A Lie conformal superalgebra and duality of representations for E(4,4)

We construct a duality functor in the category of continuous representations of the Lie superalgebra E(4, 4), the only exceptional simple linearly compact Lie superalgebra, for which it wasn’t known. This is achieved by constructing a Lie conformal superalgebra of type (4, 4) for which E(4, 4) is the annihilation algebra. Along the way we obtain an explicit realization of E(4, 4) by vector fields on a (4|4)-dimensional supermanifold.