$Y$ systems, $Q$ systems, and 4D $\mathcal{N}=2$ supersymmetric QFT

Abstract: We review the connection of $Y$- and $Q$-systems with the BPS spectra of $4D$ $\mathcal{N}=2$ supersymmetric QFTs. For each finite BPS chamber of a $\mathcal{N}=2$ model which is UV superconformal, one gets a periodic $Y$-system, while for each finite BPS chamber of an asymptotically-free $\mathcal{N}=2$ QFT one gets a $Q$-system i.e. a rational recursion all whose solutions satisfy a linear recursion with constant coefficients (depending on the initial conditions). For instance, the classical $ADE$ $Y$-systems of Zamolodchikov correspond to the $ADE$ Argyres-Douglas $\mathcal{N}=2$ SCFTs, while the usual $ADE$ $Q$-systems to pure $\mathcal{N}=2$ SYM. After having motivated the correspondence both from the QFT and the TBA sides, and having introduced the basic tricks of the trade, we exploit the connection to construct and SOLVE new $Y$- and $Q$-systems. In particular, we present the new $Y$-systems associated to the $E_6,E_7,E_8$ Minahan-Nemeshanski SCFTs and to the $D_2(G)$ SCFTs. We also present new $Q$-system corresponding to SYM coupled to specific matter systems such that the YM $\beta$-function remains negative.