  We study symmetries of quantum field theories involving topologically
distinct sectors of the field space. To exhibit these symmetries we define
special gauge invariant observables, which we call the $qq$-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 of $\Omega$-deformation and
defects obey the Ward identities of two dimensional conformal field theory and
its $q$-deformations. The details will be discussed in the companion papers.
