Integrable Bootstrap Program

• Integrable bootstrap program is a nonperturbative framework that employs analytic, unitary, crossing, and integrability constraints to construct exact S-matrices, form factors, and correlation functions in QFTs.
• It reduces complex quantum field problems to a convex optimization scheme, enabling precise vertex identification and robust control over scattering, finite density, and deformed systems.
• Extensions of the program include applications to Floquet dynamics, higher-dimensional gauge/string theories, and conformal Feynman integrals, providing broad practical insights and rigorous physical validation.

The Integrable Bootstrap Program refers to a suite of nonperturbative methods for the determination of exact observables in integrable quantum field theories (QFTs), based on exploiting constraints arising from analyticity, unitarity, crossing symmetry, and integrability. In its various incarnations, the program enables the construction of $S$-matrices, form factors, correlation functions, and related quantities, often without any explicit use of Lagrangian dynamics. Classic examples include the 2D $O(N)$ bosonic model, the sinh-Gordon model, and integrable sectors of planar $\mathcal{N}=4$ SYM, as well as recent extensions to finite-density, Floquet-driven, deformed, and higher-dimensional scenarios.

1. S-Matrix Bootstrap: Geometry and Vertex Characterization

The $S$-matrix bootstrap principle enforces a set of robust analytic and algebraic constraints on the $2\to2$ scattering amplitudes $S_a(\theta)$ of integrable 1+1d models:

• Analyticity: The amplitudes are meromorphic, with the physical sheet free of singularities except at physical thresholds or bound-state locations.
• Unitarity: $|S_a(s)|\le1$ on the physical cut, with strict saturation ($|S_a(s)|=1$) in the absence of particle production.
• Crossing symmetry: Linear relations exist between $s$- and $t$-channel amplitudes, such as $S_a(i\pi-\theta)=\sum_b C_{ab} S_b(\theta)$ for $O(N)$.
• Real analyticity: $S_a(s + i 0^+)=[S_a(s - i 0^+)]^*$.

These constraints carve out a convex set ("curved polytope") in the function space of allowed $S$-matrices. Integrable models such as the $O(N)$ bosonic model are realized as vertices of this convex body, simultaneously saturating the unitarity bounds across channels, i.e., $|S_a(\theta)|=1$ everywhere. The program reduces to identifying such vertices by maximizing a generic linear functional $$F_w[S]=\sum_a \int w_a(\theta) S_a(\theta) d\mu(\theta)$$, whose gradient lies within the cone defined by the normals to saturated constraints at the vertex. This approach translates the bootstrap into an infinite-dimensional convex optimization problem, which is numerically tractable via semidefinite programming. Notably, the characteristic zero and pole structure found numerically coincides with analytical S-matrix solutions, e.g., the Zamolodchikov-Zamolodchikov factorized $O(N)$ S-matrix, without explicitly imposing the Yang-Baxter equation (He et al., 2018).

2. Form Factor Bootstrap and Wightman Reconstruction

The form factor bootstrap—developed by Karowski, Weisz, and Smirnov—axiomatizes the matrix elements of local operators ${\cal O}(x)$, i.e.,

$$F_n^{\cal O}(\theta_1,\dots,\theta_n)=\langle0|{\cal O}(0)|\theta_1,\dots,\theta_n\rangle,$$

through:

• Watson equations (scattering/braiding): exchange of arguments yields acting with the S-matrix.
• Periodicity/crossing: relating $F_n(\theta_1+2\pi i,...)$ to cyclic permutations.
• Kinematic/bound-state pole structure: residues at physical rapidity differences correspond to lower-particle-number form factors.
• Lorentz covariance: under rapidity shifts.
• Growth bounds: at large rapidities.

The infinite series expansion for Wightman n-point functions,

$$W_n(x_1,...,x_n) = \sum_{k=0}^\infty \frac1{k!} \int \prod_{j=1}^k \frac{d\theta_j}{2\pi} F_{k+n}(\theta_1+i\pi,...,\theta_k+i\pi;...) e^{-i\sum_j p(\theta_j)\cdot x_j},$$

is conditional upon convergence: proved for the sinh-Gordon two-point case, but still open for higher correlators (Kozlowski et al., 13 Nov 2025). Under convergence, the bootstrap construction yields distributions satisfying all Wightman axioms: covariance, locality, spectrum, positive definiteness, and temperedness.

