Symmetry extension by condensation defects in five-dimensional gauge theories

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.

💡 Research Summary

The paper investigates the intricate web of higher‑form symmetries that appear in five‑dimensional Yang–Mills (YM) theories with gauge algebra su(N) and explores how these symmetries are modified when one changes the global form of the gauge group. The authors begin by recalling that any 5d gauge theory possesses a U(1) instanton 0‑form symmetry generated by the current J⁽ᴵ⁾∝⋆Tr(F∧F). In the standard SU(N) theory this symmetry coexists with a ℤₙ electric 1‑form symmetry (acting on Wilson lines) and, depending on the global form, a magnetic 2‑form symmetry (acting on ’t Hooft surfaces).

A mixed ’t Hooft anomaly between the instanton 0‑form symmetry and the electric ℤₙ¹ symmetry was previously identified; its anomaly‑inflow action can be written schematically as
A = exp