Gauged Center One-Form Symmetries

• Gauged center one-form symmetries are higher-form operations that adjust a gauge theory’s global structure by promoting a subgroup of the center symmetry to a gauge redundancy.
• Gauging these symmetries via background 2-form fields reorganizes the spectrum of Wilson and ’t Hooft lines and introduces new dual symmetries and mixed ’t Hooft anomalies.
• Frameworks like BF-type SymTFTs, defect condensation, and orbifold constructions provide concrete methods to implement gauging and study its impact across various dimensions.

Gauged center one-form symmetries are the higher-form symmetry operations obtained by promoting a subgroup of the center one-form symmetry of a gauge theory to a gauge redundancy. In the modern generalized-symmetry framework, the electric one-form symmetry is typically a subgroup of the center of the gauge group, acts on line operators, and is coupled to a background 2-form gauge field. Gauging such a symmetry changes the global form of the gauge group, reorganizes the spectrum of genuine Wilson and ’t Hooft lines, and often produces new dual higher-form symmetries, mixed ’t Hooft anomalies, or noninvertible symmetry operators. Across dimensions, the subject is formulated using background 2-cocycles and obstruction classes, BF-type SymTFTs, defect condensation, orbifolds by higher-form symmetries, and, in special settings, extensions of current algebras by simple currents (Bhardwaj et al., 2022).

1. Definition and general mechanism

A center one-form symmetry is a generalized global symmetry whose charged objects are line operators and whose generators are codimension-two topological defects. In gauge theories with gauge group $G$, the electric one-form symmetry is typically a subgroup of the center $Z_G$. When matter fields are present, only the subgroup acting trivially on all matter survives as an exact one-form symmetry. In the notation used for a compact connected gauge group $G$, if $\widehat Z_G=\mathrm{Hom}(Z_G,U(1))$ labels Wilson-line center charges and $M_G\subset \widehat Z_G$ is generated by matter charges, then the electric one-form symmetry is $\Gamma^{(1)}=\widehat Z_G/M_G$, while its Pontryagin dual is the subgroup of the center acting trivially on matter (Bhardwaj et al., 2022).

Gauging a center one-form symmetry means turning on a 2-form gauge field background and then summing over it. In gauge-theory language, this is equivalent, in suitable situations, to replacing $G$ by a quotient $\bar G=G/\widehat{\Gamma^{(1)}}$, and summing over $\bar G$-bundles whose obstruction to lifting back to $G$-bundles is fixed by the background class $Z_G$0. The obstruction is encoded by a class $Z_G$1, and the condition on a bundle $Z_G$2 is $Z_G$3 (Bhardwaj et al., 2022).

In four-dimensional $Z_G$4 gauge theories, this operation is the standard passage from $Z_G$5 to $Z_G$6, with $Z_G$7 a divisor of $Z_G$8. The gauging is implemented by introducing a pair of background fields $Z_G$9 obeying $G$0, together with a 1-form gauge symmetry $G$1, $G$2. This changes which Wilson lines remain genuine and allows fractional instanton sectors, which are central to the mixed anomalies studied in these theories (Bolognesi et al., 2019).

A recurring consequence is that gauging a center one-form symmetry changes both the global topology of the gauge bundle and the operator content. Genuine line operators are reduced to those invariant under the gauged subgroup, while new dual higher-form symmetries may appear. In $G$3 dimensions, gauging a finite 1-form symmetry $G$4 produces a dual $G$5-form symmetry $G$6, with anomalies controlled by the Postnikov data of the original theory (Bhardwaj et al., 2022).

2. Global form, backgrounds, and higher-group structures

The gauging problem is not exhausted by the choice of center subgroup. When continuous or finite zero-form symmetries are also present, the center one-form symmetry may participate in a connected or disconnected 2-group. In the connected case, the symmetry data are $G$7, where $G$8 is the connected zero-form symmetry and $G$9 is the Postnikov class. The corresponding backgrounds satisfy $\widehat Z_G=\mathrm{Hom}(Z_G,U(1))$0. In gauge theories this often arises from a short exact sequence $\widehat Z_G=\mathrm{Hom}(Z_G,U(1))$1, with $\widehat Z_G=\mathrm{Hom}(Z_G,U(1))$2 a subgroup of the flavor center and $\widehat Z_G=\mathrm{Hom}(Z_G,U(1))$3 given by the Bockstein of an obstruction class $\widehat Z_G=\mathrm{Hom}(Z_G,U(1))$4 (Bhardwaj et al., 2022).

