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We review the geometric approach to BPS states in four-dimensional \(\mathcal N=2\) theories, with a focus on class \(\mathcal S\) theories. Owing to the advanced nature of this material, our treatment is intended as a guided survey of the literature rather than a comprehensive introduction. We begin with the description of BPS states as finite string webs of variable tension on the ultraviolet curve \(\mathrm {C}_{\mathrm {UV}}\) associated with class \(\mathcal S\) theories, together with the notion of WKB triangulations in the case of type \(A_1\) theories, following Gaiotto et al. (Commun Math Phys 299:163–224, 2010; Adv Math 234:239–403, 2013). We then turn to the framework of spectral networks, introduced in Gaiotto et al. (Annales Henri Poincare 14:1643–1731, 2013), which builds on the theory of 2d–4d framed BPS states developed in Gaiotto et al. (Adv Theor Math. Phys 17(2):241–397, 2013; JHEP 12:082, 2012).

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BPS States in Class \(\mathcal S\) Theories

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  • 发表日期：

    2026年

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  • Clarence Kineider,
  • Georgios Kydonakis,
  • Eugen Rogozinnikov,
  • Valdo Tatitscheff,
  • Alexander Thomas

摘要

We review the geometric approach to BPS states in four-dimensional \(\mathcal N=2\) theories, with a focus on class \(\mathcal S\) theories. Owing to the advanced nature of this material, our treatment is intended as a guided survey of the literature rather than a comprehensive introduction. We begin with the description of BPS states as finite string webs of variable tension on the ultraviolet curve \(\mathrm {C}_{\mathrm {UV}}\) associated with class \(\mathcal S\) theories, together with the notion of WKB triangulations in the case of type \(A_1\) theories, following Gaiotto et al. (Commun Math Phys 299:163–224, 2010; Adv Math 234:239–403, 2013). We then turn to the framework of spectral networks, introduced in Gaiotto et al. (Annales Henri Poincare 14:1643–1731, 2013), which builds on the theory of 2d–4d framed BPS states developed in Gaiotto et al. (Adv Theor Math. Phys 17(2):241–397, 2013; JHEP 12:082, 2012).