Witten Index and Wall Crossing

Abstract

We compute the Witten index of one-dimensional gauged linear sigma models with at least $\mathcal{N}=2$ supersymmetry. In the phase where the gauge group is broken to a finite group, the index is expressed as a certain residue integral. It is subject to a change as the Fayet-Iliopoulos parameter is varied through the phase boundaries. The wall crossing formula is expressed as an integral at infinity of the Coulomb branch. The result is applied to many examples, including quiver quantum mechanics that is relevant for BPS states in $d=4$ $\mathcal{N}=2$ theories.

123 pages, v3: the discussion on the smooth transition (Section 4.4) is improved, more minor corrections made; v2: references added, minor corrections made

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• Gauge Fields and D-branes
• Elliptic genera of two-dimensional N=2 gauge theories with rank-one gauge groups
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• Quantum Quivers and Hall/Hole Halos
• D-branes, Categories and N=1 Supersymmetry
• Aspects of Non-Abelian Gauge Dynamics in Two-Dimensional N=(2,2) Theories
• The Strong-Coupling Spectrum of the Seiberg-Witten Theory
• Witten Index and Wall Crossing
• (0,2) Landau-Ginzburg Theory
• Aspects of 3d N=2 Chern-Simons-Matter Theories
• Phases Of N=2 Theories In 1+1 Dimensions With Boundary
• Wall-Crossing from Boltzmann Black Hole Halos
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• D-Brane Bound States Redux
• Witten Index and Threshold Bound States of D-Branes
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• Arithmetic of Quantum Entropy Function
• Dyons in N=4 Supersymmetric Theories and Three-Pronged Strings
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• Low Energy Dynamics for 1/4 BPS Dyons
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• Ab Initio Wall-Crossing
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• Lectures on AdS Black Holes, Holography and Localization
• Holographic microstate counting for AdS${}_4$ black holes in massive IIA supergravity
• Topologically twisted indices in five dimensions and holography
• Line defects and 5d instanton partition functions
• Three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories and partition functions on Seifert manifolds: A review
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• Mirror symmetry and line operators
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• Twisted Hilbert Spaces of 3d Supersymmetric Gauge Theories
• Fundamental Vortices, Wall-Crossing, and Particle-Vortex Duality
• Twisted Indices of 3d $\mathcal{N}=4$ Gauge Theories and Enumerative Geometry of Quasi-Maps
• The Coulomb Branch Formula for Quiver Moduli Spaces
• On 't Hooft Defects, Monopole Bubbling and Supersymmetric Quantum Mechanics
• 6d strings and exceptional instantons
• Integrability from 2d N=(2,2) Dualities
• 't Hooft anomalies and the holomorphy of supersymmetric partition functions
• Quantum Spin Systems and Supersymmetric Gauge Theories, I
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• Wilson surfaces in M5-branes
• Localization of supersymmetric field theories on non-compact hyperbolic three-manifolds
• Line Defects in Three Dimensional Mirror Symmetry beyond Linear Quivers
• 5D BPS Quivers and KK Towers
• Wall-crossing made smooth
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• Supersymmetric indices on $I\times T^2$, elliptic genera and dualities with boundaries
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• Witten Index for Noncompact Dynamics
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• The Origin of Calabi-Yau Crystals in BPS States Counting
• ABCD of 't Hooft operators
• M5-branes, orientifolds, and S-duality
• On the BPS Spectrum of the Rank-1 Minahan-Nemeschansky Theories
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• Counting Trees in Supersymmetric Quantum Mechanics
• Surface operators, dual quivers and contours
• Double Quiver Gauge Theory and BPS/CFT Correspondence
• Discrete Integrable Systems, Supersymmetric Quantum Mechanics, and Framed BPS States - I
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• Aspects of 5d Seiberg-Witten Theories on ${\mathbb{S}}^1$
• Recent developments in 2d $\mathcal{N}=(2,2)$ supersymmetric gauge theories
• Evidence for an Algebra of ${\mathbf{G}}_2$ Instantons
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• Wall-crossing and operator ordering for 't Hooft operators in N=2 gauge theories
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• Quantum Geometry and $\theta$-Angle in Five-Dimensional Super Yang-Mills
• Mock modularity from black hole scattering states
• Supersymmetric Wilson Loops, Instantons, and Deformed $\mathcal{W}$-Algebras
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• D-Particles on Orientifolds and Rational Invariants
• Singular BPS boundary conditions in $\mathcal{N}=(2,2)$ supersymmetric gauge theories
• Quantized Coulomb Branches, Monopole Bubbling and Wall-Crossing Phenomena in 3d $\mathcal{N}=4$ Theories
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• Pure-Higgs states from the Lefschetz-Sommese theorem
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• Hamiltonian Truncation Study of Supersymmetric Quantum Mechanics: S-Matrix and Metastable States
• An Overview of Crystals and Double Quiver Yangians
• Spectral Sequences and Vacua in N = 2 Gauged Linear Quantum Mechanics with Potentials
• Gauge origami and quiver W-algebras IV: Pandharipande--Thomas $qq$-characters
• Schubert defects in Lagrangian Grassmannians
• Wall-crossing for vortex partition function and handsaw quiver varierty
• Non-Perturbative Schwinger-Dyson Equations for 3d $\mathcal{N}=4$ Gauge Theories
• Supersymmetric Yang--Mills Matrix Integrals Revisited
• Schubert line defects in 3d GLSMs, part II: Partial flag manifolds and parabolic quantum polynomials
• Mutation, Witten Index, and Quiver Invariant
• The 4-fold Pandharipande--Thomas vertex and Jeffrey--Kirwan residue
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• OPE of line defects in 5d $E_n$ SCFT
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