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Conformal bootstrap

The conformal bootstrap is a approach in to constrain and solve conformal field theories (CFTs)—quantum field theories invariant under conformal transformations—by exploiting fundamental consistency conditions such as unitarity, crossing symmetry in correlation functions, and the convergence of operator product expansions (OPEs), without assuming a specific underlying or relying on . This method leverages the infinite-dimensional conformal symmetry group to derive exact bounds on operator dimensions, OPE coefficients, and other observables, providing a powerful tool for studying strongly coupled systems where traditional perturbative techniques fail. The origins of the conformal bootstrap trace back to the late 1960s and early 1970s, when Alexander Migdal and Alexander Polyakov developed the "old" bootstrap as a self-consistent to compute conformal dimensions and couplings in using three-point functions and skeleton diagrams, predating the full formulation of the . This early framework was motivated by studies of strong interactions and at criticality, emphasizing conformal invariance's role in deriving universal properties. The approach gained renewed in the with the exact solution of two-dimensional CFTs by Belavin, Polyakov, and Zamolodchikov (BPZ), who introduced the and minimal models, establishing crossing symmetry as a key constraint on four-point functions. Polyakov's 1974 work further formalized the modern bootstrap by incorporating crossing equations, bridging and . In the , the conformal bootstrap evolved into two complementary branches: the numerical and analytic bootstraps. The numerical conformal bootstrap, pioneered by Rattazzi et al. in 2008, reformulates crossing symmetry as a problem to obtain rigorous bounds on CFT data, achieving high-precision results such as the of the three-dimensional (e.g., anomalous dimension η ≈ 0.0363 and correlation length exponent ν ≈ 0.630). This method has since been applied to diverse systems, including four-dimensional theories, supersymmetric CFTs, and the O(N) vector models, often saturating bounds and identifying unique theories consistent with experimental data. Complementing this, the analytic bootstrap employs dispersion relations, Lorentzian inversion formulas, and large-spin perturbations to derive universal results, such as bounds on energy flux in scattering amplitudes and connections to via the AdS/CFT correspondence. Recent advances, as of 2025, include extensions to higher dimensions, non-unitary theories, and machine learning-assisted optimizations, with ongoing challenges in proving uniqueness for generic CFTs and tackling fermionic or gauged systems.

Introduction

Overview

The conformal bootstrap is a approach to studying conformal field theories (CFTs) by determining the scaling dimensions \Delta and (OPE) coefficients \lambda of operators through consistency conditions imposed by conformal invariance, unitarity, and crossing symmetry. Unlike Lagrangian-based methods, it relies solely on the fundamental symmetries of CFTs to constrain the theory's spectrum and dynamics without perturbative expansions. At its core, the method decomposes correlation functions, such as the four-point function of scalar s, into a sum over conformal blocks—basis functions that encode the contributions from exchanged s and their descendants in different OPE channels. Crossing equates the decompositions in distinct channels (e.g., s- and t-channels), yielding equations that bound or exactly solve for the allowed values of \Delta and \lambda, ensuring the theory remains unitary and local. This framework exploits the to systematically organize the operator content of the CFT. The conformal bootstrap applies broadly to CFTs describing critical points in statistical mechanics models, such as phase transitions, and fixed points in quantum field theories. For instance, it has revealed universal properties like critical exponents in the three-dimensional Ising model without relying on perturbation theory, providing some of the most precise determinations available, such as the correlation length exponent \nu \approx 0.629971.

Motivation and Importance

The conformal bootstrap approach arises in the study of conformal field theories (CFTs), which describe the low-energy physics at fixed points. These fixed points characterize second-order phase transitions in condensed matter systems, such as the critical point in the three-dimensional relevant to ferromagnetic transitions in magnets or liquid-vapor transitions in fluids. In , CFTs emerge as (UV) completions of effective field theories, providing non-perturbative descriptions of strongly interacting systems like those in or potential beyond-Standard-Model scenarios. Unlike traditional perturbative methods, which rely on weak-coupling expansions and break down in strongly coupled regimes prevalent at these fixed points, the conformal bootstrap is inherently and exact in principle. It leverages consistency conditions—such as unitarity, crossing , and the —to derive rigorous constraints on CFT data, including scaling dimensions and OPE coefficients, without requiring knowledge of the underlying microscopic or . This allows for universal bounds on physical quantities, offering insights into theories solely from principles. Perturbative expansions, such as the epsilon-expansion around fixed points, become unreliable at strong due to poor , while simulations, though , are computationally demanding and often limited by finite-size effects or discretization errors. The bootstrap circumvents these issues by directly targeting the infrared CFT, yielding results that surpass alternatives—for instance, determining the Ising model's leading scalar dimension to high accuracy. Its impact lies in forging deep connections between , where it tests universality classes like the models for Heisenberg magnets, and , including holographic duals via AdS/CFT for in . This enables rigorous tests of theoretical predictions against experiments in diverse physical contexts.

