A note on the large charge expansion in 4d CFT

In this letter, we discuss certain universal predictions of the large charge expansion in conformal field theories with $U(1)$ symmetry, mainly focusing on four-dimensional theories. We show that, while in three dimensions quantum fluctuations are responsible for the existence of a theory-independent $Q^0$ term in the scaling dimension $\Delta_Q$ of the lightest operator with fixed charge $Q\gg 1$, in four dimensions the same mechanism provides a universal $Q^0\log Q$ correction to $\Delta_Q$. Previous works discussing four-dimensional theories failed in identifying this term. We also compute the first subleading correction to the OPE coefficient corresponding to the insertion of an arbitrary primary operator with small charge $q\ll Q$ in between the minimal energy states with charge $Q$ and $Q+q$, both in three and four dimensions. This contribution does not depend on the operator insertion and, similarly to the quantum effects in $\Delta_Q$, in four dimensions it scales logarithmically with $Q$.Comment: 6 pages v2 7 pages, added ref.s and results in d=5,6, paper matches journal versio

General Bootstrap Equations in 4D CFTs

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.Comment: 57 page

Spontaneous symmetry breaking on surface defects

Coleman's theorem states that continuous internal symmetries cannot be spontaneously broken in two-dimensional quantum field theories (QFTs). In this work we consider surface (i.e. two-dimensional) defects in $d$-dimensional conformal field theories (CFTs) invariant under a continuous internal symmetry group $G$. We study under which conditions it is possible for a surface defect to break spontaneously a continuous internal symmetry. We find that spontaneous symmetry breaking (SSB) is disallowed under reasonable assumptions on the defect Renormalization Group (RG) flow. Counterexamples are possible only for exotic RG flows, that do not terminate at a fixed-point. We discuss an example of this kind. We also illustrate the mechanism with which quantum effects prevent SSB for generic defect RG flows. We find a generic weakly coupled defect universality class, where correlation functions decay logarithmically. Our analysis generalizes the recent discovery by Metlitski of the extraordinary-log boundary universality class in the $O(N)$ model.Comment: 37 pages+appendices, 3 figure

Giant Vortices and the Regge Limit

In recent years it has been shown that strongly coupled systems become analytically tractable in the regime of large quantum numbers, such as large spin or large charge. The effective theories that emerge in these two limits are Regge theory and superfluid theory, respectively. Here we make a proposal for a new phase, the ``giant vortex,'' describing an intermediate regime with large spin and charge. The new phase connects superfluid theory with the large-spin expansion. The giant vortex admits a semi-classical effective theory description with peculiar chiral excitations (moving at the speed of light) and a Fock space of states that is reminiscent of the multi-twist operators in Regge theory, including the leading and daughter Regge trajectories. A similar giant vortex phase appears for Bose-Einstein condensates in a rotating trap, and our results should be applicable in that context as well. We show that the transition from the giant vortex to the Regge regime is accompanied by the scaling dimension turning from being larger than to being smaller than the mean field theory value, i.e. gravity switches from being the weakest force at small AdS distance to being the strongest force at large AdS distance.Comment: 23 pages + appendices, 3 figures, v2 references added, small correction

Phases of Wilson Lines in Conformal Field Theories

We study the low-energy limit of Wilson lines (charged impurities) in conformal gauge theories in 2+1 and 3+1 dimensions. As a function of the representation of the Wilson line, certain defect operators can become marginal, leading to interesting renormalization group flows and for sufficiently large representations to complete or partial screening by charged fields. This result is universal: in large enough representations, Wilson lines are screened by the charged matter fields. We observe that the onset of the screening instability is associated with fixed-point mergers. We study this phenomenon in a variety of applications. In some cases, the screening of the Wilson lines takes place by dimensional transmutation and the generation of an exponentially large scale. We identify the space of infrared conformal Wilson lines in weakly coupled gauge theories in 3+1 dimensions and determine the screening cloud due to bosons or fermions. We also study QED in 2+1 dimensions in the large $N_f$ limit and identify the nontrivial conformal Wilson lines. We briefly discuss 't Hooft lines in 3+1-dimensional gauge theories and find that they are screened in weakly coupled gauge theories with simply connected gauge groups. In non-Abelian gauge theories with S-duality, this together with our analysis of the Wilson lines gives a compelling picture for the screening of the line operators as a function of the coupling.Comment: 5 pages, 2 figures, v2 references updated, v3 minor changes, journal versio

