Seiberg-Witten Theory of Rank Two Gauge Groups and Hypergeometric Series

TL;DR

This paper explores the connection between rank two Seiberg-Witten gauge theories and hypergeometric functions, extending known relations for SU(2) and SU(3) to other classical groups and examining their analytic properties.

Contribution

It demonstrates that fields in rank two gauge theories can be expressed via specific hypergeometric functions, revealing new relations for classical groups and clarifying the case of the G_2 group.

Findings

SU(2) fields are hypergeometric functions

SU(3) fields relate to Appell F_4 functions

Classical groups B_2, C_2, D_2 expressed with Appell functions

Abstract

In SU(2) Seiberg-Witten theory, it is known that the dual pair of fields are expressed by hypergeometric functions. As for the theory with SU(3) gauge symmetry without matters, it was shown that the dual pairs of fields can be expressed by means of the Appell function of type F_4. These expressions are convenient for analyzing analytic properties of fields. We investigate the relation between Seiberg-Witten theory of rank two gauge group without matters and hypergeometric series of two variables. It is shown that the relation between gauge theories and Appell functions can be observed for other classical gauge groups of rank two. For B_2 and C_2, the fields are written in terms of Appell functions of type H_5. For D_2, we can express fields by Appell functions of type F_4 which can be decomposed to two hypergeometric functions, corresponding to the fact SO(4)\sim SU(2)\times SU(2). We also consider the integrable curve of type C_2 and show how the fields are expressed by Appell functions. However in the case of exceptional group G_2, our examination shows that they can be represented by hypergeometric series which does not correspond to the Appell functions.