We study aspects of the quantum mechanics of nonlinear $\sigma$-models with superconformal invariance. The connection between the differential geometry of the target manifold and symmetries of the quantum mechanics is explored, resulting in a classification of spaces admitting $N=(n,n)$ superconformal invariance with $n=1,2,4$. We construct the corresponding superalgberas $\mathrm{su}(1,1|1), u(1,1|2)$ and $\mathrm{osp}(4\ast |4)$ explicitly. The low-energy dynamics of Yang-Mills instantons is an example of the latter and arises naturally in the discrete light-cone quantisation (DLCQ) of certain superconformal field theories. In particular, we study in some detail the quantum mechanics arising in the DLCQ of the six-dimensional (2,0) theory and four-dimensional $N=4$ SUSY Yang-Mills.

In the (2,0) case we carry out a detailed study of the representation theory of the light-cone superalgebra $\mathrm{osp}(4\ast |4)$. We give a complete classification of the unitary irreducible representations and their branching at the unitarity bound, and use this information to construct the superconformal index for $\mathrm{osp}(4\ast |4)$. States contribute to the index if and only if they are in the cohomology of a particular supercharge, which we identify as the $L2$ Dolbeault cohomology of instanton moduli space with values in a real line bundle.

In the SUSY Yang-Mills case the target space is the Coulomb branch of an elliptic quiver gauge theory, and as such is a scale-invariant special Kähler manifold. We describe a new type of $\sigma$-model with $N=(4,4)$ superconformal symmetry and $U(1)\times SO(6)$ R-symmetry which exists on any such manifold. These models exhibit $\mathrm{su}(1,1|4)$ invariance and we give an explicit construction of the superalgebra in terms of known functions. Consideration of the spectral problem for the dilatation operator in these models leads to a deformation which we interpret, via an extension of the moduli space approximation, as an anti-self-dual spacetime magnetic field coupling to the topological instanton current.