Wall-crossing, Hitchin Systems, and the WKB Approximation

Abstract: We consider BPS states in a large class of d=4, N=2 field theories, obtained by reducing six-dimensional (2,0) superconformal field theories on Riemann surfaces, with defect operators inserted at points of the Riemann surface. Further dimensional reduction on S^1 yields sigma models, whose target spaces are moduli spaces of Higgs bundles on Riemann surfaces with ramification. In the case where the Higgs bundles have rank 2, we construct canonical Darboux coordinate systems on their moduli spaces. These coordinate systems are related to one another by Poisson transformations associated to BPS states, and have well-controlled asymptotic behavior, obtained from the WKB approximation. The existence of these coordinates implies the Kontsevich-Soibelman wall-crossing formula for the BPS spectrum. This construction provides a concrete realization of a general physical explanation of the wall-crossing formula which was proposed in 0807.4723. It also yields a new method for computing the spectrum using the combinatorics of triangulations of the Riemann surface.

• The paper introduces an algorithmic method to compute BPS spectra, bridging quantum field theory and algebraic geometry via the WKB approximation.
• It constructs canonical Darboux coordinate systems on Hitchin moduli spaces, validating the Kontsevich-Soibelman wall-crossing formula using triangulations.
• The approach unifies gauge theory and integrable systems perspectives, offering insights for managing irregular singularities in supersymmetric models.

Overview of Wall-Crossing, Hitchin Systems, and the WKB Approximation

This paper by Gaiotto, Moore, and Neitzke essentially explores the complexities of BPS states within a wide class of four-dimensional (4D) $\mathcal{N} = 2$ supersymmetric field theories. The crux of their work lies in connecting the study of certain quantum field theories to well-understood mathematical structures, specifically Hitchin systems and their associated canonical coordinates. By employing the WKB approximation, the authors demonstrate a concrete realization of the Kontsevich-Soibelman wall-crossing formula for the BPS spectra.

Central Paradigm

The central theme is the analysis of BPS states, achieved through a reduction framework of six-dimensional (6D) $(2,0)$ superconformal field theories compactified on Riemann surfaces. The Hitchin systems then arise from a further reduction on a circle, $S^1$, yielding sigma models characterized by moduli spaces of Higgs bundles.

The BPS states, crucial to understanding the dynamical aspects of these theories, are particularly intriguing due to their sensitivity to changes in physical parameters, a phenomenon termed wall-crossing. The authors provide a robust combinatorial method to compute the BPS spectrum directly through triangulations of the Riemann surfaces, effectively using this approach to illustrate that the physical insight into wall-crossing is consistent with its mathematical counterpart as formulated by Kontsevich and Soibelman.

Hitchin Systems and Their Coordinate Systems

The study explores the complex geometry of moduli spaces associated with Hitchin systems. In particular, when the Higgs bundles have rank two, the authors construct canonical Darboux coordinate systems on their moduli spaces, based on the WKB approximation. These coordinates display controlled asymptotic behavior vital for the consistency of wall-crossing phenomena.

The significance of these developments lies in the coordinate systems reflecting the intricate structure of the moduli spaces and embodying the stability chambers of BPS states, thus directly linking the moduli space geometry to physical quantities like central charges and BPS spectra.

Novel Insights and Results

1. Algorithmic Computation of BPS States: The authors introduce an algorithm to compute the BPS spectrum for a class of infinite $\mathcal{N} = 2$ theories. This is a significant step, expanding the repertoire of techniques available for tackling notoriously difficult computations in supersymmetric theories.
2. Constructive Realization of KS Formula: Through their triangulation-based combinatorial approach, the authors provide a constructive realization of the Kontsevich-Soibelman wall-crossing formula. This manifests through transformations described by their canonical coordinates, revealing fundamental aspects of hyperkähler geometry inherent in these systems.
3. Dual Perspective on Energy Scales: The constructions provide dual insights by allowing interpretations both as gauge theories and integrable systems, hence seamlessly bridging physics and geometry.
4. Inclusion of Irregular Singularities: The extension to handle irregular singularities in Higgs bundles is noteworthy, addressing more general configurations that naturally arise in these gauge theories.

Implications and Future Directions

The outcomes of this work have profound implications in both theoretical physics and mathematics. Practically, they furnish a new algorithmic framework expandably applicable across a multitude of theories, potentially reshaping how BPS spectra are explored. Theoretically, they reinforce the ties between high-energy physics and algebraic geometry, enhancing understanding of dualities.

Future progress could focus on generalizing these methods to higher-rank gauge groups, exploring the extension of wall-crossing phenomena to systems tied with diverse mathematical structures, and examining deeper intersections with the geometric Langlands program.

The paper underscores the potential for new symbiotic developments between physics and mathematics, inspiring exploration across disciplines and suggesting rich terrain for continued inquiry into the nature of supersymmetric theories.