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Letters in Mathematical Physics

, Volume 109, Issue 3, pp 579–622 | Cite as

BPS/CFT correspondence IV: sigma models and defects in gauge theory

  • Nikita NekrasovEmail author
Article

Abstract

Quantum field theory \(L_1\) on spacetime \(X_{1}\) can be coupled to another quantum field theory \(L_2\) on a spacetime \(X_{2}\) via the third quantum field theory \(L_{12}\) living on \(X_{12} = X_{1} \cap X_{2}\). We explore several such constructions with two- and four-dimensional \(X_{1}, X_{2}\)’s and zero- and two-dimensional \(X_{12}\)’s, in the context of \({\mathcal {N}}=2\) supersymmetry, non-perturbative Dyson–Schwinger equations, and BPS/CFT correspondence. The companion paper (Nekrasov, “BPS/CFT correspondence V: BPZ and KZ equations from qq-characters”, 2017. arXiv:1711.11582 [hep-th]) will show that the BPZ and KZ equations of two-dimensional conformal field theory are obeyed by the half-BPS surface defects in quiver \({\mathcal {N}}=2\) gauge theories.

Keywords

Supersymmetry BPS/CFT correspondence Hidden symmetry Defects in quantum field theory 

Mathematics Subject Classification

81T60 81T13 81T45 81T30 

Notes

Acknowledgements

Research was partly supported by the National Science Foundation under Grant No. NSF-PHY/1404446. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. The author thanks A. Losev for discussions about two-dimensional topological theories in 1992–1994, E. Frenkel for the invitation to the DARPA program ‘Langlands Program and Physics’ conference at the IAS in March 8–10, 2004, and for the opportunity to present there some of the ideas developed in this paper, as well as for numerous patient explanations on [35] and other topics. The author is grateful to A. Okounkov, V. Pestun and A. Rosly for numerous useful discussions concerning the topics of this paper over the recent years, as well as to S. Jeong and O. Tsymbaliuk for their collaboration of the related projects. The paper was finished while the author visited the IHES (Bures-sur-Yvette). We thank this remarkable institution for its hospitality.

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© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Simons Center for Geometry and PhysicsStony Brook UniversityStony BrookUSA

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