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An $\mathcal{N}=1$ 3d-3d Correspondence

Julius Eckhard, Sakura Schafer-Nameki, Jin-Mann Wong

TL;DR

This paper proposes an $ ext{N}=1$ 3d--3d correspondence arising from M5-branes on associative three-cycles in $G_2$-manifolds, relating $T_{ ext{N}=1}[M_3]$ to two topological theories: BFH on $M_3$ for the Witten index and CS–Dirac on $M_3$ for the $S^3$-partition function. The Witten index is computed as the Euler character of the moduli space of generalized Seiberg–Witten solutions, with abelian checks and Lens-space refinements; the $S^3$-partition function is captured by CS–Dirac theory, which reduces to real Chern–Simons or WRT-type invariants depending on twisted harmonic spinors. A non-abelian generalization is explored via circle reductions to 2d sigma-models and Lens-space reductions, while the Lens-space case yields tractable checks against known invariants. The work connects M5-brane dynamics, associative-calibrated geometry, and topological field theories to provide a framework for $ ext{N}=1$ observables in three dimensions, with potential extensions to holography and $ ext{N}=1$ AGT-type correspondences.

Abstract

M5-branes on an associative three-cycle $M_3$ in a $G_2$-holonomy manifold give rise to a 3d $\mathcal{N}=1$ supersymmetric gauge theory, $T_{\mathcal{N}=1} [M_3]$. We propose an $\mathcal{N}=1$ 3d-3d correspondence, based on two observables of these theories: the Witten index and the $S^3$-partition function. The Witten index of a 3d $\mathcal{N}=1$ theory $T_{\mathcal{N}=1} [M_3]$ is shown to be computed in terms of the partition function of a topological field theory, a super-BF-model coupled to a spinorial hypermultiplet (BFH), on $M_3$. The BFH-model localizes on solutions to a generalized set of 3d Seiberg-Witten equations on $M_3$. Evidence to support this correspondence is provided in the abelian case, as well as in terms of a direct derivation of the topological field theory by twisted dimensional reduction of the 6d $(2,0)$ theory. We also consider a correspondence for the $S^3$-partition function of the $T_{\mathcal{N}=1} [M_3]$ theories, by determining the dimensional reduction of the M5-brane theory on $S^3$. The resulting topological theory is Chern-Simons-Dirac theory, for a gauge field and a twisted harmonic spinor on $M_3$, whose equations of motion are the generalized 3d Seiberg-Witten equations. For generic $G_2$-manifolds the theory reduces to real Chern-Simons theory, in which case we conjecture that the $S^3$-partition function of $T_{\mathcal{N}=1}[M_3]$ is given by the Witten-Reshetikhin-Turaev invariant of $M_3$.

An $\mathcal{N}=1$ 3d-3d Correspondence

TL;DR

This paper proposes an 3d--3d correspondence arising from M5-branes on associative three-cycles in -manifolds, relating to two topological theories: BFH on for the Witten index and CS–Dirac on for the -partition function. The Witten index is computed as the Euler character of the moduli space of generalized Seiberg–Witten solutions, with abelian checks and Lens-space refinements; the -partition function is captured by CS–Dirac theory, which reduces to real Chern–Simons or WRT-type invariants depending on twisted harmonic spinors. A non-abelian generalization is explored via circle reductions to 2d sigma-models and Lens-space reductions, while the Lens-space case yields tractable checks against known invariants. The work connects M5-brane dynamics, associative-calibrated geometry, and topological field theories to provide a framework for observables in three dimensions, with potential extensions to holography and AGT-type correspondences.

