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There seems to have some (deep) relation between Seiberg-Witten theory and superconductivity. e.g. this Witten paper.

Q: Could someone introduce the relations between the twos both physically in terms of intuition? and mathematically in terms of formalism? How exactly is the relation?

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2 Answers 2

The connection between superconductivity and Seiberg-Witten theory can be understood through the observation that superconductivity is related to the Meissner effect, which is the exclusion of magnetic field lines from a superconductor. Seiberg-Witten theory is based on the analysis of the moduli space of an $\mathcal{N}=2$ supersymmetric Yang-Mills theory. It turns out that the theory contains monopoles that acquire a non-zero vacuum expectation value, which can be interpreted as a version of the Meissner effect. I believe that a thorough mathematical explanation cannot be given within one answer, I would rather refer to the literature. The book "Modern Supersymmetry" by John Terning gives a nice overview of Seiberg-Witten theory; the Meissner effect is discussed as well.

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There is a very direct relationship which answers your question, and I'll state it in the way I first learned about it (but you can derive a different connection by passing between dimensions):

The 2-dimensional reduction of the Seiberg-Witten equations are the (abelian) vortex equations.

The $SU(2)$-vortex equations on $\mathbb{R}^2$ are a Yang-Mills-Higgs equation, and is a 2-dimensional version of superconductivity, which is actually defined on $\mathbb{R}^3$ with $G=U(1)\subset SU(2)$. Here the YMH-equations are precisely the Landau-Ginzburg equations and $\phi$ represents a Cooper pair (a bound state of two electrons). Minimal solutions to this have $0=D_A\phi=d\phi+A\phi$ and hence $0=D_A^2=F_A$ which physically represents the Meissner effect (the expulsion of magnetic fields from the bulk of a superconductor). Here $\phi$ takes on constant value $|\phi|=1$; perturbing this minimum $\phi=1+h$ and expanding the LG-equations to 1st-order in $h$ yields two damped ODEs (one for $h$, one for $A$) whose solutions provide the correlation length (of the Cooper pair) and penetration depth (of the magnetic field).

You may have heard of "monopoles" in relation to SW-theory. That's because the 3-dimensional reduction of the Seiberg-Witten equations are the (abelian) Bogolmony equations which define monopoles. As above, $SU(2)$ vortices and monopoles are inherently related, and are dictated by an $SU(2)$ Yang-Mills-Higgs theory on $\mathbb{R}^n$ for $n=2,3,4$ (the $n=4$ case exhibits a relation to the "Donaldson instantons"). The exact relations with everything I have mumbled will take more time to discuss (for instance, the 3-dimensional equations describing monopoles and also superconductors are slightly different, depending on the the existence of a potential and the type of representation (for the gauge group) you use).

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