Braids, Walls, and Mirrors

TL;DR

The paper develops a unifying framework linking 4d N=2 BPS data to 3d N=2 theories via an R-flow that preserves BPS phase order. By engineering the 3d theories through M5-branes wrapped on three-manifolds, the authors map each 4d BPS chamber to a dual 3d description whose spectrum is captured by tetrahedral decompositions and branched-cover geometries; wall-crossing in 4d becomes 3d mirror symmetry, with KS monodromy ensuring partition-function equivalence. The work provides explicit realizations for ADE Argyres-Douglas theories, including A2 yielding N_f=1 SQED and the XYZ model, and extends to general A_n and E_n cases, connecting to SL(N) Chern-Simons theory and non-perturbative dilogarithm structures. It offers a comprehensive toolkit—Seifert matrices, Tait graphs, braids, and Pachner moves—for translating 4d BPS data into 3d Lagrangians and their exact partition functions, with concrete predictions across chambers. The results illuminate deep ties between 4d BPS wall-crossing, 3d dualities, and geometric/topological invariants of brane configurations, potentially guiding future non-perturbative checks and holographic interpretations.

Abstract

We construct 3d, N=2 supersymmetric gauge theories by considering a one-parameter `R-flow' of 4d, N=2 theories, where the central charges vary while preserving their phase order. Each BPS state in 4d leads to a BPS particle in 3d, and thus each chamber of the 4d theory leads to a distinct 3d theory. Pairs of 4d chambers related by wall-crossing, R-flow to mirror pairs of 3d theories. In particular, the 2-3 wall-crossing for the A_2 Argyres-Douglas theory leads to 3d mirror symmetry for N_f=1 SQED and the XYZ model. Although our formalism applies to arbitrary N=2 models, we focus on the case where the parent 4d theory consists of pairs of M5-branes wrapping a Riemann surface, and develop a general framework for describing 3d N=2 theories engineered by wrapping pairs of M5-branes on three-manifolds. Each 4d chamber, which corresponds to a dual 3d description, maps to a particular tetrahedral decomposition of the UV 3d geometry. In the IR the physics is captured by a single recombined M5-brane which is a branched double cover of the original UV three-manifold. The braiding of branch loci and the geometry of branch sheets play a key role in encoding the physics.

Figures (58)

• Figure 1: A BPS M2 brane. A three-manifold $M$, shown in green, sits inside the ambient Calabi-Yau $Q$, shown in white. The three-manifold supports a non-trivial one-cycle $\Gamma$ shown in blue. A minimal M2 disc, shown in red, can end on this cycle and describes a BPS particle in $\mathbb{R}^{1,2}$.
• Figure 2: M2 brane contributions to the superpotential. In (a), we have four massive BPS states described by the pink M2 discs ending on blue one-cycles in the three-manifold. A three-dimensional closed M2 has boundary on these discs and along a two-dimensional locus in $M$ and mediates a quartic interaction between the BPS particles. In (b), the BPS states become massless and the membrane geometry degenerates to a solid ball with four marked points, whose boundary lies entirely in $M$.
• Figure 3: Knots and associated Siefert surfaces. In (a) and (c) we see non-trivial knots. In (b) and (d), we see Seifert surfaces whose boundaries are the given knots. The knots are an example of possible branch loci for a double cover of $S^{3}$. The Seifert surfaces are then the branch sheets defining the cover geometry.
• Figure 4: A Seifert surface $F$ for the unknot. The green torus minus a disc bounds the unknot shown in blue. The red and black cycles, $a$ and $b$, are a basis for the homology of $F$. The pushoff $a^{+}$ is linked with $b$.
• Figure 5: Local definition of checkerboard coloring. In (a) a planar projection of a crossing in a knot diagram. In (b) a checkerboard coloring at the crossing.