⚛️ high-energy theory

Bootstrapping Holographic Theories in the Planar Limit

This paper combines the numerical conformal bootstrap with supersymmetric localization to derive upper bounds on the scaling dimensions of the lowest scalar single-trace operators in planar $\mathcal{N}=4$ $SU(N)$ and $\mathcal{N}=2$ $U\!Sp(2N)$ super-Yang-Mills theories across all 't Hooft couplings, yielding results that align closely with known integrability predictions and weak/strong coupling limits.

Original authors: Shai M. Chester, Daniele R. Pavarini, Alessandro Piazza

Published 2026-09-28

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Shai M. Chester, Daniele R. Pavarini, Alessandro Piazza

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of theoretical physics, there exists a powerful idea known as the holographic principle. It suggests that a universe with gravity, like the one we inhabit, can be described entirely by a quantum theory living on its boundary, a sort of cosmic mirror where the complexity of three-dimensional space is encoded in two dimensions. For decades, physicists have struggled to understand the precise rules governing these boundary theories, especially when the forces involved are neither weak nor strong, but somewhere in between. To solve this, researchers have developed three distinct, non-perturbative tools: one that treats the system like a solvable puzzle using a technique called integrability, another that calculates specific properties by freezing the system into a simpler state through supersymmetric localization, and a third that deduces the rules of the game by demanding that the theory remains consistent with itself, a method known as the conformal bootstrap. While each tool has its strengths, none could previously map the entire spectrum of these theories across all possible force strengths without help from the others.

A team of researchers has now combined these three approaches to create a more complete map of these holographic worlds. They focused on two specific types of theoretical universes: one that mimics the behavior of closed strings, which are fundamental loops in string theory, and another that mimics open strings, which are like strands with endpoints. In the first case, they studied a theory known as N=4 super-Yang-Mills, which is a highly symmetric model often used as a testing ground for physics. In the second, they examined a different theory involving a specific gauge group and flavor symmetry, a model for which no complete solution was previously known. The goal was to determine the "scaling dimensions" of the lightest particles in these theories. In simple terms, a scaling dimension is a number that tells us how a particle's properties change as we zoom in or out on the universe; it is a fundamental fingerprint of the particle's identity.

The researchers faced a significant hurdle: in the limit where the number of particles becomes very large, the mathematical description of these theories becomes cluttered with "double-trace" operators. These are composite structures that act like noise, obscuring the signal of the fundamental "single-trace" particles that the scientists actually wanted to study. To cut through this noise, the team employed a technique called dispersion relations. Imagine trying to hear a specific instrument in a noisy orchestra; instead of listening to the whole sound, they developed a mathematical filter that isolates the unique signature of the fundamental instruments while ignoring the composite noise. By applying this filter, they could rewrite the theory's equations in a way that only the fundamental particles appeared, restoring the ability to use the bootstrap method effectively.

With this new clarity, they combined their filtered equations with constraints derived from supersymmetric localization. This step allowed them to inject the strength of the force, known as the 't Hooft coupling, directly into their calculations. By running a massive numerical optimization process, they searched for the highest possible value for the scaling dimension of the lightest particle that would still be consistent with all the laws of physics. They did this for every possible strength of the force, from very weak to very strong. For the first theory, where the answer was already known from a different method called integrability, their new bounds matched the known results with remarkable precision. This confirmed that their method was working correctly and that they had successfully isolated the correct physics.

For the second theory, where no such prior solution existed, the results were equally compelling. At weak forces, their calculated bounds aligned perfectly with predictions made by standard perturbation theory. At strong forces, where the theory behaves like a string moving through a curved space, their bounds matched predictions derived from the AdS Veneziano amplitude, a formula describing string scattering. In the middle ground, where neither weak nor strong approximations work, their method provided the first rigorous upper limits on the particle's properties. The researchers found that for the second theory, the bounds were tightest when they used a specific combination of mathematical tools, though they noted that the numerical stability became more difficult to manage as the force strength increased.

The significance of this work lies in its ability to navigate the "middle" of the force spectrum without needing to know the answer in advance. In the first theory, the team showed that they could recover the known integrability results without ever inputting them, proving that the bootstrap method alone, when aided by localization and dispersion relations, is powerful enough to solve the problem. For the second theory, they provided a reliable map where none existed before, showing that the lightest particle's properties transition smoothly from weak to strong coupling. This approach offers a new way to explore holographic theories that do not possess the special symmetries required for integrability, opening the door to understanding a wider class of quantum systems. The study concludes that by weaving together these different mathematical threads, physicists can now constrain the behavior of these complex universes with a level of accuracy that was previously out of reach, providing a solid foundation for future explorations of the quantum nature of gravity.

Technical Summary: Bootstrapping Holographic Theories in the Planar Limit

Problem Statement
The paper addresses the challenge of constraining the spectrum of holographic superconformal field theories (CFTs) in the planar large $N$ limit at finite 't Hooft coupling $\lambda$ . While integrability has successfully computed scaling dimensions for single-trace operators in specific theories (e.g., 4d $\mathcal{N}=4$ SYM and 3d $\mathcal{N}=6$ ABJM) across all couplings, it has not yet been able to compute Operator Product Expansion (OPE) coefficients for the full spectrum or extend to theories where integrability is unavailable. Conversely, previous applications of the numerical conformal bootstrap combined with supersymmetric localization were limited to finite $N$ or could not isolate single-trace operator data at strong coupling, as double-trace operators dominate the low-dimension spectrum in that regime. The central problem is to develop a non-perturbative method to bound the scaling dimensions of the lowest single-trace scalar operators for all $\lambda$ in the planar limit, without relying on integrability inputs, particularly for theories where integrability is not yet known.

