Parent formulation at the Lagrangian level
Maxim Grigoriev
TL;DR
The paper develops a Lagrangian version of the parent formulation, showing that for diffeomorphism-invariant theories the construction yields an AKSZ-type sigma model whose target is the BV jet space of the starting theory. It provides a BV master action $S^P$ and demonstrates, via a truncation scheme, that the parent theory is classically equivalent to the original Lagrangian theory through elimination of generalized auxiliary fields. The approach unifies BV-BRST with unfolded formalisms and connects to BFV-BRST via a multidimensional extension, illustrated through mechanics, the relativistic particle, Yang–Mills-type theories, and metric gravity, where reductions reproduce familiar first-order or Hamiltonian formulations. The framework promises a systematic path to derive frame-like and higher-spin actions from metric-like theories and offers a potential bridge to Vasiliev’s higher-spin system and related geometric formalisms. Overall, the work provides a rigorous, covariant method to generate and analyze parent formulations at the Lagrangian level with broad applicability across gauge theories.
Abstract
The recently proposed first-order parent formalism at the level of equations of motion is specialized to the case of Lagrangian systems. It is shown that for diffeomorphism-invariant theories the parent formulation takes the form of an AKSZ-type sigma model. The proposed formulation can be also seen as a Lagrangian version of the BV-BRST extension of the Vasiliev unfolded approach. We also discuss its possible interpretation as a multidimensional generalization of the Hamiltonian BFV--BRST formalism. The general construction is illustrated by examples of (parametrized) mechanics, relativistic particle, Yang--Mills theory, and gravity.