Reduced superblocks at next-to-next-to-extremality for all maximally supersymmetric CFTs

1. Problem Statement

The paper addresses the challenge of analyzing mixed four-point correlation functions of 1/2-BPS operators, denoted as $\langle \mathcal{O}_{k_1} \mathcal{O}_{k_2} \mathcal{O}_{k_3} \mathcal{O}_{k_4} \rangle$, in maximally supersymmetric Conformal Field Theories (SCFTs). Specifically, it targets theories in dimensions $d=3, 4, 6$ with 16 real supercharges:

• 3d: $\mathcal{N}=8$ ($\mathfrak{osp}(8|4)$)
• 4d: $\mathcal{N}=4$ ($\mathfrak{psu}(2,2|4)$)
• 6d: $\mathcal{N}=(2,0)$ ($\mathfrak{osp}(8^*|4)$)

The focus is on Next-to-Next-to-Extremal (NNTE) correlators, where the extremality $E$ is defined by the relation $k_4 = k_1 + k_2 + k_3 - 2E$ with $E=2$.

• The Challenge: Constructing explicit superconformal blocks (superblocks) for these mixed correlators is technically difficult. Unlike identical correlators or extremal cases, mixed correlators involve a proliferation of R-symmetry channels and superdescendant operators.
• The Obstacle: Previous work by Dolan, Gallot, and Sokatchev [1] expressed these correlators using "reduced correlators" acted upon by a differential operator $\Delta_\epsilon$. However, for $d=3$ (odd dimensions) and $d=6$, $\Delta_\epsilon$ is either non-local or has a non-trivial kernel, making it difficult to invert the operator to extract explicit block decompositions for the reduced functions.

2. Methodology

The author employs a systematic approach combining representation theory, differential operators, and special functions:

• Generalization of Reduced Correlators: The paper generalizes the formalism of [1] to mixed correlators with $E=2$. The four-point function $G_{\{k_i\}}$ is decomposed into a two-variable reduced correlator $T_{\{k_i\}}(U, V)$ and single-variable functions $h_{\{k_i\}}(z)$, acted upon by the operator $\Delta_\epsilon$.
• R-symmetry Channel Equations: The author derives a system of six coupled differential equations (R-symmetry channel equations) relating the superblocks to the reduced correlators. These equations are organized by the R-symmetry representations exchanged in the Operator Product Expansion (OPE).
• Jack Polynomial Expansion: To handle the non-locality of $\Delta_\epsilon$ in odd dimensions and the kernel issues in even dimensions, the author expands conformal blocks in terms of Jack polynomials $P^{(\epsilon)}_{a,b}(z, \bar{z})$. These polynomials are eigenfunctions of $\Delta_\epsilon$.
• Inversion Strategy: By utilizing the eigenfunction property of Jack polynomials and specific recurrence relations (derived in Appendices C and D), the author inverts the operator $\Delta_\epsilon$. This allows the extraction of explicit expressions for the reduced correlators $T_{\{k_i\}}$ and $h_{\{k_i\}}$ in terms of standard bosonic conformal blocks with shifted kinematic parameters.
• Recursive Generation: Once the "reduced blocks" are identified, the full spectrum of superdescendant conformal blocks within a supermultiplet is generated recursively using the derived recurrence relations, bypassing the need for case-by-case Racah-Speiser algorithm implementations for every specific correlator.

3. Key Contributions

A. Derivation of Reduced Superblocks for $E=2$

The paper provides the first explicit construction of reduced superblocks for NNTE correlators in all three maximally supersymmetric dimensions.

• Reduced Blocks: The author demonstrates that the two-variable reduced correlator $T_{\{k_i\}}$ can be expanded in terms of "reduced blocks" $T^{\{k_i\}}_{\Delta, \ell}$, which are proportional to standard bosonic conformal blocks $G_{\Delta, \ell}$ but with shifted external kinematics:
  $$(\tilde{\Delta}_{12}, \tilde{\Delta}_{34}) = (\Delta_{12}, \Delta_{34} - 2(\epsilon - 1))$$
  where $\epsilon = d/2 - 1$.
• Unified Formalism: This result unifies the treatment of 3d, 4d, and 6d theories, generalizing known 4d results and extending them to 6d and 3d.

