# 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
recategorized Feb 25, 2016

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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