Numerical Four-Point Bootstrap

Updated 23 August 2025

Numerical Four-Point Bootstrap is a method that uses crossing symmetry and unitarity to impose rigorous, model-independent constraints on conformal field theories.
It employs both derivative and multipoint techniques to convert four-point function equations into tractable optimization problems via linear or semidefinite programming.
Extensions to incorporate spinning correlators and global symmetries allow precise determinations of critical exponents and operator dimensions across various CFT frameworks.

The numerical four-point bootstrap refers to a class of computational and analytic techniques for extracting nonperturbative, model-independent constraints on data of conformal field theories (CFTs) by enforcing crossing symmetry and unitarity at the level of four-point correlation functions. These methods leverage convex optimization—especially linear and semidefinite programming, often accompanied by analytic input or group-theoretic decompositions—to place rigorous bounds on operator dimensions, OPE coefficients, and critical exponents. The approach is now central to modern CFT across spacetime dimensions, global symmetry groups, and supersymmetric or non-supersymmetric settings.

1. Crossing Symmetry and Four-Point Function Decomposition

At the foundation of the numerical bootstrap is the exploitation of crossing symmetry for four-point functions. For example, for four scalars ϕ, the correlator can be reduced (by conformal invariance) to a function of cross-ratios: $\langle\phi(x_1)\phi(x_2)\phi(x_3)\phi(x_4)\rangle = \frac{g(u,v)}{x_{12}^{2d} x_{34}^{2d}},$ where $u$ and $v$ are the conformally invariant cross-ratios. The function $g(u,v)$ is expanded in conformal blocks: $g(u,v) = 1 + \sum_k p_k \, g_k(u,v), \quad p_k = (C_{\phi\phi}^k)^2 \geq 0.$ Crossing symmetry enforces,

$v^d g(u,v) = u^d g(v,u),$

which upon isolating the identity operator yields a "sum rule" such as

$u^d - v^d = \sum_k p_k [v^d g_k(u,v) - u^d g_k(v,u)].$

This equation forms the core constraint used in the numerical bootstrap and is subject to the further requirement $p_k \geq 0$ by OPE coefficient positivity.

For four-point functions of operators charged under global symmetries or supersymmetry, the tensorial or superblock decomposition is essential. For instance, in nonabelian $SU(N_f)_L \times SU(N_f)_R$ symmetric theories, the OPE decomposes into several group-theoretic channels, leading to a coupled system of crossing equations (e.g., $G_S$ , $G_L$ , $G_R$ , $G_A$ for singlet, left-adjoint, right-adjoint, and bi-adjoint, respectively) (Antipin et al., 2014).

2. Numerical Implementation: Optimization Strategies

To cast these functional equations as tractable computational problems, the bootstrap reduces them to finite-dimensional systems. Two primary schemes are widely used:

Derivative (Functional) Method: Crossing equations are Taylor-expanded at a symmetric point (e.g., $z = \bar{z} = 1/2$ ), converting the problem into a set of linear constraints on operator data derivatives. A search for positive linear functionals $\alpha$ (combinations of derivatives) is executed via linear or semidefinite programming (Antunes, 2017).
Multipoint (Effective) Bootstrap: Instead of derivatives, the crossing equations are sampled at many points in the $(z, \bar{z})$ -plane. For a discretized operator spectrum up to cutoff $\Delta^*$ , the contribution from high-dimensional operators above the cutoff is rigorously bounded using Tauberian theorems, yielding a known error term. The resulting matrix inequality $M\vec{\rho} \geq \vec{\sigma} - \vec{\epsilon}_{\max}$ is solved as a linear program (Echeverri et al., 2016).

In each case, the solution space is explored for maximal gaps or bounds. At the boundary (when a bound is saturated), the extremal functional method can sometimes reconstruct a sparse "extremal" operator spectrum.

3. Extensions and Global/Local Symmetries

The bootstrap technique generalizes straightforwardly to four-point functions involving operators with spin, supersymmetry, or non-trivial global symmetry. When dealing with matrix or tensor operators, the OPE must be decomposed according to group representations. For example, in SU $(N_f)_L \times$ SU $(N_f)_R$ , the bootstrap equations become coupled via explicit crossing sum rules relating invariants in each channel (Antipin et al., 2014). For theories with product-group symmetry (e.g., O(15) $\otimes$ O(3)), the mixed-correlator bootstrap considers sets of correlators transforming in different symmetry sectors, leading to higher-dimensional vectorial sum rules enforced by SDP (Dowens et al., 2020).

In two-dimensional theories, the full Virasoro symmetry or its extensions (e.g., super-Virasoro, interchiral algebra) can be implemented by decomposing four-point functions into Virasoro conformal blocks, whose analytic and crossing properties become central (Picco et al., 2016, Jacobsen et al., 2018, He et al., 2020).

