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K-Decompositions and 3d Gauge Theories

Tudor Dimofte, Maxime Gabella, Alexander B. Goncharov

TL;DR

K-Decompositions and 3d Gauge Theories develops a comprehensive framework uniting the geometry of framed flat $PGL(K,\mathbb{C})$-connections on admissible 3-manifolds with 3d $\mathcal{N}=2$ gauge theories $T_K[M]$. The authors introduce hypersimplicial $K$-decompositions to parametrize bulk-boundary data via octahedra, construct $K_2$-Lagrangian subvarieties $\mathcal{L}_K(M)\subset \mathcal{X}^{\mathrm{un}}_K(\partial M)$, and formulate a Symplectic Gluing framework consistent with 2--3 moves. They connect these geometric constructions to a physical dictionary of cluster-like UV Lagrangians for $T_K[M]$, exhibit explicit higher-$K$ tetrahedron theories with growing combinatorial complexity, and explore knot-complement theories where $K^3$ scaling and flavor/marginal structures emerge. The paper also develops a Bloch/$K_2$-theoretic viewpoint, giving isotropy/Lagrangian statements and variation formulas for framed flat connections and motivating a large-$K$ holographic interpretation via Chern-Simons volumes. Overall, it provides a detailed, constructive path from 3-manifold triangulations to quantum field theories, their partition functions, and their large-$K$ behavior, underpinning the 3d-3d correspondence in the $K>2$ regime.

Abstract

This paper combines several new constructions in mathematics and physics. Mathematically, we study framed flat PGL(K,C)-connections on a large class of 3-manifolds M with boundary. We define a space L_K(M) of framed flat connections on the boundary of M that extend to M. Our goal is to understand an open part of L_K(M) as a Lagrangian in the symplectic space of framed flat connections on the boundary, and as a K_2-Lagrangian, meaning that the K_2-avatar of the symplectic form restricts to zero. We construct an open part of L_K(M) from data assigned to a hypersimplicial K-decomposition of an ideal triangulation of M, generalizing Thurston's gluing equations in 3d hyperbolic geometry, and combining them with the cluster coordinates for framed flat PGL(K)-connections on surfaces. Using a canonical map from the complex of configurations of decorated flags to the Bloch complex, we prove that any generic component of L_K(M) is K_2-isotropic if the boundary satisfies some topological constraints (Theorem 4.2). In some cases this implies that L_K(M) is K_2-Lagrangian. For general M, we extend a classic result of Neumann-Zagier on symplectic properties of PGL(2) gluing equations to reduce the K_2-Lagrangian property to a combinatorial claim. Physically, we use the symplectic properties of K-decompositions to construct 3d N=2 superconformal field theories T_K[M] corresponding (conjecturally) to the compactification of K M5-branes on M. This extends known constructions for K=2. Just as for K=2, the theories T_K[M] are described as IR fixed points of abelian Chern-Simons-matter theories. Changes of triangulation (2-3 moves) lead to abelian mirror symmetries that are all generated by the elementary duality between N_f=1 SQED and the XYZ model. In the large K limit, we find evidence that the degrees of freedom of T_K[M] grow cubically in K.

K-Decompositions and 3d Gauge Theories

TL;DR

K-Decompositions and 3d Gauge Theories develops a comprehensive framework uniting the geometry of framed flat -connections on admissible 3-manifolds with 3d gauge theories . The authors introduce hypersimplicial -decompositions to parametrize bulk-boundary data via octahedra, construct -Lagrangian subvarieties , and formulate a Symplectic Gluing framework consistent with 2--3 moves. They connect these geometric constructions to a physical dictionary of cluster-like UV Lagrangians for , exhibit explicit higher- tetrahedron theories with growing combinatorial complexity, and explore knot-complement theories where scaling and flavor/marginal structures emerge. The paper also develops a Bloch/-theoretic viewpoint, giving isotropy/Lagrangian statements and variation formulas for framed flat connections and motivating a large- holographic interpretation via Chern-Simons volumes. Overall, it provides a detailed, constructive path from 3-manifold triangulations to quantum field theories, their partition functions, and their large- behavior, underpinning the 3d-3d correspondence in the regime.

