A superconformal index for hyperKähler cones

In this talk, I will define conformal and supersymmetric quantum mechanics, giving geometric interpretations for both. In order for these symmetries to give a superconformal quantum mechanics (with N=(4,4) supersymmetry), it is necessary that the target space is a hyperKähler cone. This leads to problems due to singularities that appear on all non-trivial such cones. I will show how to deal with these singularities and compute a protected part of the spectrum using a superconformal index. This index has limits that compute the homology, the coordinate ring and the index of fixed point manifolds of the space. Issues about the dependence on the choice of regularisation and wall crossing will be discussed.