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

+ 4 like - 0 dislike
697 views

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!!

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:$\varnothing\hbar$ysicsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.