Methodology
The authors combine the numerical conformal bootstrap with supersymmetric localization and dispersion relations derived from Regge boundedness. The methodology proceeds through the following steps:

Correlator Setup: The study focuses on two specific holographic CFTs in the planar limit:
- $\mathcal{N}=4$ SYM: The stress tensor multiplet four-point function, dual to closed string scattering.
- $\mathcal{N}=2$ USp(2N) Theory: The four-point function of SO(8) flavor multiplets, dual to open string scattering on D7 branes.
  In the planar limit, the color structure simplifies, allowing the correlator to be decomposed into a single color-stripped amplitude.
Elimination of Double-Trace Operators: A primary obstacle in the planar bootstrap is the presence of double-trace operators, which enter with indefinite signs in the OPE expansion, violating the positivity required for standard bootstrap bounds. The authors overcome this by utilizing dispersion relations. By taking the double-discontinuity ( $d\text{Disc}$ ) of the correlator, double-trace contributions (which have integer twists) vanish due to double zeros in the dispersion kernel. This isolates the single-trace spectrum.
Polyakov-Regge Block Expansion: The correlator is expanded in terms of Polyakov-Regge blocks, which are dispersive transforms of conformal blocks. In the planar limit, these blocks receive contributions only from single-trace operators. The expansion includes a "protected" part (capturing the free theory and strong coupling limits, such as gluon or graviton exchange) and a sum over single-trace operators.
Sum Rules and Constraints:
- Crossing Symmetry: Imposed on the Polyakov-Regge expansion.
- Anti-Subtracted Dispersion Relations: Derived from the soft Regge behavior of the dual string scattering amplitudes (e.g., $M(s,t) = o(s^{-2})$ for $\mathcal{N}=2$ and $o(s^{-2})$ for $\mathcal{N}=4$ ). These provide additional sum rules that constrain the single-trace data.
- Supersymmetric Localization: Integrated constraints derived from derivatives of the sphere free energy with respect to mass parameters are injected into the bootstrap. These constraints depend explicitly on the coupling $\lambda$ , allowing the bounds to vary continuously with $\lambda$ .
Numerical Implementation: The authors formulate a linear programming problem (solved via SDPB) to maximize the gap in the scaling dimension of the lowest scalar single-trace operator. The constraints are enforced by a set of functionals acting on the OPE data, including crossing functionals ( $X$ ), anti-subtracted functionals ( $B$ for $\mathcal{N}=4$ , $R$ for $\mathcal{N}=2$ ), and localization functionals ( $I$ ).

Key Contributions

Extension to $\mathcal{N}=2$ Theory: The paper derives new dispersion relations and Polyakov-Regge block expansions for the planar limit of a 4d $\mathcal{N}=2$ USp(2N) gauge theory with SO(8) flavor symmetry. This is a significant extension of previous work which was largely restricted to $\mathcal{N}=4$ SYM.
Planar Bootstrap without Integrability: The authors demonstrate that it is possible to compute accurate upper bounds on scaling dimensions for all $\lambda$ in the planar limit without inputting scaling dimensions from integrability. This is achieved by combining localization constraints with the dispersive sum rules that isolate single-trace operators.
Refined Functionals: For the $\mathcal{N}=2$ case, the authors construct specific functionals ( $\Psi_\ell$ ) derived from the anti-subtracted dispersion relations to improve numerical convergence, noting that the direct analog of the $\Phi$ functional used in $\mathcal{N}=4$ cannot be constructed due to the completeness of the basis in the $\mathcal{N}=2$ case.

Results

$\mathcal{N}=4$ SYM: The computed upper bounds on the scaling dimension of the Konishi operator ( $\Delta_K$ ) are found to be very close to the known results from the Quantum Spectral Curve (integrability) across the entire range of $\lambda$ . The bounds successfully interpolate between weak coupling (Feynman diagram results) and strong coupling (AdS Virasoro-Shapiro amplitude results).
$\mathcal{N}=2$ USp(2N) Theory: In the absence of integrability results, the bounds on the lowest scalar single-trace operator are shown to be consistent with weak coupling perturbative calculations and strong coupling predictions derived from the AdS Veneziano amplitude. The bounds effectively constrain the spectrum in regimes where integrability is not yet available.
Convergence Issues: The authors note that the bounds are sensitive to the spin truncation ( $J_{\text{max}}$ ), particularly at weak coupling where the leading twist-2 trajectory has small twist. They observe that for certain parameters, the bounds stabilize within a "window" of $J_{\text{max}}$ before deteriorating, a phenomenon attributed to the oscillatory nature of the functionals at large spin.

Significance and Claims
The paper claims that the combination of numerical bootstrap, supersymmetric localization, and dispersive sum rules provides a robust, non-perturbative framework for studying holographic CFTs in the planar limit.

Validation: For $\mathcal{N}=4$ SYM, the method serves as a non-trivial check, reproducing integrability results without using them as input.
Predictive Power: For theories like the $\mathcal{N}=2$ USp(2N) model, the method provides the first non-perturbative constraints on the single-trace spectrum at finite coupling, bridging the gap between weak and strong coupling predictions.
Future Directions: The authors suggest that while their bounds are close to physical predictions, they do not achieve the precise saturation seen in finite- $N$ bootstrap or the tight OPE coefficient bounds found when integrability data is inserted. They propose that future improvements could come from considering mixed correlators (e.g., involving chiral or SU(2) flavor multiplets) or applying the derived dispersion relations to finite $N$ bootstrap problems. They also highlight the potential for deriving a Quantum Spectral Curve for the open string sector of the $\mathcal{N}=2$ theory, which would further refine these bounds.

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