Zero-Form Non-Invertible Symmetries

• Zero-form non-invertible symmetries are codimension-one topological operators that lack inverses, creating non-group-like fusion rules in two-dimensional quantum field theories.
• They are characterized by a commutative algebra of dimension-zero twist fields whose idempotents project onto distinct universes in orbifold and gerbe models.
• The interplay between Wilson lines and projector operators reveals a representation-theoretic framework that distinguishes central (invertible) from non-central (non-invertible) symmetry sectors.

Zero-form non-invertible symmetries are generalized symmetry structures realized by codimension-one topological operators that lack inverses under fusion. Unlike standard global (invertible) symmetries whose operators implement group actions with strict inverses, these non-invertible symmetry defects fuse according to more elaborate algebraic rules—often involving additional topological sectors or summing over universes—and play a central role in the structure, decomposition, and constraint mechanisms in quantum field theories, particularly in two dimensions.

1. Dimension-Zero Operators and Decomposition

In two-dimensional orbifold and gerbe-type quantum field theories, zero-form non-invertible symmetries emerge as properties of the ring of dimension-zero operators—commutative, finite-dimensional algebras generated by topological “twist fields.” These twist fields, built from elements of the (twisted) group algebra of the orbifold group or its trivially-acting subgroup $K$, satisfy

$$\mathcal{O}[g] = \frac{1}{|[g]|} \sum_{h} w(h, g) w(hg, h^{-1}) T_{hgh^{-1}},$$

where $w$ encodes discrete torsion and $T_{k}$ denotes group algebra elements. The twist fields are central and thus generate a commutative algebra with non-singular OPE.

Decomposition is encoded in the spectrum of this ring: the ring can be written as $C[x_0, ..., x_n]/I$, where the ideal $I$ has finite zero locus, corresponding to “universes” or decoupled sectors. In prototypical examples, as in banded abelian gerbes, the algebra takes the form $\mathbb{C}[y]/(y^k - 1)$, so the $k$ roots of unity label $k$ universes. Each dimension-zero idempotent,

$$\Pi_R = \frac{\dim R}{|K|} \sum_{k \in K} \chi_R(k^{-1}) w(k, k^{-1}) T_k,$$

associated to an irreducible representation $R$ of the trivially-acting subgroup $K$, projects onto one universe; these satisfy $\Pi_R \Pi_S = \delta_{R, S} \Pi_R$, $\sum_R \Pi_R = 1$. The OPE/fusion algebra of dimension-zero operators operationalizes the theory’s decomposition into disjoint summands.

2. Non-Invertible Symmetries—Representation-Theoretic Characterization

The ring structure of dimension-zero operators determines the higher-form and, crucially, zero-form symmetry content. The key insight is that universes corresponding to inequivalent irreducible representations of $K$ encode non-invertible symmetry sectors. More precisely:

• If all representations are one-dimensional (the central/“banded” case), the associated one-form symmetry is invertible and group-like, and universes differ only by phase.
• If some representations are higher-dimensional (the non-central/nonabelian case), the corresponding fusion rules for twist fields or projectors are non-group-like and non-invertible: they do not admit inverses. Such sectors manifest intrinsic noninvertibility in the associated higher-form symmetry.

Thus, the relation between invertibility and the algebraic/representation-theoretic structure of the orbifold group (and its trivially-acting subgroups) directly determines the emergence of non-invertible zero-form (and higher-form) symmetries.

3. Topological Operators, Wilson Lines, and Fusion Structure

Topological, dimension-zero operators (twist fields) act as defect insertions that “probe” or “relabel” universes in the decomposed theory. Bulk Wilson lines constructed in the orbifold or gerbe gauge theory correspond to topological defects sensitive to the universes they pierce. Transporting a Wilson line through a twist field effect induces a “clock shift”—a discrete phase determined by the underlying (projective) representation data. The explicit commutation relations between Wilson lines and projectors realize the structure:

• Bulk Wilson lines shift brane labels between universes.
• Boundary Wilson lines (Chan–Paton factors) are diagonalized by projectors and thus tied to individual universes.

The OPE algebra of these topological operators matches the commutative ring of dimension-zero operators. The projectors act as universal probes, defining and isolating sectors in the decomposed Hilbert space and their action on both bulk and boundary operators implements the direct sum structure at the level of correlation functions.

4. One-Form and Higher-Form Symmetry Structure

Symmetry content, including invertibility or non-invertibility, is reflected in the multiplication rules for twist fields:

• In the central case, $\mathcal{O}(g) \mathcal{O}(g') = \mathcal{O}(gg')$, so the ring is isomorphic to $\mathbb{C}[y]/(y^k-1)$. The $B\mathbb{Z}_k$ one-form symmetry acts as $y \mapsto \exp(2\pi i/k)\,y$, preserving the algebra; these are invertible.
• In the non-central or nonabelian case, twist fields and associated projectors do not close under group multiplication but have more general fusion rules. In these settings—precisely when universes correspond to non-isomorphic, higher-dimensional representations—the higher-form symmetry is non-invertible.

The explicit algebraic relations—especially the formulas for idempotent projectors—enable case-by-case determination of invertibility. Projectors commute with all line operators but, crucially, can shift or relabel boundary data, further reflecting the underlying non-group-like symmetry operation in the non-invertible case.

5. Geometric and Commutative Algebraic Perspective

From commutative algebra, the vacuum spectrum of the ring of dimension-zero operators forms a finite set of points, echoing the geometric decomposition into universes. Decomposition is thus geometrically encoded in the spectrum of the commutative fusion algebra and algebraically in the isolated zeros of the ideal $I$. Non-invertibility can also be detected by examining the vanishing of the determinant of the fusion matrix. This formalizes the appearance of codimension-one loci where invertibility fails—directly matching the loci where the associated representations of $K$ become higher-dimensional.

6. Summary and Significance

The structure of zero-form non-invertible symmetries in two-dimensional orbifold and gerbe quantum field theories is determined by the commutative algebra of topological, dimension-zero twist fields. The ring’s idempotents (projector operators) label disjoint universes in the decomposition, and their algebra precisely delineates when higher-form symmetries are invertible or non-invertible. Non-invertibility arises exactly when universes correspond to inequivalent, higher-dimensional representations of the trivially-acting subgroup. The interplay between Wilson lines, projectors, and twist fields operationalizes this decomposition at the level of concrete correlation functions and operator actions.

This framework provides both a representation-theoretic and geometric understanding of how non-invertible symmetries arise and are encoded in two-dimensional field theory, and it supplies explicit calculational tools (notably, the general formulas for projection operators) for analyzing and detecting non-invertible symmetry sectors (Sharpe, 2021).