Factorisation of N = 2 theories on the squashed 3-sphere

Abstract

Partition functions of N=2 theories on the squashed 3-sphere have been recently shown to localise to matrix integrals. By explicitly evaluating the matrix integral we show that abelian partition functions can be expressed as a sum of products of two blocks. We identify the first block with the partition function of the vortex theory, with equivariant parameter hbar=2 Pi i b^2, defined on R^2 x S_1 corresponding to the b->0 degeneration of the ellipsoid. The second block gives the partition function of the vortex theory with equivariant parameter hbar^L=2 Pi i/b^2, on the dual R^2 x S_1 corresponding to the 1/b ->0 degeneration. The ellipsoid partition appears to provide the hbar -> hbar^L modular invariant non-perturbative completion of the vortex theory.

17 pages, 2 figures. v2: added references, minor changes

References in corpus (3)

• SUSY Gauge Theories on Squashed Three-Spheres
• Remodeling the B-model
• SL(2,R) Chern-Simons, Liouville, and Gauge Theory on Duality Walls

Cited by in corpus (108)

• Partition functions of N=(2,2) gauge theories on S^2 and vortices
• Exact Results in D=2 Supersymmetric Gauge Theories
• A topologically twisted index for three-dimensional supersymmetric theories
• Holomorphic Blocks in Three Dimensions
• Defects and Quantum Seiberg-Witten Geometry
• Walls, Lines, and Spectral Dualities in 3d Gauge Theories
• Down the rabbit hole with theories of class S
• Higgs branch localization in three dimensions
• Dual Boundary Conditions in 3d SCFT's
• The equivariant A-twist and gauged linear sigma models on the two-sphere
• Seifert fibering operators in 3d $\mathcal{N}=2$ theories
• Lectures on AdS Black Holes, Holography and Localization
• BPS Degeneracies and Superconformal Index in Diverse Dimensions
• 3d Chern-Simons Theory from M5-branes
• 3d & 5d gauge theory partition functions as q-deformed CFT correlators
• 5D partition functions, q-Virasoro systems and integrable spin-chains
• Gluing gravitational blocks for AdS black holes
• Higgs branch localization of 3d N=2 theories
• Localization on Three-Manifolds
• Microstate Counting of $Ad{S}_4$ Hyperbolic Black Hole Entropy via the Topologically Twisted Index
• Factorisation and holomorphic blocks in 4d
• The supersymmetric index in four dimensions
• Global Properties of Supersymmetric Theories and the Lens Space
• Localization of 3d $\mathcal{N}=2$ Supersymmetric Theories on $S^1\times D^2$
• tt* Geometry in 3 and 4 Dimensions
• Higgs branch localization of $\mathcal{N}=1$ theories on $S^3\times S^1$
• Localization on three-dimensional manifolds
• Flipping the head of T[SU(N)]: mirror symmetry, spectral duality and monopoles
• Three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories and partition functions on Seifert manifolds: A review
• Universal 3d Cardy Block and Black Hole Entropy
• The Superconformal Index of the (2,0) Theory with Defects
• Peacock patterns and new integer invariants in topological string theory
• Spectral zeta function and non-perturbative effects in ABJM Fermi-gas
• On matrix models and their $q$-deformations
• Factorization of S^3/Z_n partition function
• A slow review of the AGT correspondence
• Holomorphic blocks and the 5d AGT correspondence
• ABJM on ellipsoid and topological strings
• An elliptic Virasoro symmetry in 6d
• How to resum perturbative series in 3d N=2 Chern-Simons matter theories
• Boundaries, Vermas, and Factorisation
• From S-confinement to 3D Chiral Theories: Dressing the Monopoles
• 4d N=2 supergravity observables from Nekrasov-like partition functions
• A 5d/3d duality from relativistic integrable system
• Fundamental Vortices, Wall-Crossing, and Particle-Vortex Duality
• Localization of 4d $\mathcal{N}=1$ theories on ${\mathbb{D}}^2\times {\mathbb{T}}^2$
• Sphere correlation functions and Verma modules
• 4d Quantum Geometry from 3d Supersymmetric Gauge Theory and Holomorphic Block
• Higgsed network calculus
