Characteristic cohomology of $p$-form gauge theories

TL;DR

This work determines the characteristic cohomology $H^k_{char}(d)$ for a system of free $p$-form gauge fields in $n$-dimensional spacetime, proving it is finite-dimensional for all $k<n-1$ and completely generated by the dual forms ${\overline H}^a$ to the field strengths. It establishes the invariant characteristic cohomology, generated by $H^a$ and ${\overline H}^a$, and proves the fundamental isomorphism $H^k(\Delta)\simeq H^k_{char}(d)$ with $\Delta=\delta+d$, relating the characteristic cohomology to the Koszul-Tate and BRST frameworks. The paper also provides a complete analysis for a single $p$-form and extends to arbitrary systems of $p$-forms, detailing the structure of $H^n_{\;j}(\delta|d)$ and the consequences for BRST cohomology and potential gauge-invariant interactions. Overall, the results constrain higher-degree conservation laws, demonstrate the robustness of the characteristic cohomology under gauge-invariant deformations, and inform the construction of consistent interactions and anomalies in BRST cohomology.

Abstract

The characteristic cohomology $H^k_{char}(d)$ for an arbitrary set of free $p$-form gauge fields is explicitly worked out in all form degrees $k<n-1$, where $n$ is the spacetime dimension. It is shown that this cohomology is finite-dimensional and completely generated by the forms dual to the field strengths. The gauge invariant characteristic cohomology is also computed. The results are extended to interacting $p$-form gauge theories with gauge invariant interactions. Implications for the BRST cohomology are mentioned.

Theorems & Definitions (19)

• Theorem 2.1
• Theorem 2.2
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 4.1
• Theorem 5.1
• Theorem 6.1
• Theorem 6.2
• Theorem 6.3