3. Thermodynamic Bootstrap and Finite Density Correlators

The thermodynamic bootstrap program (TBP) generalizes the form-factor framework to finite-energy-density backgrounds characterized by a filling function $\vartheta(\theta)$. In this context:

• Excitations are labeled as particle or hole states atop $|\vartheta\rangle$, and the finite-density form factors $$f_\vartheta^{\cal O}(\theta_1,...,\theta_n)_{\sigma_1,...,\sigma_n}$$ are normalized matrix elements between such states (Cubero et al., 2018).
• The axioms are modified to include “background dressing” factors, defining generalized scattering, periodicity (crossing), and annihilation-pole relations, as well as reparameterization invariance (Panfil et al., 2023).
• Minimal solutions are constructed via modified functional equations involving the thermodynamic back-flow $F(\theta|\theta')$. Operator dependence enters via polynomials fixed by clustering.
• The expansion of finite-density $n$-point functions (including a dressed regularization of singularities) provides a practical basis for computing finite-temperature and non-equilibrium correlations.

In the hydrodynamic limit, only the lowest particle-hole excitation contributes at leading order, yielding explicit connections with generalized hydrodynamics (GHD), including ballistic and diffusive scaling regimes. Multi-ph contributions allow systematic access to subleading corrections, reproducing GHD diffusion matrices directly from TBP axioms without phenomenological inputs (Cubero et al., 2019).

4. Extensions: Deformations, Floquet Integrability, and Higher Dimensions

The bootstrap paradigm robustly accommodates several extensions:

• Irrelevant deformations: Models deformed by $T\bar{T}$ or higher-spin currents preserve integrability via multiplication of the S-matrix by a CDD factor. Deformed form factors factorize into undeformed ones and a universal "dressing" factor, leading to explicit expressions for all $n$-point correlators and their UV/IR asymptotics (Castro-Alvaredo et al., 2023).
• Floquet integrability: In periodically driven systems with an integrable Floquet Hamiltonian, the stroboscopic bootstrap prescribes a new set of axioms for the evolution phases $F(\theta)$, which must satisfy discrete analogs of unitarity, crossing, Yang-Baxter, and bound-state consistency (Cubero, 2018).
• Higher-dimensional and planar gauge theory contexts: The program extends to non-relativistic models (e.g., the Lieb–Liniger model via a non-relativistic limit of sinh-Gordon (Panfil et al., 2023)) and to large-$N$ matrix-valued models. In the latter, all form factors and correlation functions of the $(1+1)$-dimensional principal chiral sigma model have been solved in the planar limit, enabling analytic control over observables in higher-dimensional $SU(\infty)$ Yang–Mills via a current–current coupled array approach (Cubero, 2013).

5. Integrability in Gauge/String Theory and the Planar SYM Bootstrap

In planar $\mathcal{N}=4$ SYM, the integrable bootstrap acquires a specific realization:

• The three-point function structure constants are constructed from fundamental "hexagon" form factors, built using $SU(2|2)$ symmetry and Watson/decoupling conditions, organized into dynamic and matrix parts (Basso et al., 2015).
• The hexagon bootstrap formalism captures weak-coupling (spin-chain) to strong-coupling (string worldsheet) limits, includes finite-size "wrapping" corrections, and is organized for generalization to higher-point functions via polygonal tessellations.
• This approach reveals deep connections between integrable bootstrap methods, algebraic-geometry/conformal bootstrap, and the quantum spectral curve formalism.

6. Integrable Bootstrap for Conformal Feynman Integrals

Beyond traditional QFT observables, bootstrap techniques have been adapted to the computation of conformal Feynman integrals possessing Yangian symmetry (Loebbert et al., 2019):

• Level-zero (conformal) and level-one (Yangian) symmetries enforce powerful partial differential equations (PDEs) on the integrals.
• The solution space reduces to linear combinations of multivariate hypergeometric series (Appell $F_4$, Lauricella functions), uniquely fixed by permutation symmetries and limit conditions.
• Mellin-Barnes representations provide alternative constructions, with each residue family corresponding to a Yangian-invariant solution.
• This integrability-based bootstrap is algorithmic, structurally akin to the S-matrix and form-factor programs.

7. General Features, Open Problems, and Outlook

The integrable bootstrap program in all its aspects is fundamentally nonperturbative, algebraic, and non-Lagrangian. Its key features are:

• Vertex identification: Integrable theories are recovered as extremal (often unique) points of convex sets defined by bootstrap constraints.
• Systematic construction: All dynamical information is encoded in analytic, algebraic, and geometric properties of $S$-matrices or form factors, typically without recourse to explicit action formulations.
• Numerical and analytic synergy: Techniques such as semidefinite programming, functional equation analysis, and Mellin-Barnes summation allow for explicit realization.
• Rigorous physical consequences: Once convergence is established (often only for two-point functions), the program yields QFTs satisfying all axiomatic requirements.

Outstanding challenges include:

• Full convergence analysis for general $n$-point Wightman functions.
• Extension to theories with bound states, resonances, or higher-dimensional kinematics.
• Deeper connections between the bootstrap, quantum spectral curve, and octagon/polygon tessellation programs in gauge and string theory.

The integrable bootstrap remains an active area at the intersection of mathematical physics, algebraic geometry, and rigorous quantum field theory.