Conformal Bootstrap Methods

• Conformal bootstrap is a non-perturbative framework that uses symmetry and crossing constraints to determine operator spectrum and OPE coefficients in conformal field theories.
• The method employs analytic, algebraic, and numerical techniques to yield high-precision scaling dimensions, as demonstrated in studies like the Yang-Lee edge singularity and 3D Ising model.
• Fusion rules and truncation strategies simplify the infinite operator algebra into a manageable system of equations that robustly constrain the CFT data.

The conformal bootstrap is a non-perturbative program that leverages conformal symmetry, unitarity, and crossing symmetry to constrain and, in many cases, precisely determine the operator spectrum and operator product expansion (OPE) coefficients of conformal field theories (CFTs) in various dimensions. Rather than relying on Lagrangian formulations, the bootstrap encodes physical consistency conditions—most notably the associativity of the OPE and the crossing symmetry of multi-point correlation functions—into a set of equations whose solutions (possibly subject to further constraints such as unitarity or symmetry selection rules) characterize the space of physically admissible CFTs. The modern conformal bootstrap program encompasses analytic, algebraic, and powerful numerical techniques that have produced high-precision results for critical exponents in statistical and quantum field theories, resolved long-standing questions regarding critical phenomena, and opened a new path toward the classification and solution of strongly interacting QFTs.

1. Theoretical Foundations and Key Equations

The conformal bootstrap rests on several structural pillars. At the heart is the existence of an operator algebra formed by primary fields of the CFT and their descendants under the conformal group. Each local primary operator $\mathcal{O}_i$ has a scaling dimension $\Delta_i$ and transforms under specific representations of the conformal group. Conformal invariance completely determines the form of two- and three-point functions of local operators: $$\langle \mathcal{O}_1(x_1)\mathcal{O}_2(x_2)\rangle = \frac{C_{12}}{|x_{12}|^{2\Delta_1}} \delta_{12}$$

$$\langle \mathcal{O}_1(x_1)\mathcal{O}_2(x_2)\mathcal{O}_3(x_3)\rangle = \frac{C_{123}}{|x_{12}|^{\Delta_1+\Delta_2-\Delta_3}\,|x_{13}|^{\Delta_1+\Delta_3-\Delta_2}\,|x_{23}|^{\Delta_2+\Delta_3-\Delta_1}}$$

while higher-point functions, starting with the four-point function, contain nontrivial dynamical data.

A central structure is the OPE, which asserts that the product of two local operators can be expanded as

$$\mathcal{O}_i(x)\mathcal{O}_j(0) = \sum_k C_{ijk}(x,\partial) \mathcal{O}_k(0)$$

where the OPE coefficients $C_{ijk}$ (and the associated three-point structure constants) encode the full dynamical content.

For four-point functions, conformal invariance reduces the correlator to a function of cross-ratios $u$ and $v$: $$\langle \phi(x_1)\phi(x_2)\phi(x_3)\phi(x_4)\rangle = \frac{g(u,v)}{|x_{12}|^{2\Delta_\phi}|x_{34}|^{2\Delta_\phi}}$$ with

$$u = \frac{x_{12}^2 x_{34}^2}{x_{13}^2 x_{24}^2}, \qquad v = \frac{x_{14}^2 x_{23}^2}{x_{13}^2 x_{24}^2}$$

The function $g(u,v)$ admits a conformal block decomposition: $$g(u,v) = 1 + \sum_{\Delta, L} p_{\Delta, L} \, G_{\Delta, L}(u,v)$$ where $p_{\Delta, L}$ are related to the squared OPE coefficients and $G_{\Delta, L}(u, v)$ are conformal blocks, capturing the contribution of a primary of dimension $\Delta$, spin $L$ and all its descendants.

Crossing symmetry demands that the four-point function must be invariant under the interchange of operator insertions, yielding functional equations such as

$$\sum_{\Delta,L} p_{\Delta,L} \frac{v^{\Delta_\phi} G_{\Delta,L}(u,v) - u^{\Delta_\phi} G_{\Delta,L}(v,u)}{u^{\Delta_\phi} - v^{\Delta_\phi}} = 1$$

which encode the associativity of the OPE and nontrivial consistency constraints on the operator data.

