[Paper Review] The full spectrum of AdS$_5$/CFT$_4$ II: Weak coupling expansion via the quantum spectral curve

Christian Marboe, Dmytro Volin|arXiv (Cornell University)|Dec 21, 2018

Black Holes and Theoretical Physics41 references4 citations

TL;DR

This paper presents a systematic perturbative method to compute the full spectrum of anomalous dimensions in planar $σ$-model via the Quantum Spectral Curve (QSC), focusing on the ${\bf P}\mu$-system. It introduces a robust algorithm implemented in Mathematica, enabling high-loop-order computations up to 11 loops for the Konishi multiplet, and reveals a novel class of degenerate solutions where Bethe roots merge with branch points at weak coupling, lifted by higher-order corrections.

ABSTRACT

We continue the effort to optimise and generalise the solution of the spectral problem of AdS$_5$/CFT$_4$ in the planar limit via integrability. We present a simple strategy to solve the quantum spectral curve perturbatively for general states by focusing on the $\mathbf{P}μ$-system. A Mathematica notebook with an implementation of this algorithm is provided, as well as an extensive database with a user-friendly interface containing more than 8.000 solutions of the QSC. When investigating the solution space, we observe a curious phenomenon: existence of solutions for which the Q-system degenerates in the limit $g o 0$. These degeneracies are lifted at higher orders in perturbation theory. The degenerating solutions have auxiliary Bethe roots merging with branch points at weak coupling.

Motivation & Objective

• To develop a general, efficient perturbative algorithm for solving the QSC in the weak coupling regime of planar AdS5/CFT4.
• To address the challenge of degenerate solutions in the leading-order Q-system, where auxiliary Bethe roots coalesce with branch points.
• To compute high-loop anomalous dimensions for single-trace operators, particularly the Konishi multiplet, up to 11 loops.
• To provide a publicly accessible Mathematica tool and a database of over 8,000 QSC solutions for broader research use.

Proposed method

• The method is based on a perturbative expansion of the ${\bf P}\mu$-system in powers of the coupling $g$, with explicit ansätze for ${\bf P}$ and $\mu$ functions.
• The algorithm solves coupled difference equations for $\mu$ functions using discrete integration ($\Psi$-operator) and matches solutions to the ${\tilde{\bf P}}$ ansatz via analytic continuation.
• It exploits the structure of Hurwitz $\eta$-functions and multiple zeta values (MZVs) to express solutions in terms of known transcendental constants.
• The approach is implemented in a Mathematica notebook, QSCsolver.nb, automating the solution process for arbitrary states.
• The method handles both even and odd powers of $g$, though only even powers typically contribute to the spectrum.
• A companion database, QSCdata.nb, provides a user-friendly interface to access over 8,000 precomputed QSC solutions.

Experimental results

Research questions

• RQ1 How can the QSC be systematically solved perturbatively for general states in the weak coupling limit of AdS5/CFT4?
• RQ2 What causes degeneracies in the leading-order Q-system, and how are they resolved at higher orders in $g$?
• RQ3 What is the structure of the $\mu$-functions in the weak coupling limit, and how can they be expanded consistently?
• RQ4 How can the QSC be efficiently computed to high loop orders, such as 11 loops, for the Konishi multiplet?
• RQ5 What role do single-valued MZVs play in the perturbative expansion of anomalous dimensions?

Key findings

• The 11-loop anomalous dimension of the Konishi multiplet is computed as a rational combination of multiple zeta values and single-valued MZVs, including terms up to $\zeta_{19}$ and $Z_{17}^{(5)}$.
• The method successfully computes the full spectrum up to 11 loops for the Koniali multiplet, with the result expressed in terms of 118 distinct MZV and single-valued MZV terms.
• A new class of degenerate solutions is identified where auxiliary Bethe roots merge with branch points at $g=0$, leading to a singular leading-order Q-system.
• These degeneracies are lifted at higher orders, with the first non-trivial corrections appearing at order $g^4$, indicating a non-analytic structure in the weak coupling expansion.
• The algorithm is robust and general, enabling the computation of over 8,000 distinct QSC solutions with a user-friendly Mathematica interface.
• The implementation in QSCsolver.nb and QSCdata.nb allows reproducible, high-precision computations of anomalous dimensions up to high loop orders.