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A<sc>bstract</sc> We study anO(N) invariant surface defect in the Wilson-Fisher conformal field theory (CFT) ind= 4 –ϵdimensions. This defect is defined by mass deformation on a two-dimensional surface that generates localized disorder and is conjectured to factorize into a pair of ordinary boundary conditions ind= 3. We determine defect CFT data associated with the lightestO(N) singlet and vector operators up to the third order in theϵ-expansion, find agreements with results from numerical methods and provide support for the factorization proposal ind= 3. Along the way, we observe surprising non-renormalization properties for surface anomalous dimensions and operator-product-expansion coefficients in theϵ-expansion. We also analyze the full conformal anomalies for the surface defect. 

A<sc>bstract</sc> It has been proposed that the Ginzburg-Landau description of the non-unitary conformal minimal modelM(3, 8) is provided by the Euclidean theory of two real scalar fields with third-order interactions that have imaginary coefficients. The same lagrangian describes the non-unitary modelM(3, 10), which is a product of two Yang-Lee theoriesM(2, 5), and the Renormalization Group flow from it toM(3, 8). This proposal has recently passed an important consistency check, due to Y. Nakayama and T. Tanaka, based on the anomaly matching for non-invertible topological lines. In this paper, we elaborate the earlier proposal and argue that the two-field theory describes theDseries modular invariants of bothM(3, 8) andM(3, 10). We further propose the Ginzburg-Landau descriptions of the entire class ofDseries minimal modelsM(q, 3q– 1) andM(q, 3q+ 1), with odd integerq. They involve$$ \mathcal{PT} $$ $\mathrm{PT}$symmetric theories of two scalar fields with interactions of orderqmultiplied by imaginary coupling constants. 

A<sc>bstract</sc> We study two-point functions of symmetric traceless local operators in the bulk of de Sitter spacetime. We derive the Källén-Lehmann spectral decomposition for any spin and show that unitarity implies its spectral densities are nonnegative. In addition, we recover the Källén-Lehmann decomposition in Minkowski space by taking the flat space limit. Using harmonic analysis and the Wick rotation to Euclidean Anti de Sitter, we derive an inversion formula to compute the spectral densities. Using the inversion formula, we relate the analytic structure of the spectral densities to the late-time boundary operator content. We apply our technical tools to study two-point functions of composite operators in free and weakly coupled theories. In the weakly coupled case, we show how the Källén-Lehmann decomposition is useful to find the anomalous dimensions of the late-time boundary operators. We also derive the Källén-Lehmann representation of two-point functions of spinning primary operators of a Conformal Field Theory on de Sitter. 

A<sc>bstract</sc> We explore a new approach to boundaries and interfaces in theO(N) model where we add certain localized cubic interactions. These operators are nearly marginal when the bulk dimension is 4 −ϵ, and they explicitly break theO(N) symmetry of the bulk theory down toO(N− 1). We show that the one-loop beta functions of the cubic couplings are affected by the quartic bulk interactions. For the interfaces, we find real fixed points up to the critical valueN_crit≈ 7, while forN >4 there are IR stable fixed points with purely imaginary values of the cubic couplings. For the boundaries, there are real fixed points for allN, but we don’t find any purely imaginary fixed points. We also consider the theories ofMpairs of symplectic fermions and one real scalar, which have quartic OSp(1|2M) invariant interactions in the bulk. We then add the Sp(2M) invariant localized cubic interactions. The beta functions for these theories are related to those in theO(N) model via the replacement ofNby 1 − 2M. In the special caseM= 1, there are boundary or interface fixed points that preserve the OSp(1|2) symmetry, as well as other fixed points that break it. 

Abstract In this note, we present a synopsis of geometric symmetries for (spin 0) perturbations around (4D) black holes and de Sitter space. For black holes, we focus on static perturbations, for which the (exact) geometric symmetries have the group structure of SO(1,3). The generators consist of three spatial rotations, and three conformal Killing vectors obeying a specialmelodiccondition. The static perturbation solutions form a unitary (principal series) representation of the group. The recently uncovered ladder symmetries follow from this representation structure; they explain the well-known vanishing of the black hole Love numbers. For dynamical perturbations around de Sitter space, the geometric symmetries are less surprising, following from the SO(1,4) isometry. As is known, the quasinormal solutions form a non-unitary representation of the isometry group. We provide explicit expressions for the ladder operators associated with this representation. In both cases, the ladder structures help connect the boundary condition at the horizon with that at infinity (black hole) or origin (de Sitter space), and they manifest as contiguous relations of the hypergeometric solutions. 

A bstract A pair of the 2D non-unitary minimal models M (2 , 5) is known to be equivalent to a variant of the M (3 , 10) minimal model. We discuss the RG flow from this model to another non-unitary minimal model, M (3 , 8). This provides new evidence for its previously proposed Ginzburg-Landau description, which is a ℤ 2 symmetric theory of two scalar fields with cubic interactions. We also point out that M (3 , 8) is equivalent to the (2 , 8) superconformal minimal model with the diagonal modular invariant. Using the 5-loop results for theories of scalar fields with cubic interactions, we exhibit the 6 − ϵ expansions of the dimensions of various operators. Their extrapolations are in quite good agreement with the exact results in 2D. We also use them to approximate the scaling dimensions in d = 3 , 4 , 5 for the theories in the M (3 , 8) universality class.