The study of supersymmetric quantum field theories (SQFTs) has been of high interest in theoretical physics for many years now. It has led to developments in the understanding of many topics such as conformal field theories, the AdS/CFT correspondence and non-supersymmetric QFTs, to name just a few. SQFTs may admit continuous families of vacua, and the continuous vacuum expectation values (VEVs) trace out a geometric space. The space traced out by physically distinct VEVs is termed the moduli space of the theory. Geometric properties of a point on the moduli space of a theory relate to physical properties of the theory in the corresponding vacuum state; one can explore the physics of a theory by studying its moduli space as a mathematical object. The properties of the moduli space that this note contributes to the understanding of are its global symmetry, and its Poisson bracket.
We are concerned with the Coulomb branches of 3d N = 4 quiver gauge theories, at their IR fixed point. In particular, the Coulomb branches we focus on have a global symmetry which is a product of the SU(2)R R-symmetry and the topological global symmetry GS, and are symplectic singularities. The symplectic form induces a Poisson structure, and degenerates at the singularities. The Coulomb branch global symmetry and Poisson bracket are of interest because, as mentioned above, they cor- respond to physical properties of the theory. In SQFT, particles are understood as excitations of the vacuum state and are labelled by their charges under the global symmetries of the theory. The more massless states that have been integrated out in a particular vacuum state, the more the Poisson bracket degenerates at the corresponding point on the moduli space. This thesis is devoted to recent developments made in the aid of being able to read off these properties from a given 3d N = 4 quiver using only simple graph theory operations. In the case of the global symmetry, an attempt at such an algorithm exists, but it does not work unanimously. We help to understand its failure, and suggest an amendment which works on a wider set of quivers. In the case of the Poisson bracket, we provide a conjecture for computing it for quivers of high rank, and in particular for magnetic quivers for certain 5 and 6d Higgs branches at infinite coupling.