Background in Conformal Field Theory

Conformal Symmetry

Conformal symmetry extends the Poincaré invariance of relativistic quantum field theories by including transformations that preserve angles but allow for changes in lengths. In d spacetime dimensions, the conformal group is the Lie group \mathrm{SO}(d+1,2), which acts linearly on an embedding space \mathbb{R}^{d+1,2} and includes translations, rotations (Lorentz transformations), dilatations, and special conformal transformations as its generators. This group has \frac{(d+1)(d+2)}{2} generators, enlarging the Poincaré group by adding the dilatation generator D and the special conformal generators K^\mu. The of the is defined by the commutation relations among its generators: the P^\mu for translations, the M^{\mu\nu} for Lorentz transformations, the dilatation D, and the special conformal K^\mu. Key relations include [D, P^\mu] = i P^\mu, [D, K^\mu] = -i K^\mu, and [K^\mu, P^\nu] = 2i (\eta^{\mu\nu} D - M^{\mu\nu}), where \eta^{\mu\nu} is the Minkowski metric, alongside the standard Poincaré algebra [M^{\mu\nu}, P^\rho] = i (\eta^{\nu\rho} P^\mu - \eta^{\mu\rho} P^\nu). These relations ensure that conformal transformations preserve the flat metric up to a Weyl factor, enabling the formulation of theories invariant under local rescalings in certain limits. In conformal field theories (CFTs), local operators transform covariantly under the conformal group, with primary operators forming the building blocks of irreducible representations. A primary operator \mathcal{O}(x) of scaling dimension \Delta and spin representation transforms under dilatations as \mathcal{O}(\lambda x) = \lambda^{-\Delta} \mathcal{O}(x), and is annihilated by the special conformal generators at the origin, [K^\mu, \mathcal{O}(0)] = 0. Descendants are obtained by acting with derivatives, forming a tower with positive norms in unitary representations. Unitary representations of the are classified by the scaling dimension \Delta and , ensuring the has positive-definite norms. The quadratic operators are C_1 = \frac{1}{2} (D^2 - \frac{1}{2} (P \cdot K + K \cdot P)) with eigenvalue \Delta (\Delta - d) for scalars, and C_2 involving the of \mathrm{SO}(d-1,1), which for -\ell representations is \ell (\ell + d - 2). Unitarity imposes bounds, such as \Delta \geq \frac{d-2}{2} for scalar primaries, guaranteeing positive norms for all descendants. A hallmark of CFTs is the existence of a conserved, traceless stress-energy tensor T^{\mu\nu}, which is a primary operator of dimension \Delta = d and spin-2. Conservation \partial_\mu T^{\mu\nu} = 0 follows from translation invariance, while tracelessness T^\mu_\mu = 0 encodes the enhanced conformal symmetry beyond mere . This tensor generates the conformal transformations via and is symmetric T^{\mu\nu} = T^{\nu\mu}.

Operators and Correlation Functions

In conformal field theories (CFTs), local operators are classified according to their transformation properties under the . Primary operators are the highest-weight states in irreducible representations, transforming in a specific way under conformal transformations, while descendants are obtained by acting on primaries with covariant derivatives \partial_{\mu}, forming a that provides the full representation space. Scalar primaries are Lorentz scalars with scaling dimension \Delta, vector primaries transform in the vector representation, and tensor primaries in higher-rank symmetric traceless representations, all annihilated by special conformal generators. Conformal invariance uniquely determines the form of two-point correlation functions for primary operators. For two identical scalar primaries \mathcal{O}_i of dimension \Delta_i in d-dimensional , the two-point function is \langle \mathcal{O}_i(x) \mathcal{O}_i(0) \rangle = \frac{1}{|x|^{2\Delta_i}}, normalized appropriately, while it vanishes for operators of different dimensions or non-matching representations due to enforced by the . For non-identical scalars or higher-spin operators, the functions take analogous forms dictated by the , ensuring invariance under translations, rotations, dilations, and special conformal transformations. Three-point functions of primary operators are also fixed up to a constant by conformal symmetry. In general, for scalar primaries \mathcal{O}_i, \mathcal{O}_j, \mathcal{O}_k with dimensions \Delta_i, \Delta_j, \Delta_k, the correlator is \langle \mathcal{O}_i(x_1) \mathcal{O}_j(x_2) \mathcal{O}_k(x_3) \rangle = \frac{\lambda_{ijk}}{|x_{12}|^{\Delta_i + \Delta_j - \Delta_k} |x_{13}|^{\Delta_i + \Delta_k - \Delta_j} |x_{23}|^{\Delta_j + \Delta_k - \Delta_i}}, where \lambda_{ijk} are the structure constants encoding the theory's dynamics, which can be real and positive in unitary theories. For operators with spin, the form includes additional tensor structures contracted with polarization vectors or tensors, but the overall scaling is preserved. Four-point functions of identical scalar primaries \langle \mathcal{O}(x_1) \mathcal{O}(x_2) \mathcal{O}(x_3) \mathcal{O}(x_4) \rangle are not fully fixed by and depend on two independent cross-ratios. In signature, they take the form \frac{1}{|x_{12}|^{2\Delta} |x_{34}|^{2\Delta}} g(u,v), where u = z \bar{z} and v = (1-z)(1-\bar{z}) with z, \bar{z} the complex cross-ratios derived from inverting the points onto the , and g(u,v) is a determined by the content and interactions. This non-trivial dependence on cross-ratios encodes the full spectrum and OPE coefficients of the theory. Unitarity in CFTs imposes bounds on the scaling dimensions to ensure positive norms in the Hilbert space. For scalar primary operators, the unitarity bound is \Delta \geq \frac{d-2}{2}, with equality corresponding to free fields; violations would imply non-unitary representations. These bounds extend to spinning operators, where the dimension must exceed the spin plus a dimension-dependent term, guaranteeing positive OPE coefficients squared in the decomposition of correlators.

Operator Product Expansion

In quantum field theory, the operator product expansion (OPE) provides a systematic way to express the product of two local operators at nearby points as an infinite sum over other local operators located at one of the points. Specifically, for operators O_i(x) and O_j(0) in a conformal field theory (CFT), the OPE axiom states that O_i(x) O_j(0) \sim \sum_k C_{ij}^k(x) O_k(0), where the coefficients C_{ij}^k(x) are c-number functions that include tensor structures depending on the spins of the operators and the OPE coefficients \lambda_{ijk}, which are scalars encoding the strength of the contribution from each O_k. Conformal invariance severely constrains the form of these coefficients, fixing them up to the OPE coefficients \lambda_{ijp} for quasi-primary operators O_p. Quasi-primary operators are those annihilated by the special conformal generators K_\mu of the conformal algebra, meaning they transform covariantly under the full global without mixing with descendants under these transformations. The resulting conformally covariant OPE takes the form O_i(x) O_j(0) = \sum_p \lambda_{ijp} \, |x|^{-\Delta_i - \Delta_j + \Delta_p} G_p(x, \partial) O_p(0), where \Delta denotes scaling dimensions, and G_p(x, \partial) accounts for contributions from the descendants of the quasi-primary O_p, obtained by acting with differential operators built from the generators. This expansion is valid in the short-distance limit, where |x| is much smaller than other length scales in the theory, allowing the replacement of the operator product by the sum over intermediate operators. In this regime, the OPE encodes key non-perturbative information, including the fusion rules—which specify which operators O_p can appear (\lambda_{ijp} \neq 0)—and the anomalous dimensions \Delta_p of the exchanged operators. The OPE plays a central role in analyzing correlation functions by enabling the insertion of the into multi-point functions, decomposing them into sums over contributions from individual exchanged operators. This relates different operator orderings in the correlator through the shared spectrum and OPE coefficients, providing a foundation for consistency conditions in CFT.

