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

Jialiang Dai<sup>ab

a</sup> Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou, China
^b Center of Mathematical Sciences, Zhejiang University, Hangzhou, China

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 $N_\mathrm{f}$ 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.

Keywords: Picard–Fuchs equation, Whitham hierarchy, meromorphic differential, Euler operator, instanton correction.

Received: 02.07.2019
Revised: 02.07.2019

DOI: 10.4213/tmf9772

 English version: Theoretical and Mathematical Physics, 2020, 202:2, 150–164

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