# How can we get the Seiberg-Witten curve from M-theory?

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Have people developed a systematic way to get the Seiberg-Witten curve from some M-theoretical (thus non-perturbative) configuration? If so what exactly is this mechanism and does it yield the same results as the "standard" way?

retagged Apr 16, 2015

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The fundamental initial paper on the study of N=2 four dimensional gauge theories from the M-theory point of view was written by Witten:

http://arxiv.org/abs/hep-th/9703166

The 4d theory is essentially realized on the worldvolume of a stack of M5 branes compactified on the Seiberg-Witten curve. In particular, in this picture, the Seiberg-Witten curve has a clear geometric interpretation. This M-theory description emerges naturally as the strong coupling limit of a maybe more familiar construction in terms of NS5 and D4-branes in type IIA superstring.

As far as I know, it gives the same result as the "standard" way when it is possible to compare both results. One has probably to be careful about the word "systematic". Witten's paper and its many follow-ups have certainly treated large classes of theories but probably not "all" theories can be obtained in this way. The problem of classification of $N=2$ four dimensional gauge theories, the one for which we expect the existence of a Seiberg-Witten curve, is a rather active area of current research.

answered Apr 7, 2015 by (5,140 points)
edited Apr 8, 2015 by 40227
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Indeed as said above we do get the Seiberg-Witten curve from M-theory. To do this we need to consider the following brane setup in type IIA strings:

Consider $x_0, \ldots, x_{10}$ and put $M$ $NS5$-branes in $x_0, \ldots, x_5$ and $N$ $D4$-branes in $x_0, x_1, x_2, x_3, x_6$ (and $x_{10}$ in the M-theory setup this updates to an $M5$-brane)  where the $N$ $D4$-branes are suspended between the $M$ $NS$5- branes. Then introduce $2N$ flavor branes attached to those $NS5$-branes sitting in the outermost of the configuration and extended to infinity.The resulting theory is a $d=4$ $\mathcal{N}=2$ $SU(N)^{M-1}$ gauge theory (which asymptotically is conformal). $U(1)_R$ symmetry is realized by a rotation between the $x_4$ and $x_5$ while the $SU(2)_R$ one is realized by the rotation of the $x_7,x_8$ and $x_9$. The configuration I described above is a string/gauge theory interpretation. Now if we take the tension of the branes into account, the configuration has to be modified to include the quantum effects. We can uplift this configuration to M-theory (introducing a circle $x_{10}$) and minimizing the world volume of the corresponding $M5$-brane (ex-$D4$) under fixed boundary condition will yield the Seiberg-Witten curve. This curve describes a dimension two subsurface inside the space spanned by the coordinates  $x_{4},x_5, x_6,x_{10}$. Now, one is not limited to a $d=4$ theory.

It is possible to compactify in the $x_5$ to obtain a $\mathcal{N}=1$ $d=5$ theory. Once we do the compactification we T-dualize along $x_5$ to obtain a system involving $NS$5-branes and $D5$-branes in Type IIB theory. I will stop here but I think this is the reference [hep-th/9706087] to check alongside (alongside Witten's one for $d=4$ theories).

answered Apr 9, 2015 by (3,625 points)

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