Quadratic differentials as stability conditions
type: Item
instance of: scholarly article
title: Quadratic differentials as stability conditions
zbMATH Open document ID: 1328.14025
author: Tom Bridgeland
author: Ivan Smith
published in: Publications Mathématiques
publication date: 2015-07-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/1302.7030
full work available at URL: BrSm arXiv-Version2-Corrected.pdf
review text: The paper under this review is motivated by the physics papers of Gaiotto, Moore and Neitzke [\textit{D. Gaiotto} et al., Commun. Math. Phys. 299, No. 1, 163--224 (2010; Zbl 1225.81135); Adv. Math. 234, 239--403 (2013; Zbl 1358.81150)] in the references of this paper). Let us start this review with a rather long and rough explanation of what the physicists studies, since it is helpful to understand the setting and objects Bridgeland and Smith consider. The main object of the work of Gaiotto, Moore and Neitzke is the determination of BPS spectrum of the so-called ``class S theories'', which are certain 4-dimensional \(N=2\) supersymmetric gauge theories. The class S theories originate from compactification of M5-branes in the 6-dimensional theory on a punctured Riemann surface \(C\), and the theories are labelled by the punctured surface \(C\) and by the gauge group associated to the superconformal algebras. These theories are also related to the Hitchin system on \(C\) by the \(S^1\)- and \(C\)-compactification of the original 6-dimensional theory. Simply speaking, a Hitchin system on \(C\) encodes the Seiberg-Witten theory, which is the low energy effective theory of the 4-dimensional theory we started with. The Seiberg-Witten curve \(\Sigma\) is a branched cover of \(C\), and can be considered as \(\Sigma \subset T^* C\), and the canonical 1-form on \(T^* C\) recovers the Seiberg-Witten differential \(\lambda\) when restricted to \(\Sigma\). The paper [Zbl 1358.81150] described the BPS spectrum (in some examples of class S theories) in terms of Hitchin systems. If the gauge group labelling the theory is \(\mathrm{U}(2)\), then BPS states are associated to the curves on the surface \(C\) that minimize the length in terms of the metric induced by the coordinate \(w = \int_{z_0}^z \lambda\). BPS hypermultiplets are associated with curves that begin and end on branch points of \(\Sigma \to C\), and BPS vector-multiplets are associated with curves which are closed. The…
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 81T20
zbMATH DE Number: 6460465
Wikidata QID: Q96828324
reviewed by: Shintarou Yanagida
MaRDI profile type: Publication
OpenAlex ID: W2072771170
arXiv ID: 1302.7030
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DOI: 10.1007/S10240-014-0066-5
description: scientific article