Generalized Picard—Fuchs Operators From Whitham Hierarchy in $${\mathcal N} = 2$$ Supersymmetric Gauge Theory with Massless Hypermultiplets

Abstract

Using the Whitham hierarchy, we obtain the Picard—Fuchs equations in$${\mathcal N} = 2$$supersymmetric Yang—Mills theory for a classical gauge group with Nfmassless hypermultiplets. In the general case for Nf ≠ 0, there are at least r−2 Picard—Fuchs equations that can be computed exactly from the commutation relations of the meromorphic differentials defined up to a linear combination of holomorphic differentials on the Seiberg—Witten hyperelliptic curve. Using Euler operator techniques, we study the Picard—Fuchs equations, including instanton corrections. Moreover, using symbolic computer calculations, we can obtain a complete set of Picard—Fuchs equations.