Abstract
We study the hemisphere partition function of a three-dimensional mathcal{N} = 4 supersymmetric U(N) gauge theory with one adjoint and one fundamental hypermultiplet — the ADHM quiver theory. In particular, we propose a distinguished set of UV boundary conditions which yield Verma modules of the quantised chiral rings of the Higgs and Coulomb branches. In line with a recent proposal by two of the authors in collaboration with M. Bullimore, we show explicitly that the hemisphere partition functions recover the characters of these modules in two limits, and realise blocks gluing exactly to the partition functions of the theory on closed three-manifolds. We study the geometry of the vortex moduli space and investigate the interpretation of the vortex partition functions as equivariant indices of quasimaps to the Hilbert scheme of points in ℂ2. We also investigate half indices of the ADHM quiver gauge theory in the presence of a line operator and discuss their geometric interpretation. Along the way we find interesting relations between our hemisphere blocks and related quantities in topological string theory and equivariant quantum K-theory.
Highlights
Three dimensional gauge theories with N = 4 supersymmetry sit at the centre of a remarkable web of connections between physics and mathematics [1,2,3,4,5,6,7,8,9,10]
We study the hemisphere partition function of a three-dimensional N = 4 supersymmetric U(N ) gauge theory with one adjoint and one fundamental hypermultiplet — the ADHM quiver theory
We study the geometry of the vortex moduli space and investigate the interpretation of the vortex partition functions as equivariant indices of quasimaps to the Hilbert scheme of points in C2
Summary
The state-operator correspondence relates the hemisphere partition functions to a half-index counting local operators inserted at the origin of Ω-deformed R≥0 × R2 and in this picture, as detailed in [32], these particular UV boundary conditions are associated with Verma modules of the quantised Higgs and Coulomb branch chiral rings. In interesting recent work [16], the Cardy block of the ADHM quiver theory has been evaluated in the large N limit and found to reproduce the entropy of supersymmetric asymptotically AdS4 black holes. We discuss a geometric interpretation of this boundary condition as an equivariant Euler characteristic counting sections of holomorphic line bundles over a distinguished Lagrangian in the Higgs branch of the ADHM theory. We include detailed appendices covering conventions for partitions and symmetric functions A as well as novel results on evaluating Molien integrals using Macdonald polynomial methods B
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