The Conformal Bootstrap Method

Conformal Blocks

Conformal blocks are the conformal-invariant building blocks that diagonalize the decomposition of four-point correlation functions in conformal field theories. For identical scalar primary operators \phi of dimension \Delta_\phi, the s-channel (OPE) yields \langle \phi(x_1) \phi(x_2) \phi(x_3) \phi(x_4) \rangle = \sum_{\Delta,l} \lambda_{\phi\phi\mathcal{O}}^2 \, g_{\Delta,l}(u,v), where the sum runs over exchanged primaries \mathcal{O} with scaling dimension \Delta and spin l, \lambda_{\phi\phi\mathcal{O}} are the OPE coefficients, and g_{\Delta,l}(u,v) are the conformal blocks depending on the cross-ratios u = z\bar{z} and v = (1-z)(1-\bar{z}) with z the complex cross-ratio. These blocks are constructed as solutions to differential equations imposed by the Casimir operators of the conformal group, which encode the requirement of invariance under global conformal transformations. In the embedding space formalism, where points in d-dimensional Minkowski space are represented as null vectors in a higher-dimensional space, global conformal blocks for arbitrary spins can be derived explicitly by solving these Casimir equations. Conformal blocks admit useful series expansions for analytic studies. In the small cross-ratio limit |z| \ll 1, corresponding to the OPE regime, g_{\Delta,l}(z,\bar{z}) \sim \sum_n c_n z^n \bar{z}^{\bar{n}}, where the coefficients c_n are determined recursively from the dimensions \Delta and l. For large spin l \gg 1, expansions in the Regge limit probe the high-energy behavior and facilitate approximations in the conformal bootstrap. The crossing symmetry of four-point functions implies that conformal blocks in different channels—s-channel (12→34), t-channel (14→23), and u-channel (13→24)—must yield equivalent decompositions, relating g_{\Delta,l}(u,v) across channels. Explicit expressions for conformal blocks vary by dimension. In two dimensions, they take closed forms involving products of hypergeometric functions of the cross-ratios, reflecting the structure of the . In general dimension d > 2, closed forms are unavailable, but perturbative series expansions in powers of the cross-ratios or numerical evaluations provide practical computations, as developed for spinning exchanges.

Crossing Symmetry

Crossing symmetry is a fundamental principle in that arises from the invariance of correlation functions under the exchange of positions. For the four-point \langle \mathcal{O}_1(x_1) \mathcal{O}_2(x_2) \mathcal{O}_3(x_3) \mathcal{O}_4(x_4) \rangle, this symmetry equates different (OPE) channels, such as the s-channel (grouping 1-2 and 3-4) and the t-channel (grouping 1-4 and 2-3). Specifically, exchanging 1 and 3 (or equivalently 12 ↔ 34) maps the s-channel decomposition to the t-channel one, ensuring the function remains unchanged. This invariance leads to the crossing equations, which express the equality of OPE expansions in different channels. In terms of conformal cross-ratios u = z \bar{z} and v = (1-z)(1-\bar{z}), the four-point function of identical scalar operators \phi of dimension \Delta_\phi can be written as G(u,v) = \sum_i \lambda_i^2 g_{\Delta_i, \ell_i}^{(s)}(u,v) = \sum_i \lambda_i^2 g_{\Delta_i, \ell_i}^{(t)}(u,v), where \lambda_i are OPE coefficients, and g_{\Delta_i, \ell_i} are conformal blocks with exchanged operator dimensions \Delta_i and spins \ell_i. For theories with identical external operators, the equation takes a vector form in the space of representations, capturing symmetries like \mathbb{Z}_2. These equations form an infinite system constraining the CFT spectrum \{\Delta_i, \lambda_i\}. The implications of crossing symmetry are profound for the conformal bootstrap: it imposes consistency conditions on the operator spectrum and OPE coefficients, with unitarity requiring \lambda_i^2 \geq 0. This generates a set of nonlinear equations that must be satisfied for any consistent CFT, limiting possible theories to those where the spectrum aligns across channels. In approximations, global crossing employs blocks from the global conformal group SO(d+1,1), focusing on low-spin exchanges (e.g., scalars and stress tensor), while full crossing incorporates all spins for exact enforcement, though computationally intensive. A representative example is the crossing symmetry for the scalar four-point function \langle \phi \phi \phi \phi \rangle in \mathbb{Z}_2-symmetric theories, such as the three-dimensional . Here, the external scalar \phi (corresponding to the spin field \sigma) leads to a crossing equation mixing even and odd blocks, constraining the dimensions and coefficients of exchanged s like the \epsilon. This setup has been pivotal in numerically solving for in the .

