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

  • Murat GünaydinEmail author
  • Dmytro Volin


We study the unitary representations of the non-compact real forms of the complex Lie superalgebra \({\mathfrak{sl}({\rm n}|{\rm m})}\). Among them, only the real form \({\mathfrak{su}({\rm p}, {\rm q}|{\rm m})}\) with \({({\rm p} + {\rm q}= {\rm n)}}\) 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.


Lie algebras:

\(\mathfrak {g}\); \({\mathfrak {gl}}\), \({\mathfrak {sl}}\), \({\mathfrak {su}}\), ...

Universal enveloping algebra:

\(\mathcal {U}(\mathfrak {g})\)

Lie groups:

\(\mathsf {G}\); \({\mathsf {GL}}\), \({\mathsf {SL}}\), \({\mathsf {SU}}\), ...

Ranks of (sub)algebra:

\(\mathsf {p},\mathsf {q},\mathsf {m},\mathsf {n},\ldots \)

\({\mathbb {Z}}_2\)-numbers:

\({\overline{0}}\) and \({\overline{1}}\).

parity grading function:


c-grading function :


Generic index:

\(i,j,k,\ldots \)

Generic p-even index:

\(\mu ,\nu ,\ldots \)

p-even, c-odd index:

\({\dot{\alpha }},{\dot{\beta }},\ldots \)

p-even, c-even index:

\(\alpha ,\beta ,\ldots \)

Generic p-odd index:

\(a,b,\ldots \)

\({\mathfrak {gl}}(\mathsf {m}|\mathsf {n})\) generators:

\({\mathrm {E}}_{ij}\), \({\mathrm {E}}_{\mu a}\), ...

Cartan generators:

\({\mathrm {h}}_i\equiv {\mathrm {E}}_{ii}-(-1)^{p_i+p_{i+1}}{\mathrm {E}}_{i+1,i+1}\)

Generic weight:

\(m_i\) (eigenvalue of \({\mathrm {E}}_{ii}\))

Eigenvalue of \({\mathrm {E}}_{\mu \mu }\):

\(\nu _{\mu }\) (\(\nu \) without indices denotes set of all these eigenvalues)

Eigenvalue of \({\mathrm {E}}_{aa}\):

\(\lambda _a\) (\(\lambda \) without indices denotes set of all these eigenvalues)

Fundamental weight:

\([\nu ;\lambda ]\) or \([\nu _{L};\lambda ;\nu _{R}]\) (evaluated on highest-weight state)

Repetition of the same value in weights:

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Dynkin label:

\(\omega _i\) (eigenvalue of \({\mathrm {h}}_i\) on highest-weight state)

Dynkin weight:

\(\langle \omega \rangle \equiv \langle \omega _1,\omega _2,\ldots \rangle \)

Nodes of Kac–Dynkin–Vogan diagram:


\(p_i=p_{i+1}\), \(c_{i}=c_{i+1}\):

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\(p_i=p_{i+1}\), \(c_{i}\ne c_{i+1}\):

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\(p_{i}\ne p_{i+1}\), \(c_{i}= c_{i+1}\):

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\(p_{i}\ne p_{i+1}\), \(c_{i}\ne c_{i+1}\):

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colour index:

\(A,B,C,\ldots \)

Total number of colours:


Sets of colours:

\(\mathbf{A},\mathbf{B},\mathbf{F},\mathbf{A}_{\!\Delta },\mathbf{B}_{\!\Delta },\mathbf{F}_{\!\Delta }\subset \{1,2,\ldots ,P\}\)

Number of colours in a set:

\(|\mathbf{A}|\), \(|\mathbf{B}|,\ldots \)

Ordinary Young diagrams (integer partitions):

\(\mu ,\tau ,\mu _L,\mu _R\); e.g. \(\mu \equiv \{\mu _1,\mu _2,\ldots \}\) with all \(\mu _i\in {\mathbb {Z}}_{\ge 0}\) and \(\mu _i\ge \mu _{i+1}\) .

Height of partition:

\(h_{\mu }\) (number of non-zero \(\mu _i\))

Size of partition:

\(|\mu |\equiv \mu _1+\mu _2+\cdots +\mu _{h_{\mu }}\)

Bosonic oscillators:

\(a_{\alpha }^{\phantom {\dagger }},a_{\alpha }^{\dagger }\), \(b_{{\dot{\alpha }}}^{\phantom {\dagger }},b_{{\dot{\alpha }}}^{\dagger }\)

Fermionic oscillators:

\(f_{a}^{\phantom {\dagger }},f_{a}^{\dagger }\)

Fock space:

\(\mathcal {F}\)

Fock vacuum:

\(|\mathrm{0} \rangle \)

\(\gamma \)-deformed Fock space:

\(\mathcal {F}_{\gamma }\)

Irreducible modules of \(\mathfrak {u}(\mathsf {n})\):

\(V_{\mu }\) (constructed from bosonic oscillators)\(W_{\tau }\) (constructed from fermionic oscillators)


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We thank Matthias Staudacher, Carlo Meneghelli, and especially Christian Marboe for useful discussions. The research of M.G. was supported in part by the National Science Foundation under Grant Numbers PHY-1213183 and PHY-0855356 and the US Department of Energy under DOE Grant No: DE-SC0010534. The research ofD.V.was partially supported by the People Programme (Marie CurieActions) of the European Union’s Seventh Framework Programme FP7/2007-2013/ under REA Grant Agreement No 317089 (GATIS). M.G. thanks Nordita for hospitality where part of the research was done. D.V. thanks IHES and IPhT, C.E.A-Saclay for hospitality where part of the research was done.


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Authors and Affiliations

  1. 1.Institute for Gravitation and the Cosmos & Physics DepartmentThe Pennsylvania State UniversityUniversity ParkUSA
  2. 2.Nordita, KTH Royal Institute of Technology and Stockholm UniversityStockholmSweden
  3. 3.School of Mathematics and Hamilton Mathematics InstituteTrinity College DublinCollege Green, Dublin 2Ireland

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