Generalization of the higher-form-only result to arbitrary spatial dimensions

Prove that, beyond the cases established in the paper, the conclusion that only higher-form symmetries are compatible with momentum conservation without spontaneous symmetry breaking holds for theories with $SO(d)$ symmetry in general spatial dimensions.

Background

The paper derives two representation-theoretic consistency conditions, C1 and C2, for translationally invariant theories with exotic conservation laws and momentum conservation. Under these conditions, it establishes that higher-form symmetries are the only compatible theories for SO(3)SO(3) and proves a corresponding no-go theorem for sufficiently large odd spatial dimensions, subject to restrictions on the current representation and the size of the charge representation.

The authors conjecture that this classification remains valid more generally, including dimensions outside the proven regime. Establishing the conjecture would show that momentum-conserving theories with exotic conservation laws, absent spontaneous symmetry breaking, are generically restricted to higher-form symmetries.

References

We conjecture that this result holds more generally in higher dimensions, i.e., for the group $SO{d}$, and present some partial results towards this end.

— On Higher-Form Fracton Hydrodynamics with Momentum Conservation  (2610.26862 - Bobell et al., 19 Oct 2026) in Introduction; Section 6, especially Theorem 6.1 and the discussion following it

We postulate that theories with $m\geq 2$ will generically exhibit the spontaneous symmetry breaking of emergent conservation laws; we leave a detailed confirmation to future work (also see the derivative-counting nonlocality argument at the beginning of Section~\ref{sec:secondconsistencycondition}).

— On Higher-Form Fracton Hydrodynamics with Momentum Conservation  (2610.26862 - Bobell et al., 19 Oct 2026) in Section 3.1, The First Consistency Condition (C1)

We leave such a confirmation (or a counterexample) to future work.

— On Higher-Form Fracton Hydrodynamics with Momentum Conservation  (2610.26862 - Bobell et al., 19 Oct 2026) in Concluding Thoughts