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Symmetry TFTs for 3d QFTs from M-theory

Marieke van Beest, Dewi S. W. Gould, Sakura Schafer-Nameki, Yi-Nan Wang

TL;DR

This work establishes a geometric and holographic framework to derive Symmetry Topological Field Theories (SymTFTs) for 3d SUSY QFTs arising from M-theory. By combining differential cohomology with toric intersection theory, the authors compute BF- and BB-type couplings that encode global forms of gauge groups and higher-form anomalies, explicitly for ABJ(M) theories and for 3d N=2 quivers dual to AdS4×Y^{p,k}(CP^2). The approach yields concrete anomaly coefficients, constraints on allowed boundary conditions, and a refined holographic dictionary linking bulk torsion data to boundary symmetry structures. The results demonstrate how gapped boundary conditions of the SymTFT constrain Chern-Simons terms and reveal nontrivial 1-form anomalies, providing a geometric route to classify and constrain 3d QFTs in holographic and geometric engineering setups.

Abstract

We derive the Symmetry Topological Field Theories (SymTFTs) for 3d supersymmetric quantum field theories (QFTs) constructed in M-theory either via geometric engineering or holography. These 4d SymTFTs encode the symmetry structures of the 3d QFTs, for instance the generalized global symmetries and their 't Hooft anomalies. Using differential cohomology, we derive the SymTFT by reducing M-theory on a 7-manifold $Y_7$, which either is the link of a conical Calabi-Yau four-fold or part of an $\text{AdS}_4\times Y_7$ holographic solution. In the holographic setting we first consider the 3d $\mathcal{N}=6$ ABJ(M) theories and derive the BF-couplings, which allow the identification of the global form of the gauge group, as well as 1-form symmetry anomalies. Secondly, we compute the SymTFT for 3d $\mathcal{N}=2$ quiver gauge theories whose holographic duals are based on Sasaki-Einstein 7-manifolds of type $Y_7 = Y^{p,k}(\mathbb{C}\mathbb{P}^2)$. The SymTFT encodes 0- and 1-form symmetries, as well as potential 't Hooft anomalies between these. Furthermore, by studying the gapped boundary conditions of the SymTFT we constrain the allowed choices for $U(1)$ Chern-Simons terms in the dual field theory.

Symmetry TFTs for 3d QFTs from M-theory

TL;DR

This work establishes a geometric and holographic framework to derive Symmetry Topological Field Theories (SymTFTs) for 3d SUSY QFTs arising from M-theory. By combining differential cohomology with toric intersection theory, the authors compute BF- and BB-type couplings that encode global forms of gauge groups and higher-form anomalies, explicitly for ABJ(M) theories and for 3d N=2 quivers dual to AdS4×Y^{p,k}(CP^2). The approach yields concrete anomaly coefficients, constraints on allowed boundary conditions, and a refined holographic dictionary linking bulk torsion data to boundary symmetry structures. The results demonstrate how gapped boundary conditions of the SymTFT constrain Chern-Simons terms and reveal nontrivial 1-form anomalies, providing a geometric route to classify and constrain 3d QFTs in holographic and geometric engineering setups.

Abstract

We derive the Symmetry Topological Field Theories (SymTFTs) for 3d supersymmetric quantum field theories (QFTs) constructed in M-theory either via geometric engineering or holography. These 4d SymTFTs encode the symmetry structures of the 3d QFTs, for instance the generalized global symmetries and their 't Hooft anomalies. Using differential cohomology, we derive the SymTFT by reducing M-theory on a 7-manifold , which either is the link of a conical Calabi-Yau four-fold or part of an holographic solution. In the holographic setting we first consider the 3d ABJ(M) theories and derive the BF-couplings, which allow the identification of the global form of the gauge group, as well as 1-form symmetry anomalies. Secondly, we compute the SymTFT for 3d quiver gauge theories whose holographic duals are based on Sasaki-Einstein 7-manifolds of type . The SymTFT encodes 0- and 1-form symmetries, as well as potential 't Hooft anomalies between these. Furthermore, by studying the gapped boundary conditions of the SymTFT we constrain the allowed choices for Chern-Simons terms in the dual field theory.
Paper Structure (55 sections, 206 equations, 2 figures, 6 tables)

This paper contains 55 sections, 206 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. ABJ(M).
  3. $\mathcal{N}=2$ Quiver Gauge Theories.
  4. The Symmetry Topological Field Theory
  5. Brane Constructions and Inflow.
  6. Geometric Engineering.
  7. Background Fields from Cohomology.
  8. Torsional Contributions.
  9. Background Fields from Isometries.
  10. Holographic Theories.
  11. SymTFT from M-Theory on $Y_7$
  12. Reduction using the Free Part of Cohomology
  13. Review of Differential Cohomology
  14. Accounting for Torsion using Differential Cohomology
  15. Application: Holographic AdS$_4$ Backgrounds.
  16. ...and 40 more sections

Figures (2)

  • Figure 1: The blue slab ("sandwich") is the SymTFT $\mathcal{S}$, with boundary conditions on the right ($\mathcal{B}_{\text{phy}}$) which are non-topological and to the left, which are topological $\mathcal{B}_{\text{top}}$. The red line is a gapped interface separating the symmetry theory from the anomaly theory $\mathcal{A}$. The picture introduced earlier of $\mathcal{T}$ being defined as a theory relative to $\mathcal{A}$ is obtained by collapsing the blue sandwich. The anomaly theory describes anomalies for a particular choice of global structure of the QFT.
  • Figure 2: Quiver diagram for theory with gauge group $\Pi_{i=1}^3 U(N_i)_{k_i}$. The triple arrows denote the fact that the bi-fundamental matter fields transform in the fundamental representation of a flavor $SU(3)$.