Seiberg-Witten theory and Superconductivity

Question

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?

This post imported from StackExchange Physics at 2014-06-04 11:35 (UCT), posted by SE-user Idear

Answer 1

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 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.

This post imported from StackExchange Physics at 2014-06-04 11:35 (UCT), posted by SE-user Frederic Brünner

Answer 2

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).

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 $n=2,3$. 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).