New variants of $N=3,4$ superconformal mechanics

TL;DR

This work constructs new ${\cal N}=3$ and ${\cal N}=4$ superconformal mechanics inspired by the supersymmetric Schwarzian. For ${\cal N}=3$, four fermions are enough to realize the maximal symmetry ${\rm OSp}(3|2)$, with an explicit realization of generators and a Schwarzian-like superfield formulation that reduces to a $(4,4,0)$ multiplet; for ${\cal N}=4$, an extended variant achieves full bosonic realization of the $D(1,2;α)$ algebra using two bosonic ${\mathfrak{su}}(2)$ triplets and eight fermions, revealing a rich structure of currents and an ${\rm sl}_2\times S^3\times S^3$ R-symmetry. The work shows that the ${N}=3$ Schwarzian mechanics is equivalent to the constructed ${N}=3$ model at the superfield level, while the ${N}=4$ construction provides a new realization of ${\rm D}(1,2;α)$ with flexible underlying geometry via unfixed ${\mathfrak{so}}(3)$ currents. These results broaden the landscape of superconformal mechanics and point toward higher-${\cal N}$ Schwarzian generalizations and potential AdS$_2$ holographic applications.

Abstract

We construct superconformal mechanics with $N=3$ and $N=4$ supersymmetries that were inspired by analogies with the supersymmetric Schwarzian mechanics. The Schwarzian, being another system with superconformal symmetry, provides insight into the field content of supersymmetric mechanics, most notably, on the number and properties of the fermionic fields involved. Adding more fermionic fields (four in the $N=3$ case and eight in the $N=4$ case) made it possible to construct systems possessing maximal superconformal symmetries in $N=3$ and $N=4$, namely $OSp(3|2)$ and $D(1,2;α)$. In the case of $N=4$ supersymmetry, we explicitly construct a new variant of $N=4$ superconformal mechanics in which all bosonic subalgebras of $D(1,2;α)$ superalgebra have bosonic realization. In addition, the constructed systems involve $so(3)$ currents whose parametrization is not fixed, which allows to consider different underlying geometries.