Y-systems, Q-systems, and N=2 supersymmetric QFT

We review the connection between Y- and Q-systems and the BPS spectra of 4D script N = 2 supersymmetric QFTs. For each finite BPS chamber of an script N = 2 model which is UV superconformal, one gets a periodic Y-system, while for each finite BPS chamber of an asymptotically-free script 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 script N = 2 SCFTs, while the usual ADE Q-systems correspond to pure script N = 2 SYM. After having motivated the correspondence both from the QFT and the thermodynamical Bethe ansatz 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 E6, E7, E8 Minahan-Nemeshanski SCFTs and to the D2 (G ) SCFTs. We also present new Q-systems corresponding to SYM coupled to specific matter systems such that the YM β-function remains negative. © 2014 IOP Publishing Ltd.