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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
Article
  • 16 Downloads

Abstract

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.

Notations

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:

\(p_i\)

c-grading function :

\(c_i\)

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:

P

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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Notes

Acknowledgements

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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