SEIBERG–WITTEN THEORY OF RANK TWO GAUGE GROUPS AND HYPERGEOMETRIC SERIES

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 F4. These expressions are convenient for analyzing analytic properties of fields. We investigate the relation between the 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 B2 and C2, the fields are written in terms of Appell functions of type H5. For D2, we can express fields by Appell functions of type F4 which can be decomposed to two hypergeometric functions, corresponding to the fact SO (4)~ SU (2)× SU (2). We also consider the integrable curve of type C2 and show how the fields are expressed by Appell functions. However in the case of exceptional group G2, our examination shows that they can be represented by the hypergeometric series which does not correspond to the Appell functions.