Lie Groups Seminar
Braverman, Finkelberg and Nakajima have recently proposed a mathematical definition of the Coulomb branch of a 4d $N=2$ gauge theory
of cotangent type, associating to each such theory a family of associative algebras deforming the algebra of functions on an affine Poisson variety. In this talk I will discuss joint work with Alexander Shapiro (arxiv 1910.03186) in which we confirm, in the case of gauge theories determined by quivers without loops, Gaiotto's prediction that the Coulomb branch algebra should embed into the quantum cluster algebra determined by the BPS quiver of the theory. Our proof is based on identifying the so-called 'total DT-invariant' associated to this cluster algebra, which leads to an explicit identification of the monopole operators corresponding to minuscule coweights with certain cluster monomials.