We discuss consequences of the breaking of conformal symmetry by a flat or spherical extended operator. We adapt the embedding formalism to the study of correlation functions of symmetric traceless tensors in the presence of the defect. Two-point functions of a bulk and a defect primary are fixed by conformal invariance up to a set of OPE coefficients, and we identify the allowed tensor structures. A correlator of two bulk primaries depends on two cross-ratios, and we study its conformal block decomposition in the case of external scalars. The Casimir equation in the defect channel reduces to a hypergeometric equation, while the bulk channel blocks are recursively determined in the light-cone limit. In the special case of a defect of codimension two, we map the Casimir equation in the bulk channel to the one of a four-point function without defect. Finally, we analyze the contact terms of the stress-tensor with the extended operator, and we deduce constraints on the CFT data. In two dimensions, we relate the displacement operator, which appears among the contact terms, to the reflection coefficient of a conformal interface, and we find unitarity bounds for the latter.

We study the two-point function of local operators in the presence of a defect in a generic conformal field theory. We define two pairs of cross ratios, which are convenient in the analysis of the OPE in the bulk and defect channel respectively. The new coordinates have a simple geometric interpretation, which can be exploited to efficiently compute conformal blocks in a power expansion. We illustrate this fact in the case of scalar external operators. We also elucidate the convergence properties of the bulk and defect OPE decompositions of the two-point function. In particular, we remark that the expansion of the two-point function in powers of the new cross ratios converges everywhere, a property not shared by the cross ratios customarily used in defect CFT. We comment on the crucial relevance of this fact for the numerical bootstrap.

For a single free scalar field in d ≥ 2 dimensions, almost all the unitary conformal defects must be ‘trivial’ in the sense that they cannot hold interesting dynamics. The only possible exceptions are monodromy defects in d ≥ 4 and co-dimension three defects in d ≥ 5. As an intermediate result we show that the n-point correlation functions of a conformal theory with a generalized free spectrum must be those of the generalized free theory.

Abstract:We study the one-dimensional complex conformal manifold that controls the infrared dynamics of a three-dimensional N = 2 supersymmetric theory of three chiral superfields with a cubic superpotential. Two special points on this conformal manifold are the well-known XYZ model and three decoupled copies of the critical Wess-Zumino model. The conformal manifold enjoys a discrete duality group isomorphic to S 4 and can be thought of as an orbifold of CP 1 . We use the 4−ε expansion and the numerical conformal bootstrap to calculate the spectrum of conformal dimensions of low-lying operators and their OPE coefficients, and find a very good quantitative agreement between the two approaches.

We study the kinematics of correlation functions of local and extended operators in a conformal field theory. We present a new method for constructing the tensor structures associated to primary operators in an arbitrary bosonic representation of the Lorentz group. The recipe yields the explicit structures in embedding space, and can be applied to any correlator of local operators, with or without a defect. We then focus on the two-point function of traceless symmetric primaries in the presence of a conformal defect, and explain how to compute the conformal blocks. In particular, we illustrate various techniques to generate the bulk channel blocks either from a radial expansion or by acting with differential operators on simpler seed blocks. For the defect channel, we detail a method to compute the blocks in closed form, in terms of projectors into mixed symmetry representations of the orthogonal group.

We study conformal boundary conditions for the theory of a single real scalar to investigate whether the known Dirichlet and Neumann conditions are the only possibilities. For this free bulk theory there are strong restrictions on the possible boundary dynamics. In particular, we find that the bulk-to-boundary operator expansion of the bulk field involves at most a ‘shadow pair’ of boundary fields, irrespective of the conformal boundary condition. We numerically analyze the four-point crossing equations for this shadow pair in the case of a three-dimensional boundary (so a four-dimensional scalar field) and find that large ranges of parameter space are excluded. However a ‘kink’ in the numerical bounds obeys all our consistency checks and might be an indication of a new conformal boundary condition.

A four-dimensional Abelian gauge field can be coupled to a 3d CFT with a U(1) symmetry living on a boundary. This coupling gives rise to a continuous family of boundary conformal field theories (BCFT) parametrized by the gauge coupling τ in the upper-half plane and by the choice of the CFT in the decoupling limit τ → ∞. Upon performing an SL(2, Z) transformation in the bulk and going to the decoupling limit in the new frame, one finds a different 3d CFT on the boundary, related to the original one by Witten's SL(2, Z) action [1]. In particular the cusps on the real τ axis correspond to the 3d gauging of the original CFT. We study general properties of this BCFT. We show how to express bulk one and two-point functions, and the hemisphere free-energy, in terms of the two-point functions of the boundary electric and magnetic currents. We then consider the case in which the 3d CFT is one Dirac fermion. Thanks to 3d dualities this BCFT is mapped to itself by a bulk S transformation, and it also admits a decoupling limit which gives the O(2) model on the boundary. We compute scaling dimensions of boundary operators and the hemisphere free-energy up to two loops. Using an S-duality improved ansatz, we extrapolate the perturbative results and find good approximations to the observables of the O(2) model. We also consider examples with other theories on the boundary, such as large-N f Dirac fermions -for which the extrapolation to strong coupling can be done exactly order-by-order in 1/N f -and a free complex scalar.

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