Grand-Canonical Symmetric Orbifold Theories

In this letter, we study grand-canonical symmetric orbifolds of conformal field theories. We propose to define them as the direct sum of symmetric orbifolds of all degrees. The natural basis of operators is one that mixes all sectors. We describe this basis in terms of partial permutations, and explain how to define and calculate the operator product expansion in that framework. Our construction provides a conformal field theory interpretation of the central charge operator $\mathcal{I}$ in $AdS_3$ string theory.