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We consider $\mathcal N=2$ $SU(2)$ gauge theories in four dimensions (pure ormass deformed) and discuss the properties of the simplest chiral observables inthe presence of a generic $\Omega$-deformation. We compute them by equivariantlocalization and analyze the structure of the exact instanton corrections tothe classical chiral ring relations. We predict exact relations valid at allinstanton number among the traces $\langle\text{Tr}\varphi^{n}\rangle$, where$\varphi$ is the scalar field in the gauge multiplet. In theNekrasov-Shatashvili limit, such relations may be explained in terms of theavailable quantized Seiberg-Witten curves. Instead, the full two-parameterdeformation enjoys novel features and the ring relations require non trivialadditional derivative terms with respect to the modular parameter. Higher rankgroups are briefly discussed emphasizing non-factorization of correlators dueto the $\Omega$-deformation. Finally, the structure of the deformed ringrelations in the $\mathcal N=2^{\star}$ theory is analyzed from the point ofview of the Alday-Gaiotto-Tachikawa correspondence proving consistency as wellas some interesting universality properties.

Chiral trace relations in $\Omega$-deformed $\mathcal N=2$ theories