Phases of Wilson lines in conformal field theories

We study the low-energy limit of Wilson lines (charged impurities) in conformal gauge theories in 2+1 and 3+1 dimensions. As a function of the representation of the Wilson line, certain defect operators can become marginal, leading to interesting renormalization group flows and for sufficiently large representations to complete or partial screening by charged fields. This result is universal: in large enough representations, Wilson lines are screened by the charged matter fields. We observe that the onset of the screening instability is associated with fixed-point mergers. We study this phenomenon in a variety of applications. In some cases, the screening of the Wilson lines takes place by dimensional transmutation and the generation of an exponentially large scale. We identify the space of infrared conformal Wilson lines in weakly coupled gauge theories in 3+1 dimensions and determine the screening cloud due to bosons or fermions. We also study QED in 2+1 dimensions in the large Nf limit and identify the nontrivial conformal Wilson lines. We briefly discuss ‘t Hooft lines in 3+1-dimensional gauge theories and find that they are screened in weakly coupled gauge theories with simply connected gauge groups. In non-Abelian gauge theories with S duality, this together with our analysis of the Wilson lines gives a compelling picture for the screening of the line operators as a function of the coupling

Gapped Goldstones at the cut-off scale: a non-relativistic EFT

At finite density, the spontaneous breakdown of an internal non-Abelian symmetry dictates, along with gapless modes, modes whose gap is fixed by the algebra and proportional to the chemical potential: the gapped Goldstones. Generically the gap of these states is comparable to that of other non-universal excitations or to the energy scale where the dynamics is strongly coupled. This makes it non-straightforward to derive a universal ef effective field theory (EFT) description realizing all the symmetries. Focusing on the illustrative example of a fully broken SU (2) group, we demonstrate that such an EFT can be constructed by carving out around the Goldstones, gapless and gapped, at small 3-momentum. The rules governing the EFT, where the gapless Goldstones are soft while the gapped ones are slow, are those of standard nonrelativistic EFTs, like for instance nonrelativistic QED. In particular, the EFT Lagrangian formally preserves gapped Goldstone number, and processes where such number is not conserved are described inclusively by allowing for imaginary parts in the Wilson coefficients. Thus, while the symmetry is manifestly realized in the EFT, unitarity is not. We comment on the application of our construction to the study of the large charge sector of conformal field theories with non-Abelian symmetries.Comment: 32 pages+appendices, 3 figures, v2 matches journal versio

Numerical tests of the large charge expansion

We perform Monte-Carlo measurements of two and three point functions of charged operators in the critical O(2) model in 3 dimensions. Our results are compatible with the predictions of the large charge superfluid effective field theory. To obtain reliable measurements for large values of the charge, we improved the Worm algorithm and devised a measurement scheme which mitigates the uncertainties due to lattice and finite size effects.Comment: 16 pages, 12 figures. v2: Improved finite size scaling. v3: Added comparison between Monte Carlo update

Diversification of importin-α isoforms in cellular trafficking and disease states.

The human genome encodes seven isoforms of importin α which are grouped into three subfamilies known as α1, α2 and α3. All isoforms share a fundamentally conserved architecture that consists of an N-terminal, autoinhibitory, importin-β-binding (IBB) domain and a C-terminal Arm (Armadillo)-core that associates with nuclear localization signal (NLS) cargoes. Despite striking similarity in amino acid sequence and 3D structure, importin-α isoforms display remarkable substrate specificity in vivo. In the present review, we look at key differences among importin-α isoforms and provide a comprehensive inventory of known viral and cellular cargoes that have been shown to associate preferentially with specific isoforms. We illustrate how the diversification of the adaptor importin α into seven isoforms expands the dynamic range and regulatory control of nucleocytoplasmic transport, offering unexpected opportunities for pharmacological intervention. The emerging view of importin α is that of a key signalling molecule, with isoforms that confer preferential nuclear entry and spatiotemporal specificity on viral and cellular cargoes directly linked to human diseases