Symmetry algebras of stringy cosets

Abstract We find the symmetry algebras of cosets which are generalizations of the minimal-model cosets, of the specific form SU N k × SU N ℓ SU N k + ℓ $$ \frac{\mathrm{SU}{(N)}_k\times \mathrm{SU}{(N)}_{\mathrm{\ell}}}{\mathrm{SU}{(N)}_{k+\mathrm{\ell}}} $$ . We study this coset in its free field limit, with k, ℓ → ∞, where it reduces to a theory of free bosons. We show that, in this limit and at large N, the algebra W ∞ e 1 $$ {\mathcal{W}}_{\infty}^e\left[1\right] $$ emerges as a sub-algebra of the coset algebra. The full coset algebra is a larger algebra than conventional W $$ \mathcal{W} $$ -algebras, with the number of generators rising exponentially with the spin, characteristic of a stringy growth of states. We compare the coset algebra to the symmetry algebra of the large N symmetric product orbifold CFT, which is known to have a stringy symmetry algebra labelled the ‘higher spin square’. We propose that the higher spin square is a sub-algebra of the symmetry algebra of our stringy coset.