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.Comment: 36 pages plus tables; v2: minor changes, references added, ancillary
files with mathematica notebooks adde
With the formulation of the quantum spectral curve for the AdS 5 /CFT 4 integrable system, it became potentially possible to compute its full spectrum with high efficiency. This is the first paper in a series devoted to the explicit design of such computations, with no restrictions to particular subsectors being imposed.We revisit the representation theoretical classification of possible states in the spectrum and map the symmetry multiplets to solutions of the quantum spectral curve at zero coupling. To this end it is practical to introduce a generalisation of Young diagrams to the case of non-compact representations and define algebraic Q-systems directly on these diagrams. Furthermore, we propose an algorithm to explicitly solve such Q-systems that circumvents the traditional usage of Bethe equations and simplifies the computation effort.For example, our algorithm quickly obtains explicit analytic results for all 495 multiplets that accommodate single-trace operators in N = 4 SYM with classical conformal dimension up to 13 2 . We plan to use these results as the seed for solving the quantum spectral curve perturbatively to high loop orders in the next paper of the series.
In this note we propose an approach for a fast analytic determination of all possible sets of Bethe roots corresponding to eigenstates of rational GL(N|M) integrable spin chains of given not too large length, in terms of Baxter Q-functions. We observe that all exceptional solutions, if any, are automatically correctly accounted.The key intuition behind the approach is that the equations on the Q-functions are determined solely by the Young diagram, and not by the choice of the rank of the GL symmetry. Hence we can choose arbitrary N and M that accommodate the desired representation. Then we consider all distinguished Q-functions at once, not only those following a certain Kac-Dynkin path.
Abstract:We compute the general form of the six-loop anomalous dimension of twist-two operators with arbitrary spin in planar N = 4 SYM theory. First we find the contribution from the asymptotic Bethe ansatz. Then we reconstruct the wrapping terms from the first 35 even spin values of the full six-loop anomalous dimension computed using the quantum spectral curve approach. The obtained anomalous dimension satisfies all known constraints coming from the BFKL equation, the generalised double-logarithmic equation, and the small spin expansion.
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