On Quantum Aspects of 1-Form Symmetries I: BV-BRST Cohomology and Anomaly Polynomials

Abstract: We investigate the quantum aspects of gauging continuous 1-form global symmetries. In this paper, we study the BV-BRST quantization of a $U(1)$ 2-form gauge field, described geometrically by a $U(1)$ gerbe. Starting from the local Čech data of the gerbe, we construct the corresponding infinitesimal symmetry structure in terms of a Lie 2-algebroid, and show that, together with the associated exact Courant algebroid, it provides a natural geometric framework for the BV-BRST complex of this higher-form gauge theory. In this formulation, the field-ghost tower is encoded directly in the local gerbe data, and the higher Russian formula arises naturally from the relations among the connective structure, the curving, and the 3-form curvature. We further show that the resulting Čech-de Rham bicomplex provides a natural setting for anomaly descent for $U(1)$ 1-form symmetries, and illustrate the construction with explicit examples in Maxwell theory.

• The paper constructs a comprehensive BV–BRST complex for abelian 1-form symmetries using gerbes and Lie 2-algebroids.
• It develops a cohomological descent method to derive gauge-invariant anomaly polynomials, clarifying anomalies in Maxwell theories.
• The geometric reformulation through higher algebroid structures provides new tools for quantizing complex gauge systems and addressing mixed anomalies.

Quantum Aspects of 1-Form Symmetries: BV-BRST Cohomology and Anomaly Polynomials

---

Introduction

This paper develops a comprehensive geometric and cohomological framework for the quantization and anomaly analysis of theories with continuous 1-form global symmetries, focusing on the abelian $U(1)$ 2-form gauge field case (the geometric $U(1)$ gerbe). The authors construct and interpret the minimal BV–BRST complex within higher algebroid geometry, introduce the relevant higher Russian formula, and systematically relate this to the local data of gerbes and associated exact Courant algebroids. The descriptive machinery culminates in a cohomological construction of anomaly polynomials for 1-form symmetries, directly enabling the identification and characterization of potential quantum anomalies in such higher-form-symmetric systems (2606.05656).

---

Geometric Reformulation of Gauge Symmetry and BRST/Anihilation Structure

The analysis begins by revisiting the geometric interpretation of gauge symmetry in 0-form cases, utilizing the Atiyah Lie algebroid of a principal $U(1)$-bundle. In this context, the BRST complex emerges naturally as the graded exterior algebra on the algebroid with the Lie algebroid differential, whose consistent ghost-number splitting yields the standard BRST operator. The so-called Russian formula, encapsulating the cohomological descent and horizontality of the curvature, is recast in a local Čech–de Rham bicomplex—setting the stage for higher-form generalization.

Key insight is that, for 1-form symmetries, the geometric structure transitions from principal bundles to $U(1)$ gerbes. The corresponding infinitesimal symmetries are encoded not in an ordinary Lie algebroid but in a higher, 2-categorical structure: the Lie 2-algebroid, expressed explicitly in terms of Čech cocycle data. This approach facilitates a direct local understanding of the minimal BV field-ghost tower (the collection $(B,\,C,\,c)$ in the abelian 2-form case), associating ghost number to Čech degree, and replaces the bundle-theoretic structure by the local patching and gluing maps intrinsic to gerbes.

The authors detail that the Čech–de Rham bicomplex, consisting of both the de Rham differential $d$ and the Čech coboundary $\delta$ (identified physically with the BRST operator $s$), organizes the entire extended field content and gauge structure.

---

Higher Algebroid and Courant Algebroid Constructions

The central mathematical development is the systematic construction of:

1. Čech Lie 2-algebroid: The infinitesimal symmetries of a $U(1)$ gerbe are assembled into an object whose local data encodes lifts of vector fields (gauge transformations) together with the appropriate higher patching conditions determined by the gerbe cocycle.
2. Connective Structure and Curving: The connective structure (collection $C_{ij}$) and its curving (collection $U(1)$0) solve cocycle conditions expressing how the local symmetry data glue, mirroring how local 1-forms for connections glue for bundles but now at higher degree.
3. Exact Courant Algebroid: The transition to the Courant algebroid is established by showing that a $U(1)$1 gerbe with connective structure canonically yields an exact Courant algebroid, with the curving $U(1)$2 determining an isotropic split; the global curvature is assembled into a closed 3-form $U(1)$3.
4. Higher Russian Formula: The nested local patching and the cohomological structure of the extended field/gauge tower are unified in a higher Russian formula,

$U(1)$4

encoding both the BRST/BV transformations and the global curvature.

