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  Converting between hyperelliptic and quadratic differential forms of the Seiberg-Witten curve for $\mathcal{N}=2$ theories

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For $D=4$, $\mathcal{N}=2$ QFTs, there are two "standard" forms of the Seiberg-Witten curve: hyperelliptic form, and a form which looks like $y^2 = \phi\left(z\right)$, where $\phi\left(z\right)$ appears in a quadratic differential $q = \phi\left(z\right) dz^2$. Converting between these two forms is meant to be standard. For example, Tachikawa describes how to go from the former to the latter in the case of the $SU\left(2\right)$ theory with four flavours in section 9.1 of his summary paper. Let's use this theory as an example. To recap, the hyperelliptic form of the Seiberg-Witten curve looks like:

\[\Sigma : \quad f\frac{\left(\tilde{x}-\tilde{\mu}_1\right)\left(\tilde{x}-\tilde{\mu}_2\right)}{\tilde{z}} + f' \cdot\left(\tilde{x}-\tilde{\mu}_3\right)\left(\tilde{x}-\tilde{\mu}_4\right)\tilde{z}=\tilde{x}^2-u\]

To convert to "quadratic differential form", rescale $\tilde{z}$ to make $f'=1$ and collect terms in $\tilde{x}$, and complete the square in $\tilde{x}$. We find (dropping tildes):

\[x^2 = \frac{\left(f\left(\mu_1+\mu_2\right)+{z}^2\left(\mu_3+\mu_4\right)\right)^2-4\left(f+\left(z-1\right)z\right)\left(f\mu_1\mu_2 + z\left(u + z\mu_3\mu_4\right)\right)}{4\left(f+\left(z-1\right)z\right)^2}\]

As Tachikawa tells us, we have for the Seiberg-Witten differential $\lambda = x dz/z$ and $\lambda^2 = q$, so $q = x^2 {dz^2}/{z^2}$, i.e., using the above notation:

\[\phi\left(z\right) = \frac{\left(f\left(\mu_1+\mu_2\right)+{z}^2\left(\mu_3+\mu_4\right)\right)^2-4\left(f+\left(z-1\right)z\right)\left(f\mu_1\mu_2 + z\left(u + z\mu_3\mu_4\right)\right)}{4\left(f+\left(z-1\right)z\right)^2 z^2}\]

Now, to get to my question. There's meant to be a one-one mapping between these two forms of the Seiberg-Witten curve. So given a specific expression for $\phi\left(z\right)$, we should be able to work backwards to compute a specific form of the SW curve in hyperelliptic form, i.e. find the parameters $\left\{\mu_1,\mu_2,\mu_3,\mu_4,f,u\right\}$. Consider the following specific example of such a $\phi\left(z\right)$:

\[\phi\left(z\right) = - \frac{576z\left(z^3-1\right)}{4\pi^2\left(1+8z^3\right)^2}\]

Comparing with the previous general expression and equating coefficients of the numerators and denominators, we should be able to find the $\left\{\mu_1,\mu_2,\mu_3,\mu_4,f,u\right\}$. But doing so, I find multiple possible sets of values for these parameters. Moreover, plugging these parameters back into the Seiberg-Witten curve in its original "hyperelliptic form" (the first equation), they seem to give different curves.

But I thought there was meant to be a one-one correspondence between these two forms of the curves! So I feel I must have either done something wrong, or I am missing that somehow these curves look different but are nevertheless equivalent, or I just got wrong that there's a one-one correspondence between the two forms of the curves. Can anyone help me out here? Thanks in advance!!

asked Sep 30, 2015 in Theoretical Physics by Jimeree (45 points) [ revision history ]
recategorized Sep 30, 2015 by Dilaton

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