I often encountered the term "localization" in the context of instantons, as for example in the work of Nekrasov on extensions of Seiberg-Witten theory to ${\cal N}=1$ gauge theories.

Could someone give a "intuitive" explanation of the concept of localization as well as a "simple" concrete realisation of it?


Dear inovaovao, in general, localization is the fact that certain integrals over some variables $\mu$ may be proved to be equal to lower-dimensional integrals or even the sum over contributions from a discrete - e.g. finite - set of points in the $\mu$-space.

A simple conceptual example that doesn't require any advanced Seiberg-Witten-like machinery - but that is the "master setup" for everything to be explained below - is the classical limit of a path integral: the otherwise infinite-dimensional path integral de facto localizes on the classical trajectories. Although the path integral integrates over histories that are not classical solutions, the integral localizes on histories that are solutions. Here, let me emphasize, we're not changing the rules of the game of Feynman's approach to quantum mechanics: we're showing that the resulting formulae are fully equivalent to others where the integral gets simpler or lower-dimensional.

Some of the points where the path integral gets localized are just local (rather than global) extrema of the action - the instantons.

In the Seiberg-Witten case, one has to calculate contributions of instantons - topologically nontrivial solutions of the classical equations of motion (stationary points of the action which are local minima but not the global ones). The instanton solutions themselves are spanned by many parameters: for $k$-instanton solutions of $SU(N)$ gauge theory, the number of parameters, as seen via the ADHM construction, is essentially $4kn$. It remembers the embedding of the instantons in $SU(N)$, their size, and their positions.

So the instanton contributions are still integrals over the $4kN$-dimensional moduli space of the instantons. That's progress relatively to the infinite-dimensional path integral we could have started with. But the simplification may go on. Even this integral over the moduli space of instantons can be reduced to a lower-dimensional integral, and often a sum, over individual points of the instanton moduli space.

The typical instantons that contribute are usually "point-like" and singular in some sense (in other contexts, the relevant places where the integrals localize may be labeled by unusually complex values of some parameters) and a special treatment has to be made to regulate these potentially ill-defined expressions, but it can be done. It's been pursued in many papers co-authored by Nikita Nekrasov, such as


and many others. The required formulae that allow you to prove that the integrals localize are somewhat nontrivial mathematically. After all, Nikita Nekrasov and his collaborators are pretty powerful mathematicians. But in some sense, these formulae are always generalizations of the fact that integrals in the complex plane can be rewritten in terms of residues of the poles.

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    $\begingroup$ Thanks Lubos, perfect answer! Now I just need to find the time to read those papers... $\endgroup$ – bangnab Feb 15 '11 at 14:04
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    $\begingroup$ Is instanton localization analogous to the localization of electronic states in disordered materials (Anderson localization)? $\endgroup$ – user346 Feb 15 '11 at 14:19
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    $\begingroup$ Thanks, @inovaovao, for your interest: but be ready, Nikita is able to be annoyingly technical although there's clarity beneath those thoughts. ;-) Dear @space_cadet, aside from the world and the vague meaning of the word, there's no relationship here. Anderson localization is surely not a special case of localization in the path-integral sense. The Anderson (and weak) localization is just a property of the spectrum of particular systems that don't contain certain waves; the instanton localization may apply to many systems whose behavior away from the "loci" doesn't have to be "disordered". $\endgroup$ – Luboš Motl Feb 15 '11 at 15:05

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