4. Critical Exponents and Scaling Dimensions

A flagship application of the numerical four-point bootstrap is the nonperturbative determination of scaling dimensions and OPE coefficients, i.e., critical exponents. For instance, in four-dimensional gauge–Yukawa theories with interacting conformal fixed points, the bootstrap is used to bound the dimensions of relevant composite operators built from the Higgs sector. Explicitly, the anomalous dimensions $\gamma_S$ (singlet) and $\gamma_A$ (adjoint) were computed perturbatively and compared against numerical four-point bootstrap bounds; the factorization $\Delta_A = 2 d_H$ in the adjoint channel was found to respect the bootstrap limit while $\Delta_S = 2 + \gamma_S$ could approach or exceed naive bounds, especially at large $\gamma_S$ (Antipin et al., 2014).

In three-dimensional O(N) models or the Ising model, the method isolates sharp "kinks" in dimension bounds that align with physical CFTs (Echeverri et al., 2016). The bootstrap also enables extraction of scaling dimensions for higher operators (not just the lowest) in several sectors via maximization procedures applied to multipoint data.

5. Comparison with Analytic and Lattice Methods

Numerical four-point bootstrap results are frequently compared to:

Perturbative and Multiloop CFT Calculations: For tractable models (e.g., gauge–Yukawa), explicit beta functions and anomalous dimensions computed to multiple loops inform the expected location of scaling dimensions and operator structure constants for a given fixed point (Antipin et al., 2014). Numerical bootstrap bounds either support or falsify these analytic expectations.
Monte Carlo and Lattice Calculations: In two-dimensional statistical models (e.g., percolation, Potts), structure constants and four-point connectivities computed via numerical bootstrap agree with independently measured connectivities from large-scale simulations (Picco et al., 2016).
Analytic Conformal Bootstrap: Resummations in terms of twist conformal blocks and their generalizations (the "H-functions") provide perturbative CFT data (anomalous dimensions and OPE coefficients) that benchmarks or constrains the numerical solution space (Henriksson et al., 2017).

Notably, matches and mismatches between analytic large-spin expansions, perturbative exponents, and numerical bootstrap bounds have guided both physical insight and methodological advances.

6. Advanced Topics and Methodological Innovations

Several methodological innovations have been incorporated into the four-point bootstrap:

Remainder/Truncation Control: Use of Hardy–Littlewood Tauberian bounds allows rigorous integration over high-dimensional operator contributions above a known cutoff $\Delta^*$ , yielding stated error controls for truncated systems (Echeverri et al., 2016).
Group-Theoretic Tensor Decomposition: Explicit index-level treatment of global symmetry groups through tensor contractions, facilitating the construction of basis invariants for multi-channel crossing (Antipin et al., 2014).
Multipoint Sampling: Sampling over the complex $z$ -plane, rather than derivatives at a point, is now standard in certain models, especially when analytic remainder bounds are available (Echeverri et al., 2016).
Mixed-Correlator Systems: Simultaneous imposition of crossing on multiple four-point functions involving various operator types (e.g., O(N) singlet and traceless symmetric sectors) enables discrimination among candidate fixed points and exclusion of unstable or "antichiral" theories (Dowens et al., 2020).
Extremal Functional/Determinant Methods: Alternative techniques for extracting or testing the spectrum when fusion rules are assumed or derivable, including via vanishing minors for overdetermined systems (Antunes, 2017).
Relation to Full Vertex Algebras: In 2d CFT, rigorous frameworks such as full vertex algebras provide axiomatic underpinnings for the consistency and associativity structure of four-point functions computed numerically, offering a mathematically precise backbone for bootstrap approaches (Moriwaki, 2020).

7. Phenomenological and Theoretical Implications

Results from the numerical four-point bootstrap have been instrumental in advancing both theoretical understanding and phenomenological modeling. In four-dimensional models resembling the Standard Model but with Banks–Zaks-like fixed points, the anomalous dimensions of Higgs-sector composite operators show critical exponents outside the Gaussian window, opening the possibility of phenomenologically relevant CFTs with large anomalous dimensions (Antipin et al., 2014). In statistical models, the method illuminates the operator spectra controlling geometrical observables (e.g., percolation connectivities, Potts model clusters) and reveals a much richer set of contributing fields (including dense sets of conformal weights and indecomposable modules) than previously conjectured (Jacobsen et al., 2018).

In higher-dimensional and supersymmetric models, precise bootstrap bounds on the dimensions of protected and unprotected operators are shaping understanding of SCFT landscapes, the structure of holographic correlators, and the uniqueness of special theories (e.g., the 6d $(2,0)$ $A_1$ theory at minimal central charge) (Beem et al., 2015, Zhou, 2017).

The numerical four-point bootstrap constitutes a central methodology in contemporary quantum field theory and statistical physics, uniting rigorous numerical optimization with analytic insight from conformal symmetry, operator algebra, and group theory. Its ongoing extensions—to include spinning correlators, nontrivial global symmetries, multipoint/functional approaches, and interplay with integrability and holography—make it a cornerstone of the nonperturbative paper of CFTs.