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.

💡 Research Summary

The paper introduces a novel framework called the “grand‑canonical symmetric orbifold” (GCSO) that resolves long‑standing issues in the conventional symmetric‑orbifold construction when one attempts to treat an ensemble of all orbifold degrees. In the standard picture a seed two‑dimensional CFT is tensored d times and the permutation group S_d is gauged, producing twisted sectors labelled by conjugacy classes of permutations. Correlation functions at fixed degree d are well understood, but the OPE coefficients of twist fields depend explicitly on d. Consequently, when one defines a grand‑canonical ensemble by weighting each degree with a chemical potential p, the naïve two‑point function of twist operators factorises correctly, yet higher‑point functions reveal inconsistencies: the degree‑dependent combinatorial factors prevent a universal operator product expansion (OPE) in the ensemble.

To overcome this, the authors propose to define the grand‑canonical theory as the direct sum of symmetric‑orbifold CFTs for all degrees d ≥ 0. The Hilbert space becomes
 H = ⊕_{d≥0} H(d),
where each H(d) is itself a sum over twisted sectors. This construction guarantees that the resulting theory is an ordinary, unitary quantum field theory with a well‑defined OPE, because each sector is a bona‑fide CFT and the direct sum preserves associativity and locality.

The key technical innovation is the introduction of partial permutations as the labeling device for operators that mix sectors of different degrees. A partial permutation is a pair (S, ρ) where S⊂{1,…,n} is a support set and ρ∈S_{|S|} is a permutation of that set. The product of two partial permutations is defined by union of supports and composition of the underlying permutations. Conjugation by the full symmetric group groups partial permutations into orbits characterised by the size of the support and its cycle type. The authors define orbit‑sum operators A