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

Murat Günaydin, Dmytro Volin|arXiv (Cornell University)|Dec 5, 2017

Algebraic structures and combinatorial models52 references3 citations

TL;DR

This paper presents a complete classification of unitary highest- and lowest-weight representations of the non-compact Lie superalgebra $υ(\mathsf{p},\mathsf{q}|\mathsf{m})$ using a generalized oscillator formalism that extends standard Fock spaces to allow non-integer powers of oscillator determinants. The method yields all unitary representations, including those with continuous labels, and provides a diagrammatic realization via non-compact Young diagrams, with explicit application to the physically crucial $υ(2,2|4)$ conformal superalgebra and its supermultiplets with anomalous scaling dimensions.

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) (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.

Motivation & Objective

• To provide a complete classification of unitary representations for the non-compact Lie superalgebra $υ(\mathsf{p},\mathsf{q}|\mathsf{m})$.
• To extend the standard oscillator construction to generalized Fock spaces allowing non-integer powers of oscillator determinants.
• To demonstrate that the generalized formalism captures all unitary representations, including those with continuous labels.
• To establish a diagrammatic representation of these representations using non-compact Young diagrams.
• To apply the formalism to the physically relevant case of $υ(2,2|4)$, the four-dimensional conformal superalgebra, and derive its unitary supermultiplets with anomalous scaling dimensions.

Proposed method

• The authors introduce a generalized Fock space $χ_{\gamma}$ that allows the action of oscillator determinants raised to non-integer powers, extending the standard Fock space construction.
• They employ a $τ$-deformed oscillator algebra and define a Hermitian structure on the generalized Fock space to ensure unitarity.
• The method uses a $τ$-deformation of the standard $τ$-dual pair $τ(\mathsf{K})\timesτ(\mathsf{P})$ to construct irreducible representations.
• The construction is based on a Howe duality framework between $τ(\mathsf{K})$ and $τ(\mathsf{P})$, with the highest-weight states labeled by non-compact Young diagrams.
• The formalism incorporates both bosonic and fermionic oscillators, with the highest-weight states built from Fock vacua and oscillator creation operators.
• The method is applied to $υ(2,2|4)$, where it generates supermultiplets with continuous (anomalous) scaling dimensions via actions of $Δ_{f}^{†}$ and $Δ_{b}^{†}$ determinants.

Experimental results

Research questions

• RQ1 How can the unitary dual of the non-compact Lie superalgebra $υ(\mathsf{p},\mathsf{q}|\mathsf{m})$ be completely classified using oscillator methods?
• RQ2 Can the standard oscillator formalism be extended to include non-integer powers of oscillator determinants to capture continuous labels in unitary representations?
• RQ3 What is the diagrammatic representation of unitary representations of $υ(\mathsf{p},\mathsf{q}|\mathsf{m})$, and how does it generalize standard Young diagrams?
• RQ4 How do the generalized oscillator constructions yield supermultiplets with anomalous scaling dimensions in the $υ(2,2|4)$ conformal superalgebra?
• RQ5 What is the role of CPT conjugation and self-conjugacy in the spectrum of supermultiplets constructed via this formalism?

Key findings

• The generalized oscillator formalism with non-integer powers of oscillator determinants successfully constructs all unitary representations of $υ(\mathsf{p},\mathsf{q}|\mathsf{m})$, including those with continuous labels.
• The unitary representations are fully classified and realized via non-compact Young diagrams, which encode the highest-weight states and their Dynkin labels.
• For $υ(2,2|4)$, the method reproduces the full spectrum of CPT self-conjugate supermultiplets in IIB supergravity on $ττ_{5}\times S^{5}$, including the infinite tower of $(1/2,1/2)$-BPS multiplets labeled by $[0,(n,n,0),0;0,0]$ for $n \geq 2$.
• The construction yields chiral supermultiplets with anomalous scaling dimensions via actions such as $[\Delta_{b}^{\dagger}]^{\gamma_{L}}[b_{\dot{\alpha}}^{\dagger}(1)]^{m}|0\rangle$, corresponding to labels $[m,000,0;(2+\gamma_{L}),0]$.
• The method confirms that the full spectrum of IIB supergravity on $ττ_{5}\times S^{5}$ arises from repeated tensoring of the CPT self-conjugate doubleton multiplet $[0,(1,1,0),0;0,0]$, restricted to CPT-invariant states.
• The formalism provides a unified framework for constructing all unitary irreps of $υ(2,2|4)$, including those with continuous scaling dimensions, by combining multiple oscillator colors and determinant deformations.