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We study the unitary representations of the non-compact real forms of thecomplex Lie superalgebra sl(n|m). Among them, only the real form su(p,q|m)(p+q=n) admits nontrivial unitary representations, and all such representationsare of the highest-weight type (or the lowest-weight type). We extend thestandard oscillator construction of the unitary representations of non-compactLie superalgebras over standard Fock spaces to generalised Fock spaces whichallows us to define the action of oscillator determinants raised to non-integerpowers. We prove that the proposed construction yields all the unitaryrepresentations including those with continuous labels. The unitaryrepresentations can be diagrammatically represented by non-compact Youngdiagrams. We apply our general results to the physically important case offour-dimensional conformal superalgebra su(2,2|4) and show how it yieldsreadily its unitary representations including those corresponding tosupermultiplets of conformal fields with continuous (anomalous) scalingdimensions.

The complete unitary dual of non-compact Lie superalgebra su(p,q|m) via the generalised oscillator formalism, and non-compact Young diagrams