Higher-Form Symmetries

Definition

A higher-form (generalized global) symmetry is a symmetry whose charged objects are extended rather than pointlike. The organizing idea (Gaiotto–Kapustin–Seiberg–Willett, 2015) is that a symmetry is a topological operator: a $p$-form symmetry is generated by operators $U_g(M^{d-p-1})$ supported on closed codimension-$(p+1)$ submanifolds, and it acts on $p$-dimensional charged operators by linking in spacetime.

• $p=0$ recovers ordinary symmetry: codimension-1 operators acting on local operators.
• $p=1$ (a 1-form symmetry) has codimension-2 operators acting on line operators (Wilson/‘t Hooft lines).
• For $p\ge 1$ the symmetry group is necessarily abelian.

The framework gives the Landau paradigm back for phases with no local order parameter: a phase is characterized by which higher-form symmetries are realized, spontaneously broken, or anomalous. Confinement is the statement that a 1-form symmetry is unbroken (wilson-loop area law); the deconfined/Coulomb phase is its spontaneous breaking (perimeter law), with the photon as the Goldstone mode.

Role in Research

Higher-form symmetry is the language that connects the group’s historical confinement-duality work to the current research frontier. Every Semester I object of the generalized-symmetries course — center symmetry, the magnetic shift symmetry of compact QED, monopole and vortex conservation — is a higher-form symmetry; the group’s manuscript in preparation studies the wall made by gauging a subgroup of the electric 1-form symmetry on a hypersurface of four-dimensional ${\mathbb{Z}}_N$ gauge theory, and gives a finite cost to the lines that run along it (see condensation-defects).

• Gauging a discrete $p$-form symmetry (summing over backgrounds) produces a dual “quantum” ${\mathbb{Z}}_N^{(d-p-2)}$ symmetry — the rule behind kramers-wannier-duality and behind the global-form ambiguity ($SU(N)$ vs $SU(N)/{\mathbb{Z}}_N$).
• Spontaneous breaking of a 1-form symmetry is the sharp definition of a deconfined phase and of topological-order (ground-state degeneracy = broken 1-form symmetry).
• Anomalies of higher-form symmetries (e.g. Yang–Mills at $\theta =\pi$) constrain the phase diagram; see t-hooft-anomaly.

Relations

• wilson-loop — the charged line operator of a 1-form symmetry; area/perimeter law = unbroken/broken
• lattice-gauge-theory — where these symmetries are realized concretely
• topological-order — spontaneously broken 1-form symmetry; degeneracy from symmetry operators on cycles
• t-hooft-anomaly — anomalies of and mixed anomalies with higher-form symmetries
• condensation-defects — higher gauging = gauging a higher-form symmetry on a submanifold
• villain-action — modified Villain realizes higher-form symmetries exactly on the lattice

Papers

Gaiotto–Kapustin–Seiberg–Willett, JHEP 02 (2015) 172 [arXiv:1412.5148]; reviews: McGreevy, arXiv:2204.03045; Córdova et al. (Snowmass), arXiv:2205.09545; Shao (TASI), arXiv:2308.00747. See the course bibliography.

Notes

• “Symmetry = topological operator” is the whole reframing; conservation of a Noether current is the special case where the operator is codimension-1.
• The abelianness for $p\ge 1$ is geometric: two codimension-$\ge 2$ operators can be moved past each other, so their product cannot depend on order.
• Beyond invertible higher-form symmetries lie non-invertible ones (fusion is a projector, no inverse); these appear at self-dual points and as condensation-defects.