Center Higher-Form Symmetries in QFT

• Center higher-form symmetries are generalized global symmetries in gauge theories that act on extended operators, such as Wilson lines and surfaces.
• They are formulated via higher-group structures (e.g., 2-groups) with weak associativity, classified by cohomological data like H³(G, U(1)).
• Their study reveals practical insights into anomaly matching, topological defects, and phase classifications, impacting both high-energy and condensed matter systems.

Center higher-form symmetries extend the conventional notion of global (zero-form) symmetries in quantum field theory (QFT) by acting not on point-like operators but on extended operators such as lines, surfaces, or their higher-dimensional generalizations. In the context of gauge theories, “center higher-form symmetry” specifically refers to generalized global symmetries whose symmetry group is the center of the gauge group, acting on operators (e.g., Wilson lines) charged under that center. The modern theory understands these symmetries as components of higher groups—specifically 2-groups and beyond—where the group structure only holds up to controlled isomorphisms, not strict equalities. This leads to new algebraic and geometric structures with profound implications for anomaly matching, topological defects, classification of phases, dualities, and the global structure of gauge theories.

1. Mathematical Structure and Classification: Higher Groups and 2-Groups

Center higher-form symmetries are naturally formalized as instances of higher groups, most familiarly by 2-groups. Unlike ordinary groups, which have a strict associative multiplication, higher groups (q+1-groups) possess a “weak” associativity: group multiplication is only defined up to a natural isomorphism known as the associator. For a 2-group extension

$1 \to B U(1) \to \mathcal{G} \to G \to 1,$

the associator is determined by a 3-cocycle $\alpha \in H^3(G, U(1))$ and satisfies the pentagon identity. This “categorified” central extension realizes the center symmetry as a higher-form symmetry—i.e., an object acting naturally on extended, rather than local, operators. The cohomological data (e.g., $H^3(G, U(1))$) classifies such extensions; this construction unifies cases where discrete torsion or gerbe structures are present in the moduli (vacuum) spaces (Sharpe, 2015).

Table: Comparison of group extensions

Group extension type	Cohomological class	Symmetry acts on
Ordinary central	$H^2(G, U(1))$	Local (0D) operators
2-group (center higher)	$H^3(G, U(1))$	Extended ($p>0$) operators

In QFTs, examples include 2-groups in Dijkgraaf–Witten theories ($H^3(G, U(1))$ classifying discrete torsion), orbifold and gerbe theories (center trivially acting on matter), and WZW models (central extension via affine Kac–Moody currents encodes nontrivial 2-group structure).

2. Center Higher-Form Symmetry in Gauge Theories

In non-Abelian gauge theories (such as $SU(N)$), the center $Z(G)$ is always an abelian group. The center 1-form symmetry acts on Wilson lines in representations having nontrivial center charge. These symmetry transformations shift the phase of Wilson loops, formally acting nontrivially only on those operators not screened by dynamical matter (Gomes, 2023).

Field-theoretic characterization in $d$ spacetime dimensions involves a conserved $(p+1)$-form current $J^{(p+1)}$ and associated topological charge

$$Q(\Sigma_{d-p-1}) = \int_{\Sigma_{d-p-1}} \star J^{(p+1)},$$

which is supported on codimension-$(p+1)$ submanifolds. For center 1-form symmetry ($p=1$), topological generators are codimension-2 (e.g., loops in 4D). The existence and structure of these symmetries in 5d and 6d gauge theories, including their breaking by matter and BPS defects, is determined by group cohomology and intersection data in geometric engineering (Morrison et al., 2020, Bhardwaj et al., 2020).

Explicit calculation of the center symmetry proceeds via:

• Field content and group center: The maximal 1-form symmetry is the subgroup of center elements acting trivially on all (matter) representations.
• Geometric M-theory realization: The discrete 1-form symmetry arises from torsion in relative homology $H_2^\partial(M_6)/\text{im}(f_2)$ or intersection forms (Smith normal form yields the structure, e.g., $\mathbb{Z}_g$ for $SU(N)_k$ with $g = \gcd(N,k)$) (Morrison et al., 2020, Albertini et al., 2020).

3. Topological Defects and Phases of Gauge Theories

Topological defects associated with higher-form symmetries are classified by the homotopy of the moduli space $M$ or, more generally, by stacks capturing the moduli including non-trivial center actions. For higher symmetries, many observables (such as Wilson lines or surfaces) "live" on the loop space $LM$ of $M$, and the relevant defects are classified by

$\pi_k(M) \simeq \pi_{k-1}(LM),$

with "defects" for a 2-group symmetry encoded in the topology of $LM$ (Sharpe, 2015).

