Title:

The geometry and representation theory of superconformal quantum mechanics

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 $\mathcal{N}=(n,n)$ superconformal invariance with $n=1,2,4$. We construct the corresponding superalgberas $\mathfrak{su}(1,11),~\mathfrak{u}(1,12)$ and $\mathfrak{osp}(4^*4)$ explicitly. The lowenergy dynamics of YangMills instantons is an example of the latter and arises naturally in the discrete lightcone quantisation (DLCQ) of certain superconformal field theories. In particular, we study in some detail the quantum mechanics arising in the DLCQ of the sixdimensional (2,0) theory and fourdimensional $\mathcal{N}=4$ SUSY YangMills. In the (2,0) case we carry out a detailed study of the representation theory of the lightcone superalgebra $\mathfrak{osp}(4^*4)$. 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 $\mathfrak{osp}(4^*4)$. States contribute to the index if and only if they are in the cohomology of a particular supercharge, which we identify as the $L^2$ Dolbeault cohomology of instanton moduli space with values in a real line bundle. In the SUSY YangMills case the target space is the Coulomb branch of an elliptic quiver gauge theory, and as such is a scaleinvariant special Kähler manifold. We describe a new type of $\sigma$model with $\mathcal{N}=(4,4)$ superconformal symmetry and $U(1)\times SO(6)$ Rsymmetry which exists on any such manifold. These models exhibit $\mathfrak{su}(1,14)$ 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 antiselfdual spacetime magnetic field coupling to the topological instanton current.
