Witten Index and Wall Crossing (1407.2567v3)

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.

Insights into Witten Index and Wall Crossing in One-Dimensional Supersymmetric Gauge Theories

This paper explores the computation of the Witten index for one-dimensional ${\mathcal N}=2$ supersymmetric gauge theories and explores the phenomenon of wall crossing in these systems. The authors employ the method of supersymmetric localization to evaluate the index, which is indicative of the presence of supersymmetric ground states. The focus is on gauged linear sigma models (GLSMs), restricted to theories with gauge groups that include at least one $U(1)$ factor and characterized by non-trivial Fayet-Iliopoulos (FI) parameter spaces.

Theoretical Framework and Methodology

The paper centers on computing the Witten index, a topological invariant, which remains constant under continuous deformations of parameters, barring discontinuities at phase boundaries known as wall crossings. The index counts the difference between the number of bosonic and fermionic ground states and is particularly significant in identifying supersymmetric vacua configurations in quantum mechanics.

The authors explore theories at ${\mathcal N}=2$ supersymmetry, specifically gauged linear sigma models in one dimension. For the localization technique, they consider the limit where the gauge coupling $e$ tends to zero and effectively analyze the path integral over the circle, a method quite similar to elliptic genus computations in two-dimensional theories. The moduli space of vector multiplet configurations is identified as $M = (T \times it)/W$, where $T$ is a maximal torus and $W$ the Weyl group, thus leveraging the residue formulae to distill the index contributions at various singularities of the parameter space.

Key Findings

1. Witten Index in Various Phases:

In distinct FI parameter regimes, the gauge symmetry is either fully or partially broken, leading to Higgs, mixed, or Coulomb branches. The authors derive results for the index across these branches, explaining how it changes as parameters are varied across walls separating different phases.

2. Wall Crossing Phenomenon:

As parameters cross walls, states supported by the Coulomb branches emerge or vanish. The analysis presents a formula for the wall crossing behavior, captured via a change in the Witten index, and verifies it through analytic and numerical techniques.

3. Detailed Case Studies:

Among the theory's applications are studies of dressed GLSMs with complex vacua. For instance, ${\mathcal N}=4$ quiver theories and systems with multiple FI parameters illustrate how precise computation of indices illuminates the intricate landscape of vacua and the robustness of certain BPS states against wall crossing.

Implications and Future Directions

The paper's outcomes have substantial implications for understanding $\mathcal{N}=2$ systems' algebraic and geometric structures. Index computation provides insight into the nature of supersymmetric ground states related to low-energy dynamics, pertinent in string and field theory contexts.

As theorizing continues, further exploration into non-Abelian gauge theories, which the paper tentatively addresses, could yield richer phenomena. There is also potential for broader applications in higher-dimensional systems, perhaps bridging insights into two-dimensional theories and extending techniques to more complex, realistic models used in high-energy physics.

Moreover, the results bear relevance to theoretical constructs beyond quantum mechanics, such as string compactifications and mirror symmetry, where Witten indices offer a bridge to topological and geometric aspects of moduli spaces.

Overall, this work provides essential computational tools and theoretical insights, contributing to understanding phase transitions and stable vacua in supersymmetric gauge theories.