2. Numerical Techniques and Truncation Strategies

Numerical implementations of the bootstrap revolve around converting the crossing symmetry equations into tractable forms. One robust method, detailed in (Gliozzi, 2013), applies a Taylor expansion of the crossing equations in reparameterized cross-ratio variables $(a,b)$ around the symmetric point $(a=1, b=0)$. The Taylor coefficients

$$f^{(m,n)}_{\alpha,\beta} = \left[ \partial_a^m \partial_b^n \frac{v^\alpha G_\beta(u,v) - u^\alpha G_\beta(v,u)}{u^\alpha - v^\alpha} \right]_{a=1, b=0}$$

become the entries in a (possibly infinite) linear system imposed by crossing.

Due to the infinite spectrum, one typically truncates the sum over conformal blocks, retaining only a finite set dictated by knowledge of low-lying operator content—the fusion algebra. Given $N$ operators, one enforces $M \geq N$ linear (homogeneous) constraints, yielding a matrix system whose nontrivial solutions (up to overall normalization) exist only if all $N \times N$ minors vanish: $$d_i(\{\Delta\}_N) = \det\left[ f^{(2m_i, n_i)}_{\Delta_\phi, \Delta_1, ..., \Delta_N} \right] = 0$$ These transcendental equations fix the scaling dimensions in the truncated theory. An inhomogeneous equation from the zeroth derivative ($f^{(0,0)}$) in turn fixes the OPE coefficients. This is the "determinant method" (or "small minors method") and is applicable both to unitary and nonunitary CFTs, provided the fusion algebra of low-lying primaries is specified.

A major strength of this approach is its flexibility: the sign of $p_{\Delta, L}$ is not fixed, so the method treats both unitary and nonunitary cases (such as the Yang-Lee edge singularity) on equal footing, contingent on the specified operator content.

3. Role of Fusion Rules and Truncability

Central to practical bootstrap analysis is the specification of the fusion algebra—i.e., which primary operators (and their multiplicities) appear in the OPE of two given primaries, e.g.,

$$[\Delta_\phi] \times [\Delta_\phi] = \sum_{i} N_i [\Delta_i, L_i]$$

This knowledge, which can come from physical arguments, symmetry, or previous calculations, reduces the number of free parameters, organizing the expansion and ensuring that the number of unknowns matches the number of equations.

When a theory is "truncable" at a given level $N$ (i.e., the fusion algebra is such that retaining $N$ low-lying primaries is consistent with the first $M \ge N$ equations), the method gives a direct algebraic (transcendental) system for the dimensions. In essence, the fusion algebra plays for higher-dimensional CFT a role analogous to the Vafa equations in rational 2D models.

4. Case Studies: Yang-Lee Edge Singularity and 3D Ising Model

The utility of the determinant-based, fusion-algebra-guided conformal bootstrap is exemplified by concrete studies:

Yang-Lee Edge Singularity

For the nonunitary ϕ³ theory (Yang-Lee edge), the fusion rule simplifies to

$$[\Delta_\phi] \times [\Delta_\phi] = 1 + [\Delta_\phi] + [D,2] + [\Delta_4,4] + \ldots$$

Retaining the first few blocks (e.g., a $3\times3$ system with free parameters $\Delta_\phi$, $\Delta_4$) and imposing vanishing minors yields numerical estimates:

• In $D=3$, $\Delta_\phi \approx 0.213$, $\Delta_4 \approx 4.49$.
• The critical edge exponent is then $$\sigma = \Delta_\phi / (D - \Delta_\phi) \approx 0.076$$, closely matching high-precision numeric data $\sigma = 0.077(2)$.

Similarly, in $D=4$, $\Delta_\phi \approx 0.823$, $\sigma \approx 0.259$.

3D Critical Ising Model

The 3D Ising model features a richer spectrum; in the fusion algebra various operators (identity, energy, higher-spin primaries) appear. By keeping, for example, five low-lying blocks and solving the $5\times5$ determinant conditions, the bootstrap yields

• $\Delta_{\phi^2} \approx 1.447$, $\Delta_\phi \approx 0.518$, which are in close agreement with leading numerical and Monte Carlo calculations.

