Beijing Institute of Mathematical Sciences and Applications

A typical feature of quantum integrable models is the possibility to construct and/or describe Bethe Algebra—a large family of commuting operators whose simultaneous diagonalisation fully controls the spectrum of the theory. We introduce rich combinatorial and geometrical features of the Bethe Algebra which are encoded through a variety of relations between Baxter Q-functions and which are universal for a variety of different models. The relations themselves turn out to be meaningful also in the situations when their derivation from the commutative algebra is not known or even possible. Being supplemented with the ansatz on the analytical structure of Q-functions, explicit equations can then be derived to encode the spectra of concrete physical models. Two examples are discussed in detail: One is rational spin chains where the obtained techniques turned out to be an extremely powerful computation tool which also, in certain cases, can be used to rigorously prove completeness of Bethe equations. The second one is AdS/CFT integrability - we will show how our general approach allows one to derive Quantum Spectral Curve that encodes the exact spectrum of planar N=4 SYM.