Weyl-Compensated BRST Complex

• Weyl-compensated BRST complex is a cohomological structure that extends standard BRST formalism by systematically incorporating local Weyl (scale) symmetry alongside diffeomorphism invariance.
• It employs multiple nilpotent BRST differentials and composite ghosts to ensure compatible gauge fixing and to address cohomological obstructions in both Lagrangian and non-Lagrangian dynamics.
• This framework underpins advanced applications in quantum gravity, conformal field theory, and holography by refining anomaly computations and extending classical symmetry correspondences.

The Weyl-compensated BRST complex is a cohomological and algebraic structure that generalizes the standard BRST formalism to situations in which local Weyl (scale) symmetry, possibly along with diffeomorphism and other gauge symmetries, is present and must be systematically incorporated into the quantization and symmetry analysis of gauge theories and gravitation. Its development is critical for quantizing systems with scale invariance, analyzing quantum anomalies, and extending classical correspondence theorems (such as Noether’s) beyond the Lagrangian framework.

1. Formulation and Algebraic Structure

The Weyl-compensated BRST complex captures gauge symmetries, including local scale transformations (Weyl symmetry), within a combined cohomological structure. The construction employs multiple nilpotent BRST differentials, each associated with a distinct gauge invariance—typically one for diffeomorphisms and another for Weyl symmetry (Oda et al., 2022). Denote by $Q_\mathrm{GCT}$ the BRST operator for general coordinate transformations and by $Q_\mathrm{Weyl}$ the operator for Weyl rescalings. The total BRST differential is often written as $Q_\mathrm{tot} = Q_\mathrm{GCT} + Q_\mathrm{Weyl}$, with the crucial property $$Q_\mathrm{GCT}^2 = Q_\mathrm{Weyl}^2 = Q_\mathrm{GCT}Q_\mathrm{Weyl} + Q_\mathrm{Weyl}Q_\mathrm{GCT} = 0$$ in compatible formalisms (Garcia-Lopez et al., 2023).

Gauge fixing in these contexts requires compatible choices for each symmetry, such as an "extended de Donder condition" for general covariance and separate scalar gauge-fixing for the Weyl symmetry, producing a quantum Lagrangian invariant under both BRST charges. Key BRST transformations include

$$\delta_B g_{\mu\nu} = - c^\rho \partial_\rho g_{\mu\nu} - (\partial_\mu c^\rho) g_{\rho\nu} - (\partial_\nu c^\rho) g_{\mu\rho}$$

for diffeomorphisms and

$\delta_B S_\mu = -\partial_\mu c + \ldots$

for the Weyl gauge field, with further terms guaranteeing nilpotency and compatibility (Oda et al., 2022).

2. Lagrange Structures and Extension Beyond Lagrangian Dynamics

To generalize the BRST formalism to non-Lagrangian theories, the concept of "Lagrange structure" is introduced as an essential compensator that replaces the need for an action functional (Kaparulin et al., 2011). The Lagrange structure is defined as a representative in a local BRST cohomology group, specifically $H^2(\delta|d)$. It provides a way to establish a (potentially nonbijective) correspondence between rigid symmetries (global invariances not arising from gauge freedom) and conservation laws (characteristics). The fundamental mapping is

$$f_A : \mathrm{Char}(T) \longrightarrow \mathrm{RSym}(T)$$

where $\mathrm{Char}(T)$ is the space of conservation laws (identified with $H_{-1}(8|d)$) and $\mathrm{RSym}(T)$ is the space of rigid symmetries ($H^0(8|d)$). The extension of Noether’s theorem in this context is realized through the Lagrange structure, which defines derived brackets among conservation laws: $(V_1, V_2)_A = \{\{A, V_1\}, V_2\}$ generalizing the Dickey bracket and ensuring closure properties if the integrability condition (Maurer–Cartan-type, $\{A, A\}=0$) holds (Kaparulin et al., 2011).

3. Gauge Fixing, Cohomological Obstructions, and Composite Ghosts

Gauge-fixing in systems with Weyl symmetry is considerably more intricate. The dressing field method systematically reduces the gauge group to residual symmetries—in gravitational theories, this process yields composite ghosts such as

$v'_0 = v_0 + i_\xi \varpi_0 + v_\xi$

where $v_0$ is the Weyl ghost, $\varpi_0$ the dressed Cartan connection, and $v_\xi$ the diffeomorphism ghost. The composite ghost governs the transformations in the resulting Weyl-covariant tensor calculus, with equations such as

$$\varpi_0 = - \mathcal{D}_0 v_0 + \mathcal{L}_\xi \varpi_0 - dv_\xi$$

determining the cohomological structure (François et al., 2015).

The existence of a well-defined local BRST charge, particularly in non-Lagrangian settings, is subject to obstruction classes in higher-degree cohomology and to explicit conditions involving Massey products, e.g.,

$(A,A,\ldots,A)=0$

up to finite order, with $A$ the Lagrange structure class (Kaparulin et al., 2011). Ambiguities in the BRST charge can be eliminated by canonical transformations, leading to equivalence classes of BRST complexes parameterized by cohomology.

