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Quantum Wall Crossing in N=2 Gauge Theories

Tudor Dimofte, Sergei Gukov, Yan Soibelman

TL;DR

The study addresses refined and motivic wall-crossing in four-dimensional $\mathcal{N}=2$ $SU(2)$ gauge theories with $N_f<4$, testing the conjecture that refined invariants coincide with motivic DT invariants. It develops a comprehensive framework connecting BPS spectra to Calabi–Yau geometry, stable bundles on $\mathbb{F}_0$, quiver representations, and motivic DT theory, via quantum dilogarithm identities and wall-crossing factorization. The work provides explicit refined identities for $N_f=0,1,2,3$ and demonstrates how weak-coupling spectra correspond to stable bundles while strong-coupling data emerges from quiver representations, all within the geometric-engineering perspective. Overall, the results bolster the refined = motivic conjecture, illuminate deep links between BPS spectra in gauge theory and DT theory in mathematics, and offer concrete computational tools through quantum dilogarithm-based wall-crossing formulas and their motivic counterparts.

Abstract

We study refined and motivic wall-crossing formulas in N=2 supersymmetric gauge theories with SU(2) gauge group and N_f < 4 matter hypermultiplets in the fundamental representation. Such gauge theories provide an excellent testing ground for the conjecture that "refined = motivic."

Quantum Wall Crossing in N=2 Gauge Theories

TL;DR

The study addresses refined and motivic wall-crossing in four-dimensional gauge theories with , testing the conjecture that refined invariants coincide with motivic DT invariants. It develops a comprehensive framework connecting BPS spectra to Calabi–Yau geometry, stable bundles on , quiver representations, and motivic DT theory, via quantum dilogarithm identities and wall-crossing factorization. The work provides explicit refined identities for and demonstrates how weak-coupling spectra correspond to stable bundles while strong-coupling data emerges from quiver representations, all within the geometric-engineering perspective. Overall, the results bolster the refined = motivic conjecture, illuminate deep links between BPS spectra in gauge theory and DT theory in mathematics, and offer concrete computational tools through quantum dilogarithm-based wall-crossing formulas and their motivic counterparts.

Abstract

We study refined and motivic wall-crossing formulas in N=2 supersymmetric gauge theories with SU(2) gauge group and N_f < 4 matter hypermultiplets in the fundamental representation. Such gauge theories provide an excellent testing ground for the conjecture that "refined = motivic."

Paper Structure

This paper contains 11 sections, 1 theorem, 72 equations, 4 figures.

Table of Contents

  1. Introduction
  2. ${\cal H}_{BPS}$ in $SU(2)$ gauge theory with $N_f$ flavors
  3. Coulomb branches and BPS states
  4. Stable bundles on ${\mathbb{F}}_0$
  5. Realization via BPS states on a Calabi-Yau 3-fold
  6. Refined wall crossing
  7. Refined and quantum
  8. Classical limit and factorization
  9. Motivic wall crossing
  10. Wall crossing and quivers
  11. Factorization and cohomological Hall algebra

Key Result

Proposition 1

For any $d\ge 0$ there exist integers $\delta(n,m)=\delta^{(d)}(n,m)\ge 0$ for all $n\ge 1$ and $m\in (d-1)n+2{\mathbb{Z}}=(1-d)n^2+2{\mathbb{Z}}$, such that for a given number $n$ we have $\delta(n,m)\ne 0$ only for finitely many values of $m$, and

Figures (4)

  • Figure 1: Visualization of the wall crossing encoded in the pentagon identity, in terms of BPS rays in a central charge plane. The presence of the bound state $\gamma_1+\gamma_2$ depends on the relative arguments of the central charges ${\cal Z}(\gamma_1)$ and ${\cal Z}(\gamma_2)$.
  • Figure 2: The approximate structure of the Coulomb branch ($u$-plane) of $SU(2)$ Seiberg-Witten theories (cf.BF). A wall of marginal stability separates strong and weak coupling. The blue dots correspond to singularities where BPS states become massless.
  • Figure 3: The quiver $K_2$ for $SU(2)$ super-Yang-Mills with $N_f=0$.
  • Figure 4: The acyclic quiver for $N_f=1$$SU(2)$ theory.

Theorems & Definitions (3)

  • Remark 1
  • Remark 2
  • Proposition 1