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

TL;DR

This paper explores the geometric interpretation of triality in  Seiberg-Witten theory via hypergeometric functions, revealing connections between solutions of differential equations, algebraic curves, and symmetries.

Contribution

It provides a novel geometric perspective on triality in  gauge theory using hypergeometric functions and Riemann surfaces, distinct from Liouville crossing symmetry.

Findings

Triality corresponds to permutation of six Kummer solutions.

A thrice-punctured sphere emerges from hypergeometric equations via WKB.

Permutation symmetry relates to the outer automorphism of the associated curve.

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.