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

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 \v{C}ech 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 \v{C}ech-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.