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SU(2) channels the cancellation of K3 BPS states

The conformal field theoretic elliptic genus, an invariant for N=(2,2) superconformal field theories, counts the BPS states in any such theory with signs, according to their bosonic or fermionic nature. For K3 theories, this invariant is the source of the Mathieu Moonshine phenomenon. There, the net number of quarter BPS states is positive for any conformal dimension above the massless threshold, but it may arise after cancellation of the contributions of an equal number of bosonic and fermionic BPS states present in non-generic theories, as is the case for the class of Z2-orbifolds of toroidal SCFTs. Nevertheless, the space H-roof of all BPS states that are generic to such orbifold theories provides a convenient framework to construct a particular generic space of states of K3 theories. We find a natural action of the group SU(2) on a subspace of H-roof which is compatible with the cancellations of contributions from the corresponding non-generic states. In fact, we propose that this action channels those cancellations. As a by-product, we find a new subspace of the generic space of states in H-roof.