arXiv subjects

Dmitry Rudinsky

Publications and source records attributed to Dmitry Rudinsky.

3 recordsLinked to original sources

arXiv2026

Presymplectic BV-AKSZ and constrained DGCAs

The AKSZ construction encodes the Batalin-Vilkovisky formulation of a topological field theory in terms of finite-dimensional geometric data. There are at least two ways to extend this approach to nontopological gauge theories. The first, due to Costello, replaces the spacetime exterior algebra with a more general DGCA that is not freely generated. The second replaces the symplectic structure with a degenerate presymplectic structure, which may also be non-regular. In this work, we show that both approaches are special cases of a more general algebraic AKSZ construction, in which the underlying algebras are allowed to be constrained and the presymplectic structure may be degenerate and non-regular. Within this framework, we explicitly relate Costello's formulation of the Chalmers-Siegel model to the presymplectic formulation, the latter admitting a first-principles derivation. We also construct Costello-like formulations of higher-form analogues of the Chalmers-Siegel model and of self-dual higher-spin gauge theories of Yang-Mills type, obtaining in the latter case a remarkably concise description. In so doing we propose a simple algebraic construction for the corresponding source space DGCAs underlying these examples.

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arXiv2024

Weak Gauge PDEs

Gauge PDEs generalise the AKSZ construction when dealing with generic local gauge theories. Despite being very flexible and invariant, these geometrical objects are usually infinite-dimensional and are difficult to define explicitly, just like standard infinitely-prolonged PDEs. We propose a notion of a weak gauge PDE in which the nilpotency of the BRST differential is relaxed in a controllable way. In this approach a nontopological local gauge theory can be described in terms of a finite-dimensional geometrical object. Moreover, among the equivalent weak gauge PDEs describing a given system, a minimal one can usually be found and is unique in a certain sense. In the case of a Lagrangian system, the respective weak gauge PDE naturally arises from its weak presymplectic formulation. We prove that any weak gauge PDE determines the standard jet-bundle Batalin-Vilkovisky formulation of the underlying gauge theory, giving an unambiguous physical interpretation of these objects. The formalism is illustrated by a few examples, including the non-Lagrangian self-dual Yang-Mills theory and a finite jet-bundle. We also discuss possible applications of the approach to the characterisation of those infinite-dimensional gauge PDEs that correspond to local theories.

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arXiv2023

Notes on the $L_\infty$-approach to local gauge field theories

It is well known that a $Q$-manifold gives rise to an $L_\infty$-algebra structure on the tangent space at a fixed point of the homological vector field. From the field theory perspective this implies that the expansion of a classical Batalin-Vilkovisky (BV) formulation around a vacuum solution can be equivalently cast into the form of an $L_\infty$-algebra. In so doing, the BV symplectic structure determines a compatible cyclic structure on the $L_\infty$-algebra. Moreover, $L_\infty$ quasi-isomorphisms correspond to so-called equivalent reductions (also known as the elimination of generalized auxiliary fields) of the respective BV systems. Although at the formal level the relation is straightforward, its implementation in field theory requires some care because the underlying spaces become infinite-dimensional. In the case of local gauge theories, the relevant spaces can be approximated by nearly finite-dimensional ones through employing the jet-bundle technique. In this work we study the $L_\infty$-counterpart of the jet-bundle BV approach and generalise it to a more flexible setup of so-called gauge PDEs. It turns out that in the latter case the underlying $L_\infty$-structure is analogous to that of Chern-Simons theory. In particular, the underlying linear space is a module over the space-time exterior algebra and the higher $L_\infty$-maps are multilinear. Furthermore, a counterpart of the cyclic structure turns out to be degenerate and possibly nonlinear, and corresponds to a compatible presymplectic structure which is known to encode the BV symplectic structure and hence the full-scale Lagrangian BV formulation. Moreover, given a degenerate cyclic structure one can consistently relax the $L_\infty$-axioms in such a way that the formalism still describes non-topological models but involves only finitely-generated modules, as we illustrate in the example of Yang-Mills theory.

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