Cones, Tri-Sasakian Structures and Superconformal Invariance

TL;DR

The paper proves that rigid $N=2$ superconformal hypermultiplets require target spaces that are cones over tri-Sasakian bases, with the dilatation condition $X^ u_{\nu} = \delta^ u_{ u}$ forcing a metric $g=dr^2+r^2h_{ij}dx^i dx^j$. In the Kähler case the base $B$ is Sasakian, and in the hyperKähler case $B$ is tri-Sasakian, with the total space Ricci-flat and endowed with an $SU(2)$ action. The work clarifies when dilatations yield genuine cones (hypersurface orthogonality of $X$ is essential) and provides explicit examples (ALE cones, hyperKähler quotients) illustrating symmetry enhancement to $\mathbb{R}_+\times SU(2)$. It also connects these geometric structures to cone-branes and AdS/CFT, suggesting avenues for applying cone geometry to six-dimensional Calabi–Yau cones and world-volume theories in string/M-theory contexts.

Abstract

In this note we show that rigid N=2 superconformal hypermultiplets must have target manifolds which are cones over tri-Sasakian metrics. We comment on the relation of this work to cone-branes and the AdS/CFT correspondence.