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Localization techniques in quantum field theories. (English) Zbl 1378.00123

From the text: The focus of the collection is on the localization technique and its applications in supersymmetric gauge theories. Although the ideas of equivariant localization in quantum field theory go back 30 years, here we concentrate on the recent surge in the subject during the last ten years. This subject develops rapidly and thus it is impossible to have a fully satisfactory overview of the field. This collection is about two and a half years in the making, and during this period some important new results have been obtained, and it was difficult to incorporate all of them. However, we think that it is important to provide an overview and an introduction to this quickly developing subject. This is important both for young researchers, who have just entered the field, and to established scientists as well. We have tried to do our best to review the main results during the last ten years.
The issue has two types of chapter: some concentrate on the localization calculation in different dimensions by itself, and others concentrate on the major applications of the localization result. Obviously, such a separation is sometimes artificial. The chapters are ordered roughly according to the dimensions of the corresponding supersymmetric theories. First, we try to review the localization calculation in a given dimension, and then we move to the discussion of the major applications.
The issue covers the localization calculations for the supersymmetric theories in dimensions 2, 3, 4 and 5. The collection discusses the applications of these calculations for theories living up to dimension 6, and for string/M theories. We have to apologize in advance for omitting from the review the new and important calculations which have appeared during the last couple of years.

MSC:

00B15 Collections of articles of miscellaneous specific interest
81-06 Proceedings, conferences, collections, etc. pertaining to quantum theory
81T13 Yang-Mills and other gauge theories in quantum field theory
81T60 Supersymmetric field theories in quantum mechanics
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References:

[1] Pestun V and Zabzine M 2017 Introduction to localization in quantum field theory J. Phys. A: Math. Theor.50 443001 · Zbl 1386.81117 · doi:10.1088/1751-8121/aa5704
[2] Pestun V 2017 Review of localization in geometry J. Phys. A: Math. Theor.50 443002 · Zbl 1377.81092 · doi:10.1088/1751-8121/aa6161
[3] Benini F and Le Floch B 2017 Supersymmetric localization in two dimensions J. Phys. A: Math. Theor.50 443003 · Zbl 1377.81091 · doi:10.1088/1751-8121/aa77bb
[4] Morrison D 2017 Gromov-Witten invariants and localization J. Phys. A: Math. Theor.50 443004 · Zbl 1377.81179 · doi:10.1088/1751-8121/aa6f65
[5] Dumitrescu T 2017 An introduction to supersymmetric field theories in curved space J. Phys. A: Math. Theor.50 443005 · Zbl 1386.81138 · doi:10.1088/1751-8121/aa62f5
[6] Willett B 2017 Localization on three-dimensional manifolds J. Phys. A: Math. Theor.50 443006 · Zbl 1377.81093 · doi:10.1088/1751-8121/aa612f
[7] Mariño M 2017 Localization at large N in Chern-Simons-matter theories J. Phys. A: Math. Theor.50 443007 · Zbl 1377.81143 · doi:10.1088/1751-8121/aa5f69
[8] Pufu S 2017 The F-theorem and F-maximization J. Phys. A: Math. Theor.50 443008 · Zbl 1386.81122 · doi:10.1088/1751-8121/aa6765
[9] Dimofte T 2016 Perturbative and nonperturbative aspects of complex Chern-Simons Theory J. Phys. A: Math. Theor.50 443009 · Zbl 1386.81121 · doi:10.1088/1751-8121/aa6a5b
[10] Hosomichi K 2017 N=2 SUSY gauge theories on S4 J. Phys. A: Math. Theor.50 443010 · doi:10.1088/1751-8121/aa7775
[11] Zarembo K 2017 Localization and AdS/CFT correspondence J. Phys. A: Math. Theor.50 443011 · Zbl 1377.81099 · doi:10.1088/1751-8121/aa585b
[12] Tachikawa Y 2017 A brief review of the 2d/4d correspondences J. Phys. A: Math. Theor.50 443012 · Zbl 1377.81110 · doi:10.1088/1751-8121/aa5df8
[13] Rastelli L and Razamat S 2017 The supersymmetric index in four dimensions J. Phys. A: Math. Theor.50 443013 · Zbl 1377.81211 · doi:10.1088/1751-8121/aa76a6
[14] Qiu J and Zabzine M 2017 Review of localization for 5D supersymmetric gauge theories J. Phys. A: Math. Theor.50 443014 · Zbl 1377.81109 · doi:10.1088/1751-8121/aa5ef0
[15] Minahan J 2017 Matrix models for 5D super Yang-Mills J. Phys. A: Math. Theor.50 443015 · Zbl 1386.81129 · doi:10.1088/1751-8121/aa5cbe
[16] Pasquetti S 2017 Holomorphic blocks and the 5d AGT correspondence J. Phys. A: Math. Theor.50 443016 · Zbl 1377.81098 · doi:10.1088/1751-8121/aa60fe
[17] Kim S and Lee K 2017 Indices for 6 dimensional superconformal field theories J. Phys. A: Math. Theor.50 443017 · Zbl 1377.81107 · doi:10.1088/1751-8121/aa5cbf
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