Finite outer automorphism symmetries can further refine this structure. If a finite zero-form symmetry $\widehat Z_G=\mathrm{Hom}(Z_G,U(1))$5 acts on $\widehat Z_G=\mathrm{Hom}(Z_G,U(1))$6, the full zero-form symmetry becomes $\widehat Z_G=\mathrm{Hom}(Z_G,U(1))$7, and one obtains a disconnected 2-group $\widehat Z_G=\mathrm{Hom}(Z_G,U(1))$8, where $\widehat Z_G=\mathrm{Hom}(Z_G,U(1))$9. The Postnikov class then lives in twisted cohomology, $M_G\subset \widehat Z_G$0, and the background constraint becomes $M_G\subset \widehat Z_G$1 (Bhardwaj et al., 2022).

A particularly important special case occurs when the short exact sequence $M_G\subset \widehat Z_G$2 splits, so the ordinary Bockstein vanishes, but the finite zero-form symmetry acts nontrivially on $M_G\subset \widehat Z_G$3. Then the disconnected Postnikov class reduces to $M_G\subset \widehat Z_G$4. This means that even when the connected 2-group is trivial, a nontrivial disconnected 2-group may still obstruct or constrain gauging of the center one-form symmetry (Bhardwaj et al., 2022).

In four-dimensional $M_G\subset \widehat Z_G$5 theories with matter, mixed anomalies between discrete chiral symmetries and exact center one-form symmetries are naturally exposed precisely after gauging the latter. The background $M_G\subset \widehat Z_G$6 modifies the instanton number by terms proportional to $M_G\subset \widehat Z_G$7, producing fractional instanton sectors. These fractional sectors reduce the surviving discrete chiral symmetry and thereby constrain the infrared phase. This mechanism is worked out for adjoint QCD, self-adjoint antisymmetric matter, two-index matter, and certain chiral $M_G\subset \widehat Z_G$8 theories (Bolognesi et al., 2019).

3. Defect, orbifold, and SymTFT descriptions

A general defect-theoretic description interprets gauging as the insertion of a spacetime-filling topological defect. For any gaugeable symmetry, including higher-form symmetries, the gauging operation defines a codimension-zero operator, or “gauge defect,” and therefore a $M_G\subset \widehat Z_G$9-form symmetry operator in generalized-symmetry language. In this picture, gauging a center one-form symmetry is the condensation of its symmetry defects over the whole spacetime; in four dimensions this amounts to summing over background 2-form gauge fields $\Gamma^{(1)}=\widehat Z_G/M_G$0 and corresponds to passing, for example, from $\Gamma^{(1)}=\widehat Z_G/M_G$1 to $\Gamma^{(1)}=\widehat Z_G/M_G$2 or to a different global form (Vandermeulen, 2023).

In two-dimensional Yang–Mills theory, the gauging of center one-form symmetries is completely explicit. For $\Gamma^{(1)}=\widehat Z_G/M_G$3, the electric center symmetry is $\Gamma^{(1)}=\widehat Z_G/M_G$4, acting on Wilson lines according to the number of boxes $\Gamma^{(1)}=\widehat Z_G/M_G$5 in the Young diagram of the representation. Gauging a subgroup $\Gamma^{(1)}=\widehat Z_G/M_G$6 introduces a 2-form $\Gamma^{(1)}=\widehat Z_G/M_G$7 gauge field $\Gamma^{(1)}=\widehat Z_G/M_G$8, couples the YM theory through $\Gamma^{(1)}=\widehat Z_G/M_G$9, and yields the orbifold partition function

$G$0

with discrete $G$1-angle $G$2. This is Yang–Mills with gauge group $G$3 (Santilli et al., 2024).

The same two-dimensional analysis also exhibits the dual $G$4-form symmetry that appears after gauging the center one-form symmetry. Its generators are codimension-zero operators $G$5, and gauging a subgroup of this dual symmetry partially reverses the original one-form gauging, sending $G$6 back toward $G$7 (Santilli et al., 2024). This is an explicit instance of the general statement that gauging a $G$8-form symmetry produces a dual $G$9-form symmetry.

A higher-categorical version of the same mechanism appears in three-dimensional defect TQFT. There, generalized symmetries are encoded by functors into higher categories of topological defects, and gauging is implemented by orbifold data. For a $\bar G=G/\widehat{\Gamma^{(1)}}$0-crossed braided fusion category $\bar G=G/\widehat{\Gamma^{(1)}}$1, gauging the $\bar G=G/\widehat{\Gamma^{(1)}}$2-form $\bar G=G/\widehat{\Gamma^{(1)}}$3-symmetry on the neutral modular component $\bar G=G/\widehat{\Gamma^{(1)}}$4 yields the equivariantization $\bar G=G/\widehat{\Gamma^{(1)}}$5. If $\bar G=G/\widehat{\Gamma^{(1)}}$6 is abelian, the latter theory carries a 1-form symmetry by the Pontryagin dual $\bar G=G/\widehat{\Gamma^{(1)}}$7, and gauging that 1-form symmetry recovers $\bar G=G/\widehat{\Gamma^{(1)}}$8 (Carqueville et al., 9 Jun 2025). This gives a precise TQFT realization of “gauged center one-form symmetry” as the dual operation to gauging an abelian zero-form symmetry.

