Spaces of states of the two-dimensional O(n) and Potts models

We determine the spaces of states of the two-dimensional $O(n)$ and $Q$-state Potts models with generic parameters $n,Q\in \mathbb{C}$ as representations of their known symmetry algebras. While the relevant representations of the conformal algebra were recently worked out, it remained to determine the action of the global symmetry groups: the orthogonal group for the $O(n)$ model, and the symmetric group $S_Q$ for the $Q$-state Potts model. We do this by two independent methods. First we compute the twisted torus partition functions of the models at criticality. The twist in question is the insertion of a group element along one cycle of the torus: this breaks modular invariance, but allows the partition function to have a unique decomposition into characters of irreducible representations of the global symmetry group. Our second method reduces the problem to determining branching rules of certain diagram algebras. For the $O(n)$ model, we decompose representations of the Brauer algebra into representations of its unoriented Jones–Temperley–Lieb subalgebra. For the $Q$-state Potts model, we decompose representations of the partition algebra into representations of the appropriate subalgebra. We find explicit expressions for these decompositions as sums over certain sets of diagrams, and over standard Young tableaux. We check that both methods agree in many cases. Moreover, our spaces of states are consistent with recent bootstrap results on four-point functions of the corresponding CFTs.

💡 Research Summary

This paper presents a comprehensive determination of the spaces of states for the two-dimensional O(n) and Q-state Potts models with generic complex parameters n and Q, characterizing them as representations of their full symmetry algebras. While the representations of the conformal algebra (Virasoro) for these models were recently understood, the action of the global symmetry groups—the orthogonal group O(n) for the O(n) model and the symmetric group S_Q for the Potts model—remained an open problem. The authors resolve this using two independent and complementary methodologies.

The first method involves computing the twisted torus partition function of each model at criticality. The “twist” consists of inserting an element of the global symmetry group along one cycle of the torus. This breaks modular invariance but allows the partition function to be decomposed uniquely into products of characters of irreducible representations of the global symmetry group and characters of the conformal algebra. From this decomposition, one can extract precisely how the global symmetry acts within each sector of conformal representations, yielding families of representations denoted Λ_(r,s) for the O(n) model and Ξ_(r,s) for the Potts model.

The second, algebraic approach reduces the problem to determining specific branching rules for diagram algebras. Starting from the lattice space of states S_L = V^⊗L (where V is the defining representation of the global symmetry group), the model’s critical dynamics are described by an emergent diagram algebra A_L: the unoriented Jones-Temperley-Lieb algebra uJTL_L(n) for O(n), and the periodic Temperley-Lieb algebra PTL_2L(√Q) for Potts. This algebra A_L is a subalgebra of the commutant G*_L of the global symmetry group G acting on S_L. Using Schur-Weyl duality, the decomposition of S_L into G*L × G representations is known. The problem then translates to decomposing the representations of the larger algebra G*L (the Brauer algebra for O(n), the partition algebra for Potts) into representations of its subalgebra A_L. The authors derive explicit combinatorial formulas for these branching rules as sums over sets of diagrams and standard Young tableaux. The branching coefficients are shown to become independent of the lattice size L for sufficiently large L. In the critical limit, the A_L representations become indecomposable representations of the interchiral algebra (an extension of the conformal algebra), and the associated direct sum of global symmetry representations, governed by the branching coefficients, gives the result for Λ(r,s) or Ξ(r,s).

A key achievement of the paper is demonstrating the agreement between the results derived from the twisted partition function (physical method) and those from the algebraic branching rules (combinatorial method) in numerous cases, despite originating from different lattice realizations. This consistency strongly supports the validity of the results. Furthermore, the obtained structure of the spaces of states is shown to be consistent with recent findings from the conformal bootstrap studies of four-point functions in the corresponding CFTs.

In conclusion, by synergistically combining physical partition function techniques with deep algebraic and combinatorial analysis, this work provides a complete and rigorous description of the state spaces of these fundamental models, fully elucidating the interplay between their conformal and global symmetries.