B. Resolution of Kernel Ambiguities

The paper addresses the ambiguity arising from the non-trivial kernel of $\Delta_\epsilon$.

• It is shown that the homogeneous solutions to the operator equation (which would correspond to unphysical twists) are excluded by the requirement that the full superblock must satisfy the correct spectrum of conformal primaries in all R-symmetry channels.
• This allows for a unique determination of the reduced blocks for physical correlators.

C. Handling Mixed Correlators and Bose Symmetry

Unlike previous studies focusing on identical operators ($\langle \mathcal{O}_k \mathcal{O}_k \mathcal{O}_k \mathcal{O}_k \rangle$), this work handles mixed correlators ($k_1 \neq k_2$).

• The relaxation of Bose symmetry constraints in mixed correlators leads to a more complex spectrum of superdescendants (up to 74 bosonic blocks for a single long multiplet in 6d).
• The derived reduced blocks successfully package this complexity into a single object, significantly simplifying the bootstrap analysis.

D. Dimension-Specific Subtleties

• 6d ($\epsilon=2$): The results generalize the known $\langle \mathcal{O}_2 \mathcal{O}_2 \mathcal{O}_k \mathcal{O}_k \rangle$ case. The paper also discusses an alternative formulation involving a differential operator $\Upsilon$ that connects to holographic calculations and W-algebra data.
• 4d ($\epsilon=1$): The operator $\Delta_1$ becomes a multiplicative factor. The results align with existing literature (e.g., [13, 23]), confirming the consistency of the generalized formalism.
• 3d ($\epsilon=1/2$): This is a novel result. The author identifies a paradox: the redefinition of reduced correlators that eliminates the single-variable function $h_2$ seems to absorb BPS multiplets that should be encoded in the topological 1d sector. While the full superblocks are correct, the interpretation of the 1d topological limit in the reduced formalism remains an open puzzle.

4. Results

• Explicit Formulas: The paper provides explicit formulas for the reduced blocks $T^{\{k_i\}}_{\Delta, \ell}$ and $h^{\{k_i\}}_{1, \ell}$ for all maximally supersymmetric theories.
• Tables of Contributions: Comprehensive tables (Tables 1-4) map specific supermultiplets (Long, Semi-short, BPS) to their contributions in the two-variable ($T$) and single-variable ($h$) reduced correlators for 6d and 3d.
• Recurrence Relations: A complete set of recurrence relations (Appendix D) is provided to generate the full spectrum of superdescendants from the highest R-symmetry channel.
• Mathematica Implementation: The author provides a Mathematica notebook implementing these functions and recurrence relations, facilitating immediate use in numerical and analytic bootstrap studies.

5. Significance

• Simplification of the Bootstrap: The primary significance is the drastic simplification of the conformal bootstrap for mixed correlators. Instead of dealing with dozens of coupled conformal blocks for a single supermultiplet, researchers can now work with a single "reduced superblock."
• Applicability to Holography: These results are crucial for studying the AdS/CFT correspondence in maximally supersymmetric settings. They allow for the precise calculation of anomalous dimensions and OPE coefficients in the large-$N$ limit, which are dual to string/M-theory corrections in $AdS_4 \times S^7$, $AdS_5 \times S^5$, and $AdS_7 \times S^4$.
• Foundation for Higher Extremality: The methodology developed here serves as a blueprint for tackling higher extremality ($E > 2$) correlators, which are currently intractable due to the proliferation of reduced correlator functions.
• New Insights into 3d SCFTs: The work offers the first explicit reduced block decomposition for 3d $\mathcal{N}=8$ SCFTs at $E=2$, opening new avenues for numerical bootstrap studies in three dimensions.

In summary, this paper provides a powerful, unified technical framework for decomposing complex mixed correlators in maximally supersymmetric theories, overcoming previous obstacles related to non-local operators and kernel ambiguities, and paving the way for more efficient and precise conformal bootstrap analyses.