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

Understand

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.