The Complete Unitary Dual of Non-compact Lie Superalgebra su(p , q | m) via the Generalised Oscillator Formalism, and Non-compact Young Diagrams

Murat Gunaydin, Dmytro Volin

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Abstract

We study the unitary representations of the non-compact real forms of the complex Lie superalgebra sl(n | m). Among them, only the real form 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 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.[Figure not available: see fulltext.]

Original languageEnglish (US)
JournalCommunications In Mathematical Physics
DOIs
StatePublished - Jan 1 2019

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Young Diagram
Lie Superalgebra
Unitary Representation
diagrams
oscillators
formalism
Fock Space
Anomalous Scaling
Superalgebra
low weight
Lowest
Figure
determinants
Determinant
scaling

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

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abstract = "We study the unitary representations of the non-compact real forms of the complex Lie superalgebra sl(n | m). Among them, only the real form 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 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.[Figure not available: see fulltext.]",
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