Selected Topics in the Large Quantum Number Expansion

Abstract

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.

118 pages. Matched published version in Physics Reports. arXiv admin note: text overlap with arXiv:1711.07990 by other authors

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• Resurgence of the large-charge expansion
• Decoding stringy near-supersymmetric black holes
• Giant Vortices and the Regge Limit
• Generating functions and large-charge expansion of integrated correlators in $\mathcal{N}=4$ supersymmetric Yang-Mills theory
• Field theories with higher-group symmetry from composite currents
• Large Deviations in the Early Universe
• Defects in scalar field theories, RG flows and Dimensional Disentangling
• Anomalous dimensions at large charge in d=4 O(N) theory
• Yukawa interactions at Large Charge
• Nonrelativistic CFTs at Large Charge: Casimir Energy and Logarithmic Enhancements
• Anomalous dimensions at large charge for $U(N)\times U(N)$ theory in three and four dimensions
• Operator spectrum of nonrelativistic CFTs at large charge
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• Long Range, Large Charge, Large $N$
• RG Flows and Stability in Defect Field Theories
• The connection between nonzero density and spontaneous symmetry breaking for interacting scalars
• Large Charges on the Wilson Loop in $\mathcal{N}=4$ SYM: II. Quantum Fluctuations, OPE, and Spectral Curve
• Newton vs. Coulomb in AdS/CFT and the Weak Gravity Conjecture
• Gauge Invariance at Large Charge
• Wilson loops in Large Symmetric Representations through a Double-Scaling Limit
• Conformal Data for the O(3) Wilson-Fisher Conformal Field Theory from Fuzzy Sphere Realization of the Quantum Rotor Model
• On the six-loop scaling dimensions of the $({\varphi }^2{)}^n$ operators in $d=3$
• Charging the conformal window at nonzero $\theta$ angle
• On Convexity of Charged Operators in CFTs and the Weak Gravity Conjecture
• One-loop divergences in the six-dimensional $\mathcal{N}=(1,0)$ hypermultiplet self-coupling model
• ISCOs and the weak gravity conjecture bound in higher derivative theories of gravity
• Exact evaluation of large-charge correlation functions in non-relativistic conformal field theory
• On perturbation around closed exclusion processes
• Geometry of Classical Nambu-Goldstone Fields
• Asymptotics in the bi-Yang-Baxter Sigma Model
• Higher-form anomalies and state-operator correspondence beyond conformal invariance
• Semiclassics of three-dimensional SCFTs from holography
• Matrix models for extremal and integrated correlators of higher rank