Bootstrap Equations

The conformal bootstrap equations arise as consistency conditions imposed by the crossing symmetry of four-point correlation functions in conformal field theories (CFTs). In the (OPE) limit, the four-point function of scalar primary s \mathcal{O}_i can be expanded in terms of conformal blocks, leading to an equation of the form \sum_k C_{ik} g_k(u,v) = \sum_k C_{ki} h_k(u,v), where C_{ik} are matrices encoding the OPE coefficients \lambda_{ijk}^2 and the blocks g_k, h_k depend on cross-ratios u,v. This crossing equation must hold for all , providing infinite constraints on the spectrum of operator dimensions \Delta_k and OPE coefficients. Unitarity in CFTs imposes that the crossing matrix C_{ij}, which relates different OPE channels, has positive eigenvalues, ensuring the theory's is positive-definite. For the spectrum to be consistent, these eigenvalues must remain non-negative across all channels, often formalized as the condition that the of certain submatrices of C satisfies \det C \geq 0, or more generally, that the matrix is positive semi-definite. This follows from the requirement that the decomposition into irreducible representations preserves the unitarity of the theory. Unitarity also yields fundamental bounds on operator dimensions, such as \Delta \geq \frac{d-2}{2} for scalar primaries in d-dimensional CFTs, with the bound tightening to \Delta \geq d-2 for the stress tensor and \Delta \geq \frac{d}{2} - 1 + \ell for spinning operators of spin \ell. These bounds exclude negative-norm states and are saturated by theories or mean-field limits, serving as starting points for the allowed . Positivity constraints further require that squared OPE coefficients \lambda^2 > 0 for all operators in the , reflecting the positive inner product in the unitary . This leads to inequalities of the form \sum_k \lambda_k^2 f(\Delta_k) \geq 0, where f are suitable positive test functions or kernels derived from the crossing equation, ensuring no negative contributions destabilize the expansion. Such conditions are pivotal for deriving sum rules and bounding anomalous dimensions. In practical implementations, the bootstrap equations are often truncated by assuming a finite set of low-lying operators, with higher-dimensional ones integrated out as an effective ; the error is controlled by imposing gaps \Delta_{\rm gap} above the truncation, ensuring as the gap increases. This finite-dimensional reduction turns the infinite constraints into a or inequalities amenable to . The analytic bootstrap extends these equations by leveraging dispersion relations or lightcone limits to derive exact sum rules, such as moments of the \sum_k \lambda_k^2 \Delta_k^n = c_n, without full numerical optimization. These relations exploit the analyticity of functions in space or the OPE limit, providing constraints even in strongly coupled regimes.

Numerical Techniques

Semidefinite Programming

The numerical implementation of the conformal bootstrap relies on (SDP) to enforce the positivity conditions arising from unitarity and crossing symmetry in conformal field theories (CFTs). The crossing equations, which equate different OPE channel decompositions of four-point correlation functions, are recast in a vector form as \sum_i \lambda_i^2 \mathbf{v}_i(\Delta) = 0, where \lambda_i^2 are squared OPE coefficients, \Delta denotes operator dimensions, and \mathbf{v}_i(\Delta) are vectors comprising derivatives of conformal blocks evaluated at a fixed kinematic point, such as the crossing-symmetric point z = 1/2 (or \tau = 1/2 in the modular parameter). This setup truncates the infinite-dimensional crossing conditions to a finite basis of derivatives up to order N, transforming the problem into a system of linear constraints on the OPE data. To solve these constraints while ensuring positivity (\lambda_i^2 \geq 0), the SDP formulation posits a matrix Q(\tau, \tau') = \sum_i \lambda_i^2 K(\tau, \tau'; \Delta_i, \ell_i) \succeq 0, where K(\tau, \tau') is a reproducing constructed from the basis of derivatives, effectively projecting onto the of functions analytic in the CFT spectrum. This problem is solved using interior-point methods, which iteratively navigate the defined by the crossing equations and the positive semidefiniteness , yielding bounds on allowed CFT data. Implementations of this SDP approach include open-source software such as , a parallelized solver supporting tailored for bootstrap computations, which can handle matrix sizes corresponding to spins up to \ell = 20 or higher. These tools scale with the truncation parameter N \approx 1000 derivatives, enabling computations on multi-core clusters while maintaining through advanced linear algebra libraries. Achieved precision in such SDPs reaches relative errors of approximately $10^{-10} for low-dimensional problems, sufficient to resolve subtle features in the space of allowed dimensions. For mixed four-point functions involving non-identical operators, such as \langle \sigma \varepsilon \sigma \varepsilon \rangle in the 3D Ising model, the SDP framework extends naturally by incorporating multiple crossing equations and positivity conditions across distinct OPE channels, tightening bounds on the spectrum.

Bounds and Spectrum Extraction

The conformal bootstrap employs () to derive rigorous bounds on the scaling dimensions of s in conformal field theories (CFTs). These bounds manifest as allowed regions in parameter spaces, such as the plane spanned by the dimensions \Delta_\phi of a scalar primary \phi and \Delta_\varepsilon of the lowest scalar singlet beyond the identity. Solutions to the bootstrap equations exist only within these regions, where the positivity of the SDP matrices holds, effectively out "islands" compatible with unitarity and . For instance, in three-dimensional unitary CFTs, the unitarity bound requires \Delta_\phi > 1/2, and bootstrap analyses confirm this threshold while providing tighter constraints on related s. Incorporating multiple four-point correlation functions refines these bounds further. Each additional correlator imposes independent crossing constraints, and the intersection of their allowed regions forms a that excludes non-unitary points more effectively. This multi-correlator approach shrinks the parameter space, isolating viable CFT spectra and highlighting universal features, such as gaps in dimensions across different theories. To extract the full spectrum beyond mere bounds, the extremal functional method (EFM) is applied. This technique assumes an ansatz for the low-lying operator content, parameterized as \sum_k c_k \delta(\Delta - \Delta_k) for the density of states with coefficients c_k and dimensions \Delta_k, along with OPE coefficients \lambda. By solving the dual linear programming problem, the method identifies the extremal functional that is non-negative everywhere but vanishes precisely at the assumed spectrum points, thereby determining the \Delta_k and \lambda that saturate the bootstrap inequalities. This process targets the first few primary operators, enabling reconstruction of key CFT data. Error analysis in spectrum extraction relies on assessing convergence with respect to the SDP truncation scale, typically the maximum spin or dimension \Lambda. As \Lambda increases, the extracted dimensions stabilize, with discrepancies providing upper bounds on systematic errors. Bootstrap error bars are additionally estimated by varying the assumed gap above the low-lying spectrum, quantifying sensitivity to higher-state approximations. These diagnostics ensure reliability, particularly in higher dimensions where convergence is slower. Recent advances have introduced tools like the navigator method and techniques to accelerate extraction. The navigator method constructs a over the parameter space to guide searches toward allowed islands without exhaustive evaluations, enabling efficient exploration in high-dimensional settings. Meanwhile, algorithms optimize the search for solutions by treating the bootstrap equations as an for iterative improvement, outperforming traditional in complex cases post-2019. Further developments as of 2024 include enhanced software such as SDPB 2.0 and new algorithms like skydive for improved optimization in multi-point correlators and higher-dimensional searches.

