[Paper Review] Braids, Walls, and Mirrors

Sergio Cecotti, Clay Córdova+1 more|arXiv (Cornell University)|Oct 10, 2011

Black Holes and Theoretical PhysicsPhysics and Astronomy68 references82 citations

TL;DR

This paper constructs 3d N=2 supersymmetric gauge theories by deforming 4d N=2 theories via an R-flow that preserves BPS state phases, mapping 4d chambers to distinct 3d theories. A key result is that wall-crossing in the 4d A2 Argyres-Douglas theory induces mirror symmetry between 3d Nf=1 SQED and the XYZ model, with M5-brane compactifications on 3-manifolds encoding the duality through branched double covers and braid structures on branch loci.

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.

Motivation & Objective

• To establish a systematic map from 4d N=2 supersymmetric theories to 3d N=2 theories via an R-flow that preserves BPS state phases.
• To understand how wall-crossing in 4d chambers leads to mirror symmetry in 3d dual theories, particularly in the context of the A2 Argyres-Douglas theory.
• To develop a geometric framework for 3d N=2 theories engineered by wrapping pairs of M5-branes on 3-manifolds, using tetrahedral decompositions and branched double covers.
• To connect the BPS spectrum and duality structures in 3d to topological invariants via braid group actions and SL(2,Z) transformations on Lagrangian submanifolds.
• To derive and interpret 3d partition functions for ADE-type Argyres-Douglas theories using cluster mutations and quantum dilogarithm identities.

Proposed method

• Utilizes a one-parameter R-flow in 4d N=2 theories to continuously deform central charges while preserving phase order, thereby inducing a 3d N=2 theory via Kaluza-Klein reduction.
• Maps each 4d chamber (defined by a BPS spectrum) to a distinct 3d dual theory, with wall-crossing in 4d corresponding to mirror symmetry in 3d.
• Constructs 3d theories via compactification of M5-branes on 3-manifolds, where the 4d chamber structure corresponds to a tetrahedral decomposition of the UV geometry.
• Describes the IR physics via a single recombined M5-brane that is a branched double cover of the original 3-manifold, with branch loci and sheets encoding duality and BPS data.
• Applies braid group actions and Tait graphs to encode the dual 3d theories, with crossings in the braid corresponding to particle states and superpotential terms.
• Uses quantum dilogarithm identities and cluster mutation formalism to compute 3d partition functions, particularly for ADE-type Argyres-Douglas theories, with explicit expressions in terms of sb and φ− functions.

Experimental results

Research questions

• RQ1 How does an R-flow in 4d N=2 theories generate distinct 3d N=2 theories, and what is the role of BPS state preservation in this construction?
• RQ2 What is the precise geometric and topological mechanism by which wall-crossing in 4d leads to mirror symmetry in 3d dual theories?
• RQ3 How do M5-brane compactifications on 3-manifolds realize 3d N=2 theories, and what is the role of branched double covers in encoding the IR physics?
• RQ4 What is the significance of braid structures and Tait graphs in classifying 3d dual theories and their superpotential terms?
• RQ5 How can 3d partition functions for ADE-type Argyres-Douglas theories be computed and interpreted via cluster mutations and quantum dilogarithm identities?

Key findings

• The 2-3 wall-crossing in the 4d A2 Argyres-Douglas theory induces mirror symmetry between 3d Nf=1 SQED and the XYZ model, providing a concrete realization of 3d mirror symmetry.
• Each 4d chamber corresponds to a tetrahedral decomposition of the UV 3-manifold, and the R-flow maps these to distinct 3d dual descriptions via Pachner moves.
• The IR physics is captured by a single recombined M5-brane that is a branched double cover of the original 3-manifold, with the braiding of branch loci encoding duality and BPS data.
• The partition function for the intermediate chamber of the A4 quiver is computed via a sequence of five mutations, yielding a U(1) gauge theory with five chiral fields and a cubic superpotential term X2X3Y2.
• The partition function is expressed in terms of quantum dilogarithms and sb functions, with the final result involving a non-planar braid and a finite triangle in the Tait graph corresponding to the superpotential.
• The theory exhibits a non-trivial duality structure, with the partition function invariant under mutation sequences, and the final expression reveals a dynamical U(1) gauge group with R-charge assignments and a single invariant monomial X2X3Y2 of R-charge 2, confirming the superpotential term.