ε-deformed W algebras: 4d N=2 gauge theories, 2d CFT and (bi)spectral duality

I will present a constructive definition of ε-deformed or Yangian version of q-W algebras which feature in the brane engineering of 5d N=1 quiver gauge theories. This new type of algebras fill in the missing intermediate level between q-deformed and un-deformed W algebras, and I will explain in which sense the ε-deformed algebras are (bi)spectral duals to the ordinary ones. In particular, I will discuss how the ε-deformed conformal blocks manifestly reproduce instanton partition functions of 4d N = 2 quiver gauge theories and give dual (unconventional) integral representations of Toda conformal blocks via AGT duality. Finally, I will also consider an application to integrable many-body systems