Critical D-module reps for finite superconformal algebras and their superconformal mechanics

TL;DR

This paper explores D-module representations of finite simple Lie superalgebras at critical scaling dimensions, linking them to superconformal mechanics and harmonic functions across various dimensions.

Contribution

It introduces new D-module representations at critical values for specific superalgebras and connects these to invariant superconformal mechanics actions.

Findings

D-module reps exist at critical scaling dimensions for certain superalgebras.

Critical scalings relate to harmonic functions in various dimensions.

Invariant actions for superconformal mechanics are induced by these reps.

Abstract

The simple finite Lie superalgebras D(2,1;\alpha), G(3), D(4,1), D(2,2), A(3,1) and F(4) admit D-module representations, given by a set of differential operators of a single real variable t, at a critical value of the scaling dimension \lambda. These superalgebras are one-dimensional N-extended superconformal algebras with N=4 (D(2,1;\alpha)), N=7 (G(3)) and N=8 (the remaining ones). The critical D-module reps induce invariant actions in the Lagrangian framework for superconformal mechanics in D target dimensions. The N=8 critical scalings \lambda=1/(D-4) are linked to the D-dimensional harmonic functions with D=1,2,3,5,6,7,8. This talk is based on J. Math. Phys. 53 (2012) 043513 (arXiv:1112.0995), J. Math. Phys. 53 (2012) 103518 (arXiv:1208.3612) and some extra material.