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arxiv: 2605.26024 · v1 · pith:PNYNATQNnew · submitted 2026-05-25 · 🧮 math-ph · hep-th· math.KT· math.MP

Field theory of mathfrak{su}(n): the absence of non-zero scatterings

Eugenia Boffo , J\'an Pulmann , \v{L}ubo\v{s} Ravas This is my paper

Pith reviewed 2026-06-29 19:09 UTC · model grok-4.3

classification 🧮 math-ph hep-thmath.KTmath.MP
keywords su(n) formshomological perturbation theoryscattering amplitudestree-level diagramsgauge symmetriesfield theoryhigher productsinteraction vertex
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The pith

Homological perturbation theory applied to su(n) forms shows no scattering amplitudes from trivalent tree-level diagrams except the interaction vertex.

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

The paper treats su(n) forms as a finite-dimensional toy model for field theories that include gauge symmetries. It uses homological perturbation theory to prove that all trivalent tree diagrams beyond the basic vertex give zero scattering amplitudes, and this holds for every n. A reader would care because the result limits what perturbative interactions can occur in the model and shows how the same underlying algebra behaves differently when the field space is enlarged. The contrast with the non-vanishing higher products obtained after transfer to a larger space highlights that the choice of field space controls which interactions survive.

Core claim

Relying on homological perturbation theory, we show that there are no scattering amplitudes with trivalent tree-level diagrams, except for the interaction vertex, thus extending a known argument to arbitrary n. In contrast to this, we show how to obtain non-trivial higher products when transferring to a larger space of fields.

What carries the argument

Homological perturbation theory applied directly to the finite-dimensional su(n) forms, which isolates the vanishing of all but the three-point vertex in tree-level diagrams.

If this is right

  • Only the three-point interaction vertex contributes to tree-level scattering; all other trivalent diagrams cancel.
  • The perturbative expansion of the theory remains minimal at tree level for every value of n.
  • Transferring the algebraic structure to an enlarged field space produces non-trivial higher-order operations that were absent in the original space.
  • The same su(n) algebra yields different interaction content depending on the dimension of the field space chosen.

Where Pith is reading between the lines

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

  • The vanishing result could be checked by direct low-order calculations for small n to confirm consistency with the general argument.
  • Similar transfer techniques might be applied to other finite-dimensional Lie algebra models to generate controlled higher interactions.
  • If the pattern holds, it suggests that enlarging the field space is a systematic way to introduce complexity without changing the underlying algebra.
  • The absence of extra tree diagrams may simplify numerical or symbolic checks of the model for arbitrary n.

Load-bearing premise

Homological perturbation theory applies directly to the finite-dimensional su(n) forms without additional hidden assumptions that would alter the vanishing result for tree-level diagrams.

What would settle it

An explicit computation of any trivalent tree-level scattering amplitude for a chosen n greater than 3 that yields a non-zero value would falsify the vanishing claim.

read the original abstract

We inspect $\mathfrak{su}(n)$ forms, providing greater detail for $n=2,3$, as a toy model for a field theory in finite dimensions and with gauge symmetries. Relying on homological perturbation theory, we show that there are no scattering amplitudes with trivalent tree-level diagrams, except for the interaction vertex, thus extending a known argument of Cattaneo--Mn\"{e}v to arbitrary $n$. In contrast to this, we show how to obtain non-trivial higher products when transferring to a larger space of fields.

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 / 1 minor

Summary. The manuscript claims that homological perturbation theory applied to finite-dimensional su(n) forms yields vanishing of all non-trivial trivalent tree-level scattering amplitudes except the basic interaction vertex itself. This extends the Cattaneo-Mněv argument to arbitrary n, with explicit algebraic checks provided for n=2 and n=3; the authors further contrast the vanishing result with the appearance of non-trivial higher products after transfer to an enlarged field space.

Significance. If the central derivation holds, the work supplies a clean algebraic demonstration that tree-level scattering vanishes in this finite-dimensional gauge-theoretic toy model, crediting the direct application of the homological perturbation lemma for a parameter-free result that generalizes prior work. The explicit low-n verifications and the controlled contrast with the enlarged space strengthen the claim by showing how the vanishing depends on the choice of field space.

minor comments (1)
  1. [Sections detailing n=2,3 checks] The explicit computations for n=2 and n=3 would benefit from a short table listing the chosen basis elements for the su(n) forms and the resulting differential and product maps, to make the verification steps fully reproducible from the text alone.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their accurate summary of our manuscript and for the positive assessment leading to a recommendation of minor revision. No specific major comments or requests for changes were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; extends external prior argument

full rationale

The manuscript applies homological perturbation theory directly to finite-dimensional su(n) forms, deriving the vanishing of non-trivial trivalent tree-level scattering amplitudes (except the basic vertex) for general n via algebraic computation, with explicit verification for n=2,3 and contrast to the non-vanishing case after field-space transfer. The central claim is framed as an explicit extension of the external Cattaneo-Mněv argument rather than a self-referential fit, renormalization, or self-citation chain. No step reduces by construction to its own inputs, and the derivation remains self-contained against the stated external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of homological perturbation theory to the su(n) model and on the validity of the finite-dimensional toy model as a proxy for gauge symmetries.

axioms (1)
  • domain assumption Homological perturbation theory can be applied to su(n) forms to compute scattering amplitudes.
    The proof method stated in the abstract relies on this technique.

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discussion (0)

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