We study the scaling dimension ∆ φ n of the operator φ n where φ is the fundamental complex field of the U (1) model at the Wilson-Fisher fixed point in d = 4 − ε. Even for a perturbatively small fixed point coupling λ * , standard perturbation theory breaks down for sufficiently large λ * n. Treating λ * n as fixed for small λ * we show that ∆ φ n can be successfully computed through a semiclassical expansion around a non-trivial trajectory, resulting in We explicitly compute the first two orders in the expansion, ∆ −1 (λ * n) and ∆ 0 (λ * n). The result, when expanded at small λ * n, perfectly agrees with all available diagrammatic computations. The asymptotic at large λ * n reproduces instead the systematic large charge expansion, recently derived in CFT. Comparison with Monte Carlo simulations in d = 3 is compatible with the obvious limitations of taking ε = 1, but encouraging.

We provide a framework for generic 4D conformal bootstrap computations. It is based on the unification of two independent approaches, the covariant (embedding) formalism and the non-covariant (conformal frame) formalism. We construct their main ingredients (tensor structures and differential operators) and establish a precise connection between them. We supplement the discussion by additional details like classification of tensor structures of n-point functions, normalization of 2-point functions and seed conformal blocks, Casimir differential operators and treatment of conserved operators and permutation symmetries. Finally, we implement our framework in a Mathematica package and make it freely available.

We calculate the scaling dimensions of operators with large global charge and spin in 2+1 dimensional conformal field theories. By the state-operator correspondence, these operators correspond to superfluids with vortices and can be systematically studied using effective field theory. As the spin increases from zero to the unitarity bound, the superfluid state corresponding to the lowest dimension operator passes through three distinct regimes: (1) a single phonon, (2) two vortices, and (3) multiple vortices. We also calculate correlation functions with two such operators and the Noether current.

We consider the critical O(N) model in the presence of an external magnetic field localized in space. This setup can potentially be realized in quantum simulators and in some liquid mixtures. The external field can be understood as a relevant perturbation of the trivial line defect, and thus triggers a defect Renormalization Group (RG) flow. In agreement with the g-theorem, the external localized field leads at long distances to a stable nontrivial defect CFT (DCFT) with g < 1. We obtain several predictions for the corresponding DCFT data in the epsilon expansion and in the large N limit. The analysis of the large N limit involves a new saddle point and, remarkably, the study of fluctuations around it is enabled by recent progress in AdS loop diagrams. Our results are compatible with results from Monte Carlo simulations and we make several predictions that can be tested in the future.