In this talk I am going to present certain conjectures (due to D.Gaiotto) which provide explicit exmples of (quantum) local geometric Langlands duality for
the group $GL(n)\times GL(m)$. These conjectures lead to some interesting equivalences of categories invloving representation theory of the Lie superalgebra $\mathfrak{gl}(m|n)$. I will present some evidence for these conjectures and discuss proofs in some special cases.