[Paper Review] New realizations of the supergroup D(2,1;α) in N=4 superconformal mechanics

Sergey Fedoruk, Evgeny Ivanov|arXiv (Cornell University)|Jul 30, 2015

Black Holes and Theoretical Physics39 references13 citations

TL;DR

This paper presents new off-shell realizations of the supergroup $D(2,1;\alpha)$ in ${\cal N}=4$ superconformal mechanics using reducible multiplets $({\bf 1,4,3})\oplus({\bf 0,4,4})$, $({\bf 3,4,1})\oplus({\bf 0,4,4})$, and $({\bf 4,4,0})\oplus({\bf 0,4,4})$. By constructing manifestly supersymmetric superfield actions and applying the Noether procedure to derive $D(2,1;\alpha)$ supercharges, the authors reveal new fermionic conformal couplings through auxiliary field elimination, with universal contributions from the $({\bf 0,4,4})$ multiplet across all models.

ABSTRACT

We present new explicit realizations of the most general N=4, d=1 superconformal symmetry D(2,1;α) in the models of N=4 superconformal mechanics based on the reducible multiplets (1,4,3)\oplus(0,4,4), (3,4,1)\oplus(0,4,4) and (4,4,0)\oplus(0,4,4). We start from the manifestly supersymmetric superfield actions for these systems and then descend to the relevant off- and on-shell component actions from which we derive the D(2,1;α) (super)charges by the Noether procedure. Some peculiarities of these realizations of D(2,1;α) are discussed. We also construct a new D(2,1;α) invariant system by joining the multiplets (3,4,1) and (4,4,0) in such a way that they interact with each other through an extra (0,4,4) multiplet. New fermionic conformal couplings appear as the result of elimination of the appropriate auxiliary fields.

Motivation & Objective

• To construct new off-shell realizations of the exceptional supergroup $D(2,1;\alpha)$ in ${\cal N}=4$ superconformal mechanics.
• To extend previous on-shell treatments by using manifestly supersymmetric superfield actions with auxiliary fields preserved.
• To derive $D(2,1;\alpha)$ supercharges via the Noether procedure from component Lagrangians obtained after eliminating auxiliary fields.
• To identify new fermionic conformal couplings arising from the elimination of auxiliary fields in interacting multiplet systems.
• To propose a new $D(2,1;\alpha)$-invariant system by coupling $({\bf 3,4,1})$ and $({\bf 4,4,0})$ multiplets through an intermediate $({\bf 0,4,4})$ multiplet.

Proposed method

• Employing ${\cal N}=4,d=1$ harmonic superspace formalism to construct off-shell superfield actions for reducible multiplet pairs.
• Deriving component Lagrangians by reducing superfield actions and retaining auxiliary fields for consistent elimination.
• Applying the Noether procedure to the component Lagrangians to compute the $D(2,1;\alpha)$ supercharges explicitly.
• Eliminating auxiliary fields algebraically to obtain on-shell component actions with new fermionic interaction terms.
• Constructing a novel $D(2,1;\alpha)$-invariant system by coupling $({\bf 3,4,1})$ and $({\bf 4,4,0})$ multiplets via a shared $({\bf 0,4,4})$ multiplet.
• Analyzing the structure of supercharges to identify universal contributions from the $({\bf 0,4,4})$ multiplet across different models.

Experimental results

Research questions

• RQ1 How can $D(2,1;\alpha)$ be realized in ${\cal N}=4$ superconformal mechanics using reducible multiplets beyond irreducible ones?
• RQ2 What new fermionic conformal couplings emerge from the off-shell elimination of auxiliary fields in interacting multiplet systems?
• RQ3 How does the inclusion of the $({\bf 0,4,4})$ multiplet universally modify the supercharges across different multiplet pairs?
• RQ4 What is the role of the harmonic superspace formalism in constructing consistent off-shell superfield actions for $D(2,1;\alpha)$-invariant models?
• RQ5 Can a new $D(2,1;\alpha)$-invariant system be constructed by coupling $({\bf 3,4,1})$ and $({\bf 4,4,0})$ multiplets through a $({\bf 0,4,4})$ intermediate multiplet?

Key findings

• The component Lagrangians for $({\bf 1,4,3})\oplus({\bf 0,4,4})$, $({\bf 3,4,1})\oplus({\bf 0,4,4})$, and $({\bf 4,4,0})\oplus({\bf 0,4,4})$ multiplets exhibit distinct four-fermion terms that vanish at different $\alpha$ values: $\alpha = -1/2$, $\alpha = 1/2$, and $\alpha = 1$, respectively.
• The supercharges for all three models contain a universal term proportional to $\alpha \chi^{iA}\chi^{k}_{A}$ from the $({\bf 0,4,4})$ multiplet, indicating a consistent coupling structure.
• The three-fermion terms in the supercharges vanish at $\alpha = -1/2$ for all multiplets, suggesting a special role for this value in simplifying the superalgebra.
• New fermionic couplings emerge in the on-shell Lagrangian of the triple-coupled system $({\bf 3,4,1})\oplus({\bf 4,4,0})\oplus({\bf 0,4,4})$, absent in isolated pairs, due to auxiliary field elimination.
• The supercharges for the combined systems feature angular momentum-like terms involving $\ell^{(i}_j p^{k)j}$ and $\textsc{l}^{(i}_a p^{k)a}$, which can be reinterpreted as SU(2) currents.
• The $({\bf 0,4,4})$ multiplet contributes universally to the supercharges across all models, with the same $\alpha$-dependent coefficient $-\alpha \chi^{iA}\chi^{k}_{A}$, indicating a robust structural feature.