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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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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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