Abstract
We study the Lagrangian antifield BRST formalism, formulated in terms of exterior horizontal forms on the infinite order jet space of graded fields for topological field theories associated to \(Q\)-bundles. In the case of a trivial \(Q\)-bundle with a flat fiber and arbitrary base, we prove that the BRST cohomology are isomorphic to the cohomology of the target space differential “twisted” by the de Rham cohomology of the base manifold. This generalizes the local result of G. Barnich and M. Grigoriev, computed for a flat base manifold.
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Notes
- 1.
Hereafter one has \(x_{\scriptscriptstyle l}=\pi _{\scriptscriptstyle k,l} (x_{k})\) for all \(k\ge l\) and \(x=\pi _{\scriptscriptstyle k} (x_{k})\) for all \(k\ge 0\), unless the contrary is expressed.
- 2.
Direct limit of differential forms and embeddings induced by the projections \(\pi \) and \(\pi _{k+1,k}\).
- 3.
It is based on the fact that jet bundles \(\mathrm {J}^{k+1}(\pi )\rightarrow \mathrm {J}^{k}(\pi )\) are affine.
- 4.
Symmetric powers of the dual bundle.
- 5.
The bifunctors \(\mathrm {Hom}\) and \(\otimes \) are defined over \(\mathcal{O}_X\).
- 6.
Geometrically it means that those sections are tangent to the \(Q\)-stucture on the total space.
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Bonavolontà, G., Kotov, A. (2015). Local BRST Cohomology for AKSZ Field Theories: A Global Approach. In: Calaque, D., Strobl, T. (eds) Mathematical Aspects of Quantum Field Theories. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-09949-1_10
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