RG Domain Walls and Hybrid Triangulations

TL;DR

<3-5 sentence high-level summary> This work develops a unifying framework connecting 3d N=2 theories of class R to 4d class S theories through framed 3-manifolds built from tetrahedral triangulations. It shows how 3d domain walls encode both UV S-duality and IR RG flows as cobordism walls, with Janus configurations mapping to explicit 3d Lagrangians and superpotential couplings. The authors derive a rich dictionary between boundary 3d theories and bulk 4d dynamics, including non-abelian flavor enhancements, and validate the constructions with concrete examples such as N=2* interfaces and SU(2) with N_f=4, while linking to quantum Teichmüller theory via edge-coordinate quantization and 6j-symbols. These results illuminate the geometric structure underlying dualities and RG flows in N=2 theories and provide concrete tools for constructing and analyzing domain walls in broadly supersymmetric settings.

Abstract

This paper studies the interplay between the N=2 gauge theories in three and four dimensions that have a geometric description in terms of twisted compactification of the six-dimensional (2,0) SCFT. Our main goal is to construct the three-dimensional domain walls associated to any three-dimensional cobordism. We find that we can build a variety of 3d theories that represent the local degrees of freedom at a given domain wall in various 4d duality frames, including both UV S-dual frames and IR Seiberg-Witten electric-magnetic dual frames. We pay special attention to Janus domain walls, defined by four-dimensional Lagrangians with position-dependent couplings. If the couplings on either side of the wall are weak in different UV duality frames, Janus domain walls reduce to S-duality walls, i.e. domain walls that encode the properties of UV dualities. If the couplings on one side are weak in the IR and on the other weak in the UV, Janus domain walls reduce to RG walls, i.e. domain walls that encode the properties of RG flows. We derive the 3d geometries associated to both types of domain wall, and test their properties in simple examples, both through basic field-theoretic considerations and via comparison with quantum Teichmuller theory. Our main mathematical tool is a parametrization and quantization of framed flat SL(K) connections on these geometries based on ideal triangulations.

Figures (65)

• Figure 1: Truncated tetrahedron.
• Figure 2: Left: a 3-manifold that admits a decomposition into truncated tetrahedra. It has two big boundaries (a 4-punctured sphere and a 3-punctured sphere) with fixed 2d triangulation $\mathbf t$; as well as a small disc (filling in one puncture), three small annuli, and a small torus. Topologically, this manifold has just two connected boundary components, the torus and a surface of genus two. Right: schematic of how truncated tetrahedra are assembled around a small torus boundary.
• Figure 3: Adjusting the metric on the manifold $M$ from Figure \ref{['fig:admM']} and replacing small boundaries with line defects so that the compactification of the 6d theory on $M\times {\mathbb R}^3$ produces Seiberg-Witten theories on a half-space coupled to $T_K[M]$ as a boundary condition.
• Figure 4: Forming a genus-two boundary from two big three-punctured spheres connected by small annuli. This equivalent to a pants decomposition.
• Figure 5: Cutting the big boundary of a tetrahedron in half along a four-sided polygon (in blue).

Theorems & Definitions (2)

• Theorem 1
• Theorem 2