Abstract

M5-branes on an associative three-cycle in a -holonomy manifold give rise to a 3d supersymmetric gauge theory, . We propose an 3d-3d correspondence, based on two observables of these theories: the Witten index and the -partition function. The Witten index of a 3d theory is shown to be computed in terms of the partition function of a topological field theory, a super-BF-model coupled to a spinorial hypermultiplet (BFH), on . The BFH-model localizes on solutions to a generalized set of 3d Seiberg-Witten equations on . Evidence to support this correspondence is provided in the abelian case, as well as in terms of a direct derivation of the topological field theory by twisted dimensional reduction of the 6d theory. We also consider a correspondence for the -partition function of the theories, by determining the dimensional reduction of the M5-brane theory on . The resulting topological theory is Chern-Simons-Dirac theory, for a gauge field and a twisted harmonic spinor on , whose equations of motion are the generalized 3d Seiberg-Witten equations. For generic -manifolds the theory reduces to real Chern-Simons theory, in which case we conjecture that the -partition function of is given by the Witten-Reshetikhin-Turaev invariant of .

Paper Structure

This paper contains 38 sections, 200 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Overview and Background
  3. An $\mathcal{N}=1$ 3d--3d Correspondence
  4. $G_2$-holonomy, Associatives and Deformation Theory
  5. Generalized Seiberg-Witten Equations
  6. $T_{\mathcal{N}=1}[M_3, U(1)]$
  7. Topological Twist for Associative Three-Cycles
  8. From 6d (2,0) to 3d $\mathcal{N}=1$ $T_{\mathcal{N} = 1}[M_3, U(1)]$
  9. Witten Index of $T_{\mathcal{N} = 1}[M_3, U(1)]$
  10. Non-abelian Generalization
  11. Circle Reduction to 2d $\mathcal{N} = (1,1)$ Sigma-Model
  12. Specializing $M_3$: $T_{\mathcal{N} = 1}[L(p,q), U(N)]$
  13. BFH-Model on $M_3$
  14. Topological Twist of 3d $\mathcal{N} = 8$ SYM
  15. The Abelian BFH-Model
  16. ...and 23 more sections

Figures (4)

  • Figure 1: The setup for the $\mathcal{N}=1$ 3d--3d correspondence is the 6d $(2,0)$ theory on $M_3\times T^3$. The topologically twisted reduction along $M_3$ preserves 3d $\mathcal{N}=1$ and geometrically corresponds to an embedding of $M_3$ as an associative three-cycle in a $G_2$-holonomy manifold. The correspondence states that the Witten index of the resulting theory $T_{\mathcal{N}=1} [M_3]$ can alternatively be computed from the partition function on $M_3$ of a topological twist of the $T^3$-reduction of the 6d theory, a BF-theory coupled to a spinorial hypermultiplet (BFH).
  • Figure 2: The setup for the $\mathcal{N}=1$ 3d--3d correspondence is the 6d $(2,0)$ theory on $M_3\times S^3$. The topologically twisted reduction along $M_3$ preserves 3d $\mathcal{N}=1$ and geometrically corresponds to an embedding of $M_3$ as an associative three-cycle in a $G_2$-holonomy manifold. In this variant the $S^3$-partition function of $T_{\mathcal{N}=1} [M_3]$ can alternatively be computed from the partition function of real Chern-Simons gauge theory on $M_3$ coupled to a twisted harmonic spinor, i.e. Chern-Simons-Dirac theory. In the case when there no twisted harmonic spinors, the theory reduces to real Chern-Simons theory and the partition function is given by the Witten-Reshetikhin-Turaev (WRT) invariant.
  • Figure 3: The variation of an associative three-cycle $M(b)$ inside a $G_2$-manifold, with $G_2$-form $\Phi(a)$. The variation is shown with respect to the parameter $a$ under the constraint $b^2 = a$, required for $M(b)$ to be an associative.
  • Figure 4: To study the non-abelian theory we consider first the dimensional reduction to 5d SYM and then the reduction on $M_3$ to 2d. This should correspond to the circle-reduction of the 3d $T_{\mathcal{N}=1} [M_3]$ to a 2d $\mathcal{N}= (1,1)$ sigma-model, whose target space are the generalized Seiberg-Witten equations (gSW) in \ref{['BPSeq2d']}.