Gauge PDE and AKSZ‐type Sigma Models

Abstract

A gauge PDE is a natural notion which arises by abstracting what physicists call a local gauge field theory defined in terms of BV‐BRST differential (not necessarily Lagrangian). We study supergeometry of gauge PDEs paying particular attention to globally well‐defined definitions and equivalences of such objects. We demonstrate that a natural geometrical language to work with gauge PDEs is that of Q‐bundles. In particular, we demonstrate that any gauge PDE can be embedded into a super‐jet bundle of the Q‐bundle. This gives a globally well‐defined version of the so‐called parent formulation. In the case of reparameterization‐invariant systems, the parent formulation takes the form of an AKSZ‐type sigma model with an infinite‐dimensional target space.

Matter-free higher spin gravities in 3D: Partially-massless fields and general structure

“…Furthermore, in this approach one can introduce a general notion of equivalence of local gauge field theories [60] that covers theories related by elimination of auxiliary fields as well as by elimination of Stueckelberg fields. This notion extends [61], see also [5,62], to systems defined at the level of equations of motion in which case it also extends to a more general geometrical setting [63].…”

“…Note though that for the higher spin system under consideration negative degree Ψ are not present anyway. Then one subjects Φ A k to the following equations [62,63]:…”

“…This approach to general gauge theories was originally developed [4,5,61,62] under the name of parent formulation. Its more invariant and geometrical version was proposed recently in [63], where it was also explicitly related to the invariant approach to PDEs [48] (for a review see e.g., [49,119]). In particular, the BRST complex with total differential Q 0 can be seen as a BRST extension of the infinitely prolonged PDE.…”

“…Equations (4.5) have a simple geometrical interpretation [63]: fields are components of a section σ∶T½1X → "jetspace" (jet-space is naturally a bundle over T½1X, i.e., the spacetime X extended by θ a ), i.e., Φ A k ðx; θÞ ¼ σ Ã ðΨ A k Þ, where σ Ã is a pullback map induced by σ, while the equations of motion (4.5) say that σ is a Q-map (i.e., d X ∘ σ Ã ¼ σ Ã ∘ Q 0 , or in other words σ Ã is a map of the respective homological complexes). Gauge transformations correspond to trivial deformations of σ, i.e., those of the form δ ϵ σ Ã ¼ d X ∘ ϵ Ã þ ϵ Ã ∘ Q 0 for some map ϵ Ã of degree −1, which encodes gauge parameters.…”

“…Furthermore, the minimal BRST complex for partiallymassless or conformal fields in 3D is also of the form (4.9). The only difference is that the module A has to be replaced 15 In terms of the field theory determined by Q 0 through (4.5) elimination of such contractible pairs corresponds to fixing Stueckelberg gauge (for gauge) symmetries and elimination of auxiliary fields, see [62,63] for more details. 16 In (4.8) we do not take an advantage of dualizing the spinconnection ω a 1 …a s−1 ;b into ω a 1 …a s−1 as in (2.3).…”