In four dimensions, the symmetry topological field theory perspective packages a $\bar G=G/\widehat{\Gamma^{(1)}}$9 center one-form symmetry into a five-dimensional BF theory

$\bar G$0

Different topological boundary conditions of this BF theory correspond to different global forms of the four-dimensional gauge theory. Gauging the full $\bar G$1 center symmetry or only a subgroup $\bar G$2 becomes a change of boundary polarization in the BF Hilbert space, with discrete $\bar G$3-angles encoded by phases involving the Pontryagin square $\bar G$4 (Duan et al., 2024).

4. Consequences of gauging: dual symmetries, anomalies, and noninvertibility

A central consequence of gauging a one-form symmetry is the appearance of new anomalies. In the presence of a connected or disconnected 2-group symmetry, gauging $\bar G$5 produces a mixed ’t Hooft anomaly between the dual $\bar G$6-form symmetry and the remaining zero-form symmetry. In the disconnected case the anomaly is

$\bar G$7

where $\bar G$8 is twisted by the finite zero-form background and the action $\bar G$9 (Bhardwaj et al., 2022).

In four-dimensional $G$0 and $G$1 gauge theories, gauging a subgroup of the electric center one-form symmetry can convert an ordinary discrete chiral symmetry into a noninvertible zero-form symmetry. The Hamiltonian analysis on $G$2 constructs gauge-invariant operators by averaging the naive chiral symmetry operator over the center gauge transformations. The resulting operators act as projectors onto specific electric or magnetic flux sectors and exhibit mixed anomalies with the remaining one-form symmetries (Anber et al., 2023).

This noninvertible phenomenon is part of a broader pattern. Gauging a discrete symmetry that acts on an invertible one-form symmetry can reorganize that one-form symmetry into a noninvertible one. In gauge theories based on disconnected groups such as $G$3 or $G$4, obtained by gauging charge conjugation, the original center one-form symmetry of the connected theory becomes a noninvertible one-form symmetry in the disconnected theory. Its generators are Gukov–Witten operators with fusion rules such as

$G$5

and generic generators have quantum dimension $G$6, which precludes invertibility (Arias-Tamargo et al., 2022).

A related, but lower-dimensional, manifestation occurs in two-dimensional Yang–Mills, where noninvertible one-form symmetries can also be gauged. The paper introduces a generalized $G$7-angle for such orbifolds, now labeled by irreducible representations rather than ordinary phases. Gauging suitable noninvertible subsets can project the representation ring and still permit phenomena such as spontaneous breaking of charge conjugation (Santilli et al., 2024). This suggests that gauged center one-form symmetries are part of a larger hierarchy of gauged generalized symmetries, rather than an isolated construction.

The same logic extends to three-dimensional topological phases. To gauge a compact connected Lie group symmetry $G$8, one may first pass to a central extension $G$9 with kernel $Z_G$00, gauge $Z_G$01, and then gauge the diagonal one-form symmetry $Z_G$02 generated by center lines in the resulting Chern–Simons sector. The final modular tensor category is

$Z_G$03

Consistency requires the condensed lines to be bosonic and mutually transparent, expressed by

$Z_G$04

Failure of these conditions signals an ’t Hooft anomaly, so the obstruction to gauging a Lie-group symmetry can be rephrased as the impossibility of gauging an appropriate center one-form symmetry (Cheng et al., 2022).

5. Representative theories and dimensional realizations

The four-dimensional $Z_G$05 examples provide some of the sharpest physical applications. In adjoint QCD, the full center $Z_G$06 is exact, and gauging it reduces the discrete chiral symmetry $Z_G$07 to $Z_G$08. In several other $Z_G$09 theories with matter in self-adjoint antisymmetric or two-index representations, the exact subgroup $Z_G$10 can also be gauged, producing mixed anomalies with discrete chiral symmetries that strongly constrain the infrared phase (Bolognesi et al., 2019).

