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

TL;DR

This work develops a non-perturbative Dyson–Schwinger framework within the BPS/CFT correspondence by introducing qq-characters as local observables that organize Ward identities across distinct topological sectors of ${ m N}=2$ gauge theories. It builds ${oldsymbol{bb Y}}$- and ${oldsymbol{bb X}}$-observables from equivariant cohomology on Nakajima quiver varieties, proving a main theorem that certain ${oldsymbol{bb X}}$-observables yield pole-free, polynomial expressions in the spectral parameter $x$—the Yangian qq-characters. The paper provides explicit constructions and examples for A- and D-type quivers, formulates a rigorous integral representation of qq-characters via quiver varieties in 4D/5D, and discusses how these structures encode non-perturbative symmetries, defect relations, and potential links to $q$- and $t$-characters. The results connect gauge theory partition functions, defect sectors, and two-dimensional conformal structures, with applications to effective prepotentials, instanton fusion, and integrable systems, and point to rich future directions in string realizations and higher-dimensional generalizations.

Abstract

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 $Ω$-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.