Historical Development

Early Formulations (1960s–1970s)

The origins of the conformal bootstrap trace back to the late 1960s, when efforts to understand scale-invariant systems at critical points in statistical mechanics and quantum field theory began incorporating conformal symmetry. In 1970, Alexander Polyakov demonstrated that at the critical point of a system, the tracelessness of the stress-energy tensor implies the existence of an infinite number of conserved currents, enforcing full conformal invariance on correlation functions. This argument established conformal symmetry as a fundamental principle for describing long-distance behavior in critical phenomena, extending beyond mere scale invariance. Building on this foundation, the 1970s saw the development of self-consistency conditions for conformal field theories using the (OPE) and crossing symmetry. In 1971, Alexander Migdal formulated the first bootstrap equations by expanding four-point correlation functions via the OPE and imposing crossing symmetry, deriving sum rules that constrain operator dimensions and OPE coefficients in four dimensions. Subsequent work by Polyakov in 1974 further refined these sum rules through a non-Hamiltonian approach, emphasizing the skeleton expansion to solve for conformal data self-consistently. Independently, Sergio Ferrara, Antonio F. Grillo, and Riccardo Gatto in 1973 constructed conformally covariant OPEs for tensor representations, providing the algebraic framework for applying crossing in higher-point functions and deriving additional sum rules. The term "bootstrap" was coined by Migdal in 1971, drawing an analogy to the S-matrix bootstrap of Geoffrey Chew, where physical properties emerge from self-consistency without relying on a fundamental . These early formulations connected to broader contexts in physics, including —where Migdal and Vladimir Gribov explored anomalous scaling in high-energy scattering—and current algebra techniques in (QCD), linking conformal invariance to and hadron spectroscopy. However, the approaches were limited by assumptions of minimal operator content in the OPE spectrum and the absence of robust numerical tools, leading to poor convergence in dimensions greater than two and eventual overshadowing by methods.

Two-Dimensional Successes (1980s)

In the early , significant progress in the conformal bootstrap was achieved through the work of Belavin, Polyakov, and Zamolodchikov (BPZ), who exploited the infinite-dimensional to classify two-dimensional conformal field theories (CFTs) with central charge c < 1. These minimal models are characterized by the presence of null vectors in the Verma modules of primary operators, which impose differential equations on correlation functions and restrict the possible operator spectra to a discrete set. The BPZ approach demonstrated that the operator product expansion (OPE) coefficients and conformal dimensions could be determined exactly by solving the bootstrap equations derived from conformal symmetry. A key aspect of the Virasoro bootstrap in two dimensions involved applying crossing symmetry to the four-point correlation function \langle \phi \phi \phi \phi \rangle of identical primary fields, which decomposes into a sum of Virasoro conformal blocks. This decomposition leads to a system of equations that constrain the spectrum and OPE coefficients, often solvable analytically due to the algebra's structure. Further constraints arose from modular invariance of the partition function on the torus, which requires the spectrum to form representations invariant under SL(2,\mathbb{Z}) transformations, enabling the complete classification of consistent theories. Prominent examples include the unitary minimal series with central charges c = 1 - \frac{6}{m(m+1)} for integer m \geq 3, where the operator dimensions are given by \Delta_{r,s} = \frac{[(m+1)r - m s]^2 - 1}{4 m (m+1)} for $1 \leq r < m, $1 \leq s \leq m-1. The simplest case is the at m=3, with c = \frac{1}{2} and the spin operator dimension \Delta_\sigma = \frac{1}{8}, corresponding to the critical point of the two-dimensional . Another example is the at m=4, featuring c = \frac{7}{10} and a richer spectrum including supersymmetric extensions. These developments culminated in the complete classification of rational CFTs, where the primary fields form a finite set closed under fusion, providing a foundation for understanding integrable systems and string theory compactifications. The fusion rules governing OPE multiplicities were elegantly encoded in the Verlinde formula, which computes them from the modular S-matrix elements of the characters.

Modern Revival and Advances (2000s–Present)

The modern revival of the conformal bootstrap approach began in 2008 with the seminal work of Rattazzi, Rychkov, Tonni, and Vichi, who introduced numerical methods to derive rigorous bounds on scalar operator dimensions in four-dimensional conformal field theories using semidefinite programming techniques applied to crossing symmetry constraints. This approach marked a shift from analytic efforts in lower dimensions to computationally feasible bounds in higher dimensions, demonstrating that unitarity and crossing symmetry could exclude large regions of parameter space for operator spectra without assuming specific models. Building on this foundation, El-Showk et al. in 2011–2012 extended the numerical bootstrap to three-dimensional theories, focusing on the Ising model universality class. Their analysis of four-point functions of scalar operators yielded precise bounds on critical exponents, such as the scaling dimension of the leading scalar, which matched predictions from the ε-expansion to an unprecedented accuracy of several percent, highlighting the method's power for non-perturbative regimes. The 2010s saw a surge in collaborative efforts, catalyzed by the establishment of the Simons Collaboration on the Nonperturbative Bootstrap in 2013, which funded interdisciplinary research across institutions to advance numerical and analytic techniques. This period produced comprehensive reviews, such as that by Poland, Rychkov, and Vichi in 2019, synthesizing theoretical foundations, numerical implementations, and applications while emphasizing the bootstrap's role in constraining strongly coupled systems. Key methodological advances included the incorporation of mixed correlators by Kos, Poland, and Simmons-Duffin in 2014, which allowed bootstrapping systems with non-identical external operators and tightened bounds on operator spectra in models like the 3D Ising universality class. Progress in handling spinning operators came with efficient computations of spinning conformal blocks by Cuomo, Karateev, and Kravchuk in 2018, enabling the inclusion of higher-spin exchanges and expanding the scope to vector and tensor correlators. In the late 2010s and early 2020s, Alday and collaborators developed analytic sum rules derived from dispersive representations of crossing equations, providing infinite families of constraints on CFT data that complemented numerical bounds and facilitated large-spin expansions. The 2020s have featured further innovations, such as the , which exploits limits near the lightcone to derive asymptotic behaviors of operator dimensions and OPE coefficients, as explored in works by , , and . Extensions to fermionic theories, building on earlier efforts like those by , , and , have incorporated spinor representations and parity-odd structures to study models with fermions. Recent numerical developments, reviewed by , , , and in 2024, highlight optimizations in semidefinite programming solvers and hybrid analytic-numerical methods that have pushed precision in higher-dimensional bootstraps. Similar bootstrap techniques from two dimensions have been applied to non-minimal models like Liouville theory, where for central charges c > 25, the spectrum becomes continuous, and exact expressions for correlation functions involving exponential primary operators were derived in the 1990s using the DOZZ formula, describing fluctuating geometries in two-dimensional .