These results demonstrate that determinant-based bootstrap with fixed fusion algebra can provide high-accuracy predictions for scaling dimensions and OPE coefficients (from the inhomogeneous constraint), often matching or improving upon those from other methods.

5. Applicability Beyond Unitarity and Broader Framework

This method is not restricted by assumptions of unitarity. The crossing equations and determinant conditions do not impose positivity on $p_{\Delta,L}$, allowing bootstrap analysis of nonunitary universality classes. This is of particular importance for models describing percolation, polymers, and non-hermitian Hamiltonians, where standard unitarity bounds are inapplicable.

More broadly, this approach provides a systematic construction of approximate solutions to the conformal bootstrap in situations where the OPE content is constrained, but the spectrum is not a priori known—fulfilling the vision of "solving" CFTs from crossing symmetry and operator algebra alone. The method is reminiscent of the algebraic structure of 2D rational CFTs (as in the Vafa equations), extended to arbitrary dimensions and both unitary and nonunitary systems.

This class of methods has influenced the development of more general numerical bootstrap strategies employing convex optimization, semidefinite programming, and analytic constructions in Mellin space or embedding space (Paulos, 2014, Fortin et al., 2016, Gopakumar et al., 2016).

6. Limitations, Trade-offs, and Practical Implementation

The methodology is heavily reliant on accurate knowledge of the fusion algebra and low-lying operator content; if key operators are missed, or if truncability does not hold to sufficient accuracy, results may become unreliable. The approach is best suited to theories where the spectrum is well understood, or where high-precision experimental or numerical data guide the fusion algebra input.

In practical terms:

• Truncation level $N$ sets the number of equations and unknowns, balancing computational complexity (size of the determinant system) and accuracy. Increasing $N$ generally improves fidelity but increases algebraic complexity.
• For systems with closely spaced or degenerate scaling dimensions, numerical stability can be a challenge; careful selection of which derivatives (i.e., which $(m,n)$ pairs for Taylor expansion) enter the determinants is often needed.
• In nonunitary contexts, special care must be taken in interpreting multiple real roots and the physical branch of solutions.

Despite these caveats, the determinant/fusion-algebra bootstrap provides a practical, direct pathway—beyond the boundary-bound numerical exclusion lines of standard convex optimization methods—to "solving" (within physical and numerical error) many CFTs of interest.

7. Summary Table of Methodological Steps

Step	Description	Key Formula/Action
1. Fusion Algebra	Specify primaries and multiplicities	$$[\Delta_\phi] \times [\Delta_\phi] = \sum_i N_i [\Delta_i, L_i]$$
2. Taylor Expansion	Expand crossing at $a=1, b=0$	$f^{(m,n)}_{\alpha, \beta}$ as above
3. Truncation	Retain first $N$ conformal blocks	Restricted sum $\sum_{i=1}^N$
4. Determinant System	Form $N\times N$ minors	$d_i(\{\Delta\}_N) = 0$
5. Extract OPE	Use inhomogeneous equation	$$\sum p_{\Delta, L} f^{(0,0)}_{\Delta_\phi, \Delta, L} = 1$$
6. Solve	Find roots of determinant equations	$\{\Delta_i\}$ fixed, OPEs from inhomogeneous

These steps reduce the bootstrap, in the presence of a known fusion algebra, to a tractable transcendental algebraic problem whose solution is typically highly constrained and often unique, especially when corroborated with other physical data.

8. Broader Implications and Outlook

By transforming bootstrap constraints into a finite, fusion-algebra-guided system, this approach makes possible the direct computation of CFT data for both unitary and nonunitary models. It has provided numerical data in agreement with, and at times more precise than, alternative analytic and lattice methods (notably for the Yang-Lee edge and 3D Ising models). This suggests a direct route to classifying and characterizing a wide variety of universality classes in statistical and quantum field theories, provided sufficient input on operator content is available.

Future developments may focus on automating the identification of minimal (or near-minimal) fusion algebras required for truncability, further improvements in determinant evaluation techniques for complex spectra, and formal understanding of the convergence properties of the approximations as truncation level increases.

The determinant/fusion-algebra bootstrap thus augments and deepens the conformal bootstrap arsenal, embodying the core idea that symmetry, OPE, and self-consistency are sufficient to nonperturbatively constrain, and often solve, interacting CFTs.