Abstract: Braverman, Finkelberg, and Nakajima define the K-theoretic Coulomb branch of a 3d \mathcal N=4 SUSY gauge theory as the affine variety \mathcal M_{G,N} arising as the equivariant K-theory of certain moduli space \mathscr R_{G,N}, labelled by the complex reductive group G and its complex representation N. It was conjectured by Gaiotto, that (quantized) K-theoretic Coulomb branches bear the structure of (quantum) cluster varieties. I will outline a proof of this conjecture for quiver gauge theories, and show how the cluster structure allows to count the BPS states (aka DT-invariants) of the theory. Time permitting, I will also show how the above cluster structure relates to positive and Gelfand-Tsetlin representations of quantum groups, and higher rank Fenchel–Nielsen coordinates on moduli spaces of PGL_n local systems. This talk is based on joint works with Gus Schrader.

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