Critical scaling dimension of D-module representations of N=4,7,8 Superconformal Algebras and constraints on Superconformal Mechanics

Abstract

At critical values of the scaling dimension $\lambda$, supermultiplets of the global $\mathcal{N}$-Extended one-dimensional Supersymmetry algebra induce $D$-module representations of finite superconformal algebras (the latters being identified in terms of the global supermultiplet and its critical scaling dimension). For $\mathcal{N}=4,8$ and global supermultiplets $(k,\mathcal{N},\mathcal{N}-k)$, the exceptional superalgebras $D(2,1;\alpha )$ are recovered for $\mathcal{N}=4$, with a relation between $\alpha$ and the scaling dimension given by $\alpha =(2-k)\lambda$. For $\mathcal{N}=8$ and $k\ne 4$ all four $\mathcal{N}=8$ finite superconformal algebras are recovered, at the critical values ${\lambda }_k=\frac{1}{k-4}$, with the following identifications: D(4,1) for $k=0,8$, F(4) for $k=1,7$, A(3,1) for $k=2,6$ and D(2,2) for $k=3,5$. The $\mathcal{N}=7$ global supermultiplet $(1,7,7,1)$ induces, at $\lambda =-1/4$, a $D$-module representation of the exceptional superalgebra G(3). $D$-module representations are applicable to the construction of superconformal mechanics in a Lagrangian setting. The isomorphism of the $D(2,1;\alpha )$ algebras under an $S_3$ group action on $\alpha$, coupled with the relation between $\alpha$ and the scaling dimension $\lambda$, induces non-trivial constraints on the admissible models of $\mathcal{N}=4$ superconformal mechanics. The existence of new superconformal models is pointed out. E.g., coupled $(1,4,3)$ and $(3,4,1)$ supermultiplets generate an $\mathcal{N}=4$ superconformal mechanics if $\lambda$ is related to the golden ratio. The relation between classical versus quantum $D$-module representations is presented.

26 pages

References in corpus (7)

• Refining the classification of the irreps of the 1D N-Extended Supersymmetry
• New Model of N=8 Superconformal Mechanics
• Decomposition and Oxidation of the N-Extended Supersymmetric Quantum Mechanics Multiplets
• On non-minimal N=4 supermultiplets in 1D and their associated sigma-models
• Irreps and Off-shell Invariant Actions of the N-extended Supersymmetric Quantum Mechanics
• On the Irreps of the N-Extended Supersymmetric Quantum Mechanics and Their Fusion Graphs
• An N=8 superconformal particle in the half-plane

Cited by in corpus (3)

• SU(2|2) supersymmetric mechanics
• Note on correlation functions in conformal quantum mechanics
• The octonionically-induced $\mathcal{N}=7$ exceptional $G(3)$ superconformal quantum mechanics