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Abstract We study the O(3) critical model and the free theory of a scalar triplet in the presence of a magnetic impurity. We use analytic bootstrap techniques to extract results in the ε-expansion. First, we extend by one order in perturbation theory the computation of the beta function for the defect coupling in the free theory. Then, we analyze in detail the low-lying spectrum of defect operators, focusing on their perturbative realization when the defect is constructed as a path-ordered exponential. After this, we consider two different bulk two-point functions and we compute them using the defect dispersion relation. For a free bulk theory, we are able to fix the form of the correlator at all orders in ε. In particular, taking ε → 1, we can show that in d = 3 one does not have a consistent and non-trivial defect CFT. For an interacting bulk, we compute the correlator up to second order in ε. Expanding these results in the bulk and defect block expansions, we are able to extract an infinite set of defect CFT data. We discuss low-spin ambiguities that affect every result computed through the dispersion relation and we use a combination of consistency conditions and explicit diagrammatic calculations to fix this ambiguity.

Abstract We present a dispersion relation for defect CFT that reconstructs two-point functions in the presence of a defect as an integral of a single discontinuity. The main virtue of this formula is that it streamlines explicit bootstrap calculations, bypassing the resummation of conformal blocks. As applications we reproduce known results for monodromy defects in the epsilon-expansion, and present new results for the supersymmetric Wilson line at strong coupling in N $$ \mathcal{N} $$ = 4 SYM. In particular, we derive a new analytic formula for the highest R-symmetry channel of single-trace operators of arbitrary length.

Abstract We use the numerical bootstrap to study conformal line defects with O(2) global symmetry. Our results are very general and capture in particular conformal line defects originating from bulk CFTs with a continuous global symmetry, which can either be preserved or partially broken by the presence of the defect. We begin with an agnostic approach and perform a systematic bootstrap study of correlation functions between two canonical operators on the defect: the displacement and the tilt. We then focus on two interesting theories: a monodromy line defect and a localized magnetic field line defect. To this end, we combine the numerical bootstrap with the ε-expansion, where we complement existing results in the literature with additional calculations. For the monodromy defect our numerical results are consistent with expectations, with known analytic solutions sitting inside our numerical bounds. For the localized magnetic field line defect our plots show a series of intriguing cusps which we explore.

Abstract We use analytical bootstrap techniques to study supersymmetric monodromy defects in the critical Wess-Zumino model. In preparation for this result we first study two related systems which are interesting on their own: general monodromy defects (no susy), and the ε-expansion bootstrap for the Wess-Zumino model (no defects). For general monodromy defects, we extend previous work on codimension-two conformal blocks and the Lorentzian inversion formula in order to accommodate parity-odd structures. In the Wess-Zumino model, we bootstrap four-point functions of chiral operators in the ε-expansion, with the goal of obtaining spectral information about the bulk theory. We then proceed to bootstrap two-point functions of chiral operators in the presence of a monodromy defect, and obtain explicit expressions in terms of novel special functions which we analyze in detail. Several of the results presented in this paper are quite general and should be applicable to other setups.

Abstract We study two-point functions of single-trace half-BPS operators in the presence of a supersymmetric Wilson line in N $$ \mathcal{N} $$ = 4 SYM. We use modern analytic bootstrap in the form of an inversion formula, that reconstructs the CFT data starting from a single discontinuity of the correlator. In the planar strong coupling limit only a finite number of conformal blocks contributes to the discontinuity, which allows us to obtain elegant closed-form expressions for two-point functions of single-trace operators O $$ \mathcal{O} $$ J of weight J = 2, 3, 4. Our bootstrap approach guarantees that the correlator satisfies all the expected properties: it has the correct discontinuity, it satisfies the superconformal Ward identities, and it has a sensible expansion in both defect and bulk OPEs. The result is also consistent with available results coming from localization. The method is completely algorithmic and can be implemented to calculate correlators of arbitrary weight.

