arXiv:1512.05388v2 [hep-th] 2 Jan 2016 BPS/CFT CORRESPONDENCE: NON-PERTURBATIVE DYSON-SCHWINGER EQUATIONS ANDqq-CHARACTERS NIKITA NEKRASOV Abstract. We study symmetries of quantum field theories involving topologically dis- tinct sectors of the field space. To exhibit these symmetries we define special gauge invariant observables, which we call theqq-characters. In the context of the BPS/CFT correspondence, using these observables, we derive an infinite set of Dyson-Schwinger- type relations. These relations imply that the supersymmetric partition functions in the presence ofΩ-deformation and defects obey the Ward identities of two dimen- sional conformal field theory and itsq-deformations. The details will be discussed in the companion papers. Contents 1.Introduction4 1.1.Dyson-Schwinger equations4 1.2.Non-perturbative Dyson-Schwinger identities6 1.3.Organization of the presentation8 1.4.Acknowledgements10 2.The BPS/CFT correspondence12 2.1.N= 2partition functions12 2.2.Defect operators and lower-dimensional theories13 2.3.TheY- andX-observables14 2.4.The physics ofX-observables14 2.5.Hidden symmetries17 2.6.Some notations.17 2.7.Equivariant virtual Chern polynomials22 3.Supersymmetric gauge theories22 3.1.Quivers22 3.2.Quivers with colors22 3.3.The symmetry groups22 3.4.The parameters of Lagrangian24 3.5.The groupH25 3.6.Perturbative theory26 3.7.Realizations of quiver theories28 4.Integration over instanton moduli spaces30 1
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