Global symmetry and conformal bootstrap in the two-dimensional $O(n)$ model

Abstract

We define the two-dimensional $O(n)$ conformal field theory as a theory that includes the critical dilute and dense $O(n)$ models as special cases, and depends analytically on the central charge. For generic values of $n\in \mathbb{C}$, we write a conjecture for the decomposition of the spectrum into irreducible representations of $O(n)$. We then explain how to numerically bootstrap arbitrary four-point functions of primary fields in the presence of the global $O(n)$ symmetry. We determine the needed conformal blocks, including logarithmic blocks, including in singular cases. We argue that $O(n)$ representation theory provides upper bounds on the number of solutions of crossing symmetry for any given four-point function. We study some of the simplest correlation functions in detail, and determine a few fusion rules. We count the solutions of crossing symmetry for the $30$ simplest four-point functions. The number of solutions varies from $2$ to $6$, and saturates the bound from $O(n)$ representation theory in $21$ out of $30$ cases.

arXiv comment49 pages, v3: improved explanations on a few points

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• New Developments in the Numerical Conformal Bootstrap
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• Spaces of states of the two-dimensional O(n) and Potts models
• The tricritical Ising CFT and conformal bootstrap
• From combinatorial maps to correlation functions in loop models
• Coupled minimal models revisited II: Constraints from permutation symmetry
• Global symmetry and conformal bootstrap in the two-dimensional $Q$-state Potts model
• Critical spin chains and loop models with $PSU(n)$ symmetry
• Fusion of irreducible modules in the periodic Temperley--Lieb algebra
• Logarithmic operators in $c=0$ bulk CFTs
• On currents in the $O(n)$ loop model
• On the analytical continuation of lattice Liouville theory
• Asymptotic freedom, lost: Complex conformal field theory in the two-dimensional $O(N>2)$ nonlinear sigma model and its realization in Heisenberg spin chains
• On the Wilson-Fisher fixed point in the limit of integer spacetime dimensions
• Generalized multifractality at the spin quantum Hall transition: Percolation mapping and pure-scaling observables