dr Artur R. Pietrykowski (IFT-Wrocław; JINR-Dubna)

Irregular blocks, 4d gauge theories and Hill's equation

The Alday-Gaiotto-Tachikawa (AGT) correspondence relates two-dimensional Conformal Field Theory (2dCFT) on $n$-punctured Riemann surface with four-dimensional $mathcal{N}=2$ superconformal field theories (4d-SCFT). In particular, 4d $mathcal{N}=2$ super Yang-Mills theory with $N_{f}=4$ flavors on certain background corresponds to 2dCFT on four-punctured Riemann sphere. The extension of this correspondence to $mathcal{N}=2$ nonconformal 4dSCFT revealed a completely new object in 2dCFT -- the irregular vectors and the irregular conformal blocks. On the other hand there is a relationship between a certain limit of $mathcal{N}=2$ 4dSCFT and Quantum Integrable Systems (QIS) (the so-called Bethe/Gauge correspondence). It is, therefore, possible to exploit the nonconformal AGT relation to derive the similar connection between 2dCFT and the Quantum Integrable Systems. The talk is devoted to show the latter in specific examples, namely, for number of flavors $N_{f}=0,1,2$. The case of arbitrary number of flavors $N_{f}>2$ will be also discussed.