# The full spectrum of AdS5/CFT4 I: Representation theory and one-loop Q-system

## TL;DR

This paper develops a representation-theoretic framework and an efficient algorithm for solving the quantum spectral curve of AdS5/CFT4, enabling explicit computation of the spectrum without subsector restrictions.

## Contribution

It introduces a generalized Young diagram approach and a novel Q-system solving algorithm that bypasses Bethe equations, facilitating spectrum calculations in AdS5/CFT4.

## Key findings

- Explicit classification of states using generalized Young diagrams.
- Efficient algorithm for solving Q-systems without Bethe equations.
- Computed spectrum for 495 multiplets up to conformal dimension 13/2.

## Abstract

With the formulation of the quantum spectral curve for the AdS5/CFT4 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.