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arXiv:2506.17382 · Journal of High Energy Physics

The bootstrap of points and lines

Marco Meineri, Bharathkumar Radhakrishnan

Published Apr 2, 2026• 4 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1007/jhep04(2026)040

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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.
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.
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.
We develop a neural network bootstrap framework for reconstructing partition functions of two-dimensional conformal field theories (CFTs) based on modular invariance and the Cardy condition, which are recast as crossing equations for four-point correlators. For torus partition functions, we use the twist-field representation in the symmetric-orbifold description to map modular S-invariance to four-point crossing and focus on the diagonal kinematics of four insertions on a line. For annulus partition functions, we formulate open/closed channel duality as crossing symmetry for mixed four-point functions of defect-changing operators in interface CFT. In both cases, the reconstruction problem is formulated in the anchored-bootstrap form, where the crossing constraints are supplemented by minimal spectral input (a gap) and anchor data. We solve this under-determined problem by using lightweight feed-forward neural networks to parametrise the correlators and their corresponding partition functions. A key ingredient of this approach is the spectral bias of the neural networks in the lazy training regime, which selects specific crossing-symmetric configurations. This reformulation unifies standard modular and annulus constraints in two dimensions with the anchored neural approach for CFT correlators, providing a new way to reconstruct full partition functions from sparse data with remarkable accuracy.
The paper contributes to an ongoing effort to extend the conformal bootstrap beyond its traditional focus on systems of four-point correlation functions. Recently, it was demonstrated that semidefinite programming can be used to formulate a six-point generalisation of the numerical bootstrap, yielding qualitatively new, rigorous bounds on CFT data. However, the numerical six-point bootstrap requires solving SDPs involving infinite-dimensional matrices, which has so far limited its applicability and hindered scalability in early implementations. This work overcomes the challenges by using sparse matrix decompositions to exploit the banded structure of the underlying SDP. The result is a rewriting of one-dimensional six-point bootstrap problems as effectively two-dimensional standard mixed correlator four-point bootstrap computations. As application, novel bounds whose extremal correlators interpolate between the six-point functions of the generalised free fermion and boson are derived. The extremal interpolations are matched with perturbative deformations of the massive free boson in AdS _2 2 .
The paper contributes to an ongoing effort to extend the conformal bootstrap beyond its traditional focus on systems of four-point correlation functions. Recently, it was demonstrated that semidefinite programming can be used to formulate a six-point generalisation of the numerical bootstrap, yielding qualitatively new, rigorous bounds on CFT data. However, the numerical six-point bootstrap requires solving SDPs involving infinite-dimensional matrices, which has so far limited its applicability and hindered scalability in early implementations. This work overcomes the challenges by using sparse matrix decompositions to exploit the banded structure of the underlying SDP. The result is a rewriting of one-dimensional six-point bootstrap problems as effectively two-dimensional standard mixed correlator four-point bootstrap computations. As application, novel bounds whose extremal correlators interpolate between the six-point functions of the generalised free fermion and boson are derived. The extremal interpolations are matched with perturbative deformations of the massive free boson in AdS2_22​.
This dissertation contains two parts of study on the Friedan states and on the conformal bootstrap with symmetries. The Friedan states are potential physical boundary states of the free boson. However, they exhibited a continuous spectrum in the open string sector, in contrast to more standard examples. The explicit expressions for the density of states of the Friedan states is obtained. Some pathologies and possible contradictions of these states are explored. The ggg functions of the Friedan states are shown to be infinite, suggesting an infinite number of degrees of freedom in the theory. The modular conformal bootstrap using the moment curve approach and focused on two-dimensional conformal field theory with c≥1c\geq 1c≥1 is discussed. A refinement is made to incorporate conserved currents. Specializing to theories with symmetries, the anyon partition functions, which transform as a vector under modular transformations, are introduced. Applying the moment curve bootstrap to these leads to bounds on the minimum conformal weights in different sectors. In the analytical approach, the correlated bounds between different anyon sectors are obtained, with the abelian symmetries, Z2\mathbb{Z}_2Z2​ and Z3\mathbb{Z}_3Z3​, and with the non-abelian symmetry, S3S_3S3​. The explicit bounds on the free boson theories with the presence of the symmetries are then calculated and compared with the bootstrap bounds. Finally, some numerical bootstrap results with Z2\mathbb{Z}_2Z2​ and Z3\mathbb{Z}_3Z3​ symmetries are presented.