I would like to ask for some readable introduction or maybe review of the technique of supersymmetric localization for $\mathcal{N}=1,2$ SUSY theories. I would like a different one than the one people usually suggest, the one of Marcos Mariño: Lectures on localization and matrix models in supersymmetric Chern-Simons-matter theories, arXiv:1104.0783.


As a beginner for working on relevant topics, I just write few words about your question. I hope it helps you up.

Localization Principle has been great role in computing superconformal index also it gives the exact calculation in susy gauge theories. From some excellent works by Pestun, Kapustin, Willet and so on(about a decade ago?), many researcher now used localization principle in various topics. (relevant major work is computing index.)

If you know a lots of basic stuff(SUSY, Killing spinor equation etc), i recommend to read the papers from original Pestun's paper.

Recently many review papers appeared in the arXiv. I think the Hosmichi's recent review is good for beginner.

Also, you can find some useful references in Exact results on N=2 supersymmetric gauge theories. They recently published a lots of papers on relevant topic.

Recently many people has been working on various dimension with different space (also in different number of susy) such as $S^4$ $S^3 \times S^1$, $R^4$... $S^2$, $T^2$, squashed spheres.. and so on.


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