4. Geometric and Lie Algebroid Interpretations

Recent advances re-encode the BRST complex in the exterior algebra of an Atiyah Lie algebroid constructed from the principal bundle of the gauge theory. In this setting, ghost fields are naturally associated with vertical components, and the differential is recast as the Chevalley–Eilenberg differential (Jia et al., 2023, Jia, 2 Jul 2024). This viewpoint reveals that isomorphic algebroids (related via local diffeomorphisms and gauge transformations) possess equivalent cohomology, ensuring the geometric invariance of quantum anomalies and the robustness of anomaly computations.

The Lie algebroid approach makes explicit the compensation of Weyl symmetry by introducing connection reforms that incorporate both gauge and Weyl connections: $$\omega_\tau = b_{\text{spin}} + a_{\text{Weyl}} - (\varpi_L + \varpi_W)$$ with the Maurer–Cartan form (ghost) for Weyl symmetry appearing with the opposite sign—this guarantees that curvature tensors are "compensated" and hence ghost-free for anomaly computation. Weyl-obstruction tensors arising in holographic renormalization (e.g., as poles in the Fefferman–Graham expansion) play a central role in the geometric construction of the Weyl anomaly (Jia, 2 Jul 2024).

5. Quantum Anomalies, Cohomology, and Matching

In Weyl-compensated BRST complexes, quantum anomalies materialize both as algebraic obstructions and as geometric objects. For instance, in even bulk boundary dimension, the Weyl anomaly is built from Weyl-obstruction tensors $\hat{\Omega}^{(k)}_{ij}$, which transform covariantly under Weyl rescalings, $$\hat{\Omega}^{(k)}_{ij}\to {\cal B}(x)^{2k}\,\hat{\Omega}^{(k)}_{ij}$$ (Jia, 2 Jul 2024). These tensors become the foundation for constructing the anomaly polynomial via descent equations, and by encoding anomalies through local cohomology and Lie algebroid data, matching between consistent and covariant forms follows as simple basis changes in the complex.

The machinery also accounts for anomaly matching under spontaneous breaking of Weyl-compensated BRST invariance, as demonstrated for $R^2$ gravity: the anomaly persists in both unbroken and broken phases, with matching trace anomalies between the Weyl-invariant (pure $R^2$) phase and the Einstein phase post breaking (Edery, 2023).

6. Variants, Extensions, and Duality

The BRST symmetry in Weyl-compensated complexes can possess extensions (e.g., sp(2) symmetry (Constantinescu et al., 2011)), dual forms (dual- and anti-dual-BRST (Mandal et al., 2022)), and can be given in terms of double complexes with anti-BRST partners (Prinz, 2022). Mutual anticommutation among the differentials allows the formation of total BRST and anti-BRST operators,

$$D := Q_\mathrm{GCT} + Q_\mathrm{Weyl}, \quad \bar{D} := \bar{Q}_\mathrm{GCT} + \bar{Q}_\mathrm{Weyl}$$

whose cocomplex cohomologies are isomorphic via ghost conjugation. Physical states are thus defined equally well in terms of BRST or anti-BRST cohomology.

7. Applications and Impact

The Weyl-compensated BRST complex is pivotal in modern quantum gravity, conformal field theory, string theory (particularly pure spinor and ambitwistor formulations (Mikhailov et al., 2013, Figueroa-O'Farrill et al., 17 Jul 2024)), and holographic duality. It underlies systematic quantization and the computation of gauge and Weyl anomalies, the construction of tensor calculi, and the extension of topological field theory frameworks. Its realization as Lie algebroid cohomology offers a powerful geometric language that bridges algebraic and differential geometric formulations and suggests new avenues for treating generalized symmetries, higher gauge theory, and anomaly matching in quantum field theory.

Summary Table: Core Features

Feature	Description	Source
BRST compensation	Incorporates Weyl symmetry into cohomological structure	(Jia et al., 2023, Kaparulin et al., 2011, Jia, 2 Jul 2024)
Lagrange structure	Extends Noether theorem beyond Lagrangian dynamics	(Kaparulin et al., 2011)
Composite ghost	Combines gauge, Weyl, diffeomorphism ghosts	(François et al., 2015, Garcia-Lopez et al., 2023)
Lie algebroid interpretation	BRST complex as exterior algebra of Atiyah algebroid	(Jia et al., 2023, Jia, 2 Jul 2024)
Quantum anomaly construction	Weyl-obstruction tensors, anomaly matching	(Edery, 2023, Jia, 2 Jul 2024)
Cohomology and duality	Double complexes, anti-BRST, dualities	(Prinz, 2022, Mandal et al., 2022, Constantinescu et al., 2011)

This rigorous, multi-faceted structure generalizes BRST quantization to encompass scale invariances, accommodates non-Lagrangian field theories, and geometrizes the computation of quantum anomalies, providing essential tools for modern theoretical physics.