Topological Vector Symmetry of BRSTQFT and Construction of Maximal Supersymmetry
Laurent Baulieu, Guillaume Bossard, Alessandro Tanzini
TL;DR
This work provides a geometrical construction of scalar and vector topological BRST operators $s$ and $\delta$ via horizontality on reducible manifolds, forming a closed subalgebra that determines maximal Yang–Mills supersymmetry upon twisting. By coupling gravity and Yang–Mills and introducing background connections, the authors achieve globally well-defined transformation laws and an off-shell closed algebra, with an extended horizontality condition that incorporates a covariantly constant vector $\kappa$ to realize the vector symmetry. The action is shown to be expressible as an $s$-exact (and $s\delta$-exact) term, with a ghost–antighost duality, and the conserved Noether antecedent of the energy–momentum tensor generates the vector symmetry. Equivariant extensions on Omega backgrounds further connect the framework to twisted Super Yang–Mills theories, and dimensional reduction recovers familiar SUSY structures in 4D and 8D, providing a unified geometrical route to maximal supersymmetry through topological symmetry principles.
Abstract
The scalar and vector topological Yang-Mills symmetries determine a closed and consistent sector of Yang-Mills supersymmetry. We provide a geometrical construction of these symmetries, based on a horizontality condition on reducible manifolds. This yields globally well-defined scalar and vector topological BRST operators. These operators generate a subalgebra of maximally supersymmetric Yang-Mills theory, which is small enough to be closed off-shell with a finite set of auxiliary fields and large enough to determine the Yang-Mills supersymmetric theory. Poincaré supersymmetry is reached in the limit of flat manifolds. The arbitrariness of the gauge functions in BRSTQFTs is thus removed by the requirement of scalar and vector topological symmetry, which also determines the complete supersymmetry transformations in a twisted way. Provided additional Killing vectors exist on the manifold, an equivariant extension of our geometrical framework is provided, and the resulting "equivariant topological field theory" corresponds to the twist of super Yang-Mills theory on Omega backgrounds.