Boundaries, Vermas, and Factorisation

Abstract

We revisit the factorisation of supersymmetric partition functions of 3d $\mathcal{N}=4$ gauge theories. The building blocks are hemisphere partition functions of a class of UV $\mathcal{N}=(2,2)$ boundary conditions that mimic the presence of isolated vacua at infinity in the presence of real mass and FI parameters. These building blocks can be unambiguously defined and computed using supersymmetric localisation. We show that certain limits of these hemisphere partition functions coincide with characters of lowest weight Verma modules over the quantised Higgs and Coulomb branch chiral rings. This leads to expressions for the superconformal index, twisted index and $S^3$ partition function in terms of such characters. On the way we uncover new connections between boundary 't Hooft anomalies, hemisphere partition functions and lowest weights of Verma modules.

arXiv comment33+17 pages, 3 figures. v2: added citations, minor revision of squashed sphere factorisation section

References in corpus (40)

• Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter
• S-Duality of Boundary Conditions In N=4 Super Yang-Mills Theory
• SUSY Gauge Theories on Squashed Three-Spheres
• D-Branes And Mirror Symmetry
• The complete superconformal index for N=6 Chern-Simons theory
• Vortex Counting and Lagrangian 3-manifolds
• A topologically twisted index for three-dimensional supersymmetric theories
• Index for three dimensional superconformal field theories with general R-charge assignments
• Bootstrapping the superconformal index with surface defects
• Holomorphic Blocks in Three Dimensions
• Comments on twisted indices in 3d supersymmetric gauge theories
• Phases Of N=2 Theories In 1+1 Dimensions With Boundary
• Factorisation of N = 2 theories on the squashed 3-sphere
• Defects and Quantum Seiberg-Witten Geometry
• Generalized Superconformal Index for Three Dimensional Field Theories
• Boundaries, Mirror Symmetry, and Symplectic Duality in 3d $\mathcal{N}=4$ Gauge Theory
• Down the rabbit hole with theories of class S
• Higgs branch localization in three dimensions
• Dual Boundary Conditions in 3d SCFT's
• Exact Results In Two-Dimensional (2,2) Supersymmetric Gauge Theories With Boundary
• Exact Results in Supersymmetric Field Theories on Manifolds with Boundaries
• Ω-deformation and quantization
• Coulomb Branch Operators and Mirror Symmetry in Three Dimensions
• Higgs branch localization of 3d N=2 theories
• 3D-partition functions on the sphere: exact evaluation and mirror symmetry
• Factorisation and holomorphic blocks in 4d
• Supersymmetric Casimir Energy and the Anomaly Polynomial
• Localization of 3d $\mathcal{N}=2$ Supersymmetric Theories on $S^1\times D^2$
• Flipping the head of T[SU(N)]: mirror symmetry, spectral duality and monopoles
• SCFT/VOA correspondence via $\Omega$-deformation
• Algebra of the Infrared: String Field Theoretic Structures in Massive $\mathcal{N}=(2,2)$ Field Theory In Two Dimensions
• Coulomb Branch Quantization and Abelianized Monopole Bubbling
• Chiral algebras from Ω-deformation
• Sphere correlation functions and Verma modules
• Twisted Indices of 3d $\mathcal{N}=4$ Gauge Theories and Enumerative Geometry of Quasi-Maps
• Blocks and Vortices in the 3d ADHM Quiver Gauge Theory
• The modular properties and the integral representations of the multiple elliptic gamma functions
• Factorisation of 3d $\mathcal{N}=4$ Twisted Indices and the Geometry of Vortex Moduli Space
• 3d Mirror Symmetry and Quantum $K$-theory of Hypertoric Varieties
• Factorising the 3D Topologically Twisted Index

Cited by in corpus (13)

• Surface Defect, Anomalies and $b$-Extremization
• Non-unitary TQFTs from 3D $\mathcal{N}=4$ rank 0 SCFTs
• 3d $\mathcal{N}=4$ Gauge Theories on an Elliptic Curve
• Intersecting Defects and Supergroup Gauge Theory
• BPS Indices, Modularity and Perturbations in Quantum K-theory
• Localization and Duality for ABJM Latitude Wilson Loops
• Remarks on Berry Connection in QFT, Anomalies, and Applications
• Crystal bases and three-dimensional $\mathcal{N}=4$ Coulomb branches
• Intersecting Surface defects and 3d Superconformal indices
• ${\text{AdS}}_4$ Holography and the Hilbert Scheme
• Fermionic extensions of $W$-algebras via 3d $\mathcal{N}=4$ gauge theories with a boundary
• The colored Jones polynomials as vortex partition functions
• Stokes Phenomena in 3d $\mathcal{N}=2$ SQED${}_2$ and ${\mathbb{CP}}^1$ Models