Wall-crossing in coupled 2d-4d systems

By D. Gaiotto, G. Moore, Andrew Neitzke. 2011

148 citations recorded by the source corpus.

A bstractWe introduce a new wall-crossing formula which combines and generalizes the Cecotti-Vafa and Kontsevich-Soibelman formulas for supersymmetric 2d and 4d systems respectively. This 2d-4d wall-crossing formula governs the wall-crossing of BPS states in an $ \mathcal{N}=2 $ supersymmetric 4d gauge theory coupled to a supersymmetric surface defect. When the theory and defect are compactified on a circle, we get a 3d theory with a supersymmetric line operator, corresponding to a hyperholomorphic connection on a vector bundle over a hyperkähler space. The 2d-4d wall-crossing formula can be interpreted as a smoothness condition for this hyperholomorphic connection. We explain how the 2d-4d BPS spectrum can be determined for 4d theories of class $ \mathcal{S} $, that is, for those theories obtained by compactifying the six-dimensional (0, 2) theory with a partial topological twist on a punctured Riemann surface C. For such theories there are canonical surface defects. We illustrate with several examples in the case of A1 theories of class $ \mathcal{S} $. Finally, we indicate how our results can be used to produce solutions to the A1 Hitchin equations on the Riemann surface C.

Subfields: Computational Theory and Mathematics, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/286512166391f5066beb71b56a644fd20234501a

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Citations in the corpus

Four-dimensional SCFTs from M5-branes (2012, background): “This approach has been taken first in [2, 3] and then in [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], often with remarkable efficacy.”
General Argyres-Douglas theory (2012, background): “The BPS spectrum and wall crossing behavior of above two class of theories are studied in [35, 3, 36, 37] , extended object like line operators and surface operators are studied in [38, 39].”
Spectral Networks (2012, unknown): “It also provides a geometric argument (if not quite a proof) that the degeneracies so computed obey the wall-crossing formula of Kontsevich-Soibelman [9], as well as its extension to include the coupling between 2d and 4d BPS states, given in [5].”
Scattering Amplitudes and the Positive Grassmannian (2012, background): “Moving further afield, as alluded to in section 15, the basic mathematical structures we have encountered in scattering amplitudes have also recently made an appearance in apparently completely different physical settings, related to conformal blocks for higher Toda theories [35,45], wall-crossing [”