   associated with   $\mathfrak{g}=\mathfrak{s}\mathfrak{l}_{2|1}, \mathfrak{g}\mathfrak{l}_{2|1}$   at the critical level   $\kappa _{c}$   and prove the conjecture of A. Molev and E. Ragoucy [24] in this case. The center   $\mathfrak{z}(V^{\kappa _{c}}(\mathfrak{s}\mathfrak{l}_{2|1}))$   turns out to be isomorphic to the large level limit   $\ell \rightarrow \infty $   of a vertex subalgebra, called the parafermion vertex algebra   $K^{\ell } (\mathfrak{s}\mathfrak{l}_{2})$  , of the affine vertex algebra   $V^\ell (\mathfrak{s}\mathfrak{l}_{2})$  . The key ingredient of the proof is to understand the principal   $\mathcal{W}$  -superalgebra   $\mathcal{W}^{\kappa _{c}}(\mathfrak{s}\mathfrak{l}_{2|1})$   at the critical level. It relates the center   $\mathfrak{z}(V^{\kappa _{c}}(\mathfrak{s}\mathfrak{l}_{2|1}))$   to   $V^\infty (\mathfrak{s}\mathfrak{l}_{2})$   via the Kazama–Suzuki duality while it has a surprising coincidence with   $V^{\kappa _{c}}(\mathfrak{g}\mathfrak{l}_{1|1})$  , whose center has been recently described. Moreover, the centers   $\mathfrak{z}(V^{\kappa _{c}}(\mathfrak{s}\mathfrak{l}_{2|1}))$   and   $\mathfrak{z}(\mathcal{W}^{\kappa _{c}}(\mathfrak{s}\mathfrak{l}_{2|1}))$   are proven to coincide as a byproduct. A general conjecture is proposed, which describes the center   $\mathfrak{z}(V^{\kappa _{c}}(\mathfrak{s}\mathfrak{l}_{n|m}))$   with   $n&gt;m$   as a large level limit of “the dual side”, that is, the parafermion-type subalgebras of   $\mathcal{W}$  -algebras   $\mathcal{W}^\ell (\mathfrak{s}\mathfrak{l}_{n}, \mathbb{O}_{[n-m,1^{m}]})$   associated with hook-type partitions   $[n-m,1^{m}]$  , known also as vertex algebras at the corner.

Center of Affine 𝔰𝔩 2|1 at the Critical Level