Scalar-fermion models, such as the Gross-Neveu-Yukawa model, admit natural 1$\textit{d}$ defects given by the exponential of a scalar field integrated along a straight line. In 4 − $\textit{ε}$ dimensions the defect coupling is weakly relevant and the setup defines a non-trivial interacting defect CFT. In this work we study correlation functions on these defect CFTs to order $\textit{ε}$. We focus on 1$\textit{d}$ correlators constrained to the line, which include canonical operators like the displacement and the one-dimensional analog of the spin field. These results give access to perturbative CFT data that can be used as input in the numerical bootstrap. We also consider local operators outside the line, in particular two-point functions of scalars whose dynamics are non-trivial due to the presence of the defect.

Germany

Julien Barrat

Pedro Liendo

Philine Vliet

Line defect correlators in fermionic CFTs

We continue our study of multipoint correlators of scalar fields on the 1d defect CFT generated by inserting operators along the Maldacena-Wilson line in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM. We present a weak-coupling recursion relation that captures correlators at next-to-leading order involving an arbitrary number of the elementary scalar fields ϕ  i  and ϕ 6, the latter being unprotected. We can then build correlators of composite operators by pinching the scalar fields together. As a demonstration of our method, we give explicit results for correlators containing up to six points. We also expand some selected correlators using recently obtained conformal blocks in the comb and snowflake channel, and check that the extracted low-lying CFT data is consistent with explicit computations.

Giulia Peveri

Multipoint correlators on the supersymmetric Wilson line defect CFT. Part II. Unprotected operators

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   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM. In particular, we derive a new analytic formula for the highest R-symmetry channel of single-trace operators of arbitrary length.

Aleix Gimenez-Grau

A dispersion relation for defect CFT

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.

France

Edoardo Lauria

Bootstrapping line defects with O(2) global symmetry

We study multipoint correlators of protected scalars on the Maldacena-Wilson line in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM. Working at weak coupling in the planar limit, we derive an explicit recursion relation that captures next-to-leading order correlators with an arbitrary number of insertions of the fundamental scalar field. By pinching fundamental scalars together, we can build composite protected operators with higher values of the R-charge. Our result then encompasses arbitrary n-point correlators of protected operators with arbitrary weight. As a demonstration of our method, we give explicit formulae for correlators with up to six points. Using these results we observe that all our correlators are annihilated by a special class of differential operators. We conjecture that these differential operators are non-perturbative constraints and can be considered a multipoint extension of the superconformal Ward identities satisfied by four-point functions.

Jan Plefka

Multipoint correlators on the supersymmetric Wilson line defect CFT

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.

Bootstrapping monodromy defects in the Wess-Zumino model

We study two-point functions of single-trace half-BPS operators in the presence of a supersymmetric Wilson line in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 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   <math> <mi>O</mi> </math>  $$ \mathcal{O} $$   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.

Bootstrapping holographic defect correlators in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 super Yang-Mills

Italy

Ferdinando Gliozzi

Marco Meineri

United Kingdom

Antonio Rago

Erratum to: Boundary and interface CFTs from the conformal bootstrap

We use the 4 − ϵ expansion to search for fixed points corresponding to 2 + 1 dimensional   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$ =1 Wess-Zumino models of N Φ scalar superfields interacting through a cubic superpotential. In the N Φ = 3 case we classify all SUSY fixed points that are perturbatively unitary. In the N Φ = 4 and N Φ = 5 cases, we focus on fixed points where the scalar superfields form a single irreducible representation of the symmetry group (irreducible fixed points). For N Φ = 4 we show that the S5 invariant super Potts model is the only irreducible fixed point where the four scalar superfields are fully interacting. For N Φ = 5, we go through all Lie subgroups of O(5) and use the GAP system for computational discrete algebra to study finite subgroups of O(5) up to order 800. This analysis gives us three fully interacting irreducible fixed points. Of particular interest is a subgroup of O(5) that exhibits O(3)/Z2 symmetry. It turns out this fixed point can be generalized to a new family of models, with N Φ =   <math> <mfrac> <mrow> <mi>N</mi> <mfenced> <mrow> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfenced> </mrow> <mn>2</mn> </mfrac> </math>  $$ \frac{\mathrm{N}\left(\mathrm{N}-1\right)}{2} $$  − 1 and O(N)/Z2 symmetry, that exists for arbitrary integer N≥3.

Junchen Rong

Seeking SUSY fixed points in the 4 − ϵ expansion

We study the two-point function of the stress-tensor multiplet of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM in the presence of a line defect. To be more precise, we focus on the single-trace operator of conformal dimension two that sits in the 20′ irrep of the   <math> <mi>so</mi> </math>  $$ \mathfrak{so} $$ (6)R R-symmetry, and add a Maldacena-Wilson line to the configuration which makes the two-point function non-trivial. We use a combination of perturbation theory and defect CFT techniques to obtain results up to next-to-leading order in the coupling constant. Being a defect CFT correlator, there exist two (super)conformal block expansions which capture defect and bulk data respectively. We present a closed-form formula for the defect CFT data, which allows to write an efficient Taylor series for the correlator in the limit when one of the operators is close to the line. The bulk channel is technically harder and closed-form formulae are particularly challenging to obtain, nevertheless we use our analysis to check against well-known data of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 SYM. In particular, we recover the correct anomalous dimensions of a famous tower of twist-two operators (which includes the Konishi multiplet), and successfully compare the one-point function of the stress-tensor multiplet with results obtained using matrix-model techniques.