Irregular blocks, N = 2 gauge theory and Mathieu system

23rd International Conference on Integrable Systems and Quantum Symmetries

Piatek, M.R.^1,3 ; Pietrykowski, A.R.^2,3

Institute of Physics, University of Szczecin, ul. Wielkopolska 15, Szczecin

70-451, Poland^1

Institute of Theoretical Physics, University of Wroclaw, M. Borna 9, Wroclaw

50-204, Poland^2

Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna

141980, Russia^3

关键词: Conformal blocks;    Gauge theory;    Null vectors;    Saddle point;    Seiberg-Witten theory;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/670/1/012041/pdf
DOI  :  10.1088/1742-6596/670/1/012041

来源: IOP

【 摘 要 】

The Alday-Gayotto-Tachikawa (AGT) conjecture relates 4d N = 2, SU(2) SYM theories with Nfmatter hypermultiplets to 2d CFT. In case of pure 4d N = 2, SU(2) SYM there is a corresponding irregular conformal block in 2d CFT. The AGT correspondence may be extended within a certain limit (the Nekrasov-Shataschvili limit) to the correspondence between an effective twisted superpotentials of 2d N = 2 SUSY and the Zamolodchikov's "classical" conformal blocks. When narrowed to the pure 4d N = 2 SYM case its limit is related to an irregular classical conformal block. It will be shown that according to the triple correspondence (2dCFT/Gauge/Bethe - c.f. Piatek's talk) the irregular classical conformal block yields spectrum of Mathieu operator. The latter can be obtained as a "classical" limit of the null vector decoupling equation for three-point degenerate irregular block. It will also be shown that the Mathieu spectrum can be also obtained from the limit of the pure gauge theory as a solution of the saddle point equation as well as from the Bohr-Sommerfeld quantization of the Seiberg-Witten theory.