[Paper Review] 3d-3d Correspondence and 2d $\mathcal{N}=(0,2)$ Boundary Conditions

Hee‐Joong Chung|arXiv (Cornell University)|Jul 19, 2023

Matrix Theory and AlgorithmsComputer Science3 citations

TL;DR

This paper establishes a correspondence between quiver forms of 3-manifold invariants—such as colored Jones polynomials and homological blocks—and 3d $χ=(0,2)$ boundary conditions in 3d $χ=2$ supersymmetric theories on $D^2\times_q S^1$. By interpreting these invariants as half-indices of 3d $χ=2$ theories with Dirichlet and deformed Dirichlet boundary conditions, the authors provide explicit Lagrangian descriptions of $T[M_3]$ theories for various 3-manifolds, including knot complements and surgeries, and demonstrate dualities via quiver equivalence.

ABSTRACT

We consider quiver forms that appear in the motivic Donaldson-Thomas generating series or characters of conformal field theories and relate them to 3d $\mathcal{N}=2$ theories on $D^2 imes_q S^1$ with certain boundary conditions preserving 2d $\mathcal{N}=(0,2)$ supersymmetry. We apply this to the 3d-3d correspondence and provide a Lagrangian description of 3d $\mathcal{N}=2$ theories $T[M_3]$ with 2d $\mathcal{N}=(0,2)$ boundary conditions for 3-manifolds $M_3$ in several contexts.

Motivation & Objective

• To connect quiver forms of 3-manifold invariants to 3d $χ=2$ theories with 2d $χ=(0,2)$ supersymmetry on $D^2\times_q S^1$.
• To provide a Lagrangian description of $T[M_3]$ theories for 3-manifolds $M_3$ using boundary conditions preserving $χ=(0,2)$ supersymmetry.
• To extend the 3d-3d correspondence to include abelian flat connections and homological blocks via quiver-based half-indices.
• To demonstrate dualities between 3d $χ=2$ theories via quiver equivalence, linking equivalent quivers to dual field theories.

Proposed method

• Interprets generating functions of Donaldson-Thomas invariants and quiver forms as half-indices of 3d $χ=2$ theories with Dirichlet and deformed Dirichlet boundary conditions on vector and chiral multiplets.
• Applies the 3d-3d correspondence to map topological invariants of 3-manifolds (e.g., colored Jones polynomials, homological blocks) to half-indices of 3d $χ=2$ theories.
• Uses UV Chern-Simons levels and vortex lines to encode the global symmetry and topological data of the 3d theory.
• Constructs explicit 3d $χ=2$ Lagrangians for $T[M_3]$ by specifying matter content, gauge groups, and boundary conditions matching the quiver form.
• Applies quiver equivalence operations—unlinking, linking, removing redundant pairs, and size-preserving transpositions—to relate different quivers and infer dual 3d $χ=2$ theories.
• Translates the gluing formula of Gukov-Manolescu into 3d $χ=2$ theory language for $SL(2,\mathbb{C})$ flat connections.

Experimental results

Research questions

• RQ1 How can quiver forms of 3-manifold invariants be realized as half-indices of 3d $χ=2$ theories with 2d $χ=(0,2)$ boundary conditions?
• RQ2 What is the explicit Lagrangian description of $T[M_3]$ for 3-manifolds such as knot complements and surgeries, based on quiver forms?
• RQ3 How do 3d $χ=2$ theories transform under orientation reversal of $M_3$, corresponding to $q \to q^{-1}$?
• RQ4 What is the role of quiver equivalence in identifying dual 3d $χ=2$ theories with isomorphic half-indices?
• RQ5 Can refined invariants like the $a$,$-t$-deformed homological block for torus knots be captured by a 3d $χ=2$ theory with specific boundary conditions?

Key findings

• The generating function of the colored HOMFLY polynomial in the large $N$ limit is realized as the half-index of a 3d $χ=2$ theory with Dirichlet boundary conditions on vector multiplets and deformed Dirichlet conditions on chiral multiplets.
• The homological block $F_K(x,a,q)$ for a knot complement is matched to the half-index of a 3d $χ=2$ theory with specific matter content and Chern-Simons levels, including $U(1)_i \times U(1)_v \times U(1)_R$ gauge symmetries.
• For the twist knot $K_{-p}$ and left-handed torus knot $T^l(2,2p+1)$, explicit 3d $χ=2$ Lagrangians are constructed whose half-indices reproduce the colored Jones polynomials.
• The theory $T[J_{K_1}(n,q)]$ has a specific Lagrangian with $U(1)_1 \times U(1)_2 \times U(1)_v \times U(1)_R$ gauge group, 2 chiral multiplets with $D_c$ condition, and mixed Chern-Simons levels $\begin{pmatrix} 3/2 & 1 \\ 1 & 1/2 \end{pmatrix}$.
• The theory $T[J_{K_{-1}}(n,q)]$ is dual to $T[J_{K_1}(n,q)]$ under quiver equivalence, with different mixed Chern-Simons levels $\begin{pmatrix} 1/2 & 0 \\ 0 & -1/2 \end{pmatrix}$, confirming duality via equivalent quiver forms.
• The paper establishes that equivalent quiver forms—via unlinking, linking, or size-preserving transpositions—correspond to dual 3d $χ=2$ theories, with identical half-indices and matching boundary conditions.