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Quantum Spectral Curve and the Numerical Solution of the Spectral Problem in AdS5/CFT4

Nikolay Gromov, Fedor Levkovich-Maslyuk, Grigory Sizov

TL;DR

This work develops a fast, robust numerical solver for the Quantum Spectral Curve of planar N=4 SYM, enabling finite-coupling spectrum calculations for generic operators beyond TBA. The method solves for QSC data by constructing Q_{a|i} from P- data, closing the system via omega/monodromy relations, and recasting the remaining constraints as a Levenberg–Marquardt optimization. It yields high-precision results for the ${ rak{sl}}(2)$ twist-2 sector, including Konishi, and extends to non-integer and complex Lorentz spin, enabling precise determinations of the BFKL pomeron intercept and exploration of complex spin Riemann surfaces. The approach broadens the practical reach of integrability techniques, offering cross-checks with perturbative and strong-coupling limits and potential applications to ABJM and other observables.

Abstract

We developed an efficient numerical algorithm for computing the spectrum of anomalous dimensions of the planar ${\cal N}=4$ Super-Yang--Mills at finite coupling. The method is based on the Quantum Spectral Curve formalism. In contrast to Thermodynamic Bethe Ansatz, worked out only for some very special operators, this method is applicable for generic states/operators and is much faster and precise due to its Q-quadratic convergence rate. To demonstrate the method we evaluate the dimensions $Δ$ of twist operators in $sl(2)$ sector directly for any value of the spin $S$ including non-integer values. In particular, we compute the BFKL pomeron intercept in a wide range of the 't Hooft coupling constant with up to $20$ significant figures precision, confirming two previously known from the perturbation theory orders and giving prediction for several new coefficients. Furthermore, we explore numerically a rich branch cut structure for complexified spin $S$.

Quantum Spectral Curve and the Numerical Solution of the Spectral Problem in AdS5/CFT4

TL;DR

This work develops a fast, robust numerical solver for the Quantum Spectral Curve of planar N=4 SYM, enabling finite-coupling spectrum calculations for generic operators beyond TBA. The method solves for QSC data by constructing Q_{a|i} from P- data, closing the system via omega/monodromy relations, and recasting the remaining constraints as a Levenberg–Marquardt optimization. It yields high-precision results for the twist-2 sector, including Konishi, and extends to non-integer and complex Lorentz spin, enabling precise determinations of the BFKL pomeron intercept and exploration of complex spin Riemann surfaces. The approach broadens the practical reach of integrability techniques, offering cross-checks with perturbative and strong-coupling limits and potential applications to ABJM and other observables.

Abstract

We developed an efficient numerical algorithm for computing the spectrum of anomalous dimensions of the planar Super-Yang--Mills at finite coupling. The method is based on the Quantum Spectral Curve formalism. In contrast to Thermodynamic Bethe Ansatz, worked out only for some very special operators, this method is applicable for generic states/operators and is much faster and precise due to its Q-quadratic convergence rate. To demonstrate the method we evaluate the dimensions of twist operators in sector directly for any value of the spin including non-integer values. In particular, we compute the BFKL pomeron intercept in a wide range of the 't Hooft coupling constant with up to significant figures precision, confirming two previously known from the perturbation theory orders and giving prediction for several new coefficients. Furthermore, we explore numerically a rich branch cut structure for complexified spin .

Paper Structure

This paper contains 19 sections, 53 equations.

Table of Contents

  1. Introduction
  2. Review of the AdS/CFT Quantum Spectral Curve
  3. Description of the Method
  4. Step 1: Solving the equation for ${\cal Q}_{a|i}$
  5. Large $u$ approximation.
  6. Finite $u$ approximation.
  7. Step 2: Recovering $\omega_{ij}$
  8. Finding zero modes.
  9. Step 3: Reducing to an optimization problem
  10. Levenberg-Marquardt algorithm
  11. Implementation for the $\mathfrak{sl}(2)$ Sector and Comparison with Existing Data
  12. The ${\mathfrak{sl}}(2)$ sector in the QSC framework.
  13. Improved implementation: skipping the computation of $\omega$'s.
  14. Implementation for Konishi
  15. Extension to Non-Integer Lorentz Spin
  16. ...and 4 more sections