Geometric aspects of three dimensional N = 4 gauge theories
We study geometric aspects of three dimensional N = 4 gauge theories. We focus mainly on the factorisation property of supersymmetric partition functions of these theories and introduce hemisphere blocks that precisely realise the factorisation. We define these blocks as UV partition functions on a hemisphere S1 ×H2 with an exceptional Dirichlet boundary condition and demonstrate that the resulting object is determined by the enumerative geometry of the Higgs branch. The partition function on the hemisphere is closely related to a half-index that counts local operators of the theory on a flat spacetime with boundary. In this context, we show that the hemisphere blocks realise characters of lowest weight Verma modules of the Higgs and Coulomb branch chiral rings acting on boundary local operators. We study the geometric interpretation of the twisted index factorisation in particular and demonstrate a relationship between the twisted index and the Hilbert series of a 3d N = 4 theory. We then use factorisation to provide a novel geometric expression for the Coulomb branch Hilbert series in terms of invariants of moduli spaces of quasimaps to the Higgs branch. Finally, we apply these ideas to a particularly rich example of a non-abelian gauge theory with adjoint matter whose Higgs branch coincides with a moduli space of instantons. We compute hemisphere blocks for the theory and explicitly recover Verma module characters of the Coulomb branch chiral ring. In this example, the blocks have interesting combinatorial content and can be related to generating functions of reverse plane partitions—we discuss the interpretation of 3d mirror symmetry in this context. We also study line operators in this theory and show that half indices in the presence of a line operator exhibit an integrable structure. Along the way we find interesting connections between the twisted index gluing of hemisphere blocks and related calculations in topological string theory.