Abstract Boundaries in three-dimensional N $$ \mathcal{N} $$ = 2 superconformal theories may preserve one half of the original bulk supersymmetry. There are two possibilities which are characterized by the chirality of the leftover supercharges. Depending on the choice, the remaining 2d boundary algebra exhibits N $$ \mathcal{N} $$ = (0, 2) or N $$ \mathcal{N} $$ = (1) supersymmetry. In this work we focus on correlation functions of chiral fields for both types of supersymmetric boundaries. We study a host of correlators using superspace techniques and calculate superconformal blocks for two- and three-point functions. For N $$ \mathcal{N} $$ = (1) supersymmetry, some of our results can be analytically continued in the spacetime dimension while keeping the codimension fixed. This opens the door for a bootstrap analysis of the ϵ-expansion in supersymmetric BCFTs. Armed with our analytically-continued superblocks, we prove that in the free theory limit two-point functions of chiral (and antichiral) fields are unique. The first order correction, which already describes interactions, is universal up to two free parameters. As a check of our analysis, we study the Wess-Zumino model with a super-symmetric boundary using Feynman diagrams, and find perfect agreement between the perturbative and bootstrap results.

Abstract We apply the numerical conformal bootstrap to correlators of Coulomb and Higgs branch operators in 4d N $$ \mathcal{N} $$ = 2 superconformal theories. We start by revisiting previous results on single correlators of Coulomb branch operators. In particular, we present improved bounds on OPE coefficients for some selected Argyres-Douglas models, and compare them to recent work where the same cofficients were obtained in the limit of large r charge. There is solid agreement between all the approaches. The improved bounds can be used to extract an approximate spectrum of the Argyres-Douglas models, which can then be used as a guide in order to corner these theories to numerical islands in the space of conformal dimensions. When there is a flavor symmetry present, we complement the analysis by including mixed correlators of Coulomb branch operators and the moment map, a Higgs branch operator which sits in the same multiplet as the flavor current. After calculating the relevant superconformal blocks we apply the numerical machinery to the mixed system. We put general constraints on CFT data appearing in the new channels, with particular emphasis on the simplest Argyres-Douglas model with non-trivial flavor symmetry.

Abstract We consider a non-supersymmetric domain-wall version of N $$ \mathcal{N} $$ = 4 SYM theory where five out of the six scalar fields have non-zero classical values on one side of a wall of codimension one. The classical fields have commutators which constitute an irreducible representation of the Lie algebra so $$ \mathfrak{so} $$ (5) leading to a highly non-trivial mixing between color and flavor components of the quantum fields. Making use of fuzzy spherical harmonics on S 4, we explicitly solve the mixing problem and derive not only the spectrum of excitations at the quantum level but also the propagators of the original fields needed for perturbative quantum computations. As an application, we derive the one-loop one-point function of a chiral primary and find complete agreement with a supergravity prediction of the same quantity in a double-scaling limit which involves a limit of large instanton number in the dual D3-D7 probe-brane setup.

Abstract We study half-BPS line defects in N $$ \mathcal{N} $$ = 2 superconformal theories using the bootstrap approach. We concentrate on local excitations constrained to the defect, which means the system is a 1d defect CFT with osp(4∗ |2) symmetry. In order to study correlation functions we construct a suitable superspace, and then use the Casimir approach to calculate a collection of new superconformal blocks. Special emphasis is given to the displacement operator, which controls deformations orthogonal to the defect and is always present in a defect CFT. After setting up the crossing equations we proceed with a numerical and analytical bootstrap analysis. We obtain numerical bounds on the CFT data and compare them to known solutions. We also present an analytic perturbative solution to the crossing equations, and argue that this solution captures line defects in N $$ \mathcal{N} $$ = 2 gauge theories at strong coupling.

Abstract Via a challenging field-theory computation, we confirm a supergravity prediction for the non-supersymmetric D3-D7 probe-brane system with probe geometry AdS 4 ×S 2 ×S 2, stabilized by fluxes. Supergravity predicts, in a certain double-scaling limit, the value of the one-point functions of chiral primaries of the dual defect version of N = 4 $$ \mathcal{N}=4 $$ SYM theory, where the fluxes translate into SO(3) × SO(3)-symmetric, Lie-algebra-valued vacuum expectation values for all six scalar fields. Using a generalization of the technique based on fuzzy spherical harmonics developed for the related D3-D5 probe-brane system, we diagonalize the resulting mass matrix of the field theory. Subsequently, we calculate the planar one-loop correction to the vacuum expectation values of the scalars in dimensional reduction and find that it is UV finite and non-vanishing. We then proceed to calculating the one-loop correction to the planar one-point function of any single-trace scalar operator and explicitly evaluate this correction for a 1/2-BPS operator of length L at two leading orders in the double-scaling limit, finding exact agreement with the supergravity prediction.