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

TL;DR

This paper explores the critical scaling dimensions at which supermultiplets induce finite superconformal algebras, revealing new constraints and models in superconformal mechanics for various extended supersymmetries.

Contribution

It identifies specific critical scaling dimensions for supermultiplets that generate superconformal algebras, including new models and constraints in superconformal mechanics.

Findings

Recovery of exceptional superalgebras at critical dimensions

Identification of specific superconformal algebras for different supermultiplets

Constraints on superconformal mechanics models based on scaling dimensions

Abstract

At critical values of the scaling dimension $\lambda$, supermultiplets of the global ${\cal N}$-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 ${\cal N}=4,8$ and global supermultiplets $(k, {\cal N}, {\cal N}-k)$, the exceptional superalgebras $D(2,1;\alpha)$ are recovered for ${\cal N}=4$, with a relation between $\alpha$ and the scaling dimension given by $\alpha= (2-k)\lambda$. For ${\cal N}=8$ and $k\neq 4$ all four ${\cal N}=8$ 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 ${\cal N}=7$ 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 ${\cal N}=4$ superconformal mechanics. The existence of new superconformal models is pointed out. E.g., coupled $(1,4,3)$ and $(3,4,1)$ supermultiplets generate an ${\cal N}=4$ superconformal mechanics if $\lambda$ is related to the golden ratio. The relation between classical versus quantum $D$-module representations is presented.