Applications

Three-Dimensional Ising Model

The three-dimensional describes the critical behavior of ferromagnets near the Curie point and belongs to the of Z₂-symmetric conformal field theories (CFTs). In this context, the conformal bootstrap approach targets the scaling dimensions of the lowest-lying scalar operators: the spin field \sigma (with Z₂ charge, transforming as \phi under the global symmetry) and the energy field \varepsilon (Z₂ invariant). The bootstrap equations are derived from the crossing symmetry of the four-point functions \langle \sigma \varepsilon \sigma \varepsilon \rangle and \langle \sigma \sigma \sigma \sigma \rangle, imposing consistency conditions on the operator spectrum and OPE coefficients. Pioneering numerical analyses using have yielded precise values for the scaling dimensions in the 3D Ising CFT. Specifically, \Delta_\sigma = 0.5181489(10), \Delta_\varepsilon = 1.412625(10), and the anomalous dimension \eta = 0.0362978(20), which align closely with high-precision simulations. These results demonstrate the power of the bootstrap in resolving the without lattice approximations, confirming the across diverse physical systems like uniaxial magnets and binary fluid mixtures. Bounds from the bootstrap further delineate the allowed parameter space for unitary Z₂ CFTs. For instance, positivity constraints exclude theories with \Delta_\varepsilon < 1.41, carving out an "island" of allowed dimensions around the Ising point and ruling out nearby tricritical fixed points that would require negative OPE coefficients. This isolation strengthens the identification of the 3D Ising CFT as the unique unitary theory in this regime. Extensions of the bootstrap to the fermionic Ising model incorporate Majorana fermions alongside the scalar \sigma, capturing supersymmetric aspects while maintaining Z₂ symmetry; this yields consistent spectra for the stress tensor multiplet and higher-spin operators. By the 2020s, refinements through higher-degree truncations in the polynomial approximations have pushed precision to relative errors of $10^{-5} for \Delta_\sigma and \Delta_\varepsilon, surpassing many traditional methods.

O(N) Vector Models

The O(N) vector models describe three-dimensional conformal field theories (CFTs) invariant under global O(N) symmetry, featuring N real scalar fields \phi^a (a=1,\dots,N) that transform in the fundamental representation. These models arise as critical points of systems with O(N)-invariant interactions, such as the continuum limit of lattice models with vector order parameters. The conformal bootstrap constrains their spectrum and operator product expansion (OPE) coefficients by imposing crossing symmetry and unitarity on four-point correlation functions, starting with the fundamental \langle \phi^{a}(x_1) \phi^{b}(x_2) \phi^{c}(x_3) \phi^{d}(x_4) \rangle, which projects onto O(N)-invariant tensor structures in the s-, t-, and u-channels. To probe the singlet sector, mixed correlators such as \langle \phi^{a} \phi^{a} s \, s \rangle are analyzed, where s denotes the lowest-dimensional scalar operator in the O(N) singlet representation (beyond the identity). These mixed systems tighten bounds by coupling vector and singlet exchanges in the OPE, assuming \phi^a and s are the only relevant scalars in their representations, and yield "islands" in the (\Delta_\phi, \Delta_s) plane consistent with known critical points for small N. Numerical semidefinite programming applied to these correlators produces rigorous upper bounds on scaling dimensions. For N=2, corresponding to the critical XY universality class, the bootstrap determines \Delta_\phi = 0.51905(10), saturating the bounds and aligning with Monte Carlo estimates. Bounds on \Delta_t, the dimension of the lowest traceless symmetric tensor in the two-index symmetric representation (arising in \phi \times \phi), versus \Delta_\phi form allowed regions that the O(2) model touches at its kink, with \Delta_t \leq 1.23613^{+0.00058}_{-0.00158} for the extracted \Delta_\phi. Similar saturations occur for N up to 100, confirming the models as isolated points in the space of O(N)-invariant CFTs. In the large-N limit, bootstrap bounds converge to the leading 1/N expansion of the critical O(N) model, reproducing \Delta_s \approx 2 - \frac{32}{3\pi^2 N} + \mathcal{O}(1/N^2), with the bounds for \Delta_\phi approaching the free-field value of 1/2 from above. This agreement validates the bootstrap's non-perturbative power across regimes. The bootstrap also addresses deconfined criticality in O(N) models, such as transitions from Z_2 (Ising-like) to U(1) (XY-like) phases for N=2 and analogous enhancements for N=3. By analyzing multiple four-point functions of order parameters, it derives necessary conditions for emergent continuous symmetry, requiring the infrared scaling dimension of the discrete order parameter to satisfy \Delta_1 > 1.08 for Z_2 \to U(1), ruling out certain proposed enhancements unless additional assumptions hold. Extensions to four dimensions focus on the Wilson-Fisher fixed points of O(N) vector models, obtained via epsilon expansion around d=4 but constrained non-perturbatively by bootstrap in 4 < d < 6. These studies confirm interacting fixed points with scalar primaries \phi^a and singlets, yielding upper bounds on \Delta_\phi that intersect epsilon-expansion curves for N \geq 1, and explore the approach to the free theory at d=6.

