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

References in corpus (9)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Witten Index and Wall Crossing
• Phases Of N=2 Theories In 1+1 Dimensions With Boundary
• Wall-Crossing from Boltzmann Black Hole Halos
• Arithmetic of Quantum Entropy Function
• A fixed point formula for the index of multi-centered N=2 black holes
• An Index Formula for Supersymmetric Quantum Mechanics
• Framed BPS States, Moduli Dynamics, and Wall-Crossing
• General instanton counting and 5d SCFT

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• General instanton counting and 5d SCFT
• On the Jeffrey-Kirwan residue of BCD-instantons
• Chiral operators in two-dimensional (0,2) theories and a test of triality
• M5-branes, orientifolds, and S-duality
• Counting Trees in Supersymmetric Quantum Mechanics
• Discrete Integrable Systems, Supersymmetric Quantum Mechanics, and Framed BPS States - I
• Asymptotics of Ground State Degeneracies in Quiver Quantum Mechanics
• Recent developments in 2d $\mathcal{N}=(2,2)$ supersymmetric gauge theories
• Surface defects and instanton partition functions
• Mutation, Witten Index, and Quiver Invariant