Gauge Theories Labelled by Three-Manifolds

Abstract: We propose a dictionary between geometry of triangulated 3-manifolds and physics of three-dimensional N=2 gauge theories. Under this duality, standard operations on triangulated 3-manifolds and various invariants thereof (classical as well as quantum) find a natural interpretation in field theory. For example, independence of the SL(2) Chern-Simons partition function on the choice of triangulation translates to a statement that S^3_b partition functions of two mirror 3d N=2 gauge theories are equal. Three-dimensional N=2 field theories associated to 3-manifolds can be thought of as theories that describe boundary conditions and duality walls in four-dimensional N=2 SCFTs, thus making the whole construction functorial with respect to cobordisms and gluing.

• The paper introduces a framework linking the triangulated structure of three-manifolds with corresponding 3d N=2 supersymmetric gauge theories.
• It demonstrates that different manifold triangulations yield invariant physical outcomes, as seen through consistent S³₍b₎ partition functions.
• The study extends to manifolds with boundaries and cusps, connecting geometric invariants to topological quantum field theory and dualities.

Gauge Theories Labelled by Three-Manifolds: An Overview

The paper by Dimofte, Gaiotto, and Gukov presents an innovative framework that bridges the geometry of three-dimensional manifolds—especially their triangulated structure—and the physics of three-dimensional $\mathcal{N} = 2$ supersymmetric gauge theories. This work leverages the intriguing connections between fields of high-energy theoretical physics and low-dimensional topology, particularly as an extension of concepts emerging from the study of six-dimensional superconformal field theories (SCFTs) and their compactification.

Core Concept

At the heart of this work is the duality between the geometric operations on three-manifolds and the related transformations in the associated three-dimensional supersymmetric gauge theories. The authors propose that the structure of triangulated three-manifolds, including both classical and quantum invariants, finds natural counterparts within field theory constructs. Specifically, they suggest that the operation independence witnessed in SL(2) Chern-Simons partition functions, regarding manifold triangulation, corresponds to an equality seen in $S^3_b$ partition functions of mirror 3d $\mathcal{N}=2$ gauge theories.

The Triangulation and Field Theory Correspondence

The paper delineates a formal approach to associating a specific $\mathcal{N}=2$ field theory, $T[M, \mathfrak{g}]$, to a given three-manifold $M$. These manifold-associated theories are posited to represent boundary conditions and duality walls in four-dimensional $\mathcal{N}=2$ SCFTs, establishing a functorial relationship traversing cobordisms and manifold gluing.

1. Triangulation and Gauge Theory Construction: The authors articulate a systematic methodology to deconstruct a three-manifold into constituent tetrahedra. Each tetrahedron is then associated with a fundamental 3d gauge theory building block. This decomposition aligns with known S-dualities in the two-dimensional scenario, offering insights into the potential connections between different manifold decompositions through known mirror symmetries.
2. Independence from Triangulation: Echoing the prevalent theme within topology, different triangulations of the same three-manifold can lead to descriptions of SCFTs yielding the same physical outcomes, thus supportive of the triangulation independence.
3. Boundary and Cusp Treatment: The authors extend the scope to include manifolds exhibiting either geodesic or cusp-style boundaries. This significantly enhances the applicability of their framework in modelling various boundary phenomenon within the associated field theories.

Mathematical Rigor and Physical Implications

Employing a rigorous mathematical construction, the paper bridges the triangulated geometry directly to Chern-Simons theories, faithfully capturing the partition function characteristics and their physical analogues in $\mathcal{N}=2$ theories. Important connections are drawn between:

• The moduli spaces of flat connections on manifolds $M$ and the corresponding moduli spaces of vacua in the effective lower-dimensional theories.
• The $S^3_b$ partition functions which are shown to correspond directly with geometric invariants from Chern-Simons theory.

Future Directions and Theoretical Implications

The implications of this work are significant in both theoretical physics and mathematical topology:

1. Topological Field Theories: By providing a robust correspondence between topology and physics, this framework enriches the understanding of topological field theories (TFTs), offering pathways to extend these correspondences into quantum field theories and string theory contexts.
2. Applications in Quantum Gravity: This construction resonates with approaches in quantum gravity, particularly in studying compactifications and dualities in string theory and M-theory.
3. Enhancement of Computational Techniques: Further refinement in computational tools to examine large-dimensional SCFTs, potentially linking algebraic geometry and quantum systems exhibiting supersymmetry.

In summary, the paper by Dimofte, Gaiotto, and Gukov elucidates a profound and intricate connection between the topology of 3-manifolds and $\mathcal{N}=2$ gauge theories, extending the frontier of knowledge at the interface of geometry, topology, and high energy theoretical physics. This work not only enhances the comprehension of existing dualities but opens new avenues for exploration in both gauge theory dynamics and topological quantum field theory.