Exact 3D Conformal Blocks from Fractional Calculus

1. Problem Statement

The global conformal block is a fundamental building block of the modern conformal bootstrap program, encoding the contributions of intermediate primary operators in correlation functions.

• The Gap: While compact, closed-form analytic expressions for conformal blocks are well-established in two dimensions (2D) and four dimensions (4D), a similarly compact form for three dimensions (3D) has remained elusive.
• The Conjecture: Previous work by Hogervorst (2016) proposed that 3D conformal blocks could be decomposed into a sum of 2D blocks with remarkably simple rational coefficients. While verified for low-order terms, this formula remained a conjecture lacking a rigorous proof.
• The Challenge: Existing methods for 3D blocks rely on series expansions or recursion relations, which are computationally cumbersome and lack the analytic clarity found in even dimensions.

2. Methodology: Fractional Calculus as a Transform

The author introduces a novel mathematical framework connecting conformal blocks to fractional calculus, specifically utilizing a modified form of the half-derivative.

• The Operator $T^\delta_x$: The paper defines a specific fractional integral operator $T^\delta_x$ based on the Riemann-Liouville derivative but adapted with a coordinate transformation $x \to -1/x$.
  $$(T^\delta_x f)(x) = x^{-\delta} \frac{1}{\Gamma(\delta)} \int_0^1 f(xt) (1-t)^{\delta-1} t^{-\delta-1} dt$$
  This operator satisfies a semigroup property ($T^\alpha T^\beta = T^{\alpha+\beta}$) and specific commutation relations with the dilation operator $D_x = x\partial_x$.

• The Half-Derivative Insight: The core innovation is applying the operator with $\delta = \pm 1/2$.

  • The operator $T^{1/2}_z$ acts on the 1D conformal block $k_h(z)$ (a hypergeometric function $_2F_1$) and transforms it into a simple monomial in the radial coordinate $\rho$:
    $$T^{1/2}_z k_h(z) = \rho^{h-1/2}$$
  • Conversely, $T^{-1/2}_z$ maps the monomial back to the block.
  • This "trivializes" the 1D block, suggesting that the complexity of conformal blocks in higher dimensions can be unraveled by transforming the Casimir equation into this fractional representation.

3. Key Contributions and Derivation

A. Reformulating the Casimir Equation

Conformal blocks are eigenfunctions of the quadratic Casimir operator $D$. The author transforms the standard Casimir equation $D G = \lambda G$ into a new representation using the fractional operators:
$$\tilde{D} \tilde{G}_{\Delta, \ell} = \frac{1}{2} C_{\Delta, \ell} \tilde{G}_{\Delta, \ell}$$
where $\tilde{G} = T^{1/2} G$ and $\tilde{D} = T^{1/2} D T^{-1/2}$.

• In 2D ($\nu=0$), this transformation reduces the Casimir equation to a simple sum of second-order derivatives, recovering the known product of 1D blocks.
• In 3D ($\nu=1/2$), the transformed equation allows for a systematic solution via the Frobenius method.

B. Exact Solution for 3D Blocks

By solving the transformed Casimir equation for general spin $\ell$ in 3D, the author derives an exact, closed-form solution.

• The transformed block $\tilde{G}^{(3d)}_{\Delta, \ell}(\rho, \bar{\rho})$ is expressed as the product of two generalized hypergeometric functions $_4F_3$:
  $$\tilde{G}^{(3d)}_{\Delta, \ell} = \rho^{\frac{\Delta-1-\ell}{2}} \bar{\rho}^{\frac{\Delta-1+\ell}{2}} \times \left[ _4F_3(\dots; \rho\bar{\rho}) \right] \times \left[ _4F_3(\dots; \rho/\bar{\rho}) \right]$$
• Crucially, the second $_4F_3$ function truncates to a polynomial because one of its parameters is a negative integer ($-\ell$), ensuring the solution is well-behaved.

C. Proof of Hogervorst's Formula

By applying the inverse transform $T^{-1/2}$ to the derived product of $_4F_3$ functions, the author expands the result back into the standard basis of 2D blocks.

• The resulting expansion coefficients match exactly the rational form conjectured by Hogervorst.
• This provides the first rigorous proof of Hogervorst's formula for spinning operators in 3D.

4. Results and Numerical Validation

• Analytic Form: The paper provides the first compact analytic formula for 3D conformal blocks with arbitrary spin, replacing double summations with a product of hypergeometric functions.
• Convergence: The transformed blocks $\tilde{G}$ converge significantly faster and more smoothly than standard blocks. While standard blocks develop logarithmic divergences as $z, \bar{z} \to 1$, the $_4F_3$ representation remains finite.
• Numerical Tests:
  • The author validates the formula against standard numerical evaluations of scalar blocks, showing agreement within machine precision.
  • A crossing symmetry test was performed using 3D Ising model data. The s-channel and t-channel expansions, truncated at spin $\ell_{max}=10$, agreed to high precision, confirming the utility of the new formula in bootstrap calculations.

5. Significance and Applications

• Resolution of a Long-Standing Problem: The paper resolves a major open problem in the conformal bootstrap by providing an exact solution for 3D blocks, a dimension critical for studying physical systems like the 3D Ising model.
• New Analytical Tools: The connection between fractional calculus and conformal field theory (CFT) offers a new paradigm. The fractional derivative $T^{1/2}$ acts as a "simplifier," converting complex hypergeometric structures into monomials.
• Bootstrap Efficiency:
  • OPE Coefficients: The method offers a direct way to compute Operator Product Expansion (OPE) coefficients from power series without complex integrations (unlike the alpha-space transform).
  • Crossing Equations: The reformulation of the crossing symmetry equation using the transformed blocks and a specific "crossing kernel" simplifies the numerical implementation of the bootstrap in 3D.
• Future Directions: The framework suggests potential extensions to mixed correlators and arbitrary dimensions $d$, though this requires generalizing the integral transform to preserve the necessary algebraic properties.

Conclusion

Chaoming Song's work establishes a profound link between fractional calculus and conformal field theory. By leveraging a modified half-derivative operator, the author derives the first exact closed-form expression for 3D conformal blocks, rigorously proving Hogervorst's conjecture. This breakthrough not only fills a theoretical gap but also provides powerful, convergent tools for both analytical and numerical studies of the conformal bootstrap.