Abstract
We derive infinitely many meromorphic differentials based on the fractional powers of the superpotential arising from hyperelliptic curves. We obtain various differential equations expressed in terms of the moduli derivatives of the Seiberg–Witten differential. Taking advantage of the cross derivatives of these differentials, we can derive some Picard–Fuchs equations and use the Euler operator to obtain a complete set of Picard–Fuchs equations containing the instanton correction term. We solve the complete system of equations by expanding the moduli parameters in power series.
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References
N. Seiberg and E. Witten, “Electric-magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang–Mills theory,” Nucl. Phys. B, 426, 19–52 (1994); arXiv:hep-th/9407087v1 (1994).
K. Ito and N. Sasakura, “Exact and microscopic one-instanton calculations in N=2 supersymmetric Yang–Mills theories,” Nucl. Phys. B, 484, 141–166 (1997); arXiv:hep-th/9608054v1 (1996).
E. D’Hoker, I. M. Krichever, and D. H. Phong, “The effective prepotential of N=2 supersymmetric SU(Nc) gauge theories,” Nucl. Phys. B, 489, 179–210 (1997); arXiv:hep-th/9609041v1 (1996).
E. D’Hoker, I.M. Krichever, and D. H. Phong, “The effective prepotential of N=2 supersymmetric SO(Nc) and Sp(Nc) gauge theories,” Nucl. Phys. B, 489, 211–222 (1997); arXiv:hep-th/9609145v1 (1996).
A. Klemm, W. Lerche, and S. Theisen, “Nonperturbative effective actions of N=2-supersymmetric gauge theories,” Internat. J. Modern Phys. A, 11, 1929–1973 (1996); arXiv:hep-th/9505150v1 (1995).
J. M. Isidro, A. Mukherjee, J. P. Nunes, and H. J. Schnitzer, “On the Picard–Fuchs equations for massive N=2 Seiberg–Witten theories,” Nucl. Phys. B, 502, 363–382 (1997); arXiv:hep-th/9704174v1 (1997).
J. M. Isidro, A. Mukherjee, J. P. Nunes, and H. J. Schnitzer, “A new derivation of the Picard–Fuchs equations for effective N=2 super Yang–Mills theories,” Nucl. Phys. B, 492, 647–681 (1997); arXiv:hep-th/9609116v2 (1996).
M. Alishahiha, “On the Picard–Fuchs equations of the SW models,” Phys. Lett. B, 398, 100–103 (1997); arXiv:hep-th/9609157v2 (1996).
J. M. Isidro, “Integrability, Seiberg–Witten models, and Picard–Fuchs equations,” JHEP, 0101, 043 (2001); arXiv:hep-th/0011253v2 (2000).
K. Ito and S.-K. Yang, “Picard–Fuchs equations and prepotentials in N=2 supersymmetric QCD,” in: Frontiers in Quantum Field Theory (Osaka, Japan, 14–17December 1995, H. Itoyama, M. Kaku, H. Kunitomo, M. Nimomiya, and H. Shirohura, eds.), World Scientific, Singapore (1996), pp. 331–342; arXiv:hep-th/9603073v1 (1996).
K. Ito, “Picard–Fuchs equations and prepotential in N=2 supersymmetric G2Yang–Mills theory,” Phys. Lett. B, 406, 54–59 (1997); arXiv:hep-th/9703180v2 (1997).
T. Nakatsu and K. Takasaki, “Whitham–Toda hierarchy and N=2 supersymmetric Yang–Mills theory,” Modern Phys. Lett. A, 11, 157–168 (1996); arXiv:hep-th/9509162v1 (1995).
E. D’Hoker and D. H. Phong, “Seiberg–Witten theory and integrable systems,” in: Integrability: The Seiberg–Witten and Whitham Equations (Edinburgh, 14–18September 1998, H. W. Braden and I. M. Krichever, eds.), Gordon and Breach, Amsterdam (2000), pp. 43–68; arXiv:hep-th/9903068v1 (1999).
A. Marshakov, Seiberg–Witten Theory and Integrable Systems, World Scientific, Singapore (1999).
K. Takasaki, “Whitham deformations of Seiberg–Witten curves for classical gauge groups,” Internat. J. Modern Phys. A, 15, 3635–3666 (2000); arXiv:hep-th/9901120v2 (1999).
K. Takasaki, “Whitham deformations and tau function in N=2 supersymmetric gauge theories,” Prog. Theor. Phys. Suppl., 135, 53–74 (1999); arXiv:hep-th/9905224v2 (1999).
L. Chekhov and A. Mironov, “Matrix models vs. Seiberg–Witten/Whitham theories,” Phys. Lett. B, 552, 293–302 (2003).
A. Marshakov and N. Nekrasov, “Extended Seiberg–Witten theory and integrable hierarchy,” JHEP, 0701, 104 (2007); arXiv::hep-th/0612019v2 (2006).
A. Gorsky and A. Milekhin, “RG–Whitham dynamics and complex Hamiltonian systems,” Nucl. Phys. B, 895, 33–63 (2015); arXiv:1408.0425v3 [hep-th] (2014).
A. Gorsky, A. Marshakov, A. Mironov, and A. Morozov, “RG equations from Whitham hierarchy,” Nucl. Phys. B, 527, 690–716 (1998); arXiv:hep-th/9802007v2 (1998).
Y. Ohta, “Picard–Fuchs equation and Whitham hierarchy in N=2 supersymmetric SU(r+1) Yang–Mills theories,” J. Math. Phys., 40, 6292–6301 (1999); arXiv:hep-th/9906207v3 (1999).
H. Itoyama and A. Morozov, “Integrability and Seiberg–Witten theory: Curves and periods,” Nucl. Phys. B, 477, 855–877 (1996); arXiv:hep-th/9511126v2 (1995).
J. D. Edelstein and J. Mas, “N=2 supersymmetric Yang–Mills theories and Whitham integrable hierarchies,” AIP Conf. Proc., 484, 195–212 (2008); arXiv:hep-th/9902161v1 (1999).
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Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 198, No. 3, pp. 365–380, March, 2019.
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Dai, J., Fan, E. Whitham Hierarchy and Generalized Picard–Fuchs Operators in the N=2 Susy Yang–Mills Theory for Classical Gauge Groups. Theor Math Phys 198, 317–330 (2019). https://doi.org/10.1134/S0040577919030012
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DOI: https://doi.org/10.1134/S0040577919030012