• The Stringy Instanton Partition Function
• 5d Higgs Branch Localization, Seiberg-Witten Equations and Contact Geometry
• $q$-Virasoro modular double and 3d partition functions
• Localization of Supersymmetric Chern-Simons-Matter Theory on a Squashed $S^3$ with $SU(2)\times U(1)$ Isometry
• Ω-deformation of B-twisted gauge theories and the 3d-3d correspondence
• Heterotic Surface Defects and Dualities from 2d/4d Indices
• Basic hypergeometry of supersymmetric dualities
• Ellipsoid partition function from Seiberg-Witten monopoles
• 3d Mirror Symmetry from S-duality
• Supersymmetric solutions and Borel singularities for N=2 supersymmetric Chern-Simons theories
• On 3d Seiberg-like Dualities with Two Adjoints
• Universal AdS black holes in theories with sixteen supercharges and their microstates
• Picard-Lefschetz decomposition and Cheshire Cat resurgence in 3D $\mathcal{N}=2$ field theories
• Expanding 3d $\mathcal{N}=2$ Theories around the Round Sphere
• The joy of factorization at large $N$: five-dimensional indices and AdS black holes
• 3d Expansions of 5d Instanton Partition Functions
• Bethe/Gauge correspondence in odd dimension: modular double, non-perturbative corrections and open topological strings
• Blocks and Vortices in the 3d ADHM Quiver Gauge Theory
• Factorisation of 3d $\mathcal{N}=4$ Twisted Indices and the Geometry of Vortex Moduli Space
• On Higgs Branch Localization of Seiberg-Witten Theories on Ellipsoid
• Factorising the 3D Topologically Twisted Index
• Superconformal index on ${\mathbb{RP}}^2\times {\mathbb{S}}^1$ and mirror symmetry
• SYM on Quotients of Spheres and Complex Projective Spaces
• Connecting Mirror Symmetry in 3d and 2d via Localization
• BPS Indices, Modularity and Perturbations in Quantum K-theory
• Abelian 3d mirror symmetry on ${\mathbb{RP}}^2\times {\mathbb{S}}^1$ with $N_f=1$
• Elliptic modular double and 4d partition functions
• On Supersymmetry, Boundary Actions and Brane Charges
• On Integrable Structure and Geometric Transition in Supersymmetric Gauge Theories
• Bootstrapping the $S^5$ partition function
• Trinion Conformal Blocks from Topological strings
• Exact SUSY Wilson loops on $S^3$ from $q$-Virasoro constraints
• Seiberg-like dualities for orthogonal and symplectic 3d $\mathcal{N}=2$ gauge theories with boundaries
• Gromov-Witten invariants and localization
• Resurgence and Lefschetz thimble in 3d N=2 supersymmetric Chern-Simons matter theories
• Marginal Deformations of 3d N=4 Linear Quiver Theories
• Web of Seiberg-like dualities for 3d $\mathcal{N}=2$ quivers
• Exact WKB Analysis of ${\mathbb{CP}}^1$ Holomorphic Blocks
• Integral pentagon relations for 3d superconformal indices
• Localization and Non-Renormalization in Chern-Simons Theory
• Abelian mirror symmetry of $\mathcal{N}=(2,2)$ boundary conditions
• Elliptic Genus Derivation of 4d Holomorphic Blocks
• Mirror Symmetry and Mixed Chern-Simons Levels for Abelian 3d $\mathcal{N}=2$ theories
• Vortex counting and the quantum Hall effect
• From 5d Flat Connections to 4d Fluxes (the Art of Slicing the Cone)
• Exact Results for SYM on $Y^{p,q}$ and $S^2\times S^2$ with Conical Singularities
• 3d $\mathcal{N}=2$ Brane Webs and Quiver Matrices
• $\mathcal{I}$-extremization with baryonic charges
• Virasoro Constraint for Uglov Matrix Model
• Intersecting Surface defects and 3d Superconformal indices
• Matching 3d N=2 Vortices and Monopole Operators
• Varieties of Abelian mirror symmetry on ${\mathbb{RP}}^2\times {\mathbb{S}}^1$
• The colored Jones polynomials as vortex partition functions
• Stokes Phenomena in 3d $\mathcal{N}=2$ SQED${}_2$ and ${\mathbb{CP}}^1$ Models
• Topological phase, spin Chern-Simons theory and level rank duality on lens space
• Supersymmetric localisation of $\mathcal{N}=(2,2)$ theories on a spindle
• N=2 Chern-Simons-Matter Theories Without Vortices
• Partition functions for supersymmetric gauge theories on spheres
• Schur Connections: Chord Counting, Line Operators, and Indices