I will describe the Baxter equation whose solutions contain the non-perturbative spectrum of certain boundary integrable systems related to Maldacena-Wilson loops in N=4 supersymmetric Yang-Mills theory.
The first system that I will consider is the cusped Maldacena-Wilson line in the so-called ladders limit, with orthogonal scalar insertions at the cusp. Here we will see that this setup can be described by a dual open fishchain - a discretised open string. I will show that this system is integrable at the quantum level and derive the Baxter equation to obtain the non-perturbative spectrum of an arbitrary number of L orthogonal insertions.
The spectrum of the Baxter equation contains excited states which can be shown to correspond to parallel insertions - i.e. scalars spanned by those that couple to the line. The J^th excited state corresponds to an insertion of J parallel scalars. For the case of L = 0, i.e. the case with no orthogonal insertions, one can go away from the ladders limit, and take the straight-line limit of the Wilson line to obtain the one-dimensional defect CFT that lives on the 1/2-BPS infinite straight Maldacena-Wilson line - the second system that I consider. The excited states of the Baxter equation can be mapped to states in the defect CFT. Starting with the Quantum Spectral Curve (QSC) for these excited states, I will derive the Baxter equation that captures the spectrum of an arbitrary number of J parallel insertions. I will also display some analytical results at weak and strong coupling, and show that the numerical solution obtained from the QSC interpolates between them at finite coupling.