In this review we study quantum field theories and conformal field theories with global symmetries in the limit of large charge for some of the generators of the symmetry group. At low energy the sectors of the theory with large charge are described by a hybrid form of Goldstone’s theorem, involving its relativistic and non-relativistic forms. The associated effective field theory in the infrared allows the computation of anomalous dimensions, and operator product expansion coefficients in a well defined expansion in inverse powers of the global charge. This applies even when the initial theory does not have a reliable semiclassical approximation. The large quantum number expansion complements, and may provide an alternative approach to the bootstrap and numerical treatments. We will present some general features of the symmetry breaking patterns and the low-energy effective actions, and a fairly large number of examples exhibiting the salient features of this method.

We study the $O(4)$ Wilson–Fisher fixed point in $2+1$ dimensions in fixed large-charge sectors identified by products of two spin-$j$ representations $(j_L,j_R)$. Using effective field theory we derive a formula for the conformal dimensions $D(j_L, j_R)$ of the leading operator in terms of two constants, $c_{ 3 / 2}$ and $c_{ 1 / 2}$, when the sum $j_L + j_R$ is much larger than the difference $|j_L-j_R|$. We compute $D(j_L,j_R)$ when $j_L= j_R$ with Monte Carlo calculations in a discrete formulation of the $O(4)$ lattice field theory, and show excellent agreement with the predicted formula and estimate $c_{ 3 / 2}=1.068(4)$ and $c_{ 1 / 2}=0.083(3)$.

We study some examples of Yang-Baxter deformations of the $AdS_5 \times S^5$ superstring with non-Abelian classical $r$-matrices which satisfy the homogeneous classical Yang-Baxter equation (CYBE). All of the resulting backgrounds satisfy the generalized type IIB supergravity equations. For some of them, we derive “T-dualized” backgrounds and show that these satisfy the usual type IIB supergravity equations. Remarkably, some of them are locally identical to undeformed $AdS_5 \times S^5$.

We calculate the anomalous dimensions of operators with large global charge $J$ in certain strongly coupled conformal field theories in three dimensions, such as the $O(2)$ model and the supersymmetric fixed point with a single chiral superfield and a $W = \Phi^3$ superpotential. Working in a $1/J$ expansion, we find that the large-$J$ sector of both examples is controlled by a conformally invariant effective Lagrangian for a Goldstone boson of the global symmetry. For both these theories, we find that the lowest state with charge $J$ is always a scalar operator whose dimension is $ \Delta(J) = a J^{3 / 2} + b J^{1 / 2} - 0.093$, up to corrections that vanish at large $J$.