In four-dimensional supersymmetric gauge theory, the center one-form symmetry can play a more structural role even without being explicitly gauged. In pure $Z_G$11 super Yang–Mills in a self-dual $Z_G$12-background, surface operators associated with affine Dynkin nodes generate the center one-form symmetry. Their vacuum expectation values $Z_G$13 satisfy a non-autonomous Toda system in the RG scale

$Z_G$14

Although the work does not explicitly gauge the one-form symmetry, it shows that treating the center one-form symmetry generators as fundamental and constraining them by exact RG/Toda equations determines the full instanton series from perturbative data (Bonelli et al., 2021). A plausible implication is that center one-form symmetry data can control nonperturbative dynamics even before any explicit gauging operation is performed.

In three-dimensional Chern–Simons-matter theories with at least $Z_G$15, gauging center one-form symmetries appears as discrete quotients of gauge groups and generates new dualities. A central web involves

$Z_G$16

and its descendants obtained by gauging $Z_G$17 subgroups: $Z_G$18

$Z_G$19

The superconformal index tracks the discrete zero-form symmetries produced by these gauging operations and thereby identifies the underlying one-form structure (Beratto et al., 2021).

In two-dimensional Yang–Mills, the gauging operation is exact and geometric. The background 2-form gauge field is naturally described as a gerbe connection, and the full gauged theory can be formulated in higher-gauge terms using a non-abelian $Z_G$20-2-bundle with 2-connection $Z_G$21. The discrete $Z_G$22-angle corresponds to a coupling $Z_G$23, and gauging the center one-form symmetry yields the $Z_G$24 orbifold sectors of the theory (Santilli et al., 2024).

In gravitational contexts, the phrase “gauged center one-form symmetry” refers to the global form of the Lorentz group. In tetradic Palatini gravity, a center one-form symmetry associated with the center of the Lorentz group acts on spin holonomies. The no-global-symmetries principle then suggests that this symmetry must either be gauged, corresponding to quotienting the Lorentz group by its center, or explicitly broken by matter. The paper argues that if the center is not gauged, explicit breaking requires fermions, since spinor holonomies are the natural charged line operators (Cheung et al., 2024).

String-theoretic faithful probes sharpen this picture in quantum gravity. For theories with worldsheet current algebras realizing the gauge symmetry, gauged center one-form symmetries correspond to extensions of the left-moving Kac–Moody algebra by integral-spin simple currents. The existence of such simple-current extensions reproduces known field-theoretic and geometric constraints on gauging center one-form symmetries in six and eight dimensions and fixes the global form $Z_G$25 of the bulk gauge group (Lockhart et al., 12 May 2026).

6. Conceptual significance and common misconceptions

A common misconception is that gauging a center one-form symmetry is only a reformulation of changing the gauge group from $Z_G$26 to $Z_G$27. The literature shows that this is correct but incomplete. The operation also changes the spectrum of genuine line operators, introduces new background constraints, can mix with ordinary zero-form symmetries into connected or disconnected 2-groups, and may generate new dual higher-form symmetries or noninvertible operators (Bhardwaj et al., 2022).

Another misconception is that one-form symmetries can always be studied independently of ordinary global symmetries. When finite outer automorphisms act on the center, the appropriate symmetry object is often a disconnected 2-group rather than a direct product. In that setting, consistent gauging of the center one-form symmetry requires compatible zero-form backgrounds, and the obstruction is encoded by a twisted Postnikov class rather than by ordinary cohomology alone (Bhardwaj et al., 2022).

It is likewise misleading to regard gauging as an essentially Lagrangian operation. Defect-based, TQFT, and SymTFT formulations show that gauging is naturally an operation on topological defects or boundary conditions. In three-dimensional defect TQFT it is encoded by orbifold data (Carqueville et al., 9 Jun 2025); in the defect-condensation perspective it is a spacetime-filling gauge defect (Vandermeulen, 2023); and in four-dimensional SymTFT it is a choice of BF topological boundary condition (Duan et al., 2024). These formalisms make manifest why gauging reorganizes the full generalized-symmetry structure rather than only the local gauge algebra.

Finally, nontrivial consequences of gauged center one-form symmetries are not restricted to strongly coupled continuum gauge theory. They appear in exact orbifolds of two-dimensional Yang–Mills (Santilli et al., 2024), in modular tensor category constructions of three-dimensional topological phases (Cheng et al., 2022), in supersymmetric partition functions and duality webs of four-dimensional theories (Duan et al., 2024), in ABJ/ABJM-type Chern–Simons matter systems (Beratto et al., 2021), and in quantum-gravitational constraints on gauge group topology (Lockhart et al., 12 May 2026).

Taken together, these developments establish gauged center one-form symmetries as a unifying principle for the global form of gauge groups, the classification of line operators, the emergence of dual and higher-group symmetries, and the anomaly structure of gauge and gravitational theories across dimensions.