Generalized Picard–Fuchs operators from Whitham hierarchy in $\mathcal N=2$ supersymmetric gauge theory with massless hypermultiplets

Using the Whitham hierarchy, we obtain the Picard–Fuchs equations in $\mathcal N=2$ supersymmetric Yang–Mills theory for a classical gauge group with $N_\mathrm{f}$ massless hypermultiplets. In the general case for $N_\mathrm{f}\ne0$, 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.