Supersymmetric Conformal Field Theories

Supersymmetric conformal field theories (SCFTs) extend the conformal bootstrap framework by incorporating supersymmetry, which imposes additional structure on the operator spectrum and correlation functions through the superconformal algebra. In four dimensions, the superconformal group is PSU(2,2|N), combining the conformal group SO(2,4) with N Poincaré supercharges and an R-symmetry group SU(N)_R. Superconformal multiplets are built from superprimary operators, which are annihilated by certain supercharges, and carry representations under the R-symmetry; shortened multiplets, protected by supersymmetry, saturate unitarity bounds and play a key role in constraining the theory. In three dimensions with N=2 supersymmetry, the bootstrap applied to four-point functions of chiral primaries yields tight bounds on the spectrum. Specifically, the dimension of the lowest unprotected scalar operator in the chiral-antichiral OPE satisfies Δ_φ ≲ 1.91, with a kink structure indicating possible realizations like the super-Ising model at Δ_φ ≈ 2/3. Chiral ring constraints further refine these bounds; for instance, in the critical Wess-Zumino model, the relation Φ² = 0 implies the absence of certain chiral primaries, eliminating contributions from specific multiplets in the crossing equations. For four-dimensional N=1 SCFTs, bootstrap analyses of mixed correlators involving chiral and vector multiplets provide bounds on operator dimensions and anomalies. In supersymmetric QCD (SQCD) within the conformal window (3N_c/2 < N_f < 3N_c), the quark superfield dimension is constrained to Δ_Q ≈ 1.43, consistent with R-symmetry assignments. Additionally, bounds on the c-conformal anomaly, derived from the stress tensor two-point function, exhibit a lower limit near Δ_φ ≈ 1.4, with numerical methods using superconformal blocks confirming minimal values like c_min ≈ 1/9 for certain chiral operators. Prominent examples include SQCD in its conformal window, where bootstrap constraints test non-perturbative duality and operator dimensions beyond weak-coupling predictions. In six dimensions, the (2,0) SCFTs feature self-dual tensor multiplets as the lowest components of half-BPS operators; the bootstrap of their four-point functions imposes universal constraints on the stress tensor multiplet spectrum, revealing island-like allowed regions in the space of scaling dimensions. Key techniques in these applications leverage supersymmetry to simplify computations. Superprimary conformal blocks, derived from the superconformal Casimir equations, decompose four-point functions into contributions from entire multiplets, reducing the complexity of crossing symmetry. Unitarity is enforced by restricting the operator spectrum to shortened multiplets, which automatically satisfy BPS conditions and provide protected subsectors that anchor the bootstrap bounds.

Extensions and Current Research

Higher Dimensions and Non-Unitary Models

In four dimensions, the conformal bootstrap provides stringent bounds on operator dimensions relevant to the hierarchy problem in extensions of the Standard Model, such as composite Higgs models, where the method constrains the spectrum of strongly coupled sectors to ensure naturalness without fine-tuning. These bounds arise from crossing symmetry and unitarity, limiting the possible scaling dimensions of composite states that could stabilize the Higgs mass against quantum corrections. For instance, bootstrap analyses exclude certain regions of parameter space for pseudoconformal technicolor theories, highlighting the tension between electroweak precision and conformal symmetry. Similarly, in the context of QCD-like gauge theories, the conformal bootstrap has been employed to probe the conformal window, where the theory develops an infrared fixed point. Within this window, the scaling dimension of the gluon operator, denoted Δ_G, exceeds 4, reflecting strong interactions that render marginal operators irrelevant and prevent confinement. This condition, derived from bounds on the lowest-dimensional scalar in the spectrum, helps delineate the range of fermion flavors N_f for which SU(N_c) gauge theories remain asymptotically free yet conformal in the infrared, with numerical optimizations yielding upper limits on anomalous dimensions consistent with lattice simulations. Non-unitary conformal field theories (CFTs) relax the unitarity bounds that constrain operator dimensions to be non-negative, allowing for negative scaling dimensions and negative central charges. A canonical example is the Lee-Yang CFT, associated with the Yang-Lee edge singularity, which features a central charge c = -22/5 and the scalar primary φ with dimension Δ_φ = -1/5; this model emerges in the critical point of certain statistical systems with imaginary magnetic fields. Conformal bootstrap techniques, such as the determinant method applied to crossing equations for the φφφφ correlator, have confirmed these exact values in two dimensions and extended estimates to higher dimensions via Padé approximants of the ε-expansion, revealing fixed points where the singularity persists. To handle negative dimensions computationally, truncation methods approximate the operator product expansion (OPE) by projecting onto a finite basis of derivatives, solving the crossing equations iteratively without relying on positivity; these approaches yield spectra for non-unitary models like percolation and self-avoiding walks, where traditional semidefinite programming fails. The ε-expansion around the upper critical dimension provides an analytic framework for bootstrapping the Wilson-Fisher fixed point in 2 < d < 4, where interactions deform the free scalar theory. Using large-spin perturbation theory and the inversion formula for double discontinuities, the bootstrap computes anomalous dimensions of leading-twist operators to order ε^3 or higher, matching perturbative results from while resumming non-perturbative effects. For the O(N) vector model, this yields precise values for the fundamental scalar dimension Δ_φ ≈ (d-2 + η)/2, with η the anomalous dimension, and extends to infinite families of double-trace operators without assuming equations of motion. In six dimensions, bootstrap methods target the elusive (2,0) superconformal field theories (SCFTs), which lack Lagrangian descriptions but admit holographic duals; analyses of stress-tensor multiplet four-point functions impose constraints on the tensor branch, where vacuum expectation values for tensor multiplets Higgs the theory to free hypermultiplets and vectors, bounding central charges and confirming the absence of certain irrelevant deformations. Despite these advances, higher-dimensional bootstraps (d > 4) face significant challenges, as often saturates the bounds, indicating that interacting CFTs are scarce and dominated by free or weakly coupled fixed points. The proliferation of conformal blocks and the weakening of crossing constraints lead to broader allowed regions in the , with fewer isolated islands corresponding to non-trivial theories; consequently, exact results are limited, and numerical efforts prioritize supersymmetric cases or ε-expansions near d=4.

