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Four-manifolds and Symmetry Categories of 2d CFTs

Vladimir Bashmakov, Michele Del Zotto, Azeem Hasan

TL;DR

<3-5 sentence high-level summary>The paper develops a geometric, symmetry-TFT-based framework to understand higher and non-invertible symmetries of 2d CFTs arising from dimensional reduction of 6d (2,0) theories on 4-manifolds. It shows that the global structure of the resulting 2d theory is controlled by the presence or absence of null bordisms of the 4-manifold, with absolute theories corresponding to null-bordant manifolds and relative theories arising otherwise; non-invertible symmetries emerge as sums over topologies in the 7d symmetry TFT and can be realized as Tambara-Yamagami categories TY(Z_N). The work provides concrete demonstrations via the S^2 × S^2 case, del Pezzo surfaces, and the analysis of Lagrangian subgroups, discriminant groups, and condensation defects, linking 4-manifold topology to 2d symmetry categories. It offers a path to generalize these ideas to other algebras and manifolds, and draws connections to symmetry fractionalization in Maxwell theory and to broader fusion-category structures in low dimensions.

Abstract

In this paper we study the geometric origin of non-invertible symmetries of 2d theories arising from the reduction of 6d $(2,0)$ theories on four-manifolds. This generalizes and extends our previous results in the context of class $\mathcal S$ theories to a wider realm of models. In particular, we find that relative 2d field theories, such as the chiral boson, have a higher dimensional origin in four-manifolds that are not null cobordant. Moreover, we see that for the 2d theories with a 6d origin, the non-invertible symmetries have a geometric origin as a sum over topologies from the perspective of the 7d symmetry TFT. In particular, we show that the Tambara-Yamagami non-invertible symmetries $TY(\mathbb Z_N)$ can be given a geometric origin of this kind. We focus on examples that do not depend on spin structures, but we analyse the simplest of such cases, finding an interesting parallel between the extra choices arising in that context and symmetry fractionalization in Maxwell theories.

Four-manifolds and Symmetry Categories of 2d CFTs

TL;DR

<3-5 sentence high-level summary>The paper develops a geometric, symmetry-TFT-based framework to understand higher and non-invertible symmetries of 2d CFTs arising from dimensional reduction of 6d (2,0) theories on 4-manifolds. It shows that the global structure of the resulting 2d theory is controlled by the presence or absence of null bordisms of the 4-manifold, with absolute theories corresponding to null-bordant manifolds and relative theories arising otherwise; non-invertible symmetries emerge as sums over topologies in the 7d symmetry TFT and can be realized as Tambara-Yamagami categories TY(Z_N). The work provides concrete demonstrations via the S^2 × S^2 case, del Pezzo surfaces, and the analysis of Lagrangian subgroups, discriminant groups, and condensation defects, linking 4-manifold topology to 2d symmetry categories. It offers a path to generalize these ideas to other algebras and manifolds, and draws connections to symmetry fractionalization in Maxwell theory and to broader fusion-category structures in low dimensions.

Abstract

In this paper we study the geometric origin of non-invertible symmetries of 2d theories arising from the reduction of 6d theories on four-manifolds. This generalizes and extends our previous results in the context of class theories to a wider realm of models. In particular, we find that relative 2d field theories, such as the chiral boson, have a higher dimensional origin in four-manifolds that are not null cobordant. Moreover, we see that for the 2d theories with a 6d origin, the non-invertible symmetries have a geometric origin as a sum over topologies from the perspective of the 7d symmetry TFT. In particular, we show that the Tambara-Yamagami non-invertible symmetries can be given a geometric origin of this kind. We focus on examples that do not depend on spin structures, but we analyse the simplest of such cases, finding an interesting parallel between the extra choices arising in that context and symmetry fractionalization in Maxwell theories.
Paper Structure (28 sections, 74 equations, 4 figures)

This paper contains 28 sections, 74 equations, 4 figures.

Table of Contents

  1. Abstract
  2. Introduction
  3. Symmetry TFT from dimensional reduction
  4. Topological boundary conditions
  5. Dimensional reduction and boundary conditions
  6. Two dimensional theories
  7. Exceptional Symmetries and Non Invertible Defects
  8. Codensation Surgery Defects
  9. The 7d view of 3d Chern Simons
  10. 3d Abelian Chern-Simons theories
  11. Discriminant group and topological boundary conditions
  12. Categorical structure of Chern-Simons
  13. Topological boundary conditions from $6d$
  14. The case of $S^2 \times S^2$ and $TY(\mathbb Z_N)$
  15. Algebra and geometry of topological boundary conditions
  16. ...and 13 more sections

Figures (4)

  • Figure 1: Dimensional reduction of the 7d/6d relative field theory. Provided a maximally isotropic sublattice $\mathcal{L}$ exists in a way which is independent of the choice of $M_d$ in the splitting $X_6 = Y_{k} \times M_d$ we obtain a corresponding absolute theory associated to a gapped topological boundary condition for $D_{Y_{k}}\mathfrak F_N$.
  • Figure 2: The surgery defect ${\cal C}_F$ is given a topological boundary and one obtains a twist defect. Shrinking the slab, this twist defect becomes a duality interface ${\cal N}_F$.
  • Figure 3: The automorphism transformation operators ${\cal C}_F(\Sigma_2)$ change the anyon lables in a given way. The folding trick implies that this operators should be able to absorb lines of the form $a_{(\mathds{1}-F)\mathbf{n}}(\gamma)$, and therefore are condensates of them.
  • Figure 4: The surgery defect ${\cal C}_F$ is given a topological boundary and one obtains a twist defect. Shrinking the slab, this twist defect becomes a duality interface ${\cal N}_F$.