Dynamics of small-amplitude perturbations in the global anti-de Sitter (AdS) spacetime is restricted by selection rules that forbid effective energy transfer between certain sets of normal modes. The selection rules arise algebraically because some integrals of products of AdS mode functions vanish. Here, we reveal the relation of these selection rules to AdS isometries. The formulation we discover through this systematic approach is both simpler and stronger than what has been reported previously. In addition to the selection rule considerations, we develop a number of useful representations for the global AdS mode functions, with connections to algebraic structures of the Higgs oscillator, a superintegrable system describing a particle on a sphere in an inverse cosine-squared potential, where the AdS isometries play the role of a spectrum-generating algebra.

Using derivatives of primary fields (null or not) with respect to the conformal dimension, we build infinite families of non-trivial logarithmic representations of the conformal algebra at generic central charge, with Jordan blocks of dimension 2 or 3. Each representation comes with one free parameter, which takes fixed values under assumptions on the existence of degenerate fields. This parameter can be viewed as a simpler, normalization-independent redefinition of the logarithmic coupling. We compute the corresponding non-chiral conformal blocks, and show that they appear in limits of Liouville theory four-point functions.As an application, we describe the logarithmic structures of the critical two-dimensional O(n) and Q-state Potts models at generic central charge. The validity of our description is demonstrated by semi-analytically bootstrapping four-point connectivities in the Q-state Potts model to arbitrary precision. Moreover, we provide numerical evidence for the Delfino-Viti conjecture for the three-point connectivity. Our results hold for generic values of Q in the complex plane and beyond.

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}n∈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.

We give a solution to the long-standing problem of constructing the generators of hidden symmetries of the quantum Higgs oscillator, a particle on a d-sphere moving in a central potential varying as the inverse cosine-squared of the polar angle. This superintegrable system is known to possess a rich algebraic structure, including a hidden SU (d) symmetry that can be deduced from classical conserved quantities and degeneracies of the quantum spectrum. The quantum generators of this SU (d) have not been constructed thus far, except at d = 2, and naive quantization of classical conserved quantities leads to deformed Lie algebras with quadratic terms in the commutation relations. The nonlocal generators we obtain here satisfy the standard su(d) Lie algebra, and their construction relies on a recently discovered realization of the conformal algebra, which contains a complete set of raising and lowering operators for the Higgs oscillator. This operator structure has emerged from a relation between the Higgs oscillator Schrödinger equation and the Klein-Gordon equation in Anti-de Sitter spacetime. From such a point-of-view, constructing the hidden symmetry generators reduces to manipulations within the abstract conformal algebra so(d, 2).