Spectral problem – our main goals

• A detailed exploration of the spectrum from the existing TBA/Y-system equations.

With the exact spectral equations at hand, one can scan the spectrum of the SYM theory at the fully non-perturbative level. Due to the complexity of the TBA equations this has only been done numerically to date, and only for the simplest subsets of operators. The TBA equations certainly have larger potential and can be used to study both individual states and general characteristics of the spectrum. One particularly interesting set of operators are twist operators, direct counterparts of the QCD operators that arise in the OPE analysis of the deep inelastic scattering.

• Making the existing description more suitable for analytic and numerical analyses and improving the TBA.

The solution to the spectral problem in N = 4 SYM was originally found in the form of an infinite system of non-linear integral equations (TBA) or functional equations (Ysystem). These were recently reduced to a finite set of nonlinear equations of Destri – de Vega type, offering the prospect of rapid progress on the extraction of highprecision numerical results from the system, and the full exploration of its set of solutions.

• Derivation of the spectral equations from first principles and understanding of the origins of integrability in gauge theory.

The AdS/CFT system was basically solved by bootstrap, which is not unusual for integrable models. But for many other integrable systems a systematic, first principles derivation is also possible. The starting point is the algebraic framework of the quantum inverse scattering problem. Such an algebraic approach may uncover deep symmetries responsible for integrability and make contact with mathematical results in representation theory and geometry.