This geometric perspective gives a precise understanding of reducible gauge systems in the BV formalism and clarifies the physical hierarchy of ghosts and ghost-for-ghosts as arising from the layers of patching in the Čech bicomplex.

---

Anomaly Polynomials and Descent in Higher-Form Symmetry

With the higher Russian formula in place, the paper develops the cohomological descent formalism for anomalies associated with 1-form symmetries:

• Anomaly Polynomials: Given the globally-defined ghost-free curvature 3-form $U(1)$5, one can construct closed, gauge-invariant forms $U(1)$6—the anomaly polynomials—which serve as the topological characteristic classes diagnosing anomalies in background $U(1)$7-fields.
• Descent Equations and BV–BRST Cohomology: The expansion of associated Chern–Simons forms in the Čech–de Rham bicomplex yields a hierarchical descent structure whose ghost-number one component encodes the consistent local anomaly.

Notably, self-anomalies for a single abelian 1-form symmetry vanish at the perturbative (local polynomial) level, since $U(1)$8, but nontrivial mixed (or gauge-gravitational) anomalies can and do arise.

---

Applications: Abelian Maxwell Theories and Explicit Anomaly Computations

The formalism is illustrated in concrete physical systems:

• $U(1)$9 Maxwell Theory: The distinguished mixed 't Hooft anomaly for electric and magnetic $U(1)$0 1-form symmetries is recovered and interpreted as both a mixed anomaly and an ABJ-type nonconservation under gauging. The consistent anomaly polynomial $U(1)$1, which cannot be removed by local counterterms, is derived from the higher-form structure.
• $U(1)$2 Maxwell Theory: Gauge-gravitational anomalies with anomaly polynomial $U(1)$3 (with $U(1)$4 a closed degree-4 gravitational form) are characterized, and their implications for the structure of the theory and the possible Green–Schwarz mechanism are analyzed.

The results confirm that the geometric (higher-algebroid) perspective provides a uniform mechanism to diagnose and classify anomalies in higher-form symmetric QFTs.

---

Implications and Further Developments

The formalization of (quantum) 1-form symmetry via higher algebroid and Courant algebroid geometry reveals deep connections between cohomological quantization, local-to-global data in gauge theory, and the structure of quantum anomalies. Several directions are indicated for further research:

• Extension to Non-Abelian and Higher-Form Symmetries: While the current analysis is strictly abelian, the geometric framework is suggestive for non-abelian and more general $U(1)$5-form symmetries, provided the appropriate higher algebroid structures can be identified.
• Relation to Higher Principal Bundles and Representation Theory: The precise global geometric object underlying the local Lie 2-algebroid presentation, and how to naturally incorporate matter and nontrivial representations, remains to be developed.
• Unified Higher Structure: The interplay between the Lie 2-algebroid and its associated Courant algebroid hints at the existence of a unified cohomological structure, potentially providing a direct construction of the total BV field.
• Covariant Anomaly Structure and Higher-Group Symmetry: Further study is warranted on "covariant splitting" and the connections with higher-group symmetry (and higher gauge structures in condensed matter and string theory).

---

Conclusion

This work rigorously establishes and elucidates the geometric, cohomological, and anomaly-theoretic structure of theories with abelian continuous 1-form symmetry, by developing the BV–BRST complex in terms of gerbes, Lie 2-algebroids, and exact Courant algebroids. The resulting higher Russian formula and Čech–de Rham bicomplex organize the quantization and consistent anomaly structure in a canonical way. These insights not only clarify the landscape of quantum anomalies for higher-form symmetries but also suggest a robust geometric platform for the study of more general higher gauge theories and their quantum properties (2606.05656).