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arXiv:2111.04747 · Journal of High Energy Physics

Bootstrapping boundary-localized interactions II. Minimal models at the boundary

Connor Behan, Lorenzo Di Pietro, Edoardo Lauria, Balt C. van Rees

Published Mar 1, 2022• 35 citations
arXivHTMLPDF

Abstract

Abstract We provide evidence for the existence of non-trivial unitary conformal boundary conditions for a three-dimensional free scalar field, which can be obtained via a coupling to the m’th unitary diagonal minimal model. For large m we can demonstrate the existence of the fixed point perturbatively, and for smaller values we use the numerical conformal bootstrap to obtain a sharp kink that smoothly matches onto the perturbative predictions. The wider numerical analysis also yields universal bounds for the spectrum of any other boundary condition for the free scalar field. A second kink in these bounds hints at a second class of non-standard boundary conditions, as yet unidentified.

DOI
10.1007/jhep03(2022)146

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Abstract We provide evidence for the existence of non-trivial unitary conformal boundary conditions for a three-dimensional free scalar field, which can be obtained via a coupling to the m’th unitary diagonal minimal model. For large m we can demonstrate the existence of the fixed point perturbatively, and for smaller values we use the numerical conformal bootstrap to obtain a sharp kink that smoothly matches onto the perturbative predictions. The wider numerical analysis also yields universal bounds for the spectrum of any other boundary condition for the free scalar field. A second kink in these bounds hints at a second class of non-standard boundary conditions, as yet unidentified.
A bstract For QFTs in AdS the boundary correlation functions remain conformal even if the bulk theory has a scale. This allows one to constrain RG flows with numerical conformal bootstrap methods. We apply this idea to flows between two-dimensional CFTs, focusing on deformations of the tricritical and ordinary Ising model. We provide non-perturbative constraints for the boundary correlation functions of these flows and compare them with conformal perturbation theory in the vicinity of the fixed points. We also reproduce a completely general constraint on the sign of the TT‾T\overline{T }TT deformation in two dimensions.
For QFTs in AdS the boundary correlation functions remain conformal even if the bulk theory has a scale. This allows one to constrain RG flows with numerical conformal bootstrap methods. We apply this idea to flows between two-dimensional CFTs, focusing on deformations of the tricritical and ordinary Ising model. We provide non-perturbative constraints for the boundary correlation functions of these flows and compare them with conformal perturbation theory in the vicinity of the fixed points. We also reproduce a completely general constraint on the sign of the TTˉT\bar TTTˉ deformation in two dimensions.
Abstract The study of conformal boundary conditions for two-dimensional conformal field theories (CFTs) has a long history, ranging from the description of impurities in one-dimensional quantum chains to the formulation of D-branes in string theory. Nevertheless, the landscape of conformal boundaries is largely unknown, including in rational CFTs, where the local operator data is completely determined. We initiate a systematic bootstrap study of conformal boundaries in 2d CFTs by investigating the bootstrap equation that arises from the open-closed consistency condition of the annulus partition function with identical boundaries. We find that this deceivingly simple bootstrap equation, when combined with unitarity, leads to surprisingly strong constraints on admissible boundary states. In particular, we derive universal bounds on the tension (boundary entropy) of stable boundary conditions, which provide a rigorous diagnostic for potential D-brane decays. We also find unique solutions to the bootstrap problem of stable branes in a number of rational CFTs. Along the way, we observe a curious connection between the annulus bootstrap and the sphere packing problem, which is a natural extension of previous work on the modular bootstrap. We also derive bounds on the boundary entropy at large central charge. These potentially have implications for end-of-the-world branes in pure gravity on AdS3.
A bstract We study two-dimensional conformal field theories (CFTs) with boundaries via the conformal bootstrap. We derive a positive semi-definite program from crossing symmetry of three observables: the annulus partition function, the two-point function of identical operators in the presence of a boundary, and the four-point function of the same operators on the infinite plane. The mixed-correlator system allows the numerical bootstrap to access new data, like the bulk-to-boundary Operator Product Expansion coefficients, and to strengthen the bounds on observables already contained in the partition function on the annulus, such as the boundary entropy. We test the method on the free boson CFT; then, as a first application, we produce new non-perturbative bounds on the entropy and the gaps in boundary CFTs with central charge c = 3/2, with special emphasis on the 𝔰𝔲(2) 2 WZW model.
A bstract We propose explicit expressions for the boundary reflection matrices of the Am+(r,s){\mathcal{A}}_{m}+(r, s)Am​+(r,s) series of massive scattering theories, obtained by perturbing the Am{\mathcal{A}}_{m}Am​ unitary minimal models with ( r, s ) boundary conditions with both bulk and boundary ϕ 1 , 3 operators. We identify the vacua that live on the boundary with the allowed edges of the ( r, s ) conformal boundary conditions of the A m Andrews-Baxter-Forrester model. The boundary reflection matrices are then “direct sums” of certain pairs of A m− 1 Behrend-Pearce solutions of the boundary Yang-Baxter equation and are consistent with the boundary bootstrap and the recently-introduced crossing, as well as the Z 2 (height-reversal), Kac table and non-invertible symmetries.
This thesis explores the application of numerical bootstrap techniques to two-point functions of bulk insertions in the presence of a boundary in two-dimensional conformal field theories. The first part of this thesis is devoted to presenting a comprehensive introduction to essential concepts in the numerical bootstrap of two-dimensional CFTs. Namely, an extensive introduction to the basics of conformal field theories (CFTs), their properties arising from the algebraic structure of conformal transformations, consitency conditions on the observables in a CFT, as well as the effects of the presence of defects in two-dimensional CFTs, is given. This is followed by a quick recount of the numerical techniques employed in the literature to solve for the aforementioned consistency conditions. These chapters form an overview of the seminal works that set the backdrop of the contributions made in the second part of the thesis. In the contributions of the thesis, we discuss the progress made in the investigations in (Meineri and Radhakrishnan 2025) where two-point functions in BCFTs are bootstrapped by treating them as a part of a larger system of consistency conditions involving the partition function on an annulus, the two-point function in the presence of a boundary, and four-point functions on the infinite plane. This perspective of treating the two-point function as a part of a larger system makes accessible parts of the BCFT data that have so-far remained inaccessible, and we present results on the boundary gap in the two-point function, and improvements in existing numerical bounds on the boundary entropy.