Boundary structure of gauge fields on asymptotically AdS spaces
Maxim Grigoriev, Mikhail Markov
TL;DR
This paper develops a gauge-PDE framework to analyze the boundary structure of gauge theories on asymptotically AdS spaces, using a boundary-defining field $\Omega$ to implement Penrose's asymptotically simple boundary within a $Q$-manifold formalism. It constructs a systematic boundary calculus that yields explicit boundary equations and gauge transformations for gravity, scalar fields, and Yang–Mills fields, including the Bach/obstruction tensors and generalized conservation laws; for $d=8$ it derives a higher conformal Yang–Mills equation. The approach extends the Fefferman–Graham program by incorporating both leading and subleading bulk data into a boundary conformal geometry, clarifying the structure of subleading boundary modes as a linear system over a nonlinear leading bulk background. The results provide a general, coordinate-free method to obtain boundary dynamics in AAdS holography, with concrete formulas for odd and even boundary dimensions and new boundary theories such as higher-dimensional conformal YM. Overall, the work offers a versatile framework to study boundary gauge theories on AdS backgrounds and their holographic implications through a rigorous, recursive boundary calculus.
Abstract
We study boundary structure of asymptotically AdS gravity and (gauge) fields defined on this background by employing the gauge PDE approach. The essential step of the construction is the incorporation of the boundary-defining function among the fields of the theory, which allows us to realise the asymptotic boundary as a space-time submanifold by employing the gauge PDE implementation of Penrose's concept of asymptotically-simple space. In so doing the gauge PDE describing the boundary structure is obtained by restricting to the boundary of spacetime and simultaneously restricting to the boundary of the field space by setting the boundary defining function to zero. To implement this step systematically we introduce a notion of $Q$-boundary which seems to be new. The main concrete result of this work is the construction of the efficient boundary calculus, which gives a recursive procedure to obtain the explicit form of the equations satisfied by the boundary fields and their gauge transformations for boundary dimension $d \geq 3$. These include obstruction equations (such as Bach equation or Yang-Mills equation for $d=4$) and generalised conservation equations in the subleading sector. In particular, we derive the explicit form of the higher conformal Yang-Mills equation for $d=8$. The approach is very general and, in principle, applies to generic (gauge) fields on the Einstein gravity background producing a conformally-invariant gauge theory on the boundary, which describes their boundary structure. It can be considered as an extension of the Fefferman-Graham construction that takes into account both the leading and the subleading sector of the bulk fields.