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We find the symmetry algebras of cosets which are generalizations of theminimal-model cosets, of the specific form $\frac{SU(N)_{k} \timesSU(N)_{\ell}}{SU(N)_{k+\ell}}$. We study this coset in its free field limit,with $k,\ell \rightarrow \infty$, where it reduces to a theory of free bosons.We show that, in this limit and at large $N$, the algebra$\mathcal{W}^e_\infty[1]$ emerges as a sub-algebra of the coset algebra. Thefull coset algebra is a larger algebra than conventional$\mathcal{W}$-algebras, with the number of generators rising exponentially withthe spin, characteristic of a stringy growth of states. We compare the cosetalgebra to the symmetry algebra of the large $N$ symmetric product orbifoldCFT, which is known to have a stringy symmetry algebra labelled the `higherspin square'. We propose that the higher spin square is a sub-algebra of thesymmetry algebra of our stringy coset.

Symmetry Algebras of Stringy Cosets