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We study symmetries of quantum field theories involving topologicallydistinct sectors of the field space. To exhibit these symmetries we definespecial gauge invariant observables, which we call the $qq$-characters. In thecontext of the BPS/CFT correspondence, using these observables, we derive aninfinite set of Dyson-Schwinger-type relations. These relations imply that thesupersymmetric partition functions in the presence of $\Omega$-deformation anddefects obey the Ward identities of two dimensional conformal field theory andits $q$-deformations. The details will be discussed in the companion papers.

BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters