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arxiv: 2606.29587 · v1 · pith:CMIDHCM6new · submitted 2026-06-28 · ✦ hep-th · math-ph· math.MP

Asymptotic boundary structure of Lagrangian gauge theories

Ivan Dneprov , Maxim Grigoriev , Mikhail Markov This is my paper

Pith reviewed 2026-06-30 01:58 UTC · model grok-4.3

classification ✦ hep-th math-phmath.MP
keywords gauge PDEasymptotic boundarypresymplectic structureQ-cocycleholographic anomalyAdS/CFTnull infinityWeyl anomaly
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0 comments X

The pith

Any bulk Q-cocycle in a Lagrangian gauge theory determines both a renormalized cocycle and a lower-degree anomaly cocycle on the asymptotic boundary.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that gauge theories defined on spacetimes with asymptotic boundaries inherit a boundary gauge PDE equipped with its own presymplectic structure. Although the bulk presymplectic form diverges at the boundary, any Q-cocycle from the bulk can be mapped to a pair of compatible boundary Q-cocycles: one finite renormalized version at the original ghost degree and one anomaly version at ghost degree one lower. The map for the anomaly part works like a residue extraction that preserves the Q-cohomology. This construction recovers the holographic Weyl anomaly action in the AdS case and supplies a general procedure for null-infinity boundaries in flat space, where only isolated examples were previously known.

Core claim

Given a local gauge theory on spacetime with boundary that admits a compatible presymplectic structure, the structure diverges at an asymptotic boundary, yet any Q-cocycle in the bulk determines a pair of compatible Q-cocycles on the boundary gauge PDE: the renormalized cocycle of the same ghost degree together with the anomaly cocycle of degree one lower. The anomaly construction is analogous to the residue map in b-geometry. Explicit checks for scalar and Maxwell fields confirm that the AdS anomaly presymplectic structure reproduces the known holographic Weyl anomaly while the Minkowski null-infinity case extends earlier isolated results.

What carries the argument

The residue-like map that extracts from a divergent bulk Q-cocycle a pair of compatible boundary Q-cocycles consisting of the finite renormalized part and the anomaly part of one lower degree.

If this is right

  • The boundary gauge PDE carries a well-defined presymplectic structure obtained from the bulk via the splitting.
  • In AdS the anomaly cocycle produces the standard holographic Weyl anomaly action.
  • In flat space at null infinity the same splitting supplies a Lagrangian for the boundary theory in cases where only special examples existed before.
  • The procedure applies uniformly to any gauge PDE equipped with a compatible presymplectic structure and asymptotic boundary.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same residue construction applied at the level of the equations of motion alone may produce consistent boundary dynamics without needing the full Lagrangian data.
  • The anomaly cocycle could serve as a starting point for defining boundary counterterms or charges in a wider class of asymptotically flat holographic models.
  • Extending the construction to include gravitational theories would test whether the anomaly structure reproduces known surface terms in asymptotic symmetry analyses.

Load-bearing premise

The bulk presymplectic structure diverges at the asymptotic boundary in a manner that permits separation into a finite renormalized component and a lower-degree anomaly component through a residue-like operation that preserves the Q-cohomology.

What would settle it

Direct computation of the anomaly presymplectic structure for the Maxwell field at null infinity in Minkowski space and comparison against the handful of previously known particular cases to check whether the general formula reproduces or contradicts those examples.

read the original abstract

Given a local gauge theory on spacetime with boundary, it naturally defines another gauge theory which can be regarded as a theory of the boundary values. For Lagrangian theories, it comes equipped with the presymplectic structure which can be used to define one or another version of Hamiltonian-like formulation of the initial model. This relation is especially manifest for AKSZ sigma models and more-generally gauge PDEs with compatible presymplectic structure in which case the boundary system is again a gauge PDE with presymplectic structure. In the context of (flat space) holography one is interested in boundaries at infinity, also known as asymptotic boundaries. The gauge PDE framework naturally extends to this setup, resulting in the notion of gauge PDE with asymptotic boundaries. Although this works perfectly well at the level of equations of motion, the extension to Lagrangian systems appears quite subtle because the presymplectic structure capturing the Lagrangian is divergent at the boundary. We show that any $Q$-cocycle in the bulk (and presymplectic structure in particular) determines a pair of compatible $Q$-cocycles of the boundary gauge PDE: the renormalized one of the same ghost-degree, and the anomaly cocycle of degree one lower. For the latter, the construction is somewhat analogous to the residue map known in the context of b-geometry. The general formalism is exemplified by scalar and Maxwell fields on AdS and Minkowski spaces. It turns out that in the AdS case the natural action determined by the anomaly presymplectic structure is precisely the one known as the holographic Weyl anomaly in the AdS/CFT context while its null-infinity counterpart was known in a few very particular cases only.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript develops a framework for Lagrangian gauge theories on spacetimes with asymptotic boundaries by extending the gauge PDE formalism with compatible presymplectic structures. It claims that any bulk Q-cocycle determines a pair of compatible boundary Q-cocycles: a renormalized cocycle of the same ghost degree and an anomaly cocycle of degree one lower, obtained via a residue-like map that preserves the Q-cohomology. The construction is illustrated for scalar and Maxwell fields on AdS and Minkowski spaces, with the AdS anomaly reproducing the known holographic Weyl anomaly.

Significance. If the central mapping holds, the work supplies a systematic procedure for extracting finite renormalized structures and anomaly terms from divergent presymplectic data at asymptotic boundaries. This extends AKSZ sigma-model techniques to holographic and flat-space settings and supplies an independent consistency check through reproduction of the AdS Weyl anomaly. The residue-map analogy to b-geometry is a potentially useful technical device for handling divergences while maintaining cohomological compatibility.

minor comments (3)
  1. The abstract states the main result but supplies no explicit formulas or derivation outline; adding one or two key equations (e.g., the definition of the residue-like map) would improve accessibility without altering length.
  2. Notation for ghost degree, Q-cohomology, and the precise meaning of 'compatible' boundary cocycles should be collected in a short preliminary subsection to avoid scattered definitions.
  3. In the AdS and Minkowski examples, the manuscript should state explicitly which bulk cocycle is used and how the anomaly term matches the known Weyl anomaly expression (reference the relevant equation or table).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the summary of the central construction and the significance statement regarding its relation to AKSZ techniques and the holographic Weyl anomaly. The recommendation is for minor revision, but the report contains no major comments.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper's central claim is a direct construction: any bulk Q-cocycle (including the presymplectic structure) determines a renormalized boundary Q-cocycle of the same ghost degree and an anomaly cocycle of degree one lower via a residue-like map that preserves Q-cohomology. This is presented as a general formalism for gauge PDEs with asymptotic boundaries, with the map defined analogously to known b-geometry residues rather than by fitting or self-reference. The construction is then checked by explicit reproduction of the known holographic Weyl anomaly for AdS scalar/Maxwell fields and related null-infinity cases, supplying an external consistency benchmark independent of the paper's own parameters or prior self-citations. No load-bearing step reduces by definition, renaming, or self-citation chain to the input data; the derivation remains non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The construction rests on the assumption that the bulk theory is a gauge PDE equipped with a compatible presymplectic structure whose divergence at infinity admits a cohomological splitting; no numerical free parameters appear. The anomaly cocycle is introduced as a derived object rather than an independent postulate.

axioms (2)
  • domain assumption Local gauge theories on spacetime with boundary naturally induce boundary gauge theories equipped with presymplectic structure.
    Opening sentence of the abstract.
  • domain assumption For AKSZ sigma models and gauge PDEs the boundary system remains a gauge PDE with presymplectic structure.
    Stated explicitly in the abstract.
invented entities (2)
  • gauge PDE with asymptotic boundaries no independent evidence
    purpose: Framework for treating boundaries at infinity while preserving the gauge PDE structure.
    Defined to extend the existing gauge PDE formalism to holographic settings.
  • anomaly cocycle no independent evidence
    purpose: Captures the divergent contribution to the presymplectic structure at the boundary.
    Constructed via a residue-like map from the bulk cocycle.

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