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 $_2$ .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$_2$.

arXiv:2507.00124Jan 1, 20250 citations

Numerical Bootstrap Program for Boundary Conformal Field Theories

Bharathkumar Radhakrishnan

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.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.

Jan 1, 20250 citations

The bootstrap of points and lines

Marco Meineri, Bharathkumar Radhakrishnan

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 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.

arXiv:2506.17382Journal of High Energy PhysicsApr 2, 20264 citations

A bootstrap study of minimal model deformations

António Antunes, Edoardo Lauria, Balt C. van Rees

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 $T\overline{T }$ deformation in two dimensions.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 $$T\overline{T }$$ deformation in two dimensions.

Journal of High Energy PhysicsMay 3, 20247 citations

Weizmann lectures on the numerical conformal bootstrap

Shai M. Chester

These lectures were given at the Weizmann Institute in the spring of 2019. They are intended to familiarize students with the nuts and bolts of the numerical bootstrap as efficiently as possible. After a brief review of the basics of conformal field theory in d>2 spacetime dimensions, we discuss how to compute conformal blocks, formulate the crossing equations as a semi-definite programming problem, solve this problem using SDPB on a personal computer, and interpret the results. We include worked examples for all steps, including bounds for 3d CFTs with Z2 or O(N) global symmetries. Each lecture includes a problem set, which culminate in a precise computation of the 3d Ising model critical exponents using the mixed correlator Z2 bootstrap. A Mathematica file is included that transforms crossing equations into the proper input form for SDPB.These lectures were given at the Weizmann Institute in the spring of 2019. They are intended to familiarize students with the nuts and bolts of the numerical bootstrap as efficiently as possible. After a brief review of the basics of conformal field theory in d>2 spacetime dimensions, we discuss how to compute conformal blocks, formulate the crossing equations as a semi-definite programming problem, solve this problem using SDPB on a personal computer, and interpret the results. We include worked examples for all steps, including bounds for 3d CFTs with Z2 or O(N) global symmetries. Each lecture includes a problem set, which culminate in a precise computation of the 3d Ising model critical exponents using the mixed correlator Z2 bootstrap. A Mathematica file is included that transforms crossing equations into the proper input form for SDPB.

arXiv:1907.05147Physics ReportsNov 2, 202352 citations

Conformal bootstrap with reinforcement learning

Gergely Kántor, Vasilis Niarchos B, Constantinos Papageorgakis A

We introduce the use of reinforcement-learning (RL) techniques to the conformal-bootstrap program.We demonstrate that suitable soft Actor-Critic RL algorithms can perform efficient, relatively cheap highdimensional searches in the space of scaling dimensions and OPE-squared coefficients that produce sensible results for tens of CFT data from a single crossing equation.In this paper we test this approach in well-known 2D CFTs, with particular focus on the Ising and tricritical Ising models and the free compactified boson CFT.We present results of as high as 36-dimensional searches, whose sole input is the expected number of operators per spin in a truncation of the conformal-block decomposition of the crossing equations.Our study of 2D CFTs uses only the global soð2; 2Þ part of the conformal algebra, and our methods are equally applicable to higher-dimensional CFTs.When combined with other, already available, numerical and analytical methods, we expect our approach to yield an exciting new window into the nonperturbative structure of arbitrary (unitary or nonunitary) CFTs.We introduce the use of reinforcement-learning (RL) techniques to the conformal-bootstrap program.We demonstrate that suitable soft Actor-Critic RL algorithms can perform efficient, relatively cheap highdimensional searches in the space of scaling dimensions and OPE-squared coefficients that produce sensible results for tens of CFT data from a single crossing equation.In this paper we test this approach in well-known 2D CFTs, with particular focus on the Ising and tricritical Ising models and the free compactified boson CFT.We present results of as high as 36-dimensional searches, whose sole input is the expected number of operators per spin in a truncation of the conformal-block decomposition of the crossing equations.Our study of 2D CFTs uses only the global soð2; 2Þ part of the conformal algebra, and our methods are equally applicable to higher-dimensional CFTs.When combined with other, already available, numerical and analytical methods, we expect our approach to yield an exciting new window into the nonperturbative structure of arbitrary (unitary or nonunitary) CFTs.

arXiv:2108.09330Physical review. D/Physical review. D.Jan 24, 202236 citations

_2

T\overline{T }

deformation in two dimensions.