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Generalized Picard-Fuchs operators from Whitham hierarchy in \(\mathcal{N} = 2\) supersymmetric gauge theory with massless hypermultiplets. (English. Russian original) Zbl 1445.81038

Theor. Math. Phys. 202, No. 2, 150-164 (2020); translation from Teor. Mat. Fiz. 202, No. 2, 170-186 (2020).
Summary: Using the Whitham hierarchy, we obtain the Picard-Fuchs equations in \(\mathcal{N} = 2\) supersymmetric Yang-Mills theory for a classical gauge group with \(N_f\) massless hypermultiplets. In the general case for \(N_f \neq 0\), 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.

MSC:

81T13 Yang-Mills and other gauge theories in quantum field theory
81T60 Supersymmetric field theories in quantum mechanics
35Q40 PDEs in connection with quantum mechanics
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
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