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Symmetry extension by condensation defects in five-dimensional gauge theories

Matteo Bertolini, Lorenzo Di Pietro, Stefano C. Lanza, Pierluigi Niro, Antonio Santaniello

Abstract

We investigate the symmetry structure of five-dimensional Yang-Mills theories with $\mathfrak{su}(N)$ gauge algebra. These theories feature intertwined 0-, 1-, and 2-form symmetries, depending on the global variant one is considering. In the $SU(N)$ theory, there is a mixed 't Hooft anomaly between the instantonic 0-form symmetry and the electric 1-form symmetry. We show that in the $PSU(N)$ theory this translates into a $\mathbb{Z}_N$ extension of the instantonic symmetry, generated by an invertible condensation defect of the magnetic 2-form symmetry. We identify the charged configurations as linked 't Hooft surfaces, while pointlike instanton operators remain insensitive to the extension. We generalize our analysis to the $SU(N)/\mathbb{Z}_k$ global form and show that similar results hold, embedded now in a 3-group structure for generic $k$. We then apply our findings to $SO(3)$ supersymmetric Yang-Mills theory. We determine the global form of the enhanced instantonic symmetry of its superconformal UV completion, showing that it arises through a similar symmetry extension mechanism from the parent $E_1$ theory, which is the UV completion of $SU(2)$ supersymmetric Yang-Mills theory. Finally, we recast our results in the language of the symmetry topological field theory. As a warm-up, we also analyze Maxwell theory, highlighting analogous features involving continuous symmetries and composite currents.

Symmetry extension by condensation defects in five-dimensional gauge theories

Abstract

We investigate the symmetry structure of five-dimensional Yang-Mills theories with gauge algebra. These theories feature intertwined 0-, 1-, and 2-form symmetries, depending on the global variant one is considering. In the theory, there is a mixed 't Hooft anomaly between the instantonic 0-form symmetry and the electric 1-form symmetry. We show that in the theory this translates into a extension of the instantonic symmetry, generated by an invertible condensation defect of the magnetic 2-form symmetry. We identify the charged configurations as linked 't Hooft surfaces, while pointlike instanton operators remain insensitive to the extension. We generalize our analysis to the global form and show that similar results hold, embedded now in a 3-group structure for generic . We then apply our findings to supersymmetric Yang-Mills theory. We determine the global form of the enhanced instantonic symmetry of its superconformal UV completion, showing that it arises through a similar symmetry extension mechanism from the parent theory, which is the UV completion of supersymmetric Yang-Mills theory. Finally, we recast our results in the language of the symmetry topological field theory. As a warm-up, we also analyze Maxwell theory, highlighting analogous features involving continuous symmetries and composite currents.

Paper Structure

This paper contains 23 sections, 163 equations, 4 figures.

Table of Contents

  1. Introduction and summary of results
  2. Five-dimensional Maxwell Theory
  3. Action of the instantonic symmetry and triple-linking
  4. Instantonic symmetry defects by topological manipulations
  5. Background fields, anomalies, and 3-group
  6. Global forms of five-dimensional su(N) Yang-Mills
  7. Review of SU(N) Yang-Mills and its mixed 't Hooft anomaly
  8. Gauging the 1-form symmetry: PSU(N) Yang-Mills
  9. ZN generator as a condensation defect
  10. Generalization to SU(N)/Zk
  11. Supersymmetric theory and the global forms of the E1 SCFT
  12. Comments about the 2-group structure in the $E_1$ theory
  13. Symmetry Topological Field Theory
  14. SymTFT for su(2) Yang-Mills
  15. Global variants and boundary conditions
  16. ...and 8 more sections

Figures (4)

  • Figure 1: Three-dimensional depiction of a configuration activating the triple-linking in \ref{['Maxwell_alternativetlk']}, between the instantonic operator defined on $\Sigma_4$ (2-torus in the figure) and two double-linking 't Hooft surfaces $\gamma_2$ and $\gamma_2'$ (circles in the figure). The surface $\gamma_2$ (dashed red in the figure) is inside $\Sigma_4$, while the surface $\gamma_2'$ (thick blue in the figure) is outside $\Sigma_4$. The simplest five-dimensional configuration with triple-linking is the one in which the circles in the figure are interpreted as two-spheres, i.e. $\gamma_2$ and $\gamma_2'$ are linked two-spheres, and $\Sigma_4=S^2\times S^2$.
  • Figure 2: The evaluation of the anomaly-inflow functional on $\mathcal{Y}_6$ with $\partial \mathcal{Y}_6 = \mathcal{M}_5$, where $\mathcal{M}_5 = S^1\times \Sigma_4$ and $\mathcal{Y}_6 = D_2\times \Sigma_4$. A symmetry defect of the instantonic symmetry $U^\alpha_I$ wraps $\Sigma_4$. Shifting $\alpha\to\alpha+2\pi$ corresponds to a large gauge transformation. This activates the anomaly functional evaluated on $\mathcal{Y}_6$. As a result, $U^\alpha_I$ and $U^{\alpha+2\pi}_I$ are not equal, but rather related by a phase factor determined by the anomaly.
  • Figure 3: Three-dimensional depiction of the two 3-surfaces with intersecting fillings representing the insertion of the magnetic operators $\widetilde{W}_3^m[\Sigma_3,D_4]$ and $\widetilde{W}_3^{m'}[\Sigma_3',D_4']$ in six dimensions. When we shift their labels, we generate two Wilson lines for $A_1$, with opposite charges, at the two boundaries of the intersection of their fillings $D_4\cap D'_4$.
  • Figure 4: The configuration \ref{['W3-ending']} of the operators $\widetilde{W}_3^{\widetilde{m}N/k}[X_3,Y_4]$ ending at the topological boundary $\partial\mathcal{Y}_6^T$.