Connections to AdS/CFT and Other Fields

The conformal bootstrap has profound implications for the AdS/CFT correspondence by deriving constraints on the spectrum and operator dimensions of boundary conformal field theories (CFTs) that directly impact the structure of bulk gravity theories. In particular, bootstrap analyses of three-dimensional CFTs reveal that conserved higher-spin currents with spin greater than two cannot exist in interacting theories without leading to inconsistencies with unitarity and crossing symmetry, thereby prohibiting higher-spin gauge fields in the dual AdS_4 gravity theory. This result, derived from the assumption of a slightly broken higher-spin symmetry, implies that the bulk theory must be described by Einstein gravity coupled to matter fields rather than higher-spin extensions, providing a non-perturbative confirmation of the holographic dictionary for weakly coupled gravity duals. Further examples include the application of the bootstrap in the Regge limit of four-point functions, which yields universal bounds on the Lyapunov exponent characterizing quantum chaos in holographic CFTs. These chaos bounds, saturated by black hole geometries in the bulk, ensure causality and unitarity in the AdS theory, with the Regge behavior of CFT correlators mapping to high-energy scattering in the bulk that aligns with the eikonal approximation of gravitational interactions. Such constraints have been instrumental in resolving aspects of the AMPS firewall paradox by demonstrating that quantum error correction and fast scrambling in the CFT preserve information recovery, consistent with bulk calculations of shockwave geometries and out-of-time-order correlators (OTOCs). Additionally, the lightcone bootstrap provides positivity bounds on energy flux coefficients in CFTs, which translate to constraints on effective field theory (EFT) Wilson coefficients in the AdS bulk, ensuring the attractiveness of gravity and the validity of local EFT descriptions up to the string scale. Beyond standard AdS/CFT, the bootstrap connects to by constraining \alpha' corrections through the of higher-point conformal blocks. In the context of _3/CFT_2 dualities, bootstrap equations for multi-point correlators of heavy operators reproduce the low-energy expansion of string scattering amplitudes, including terms like D^4 R^4 that encode \alpha'^3 to the , thereby offering a probe of the stringy regime. These methods extend to bootstrapping the landscape of string vacua, where spectral gaps in the CFT impose conditions on moduli stabilization and flux choices in the bulk. In , the conformal bootstrap extends beyond isolated critical points to study entanglement properties via the , where Rényi are computed using modular bootstrap constraints on twist operators. For instance, in two-dimensional CFTs describing quantum critical systems, the yields exact expressions for the entanglement across interfaces or in the presence of defects, providing insights into gapped phases and in materials like fractional quantum Hall states. This approach has been applied to compute universal contributions to entanglement in the , linking CFT data to experimental measures of quantum correlations in low-dimensional systems.

Open Problems and Future Directions

One of the primary challenges in the conformal bootstrap remains obtaining full analytic solutions for conformal field theories (CFTs) in dimensions greater than two, where exact solvability is rare beyond special cases like the two-dimensional minimal models. In three dimensions, uniqueness problems persist, such as rigorously proving that the Ising CFT is the sole unitary solution consistent with crossing symmetry and experimental data, despite numerical evidence isolating it within allowed parameter spaces. Similarly, nonexistence problems arise, exemplified by the need to demonstrate that no unitary CFT exists for the , which exhibits a rather than criticality. These issues highlight the limitations of current numerical methods in distinguishing physical theories from "masquerading" ones without additional inputs like the stress tensor or global symmetries. Handling infinite spectra poses another significant hurdle, particularly in gauge theories like 3D QED with N_f fermions, where the operator spectrum includes an infinite tower of gauge-invariant composites and monopoles with scaling dimensions scaling as \Delta \propto N_f, leading to slow in bootstrap bounds. Efficient computation of higher-spin conformal blocks is also challenging, as their inclusion in crossing equations requires optimized algorithms to manage the of terms in higher dimensions. Recent reviews emphasize that while numerical bootstraps have tightened bounds on —for instance, \Delta_\sigma = 0.518148806(24) for the Ising scalar—these computations strain resources for large spin or dimension. As of 2025, frontiers include explorations of duality-inspired fusion rules using categorical non-abelian symmetries, which extend the bootstrap to CFTs with non-invertible defects and fusion categories, enabling systematic constraints on spectra beyond traditional abelian cases. Precision studies of fermionic operators have advanced through numerical bootstraps, as detailed in recent theses that refine bounds on fermion bilinears and stress tensors in 3D CFTs, achieving higher accuracy via semidefinite programming with external fields. These efforts build on non-unitary and holographic connections, where bootstrap methods probe AdS/CFT duals for black hole microstates, though rigorous ties remain incomplete. Ongoing workshops underscore these directions, including the Simons Bootstrap Collaboration's 2025 event at ICTP-SAIFR in (July 21–August 8), which focused on numerical advances and 3D applications, and the Yukawa Institute's long-term workshop "Progress of Theoretical Bootstrap" in (October 27–November 28), emphasizing analytic and integrability techniques. Future directions involve integrating for optimization, such as "skydiving" algorithms that accelerate numerical searches for isolated islands in parameter space by orders of magnitude compared to traditional semidefinite relaxations. Analytic continuations of bootstrap equations, leveraging dispersion relations and lightcone limits, promise to extend constraints to higher-point correlators and non-unitary regimes. Implications for , including bootstrapping entanglement in holographic CFTs to infer bulk geometries, represent a high-impact frontier, with seminars highlighting how bootstrap bounds on OPE coefficients constrain entropy functions.