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

Abstract

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.

61 pages, v3: corrected mistake in Eq. (1.1)

References in corpus (8)

• Enlarged symmetry algebras of spin chains, loop models, and S-matrices
• Geometrical four-point functions in the two-dimensional critical $Q$-state Potts model: The interchiral conformal bootstrap
• Logarithmic CFT at generic central charge: from Liouville theory to the $Q$-state Potts model
• Global symmetry and conformal bootstrap in the two-dimensional $O(n)$ model
• Geometric Algebra and Algebraic Geometry of Loop and Potts Models
• The action of the Virasoro algebra in the two-dimensional Potts and loop models at generic $Q$
• Fusion in the periodic Temperley-Lieb algebra and connectivity operators of loop models
• Heisenberg models and Schur--Weyl duality

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• Global symmetry and conformal bootstrap in the two-dimensional $Q$-state Potts model
• Critical spin chains and loop models with $PSU(n)$ symmetry
• Two-dimensional Ising and Potts model with long-range bond disorder: a renormalization group approach
• Boundary Criticality of Complex Conformal Field Theory: A Case Study in the Non-Hermitian 5-State Potts Model
• Logarithmic operators in $c=0$ bulk CFTs
• On currents in the $O(n)$ loop model
• Asymptotic freedom, lost: Complex conformal field theory in the two-dimensional $O(N>2)$ nonlinear sigma model and its realization in Heisenberg spin chains