Symmetry Algebras of Stringy Cosets

We find the symmetry algebras of cosets which are generalizations of the minimal-model cosets, of the specific form $\frac{SU(N)_{k} \times SU(N)_{\ell}}{SU(N)_{k+\ell}}$. 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. The full coset algebra is a larger algebra than conventional $\mathcal{W}$-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.