Week 27.06.2022 – 03.07.2022

In recent years, it has been understood how local operators do not paint the entire picture of a quantum field theory, but we need to introduce extended operators to understand finer details about these theories. Motivated by this, we investigate a subset of these extended operators in particular in the context of 4d N=2 superconformal field theories. I will start by introducing the setup that we use to describe the possible configurations of these extended operators. I will also compare and contrast the operators that appear in our framework with the more familiar (Wilson and 't Hooft) line operators. Time permitting, I will then introduce twists of these theories by a choice of an appropriate nilpotent supercharge. The restriction to (extended) operators living in the cohomology of this supercharge gives rise to interesting algebraic structures, that are analogous to, or rather an extension of, the 2d vertex operator algebras that have now become familiar familiar in the context of 4d N=2 theories.

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.