An Exploration of Q-Systems : From Spin Chains to Low-Dimensional AdS/CFT

Abstract: The discovery of integrability in the planar limit of the AdS5/CFT4 correspondence has led to impressive progress in the study of string theory and four-dimensional N=4 Super-Yang-Mills. In particular, with the formulation of the Quantum Spectral Curve (QSC) the spectral problem is now solved. The QSC has demonstrated its versatility and usefulness in various other applications, including the study of Wilson lines, conformal bootstrap, and the calculation of structure constants.It would be highly desirable to extend the QSC from AdS5/CFT4 to other instances of the AdS/CFT correspondence, a program so far only fully completed for AdS4/CFT3. Achieving this objective requires a solid understanding of the foundation of the QSC, a so-called analytic Q-system. In this thesis and the included papers, we investigate Q-systems and their algebraic and analytic properties. Inspired by the ODE/IM correspondence we propose Q-systems that encode the conserved charges of integrable models with a simply-laced symmetry algebra. In particular, for the case of D(r) a powerful parameterization of the Q-system using pure spinors is employed to efficiently solve compact rational spin chains and find T-functions solving Hirota equations. The extension from D(r) to the non-simply laced algebra B(2)/C(2) is explained and detailed. While many features are similar the relation between the symmetry of the Q-system and that of the integrable model becomes more intricate. By introducing a new method dubbed Monodromy Bootstrap we construct new Quantum Spectral Curves based on gl(2|2). In particular, one curve is conjectured to describe planar string theory on AdS3 with pure RR-flux. We solve the curve in a weak coupling limit both analytically and numerically.We also discuss the problem of finding the eigenvalue spectrum of operators on the squashed seven-sphere coming from the compactification of eleven-dimensional supergravity. These eigenvalues are of importance for the mass spectrum of fields in AdS4.