The spectral problem for N=4 Super Yang-Mills can be formulated as a set of quantisation conditions on a handful of functions called Q-functions. Recent analysis suggests that the Q-functions can be used as simple building blocks for 3-point correlation functions. This strongly resembles the situation in integrable spin chains where the wave functions factorise into a simple product of Q-functions in a special basis called Sklyanin’s separation of variables (SoV) basis which is one of the most powerful approaches for solving integrable systems. Unfortunately this framework has only been developed for the simplest integrable spin chains with sl(2) symmetry, far from the psu(2,2|4) needed to describe N=4 SYM. In this talk I will review recent advances in developing the SoV approach for higher rank integrable spin chains. I will explain how to construct the SoV basis in a systematic fashion and how it links to the representation theory of the system. Next, I will discuss a new approach for obtaining the measure in separated variables based on the famous Baxter TQ equation and how the approach naturally provides a large family of correlation functions as very simple determinants in Q-functions. I will briefly discuss how the approach can be applied directly to certain 4d QFTs, in particular the fishnet cousin of N=4 SYM.

"I will review recent advancements in the development of the Separation of Variables (SoV) program for rational higher rank spin chains, motivated by the recent appearance of SoV-type structures in AdS/CFT. I will discuss the main approaches for constructing a separated variable basis which include diagonalising the B-operator and the action of fused transfer matricies on a suitable vacuum state. I will explain how these approaches are linked and demonstrate how they can be unified into a single framework governed by Yangian representation theory. The outcome is that for any finite-dimensional su(n) spin chain the wave functions (Bethe vectors) factorise into an ascending product of Slater determinants in Baxter Q-functions, allowing us to immediately link this operatorial construction of states with the recently developed functional approach of computing scalar products and form factors." NOTE: this will be an online seminar using Zoom. To participate please fill the registration form on integrability-london.weebly.com.