The complete unitary dual of non-compact Lie superalgebra \(\mathfrak{su}(p,q|m)\) via the generalised oscillator formalism, and non-compact Young diagrams. (English) Zbl 1464.17013
Summary: We study the unitary representations of the non-compact real forms of the complex Lie superalgebra \(\mathfrak{sl}(n|m)\). Among them, only the real form \(\mathfrak{su}(p, q|m)\) with \((p+ q= n)\) admits nontrivial unitary representations, and all such representations are of the highest-weight type (or the lowest-weight type). We extend the standard oscillator construction of the unitary representations of non-compact Lie superalgebras over standard Fock spaces to generalised Fock spaces which allows us to define the action of oscillator determinants raised to non-integer powers. We prove that the proposed construction yields all the unitary representations including those with continuous labels. The unitary representations can be diagrammatically represented by non-compact Young diagrams. We apply our general results to the physically important case of four-dimensional conformal superalgebra \({\mathfrak{su}(2,2|4)}\) and show how it yields readily its unitary representations including those corresponding to supermultiplets of conformal fields with continuous (anomalous) scaling dimensions.
MSC:
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
| 17B81 | Applications of Lie (super)algebras to physics, etc. |
| 22E65 | Infinite-dimensional Lie groups and their Lie algebras: general properties |
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
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