In discrete lattice gauge theories, whether the Gauss law constraint and 1-form symmetry are exact or emergent affects the entanglement spectrum and topological entanglement entropy (TEE). Spontaneously broken higher-form symmetry can remove entanglement degeneracy and is required for a robust TEE (Xu et al., 2023). In the continuum, signature behaviors such as the area vs perimeter law for Wilson loops diagnose symmetry-breaking phases: an area law signals unbroken symmetry (confinement), perimeter law (or Coulomb decay) indicates spontaneous symmetry breaking (deconfined phases where the center symmetry is Goldstone-like) (Lake, 2018, Gomes, 2023).

4. Higher-Group Symmetry, Mixed Anomalies, and Transmutation

A central conceptual advance is the realization that mixed anomalies between 0-form and higher-form symmetries are signatures of a nontrivial higher-group symmetry structure, often realized as a 2-group or, in more complex cases, a 3-group or 2-crossed module. The failure of strict associativity manifests as a 3-cocycle, and anomalies that traditionally signal breakdown of symmetry are recast as "transmutation": an anomaly promotes a classical group symmetry into a higher-group quantum symmetry (Sharpe, 2015, Hidaka et al., 2020).

For example, in WZW models and bosonization, the central extension (Kac–Moody algebra level) is the infinitesimal shadow of the 2-group: anomalies encoded by the 3-cocycle in $H^3(G,U(1))$ promote $G$ to a 2-group $\mathcal{G}$. In the geometric engineering of gauge theories, turning on torsion flux backgrounds for the center symmetry leads to fractionalization of instanton number and matches 't Hooft anomalies computed both geometrically (via Lefschetz duality, Smith normal form of intersection matrices) and field-theoretically (Cvetic et al., 2021).

Table: Anomaly versus transmutation in symmetry structure

Classical theory symmetry	Quantum anomaly	Symmetry structure after quantization
Group $G$	Mixed anomaly ($H^3$)	2-group extension $\mathcal{G}$
0-form + 1-form	Mixed anomaly	3-group, crossed module, or 2-group

5. Symmetry Breaking, Goldstone Realization, and SSB Constraints

Higher-form symmetries admit their own Goldstone theorem analogues (Lake, 2018). Spontaneous breaking of a continuous $p$-form symmetry leads to gapless (massless) $p$-form gauge fields as Goldstone modes (e.g., the photon as a 1-form Goldstone boson in $D=4$). The generalization of the Coleman–Mermin–Wagner theorem constrains which symmetries can be spontaneously broken: continuous $p$-form symmetries can be broken only for $p < D-2$ (continuous.) For discrete symmetries, the threshold is $p < D-1$.

Boundary conditions and gauge-fixing become especially subtle: only with Dirichlet-type boundaries (fixed gauge field) does the theory realize definite vacua breaking the symmetry, while Neumann conditions (fixed field strength) can select different phases (Lake, 2018). Similar results control the appearance and structure of higher-form symmetry protected topological phases in lattice models (Xu et al., 2023).

6. Higher-Form Symmetries in Geometry, Stacks, and Gerbes

The geometry of center higher-form symmetries is more intricate than that of ordinary global groups. Theories where the moduli "space" is a stack (e.g., Deligne–Mumford or Artin stacks) exhibit nontrivial higher-group structure encoded in higher cohomology and homotopy data. Theories with trivial center action (such as gerbe theories) admit a 2-group action corresponding to $BZ$ for a finite central subgroup $Z$. In mathematical language, such structures are described by gerbes and higher stacks capturing both the field content and its automorphisms (Sharpe, 2015).

In geometric engineering, the center higher-form symmetry is expressed via torsion in relative homology or as arising from background discrete fluxes on boundaries of noncompact Calabi–Yau threefolds; the invariance under flop transitions and dualities is structurally guaranteed by the geometric picture, with the center symmetry matching across all dual phases (Morrison et al., 2020, Bhardwaj et al., 2020).

7. Multi-Component and Intertwined Symmetries: Higher-Group Entanglement

In systems with multiple spontaneously broken continuous symmetries (e.g., mixtures of superfluids), an emergent higher-group symmetry typically arises from the Grassmann algebra of topological currents. The higher-group structure intertwines the center higher-form symmetry with 0-form symmetries and their composites. The effective action includes sources for all composite and primitive currents; entanglement of transformation rules for background gauge fields encode the higher-group algebra present at low energies (Brauner, 2020).

This also leads to novel phase structures, hydrodynamics, and dualities in condensed matter and high energy systems, with clear mechanisms for the classification and realization of topological order, SPT phases, and transitions driven by the proliferation of topological defects or explicit symmetry breaking (Armas et al., 2023).

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By viewing center higher-form symmetries through the modern lens of higher-group theory, categorification, and their explicit realization in quantum field theory and geometry, one obtains a unified and robust framework for the classification of phases, analysis of anomalies, dualities, and topological structures in both high-energy and condensed-matter systems. The interplay between global symmetry groups, cohomological and geometric data, and the structure of extended operators underpins their fundamental role across diverse domains of theoretical physics.