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The MacMahon Master Theorem for right quantum superalgebras and higher Sugawara operators for $\widehat{\mathfrak{gl}}(m|n)$

We prove an analogue of the MacMahon Master Theorem for the right quantum superalgebras. In particular, we obtain a new and simple proof of this theorem for the right quantum algebras. In the super case the theorem is then used to construct higher order Sugawara operators for the affine Lie superalgebra $\widehat{\mathfrak{gl}}(m|n)$ in an explicit form. The operators are elements of a completed universal enveloping algebra of $\widehat{\mathfrak{gl}}(m|n)$ at the critical level. They occur as the coefficients in the expansion of a noncommutative Berezinian and as the traces of powers of generator matrices. The same construction yields higher Hamiltonians for the Gaudin model associated with the Lie superalgebra $\mathfrak{gl}(m|n)$. We also use the Sugawara operators to produce algebraically independent generators of the algebra of singular vectors of any generic Verma module at the critical level over the affine Lie superalgebra.