Abstract

This paper combines several new constructions in mathematics and physics. Mathematically, we study framed flat PGL(K,C)-connections on a large class of 3-manifolds M with boundary. We define a space L_K(M) of framed flat connections on the boundary of M that extend to M. Our goal is to understand an open part of L_K(M) as a Lagrangian in the symplectic space of framed flat connections on the boundary, and as a K_2-Lagrangian, meaning that the K_2-avatar of the symplectic form restricts to zero. We construct an open part of L_K(M) from data assigned to a hypersimplicial K-decomposition of an ideal triangulation of M, generalizing Thurston's gluing equations in 3d hyperbolic geometry, and combining them with the cluster coordinates for framed flat PGL(K)-connections on surfaces. Using a canonical map from the complex of configurations of decorated flags to the Bloch complex, we prove that any generic component of L_K(M) is K_2-isotropic if the boundary satisfies some topological constraints (Theorem 4.2). In some cases this implies that L_K(M) is K_2-Lagrangian. For general M, we extend a classic result of Neumann-Zagier on symplectic properties of PGL(2) gluing equations to reduce the K_2-Lagrangian property to a combinatorial claim. Physically, we use the symplectic properties of K-decompositions to construct 3d N=2 superconformal field theories T_K[M] corresponding (conjecturally) to the compactification of K M5-branes on M. This extends known constructions for K=2. Just as for K=2, the theories T_K[M] are described as IR fixed points of abelian Chern-Simons-matter theories. Changes of triangulation (2-3 moves) lead to abelian mirror symmetries that are all generated by the elementary duality between N_f=1 SQED and the XYZ model. In the large K limit, we find evidence that the degrees of freedom of T_K[M] grow cubically in K.

Paper Structure

This paper contains 96 sections, 5 theorems, 294 equations, 75 figures, 1 table.

Table of Contents

  1. Introduction
  2. The 3d-3d correspondence
  3. A mathematical perspective
  4. Organization
  5. Basic tools and definitions
  6. Gluing admissible 3-manifolds from truncated tetrahedra
  7. Configurations of flags
  8. The basic moduli spaces
  9. 1. The phase space for a surface with a simple lamination.
  10. 2. The phase space and the coordinate phase space for a surface.
  11. 3. The phase space for a torus $T^2$.
  12. 4. The moduli space ${\cal L}_K(M, {\bf t})$.
  13. Examples:
  14. Conclusion:
  15. From framed flat bundles to configurations of flags
  16. ...and 81 more sections

Key Result

Theorem 3.1

The octahedron parameters satisfying the equations Poct-exp--Loct-exp and the monomial gluing relations parametrize an open subset $\widetilde{\mathcal{L}}_K(M,\mathbf t_{3d})$ of the space of framed flat connections $\widetilde{\mathcal{L}}_K(M)$.

Figures (75)

  • Figure 1: The 2--3 move: decomposing a bipyramid into two or three tetrahedra.
  • Figure 2: Stacking $\frac{1}{6}K(K^2-1)$ octahedra to form an $A_{K-1}$ tetrahedron (for $K=4$).
  • Figure 3: ($a$) truncated tetrahedra, glued together to form 3-manifolds that have either ($b$) big boundaries tiled by hexagonal faces with holes filled in by small vertex triangles; or ($c$) small torus boundaries tiled exclusively by vertex triangles.
  • Figure 4: A topological surface with lamination and marked points is equivalent to a big boundary with small annuli and small discs connecting its holes. Here, two loops of a lamination are replaced by two annuli (depicted as long tubes); and two marked points are replaced by discs. The remaining big boundary consists of two 3-punctured spheres.
  • Figure 5: Parallel transport of flags to the common point $p$.
  • ...and 70 more figures

Theorems & Definitions (16)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 3.1
  • Conjecture 4.1: Symplectic Gluing
  • Corollary 4.1
  • Definition 4.1
  • ...and 6 more