I will discuss the problem of counting states in superconformal field theories, focusing on 4D N=4 SYM. The main focus will be on reviewing background material before briefly commenting on some recent developments in 2306.04693 and 2209.06728. I also hope to make some comments on the history of the problem.

The Quantum Spectral Curve (QSC) is a powerful integrability-based method capable of computing the spectrum of planar N=4 SYM. It has also been generalised in many directions, for example to cusped Wilson lines and various deformations. The success of the QSC motivates trying to extend the formalism beyond N=4 to other theories. This requires the study of the underlying structure of the QSC, a so called analytic Q-system. To construct an analytic Q-system it is necessary to specify both its algebraic structure, usually encoded into QQ-relations, and its analytic properties.


I will talk about recent work to study Q-systems beyond the ones relevant for N=4, discussing both their algebraic and analytic properties. In particular I will discuss the recent conjecture of a QSC for AdS3/CFT2 which non-trivially couples two different Q-systems. While the curve shares many features with the N=4 QSC it also offers new surprises and challenges. If this new curve can be brought under full control and further tested many interesting applications and generalisations are within reach.