Quantum spectral curve as a tool for a perturbative quantum field theory

TL;DR

The paper develops a perturbative weak-coupling expansion of the quantum spectral curve (QSC) for planar $N=4$ SYM in the $SL(2)$ sector, enabling systematic, operator-by-operator computation of conformal dimensions to high loop orders. It introduces a universal iterative procedure built on the ${f P}{oldsymbol{ m extmu}}$-system, Riemann-Hilbert structure, and a $oxed{ m extPsi}$-based machinery to solve Baxter-type equations, capturing finite-size effects and wrapping corrections. The main findings are that, at any loop order, dimensions decompose into multiple zeta-values (MZVs) with algebraic-number coefficients from the Baxter data, and that explicit results consistently involve only single-valued MZVs, with concrete ten-loop Konishi data and a broad catalog of operators provided. The work delivers a practical,Automatable method to perturbatively solve the finite-volume AdS/CFT spectral problem, enhances confidence in the QSC formalism, and points toward broader extensions beyond the $SL(2)$ sector and connections to Hopf-algebraic structures in perturbative QFT.

Abstract

An iterative procedure perturbatively solving the quantum spectral curve of planar N=4 SYM for any operator in the sl(2) sector is presented. A Mathematica notebook executing this procedure is enclosed. The obtained results include 10-loop computations of the conformal dimensions of more than ten different operators. We prove that the conformal dimensions are always expressed, at any loop order, in terms of multiple zeta-values with coefficients from an algebraic number field determined by the one-loop Baxter equation. We observe that all the perturbative results that were computed explicitly are given in terms of a smaller algebra: single-valued multiple zeta-values times the algebraic numbers.

Figures (2)

• Figure 1: History of perturbative computations of the Konishi anomalous dimension in planar $\mathcal{N}$=4 SYM.
• Figure 2: Physical kinematics and continuation to the next sheet.