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

Sadi Khodaee; Francesco Toppan

doi:10.1063/1.4758923

paper

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

arXiv:1208.3612 · doi:10.1063/1.4758923

Abstract

At critical values of the scaling dimension $λ$ , supermultiplets of the global $N$ -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 $N = 4, 8$ and global supermultiplets $(k, N, N - k)$ , the exceptional superalgebras $D (2, 1; α)$ are recovered for $N = 4$ , with a relation between $α$ and the scaling dimension given by $α = (2 - k) λ$ . For $N = 8$ and $k \neq = 4$ all four $N = 8$ finite superconformal algebras are recovered, at the critical values $λ_{k} = \frac{1}{k - 4}$ , 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 $N = 7$ global supermultiplet $(1, 7, 7, 1)$ induces, at $λ = - 1/4$ , 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 $D (2, 1; α)$ algebras under an $S_{3}$ group action on $α$ , coupled with the relation between $α$ and the scaling dimension $λ$ , induces non-trivial constraints on the admissible models of $N = 4$ superconformal mechanics. The existence of new superconformal models is pointed out. E.g., coupled $(1, 4, 3)$ and $(3, 4, 1)$ supermultiplets generate an $N = 4$ superconformal mechanics if $λ$ is related to the golden ratio. The relation between classical versus quantum $D$ -module representations is presented.

arXiv comment26 pages

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

Abstract

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