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A holomorphic property in the Seiberg-Witten solution

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In the Seiberg-Witten solution of N=2 super-Yang-Mills theory, the parameter for the moduli space is chosen as u=<tr phi^2> = sum_A<(phi_A)^2>, where A=1,2,3 is the color index of the SU(2) group.  The fact that a(u) and a_D(u) depends holomorphically on u was used in the Seiberg-Witten solution. I understand that a_D(u) should depends holomorphically on a(u), which is required by supersymmetry, but how can I see that a(u) and a_D(u) should depend holomorphically on u? From my naive understanding, u can be a function of both a and a*, and thus the function a and a_D should be a(u,u*) and a_D(u,u*). Surely I have not understood some important thing here. Thanks in advance for explanation.

asked Feb 25, 2016 in Theoretical Physics by Dirac (15 points) [ no revision ]
recategorized Feb 25, 2016 by Dilaton

In both cases it follows from supersymmetry. Why do you think that it is clear for $a_D$ but not for $a$?

 @40227  I understand that a_D should depends holomorphically on a, which is required by supersymmetry. But I don't understand why a_D or a should depends holomorphically on u.

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