Wall-crossing in coupled 2d-4d systems
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Publication:1796208
DOI10.1007/JHEP12(2012)082zbMATH Open1397.81364arXiv1103.2598OpenAlexW2168048313MaRDI QIDQ1796208
Author name not available (Why is that?)
Publication date: 16 October 2018
Published in: (Search for Journal in Brave)
Abstract: We 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 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 hyperkahler 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 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 A_1 theories of class S. Finally, we indicate how our results can be used to produce solutions to the A_1 Hitchin equations on the Riemann surface C.
Full work available at URL: https://arxiv.org/abs/1103.2598
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