Triality in SU(2) Seiberg-Witten theory and Gauss hypergeometric function

Abstract

Through AGT conjecture, we show how triality observed in \N=2 SU(2) N_f=4 QCD can be interpreted geometrically as the interplay among six of Kummer's twenty-four solutions belonging to one fixed Riemann scheme in the context of hypergeometric differential equations. We also stress that our presentation is different from the usual crossing symmetry of Liouville conformal blocks, which is described by the connection coefficient in the case of hypergeometric functions. Besides, upon solving hypergeometric differential equations at the zeroth order by means of the WKB method, a curve (thrice-punctured Riemann sphere) emerges. The permutation between these six Kummer's solutions then boils down to the outer automorphism of the associated curve.