[en] We study the ground states and left-excited states of the $A_{k-1}$ $\mathcal{N}=(2,0)$ little string theory. Via a theorem by Atiyah [1], these sectors can be captured by a supersymmetric nonlinear sigma model on $\mathbb{C}{P}^1$ with target space the based loop group of $SU(k)$. The ground states, described by $L^2$-cohomology classes, form modules over an affine Lie algebra, while the left-excited states, described by chiral differential operators, form modules over a toroidal Lie algebra. We also apply our results to analyze the 1/2 and 1/4 BPS sectors of the M5-brane worldvolume theory.