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arxiv: 2606.18331 · v1 · pith:TDAE2QXLnew · submitted 2026-06-16 · ✦ hep-th · hep-ph· math.QA

Meromorphic amplitudes from 3-dimensional supersymmetry

Federico Ambrosino , Nathan Haouzi This is my paper

Pith reviewed 2026-06-26 23:29 UTC · model grok-4.3

classification ✦ hep-th hep-phmath.QA
keywords Coon amplitudeXYZ modelhalf-indexVeneziano amplitudemeromorphic amplitudes3d N=2 supersymmetrymirror symmetryscattering amplitudes
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The pith

The Coon amplitude equals the 3d N=2 half-index of the XYZ model with nontrivial boundary conditions.

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

The paper links the Coon amplitude to a calculation in three-dimensional supersymmetric field theory. It shows this amplitude is reproduced exactly by the half-index of the XYZ model once specific boundary conditions are imposed. The ultraviolet three-dimensional theory flows in the infrared to a sigma model whose partition function recovers the Veneziano amplitude. Mirror symmetry between the XYZ model and SQED accounts for crossing symmetry. The same correspondence produces a meromorphic version of the Coon amplitude by replacing the dressing factor with an elliptic completion that remains single-valued while keeping positive residues at physical poles.

Core claim

The Coon amplitude coincides with the 3d N=2 half-index of the XYZ model with nontrivial boundary conditions. The 3d UV theory flows in the IR to a sigma model whose partition function is the Veneziano amplitude. Crossing symmetry is realized as a consequence of 3d N=2 mirror symmetry between XYZ and SQED. A meromorphic modification of the Coon amplitude is constructed by promoting the dressing factor q^{ST} to an elliptic completion thereof, showing that single-valuedness is compatible with positivity at the physical poles.

What carries the argument

the 3d N=2 half-index of the XYZ model with nontrivial boundary conditions, which equals the Coon amplitude

If this is right

  • Crossing symmetry of the amplitude follows from 3d mirror symmetry between XYZ and SQED.
  • The infrared limit of the boundary-conditioned theory reproduces the Veneziano amplitude.
  • The elliptic completion of the dressing factor yields a single-valued meromorphic amplitude.
  • Positivity of residues at physical poles is preserved under the meromorphic modification.

Where Pith is reading between the lines

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

  • The same boundary-condition construction may generate other known deformations of the Veneziano amplitude.
  • Partition functions of 3d supersymmetric theories with boundaries could serve as a source for new amplitude expressions.
  • The elliptic-completion technique may extend to higher-point or higher-spin amplitudes while retaining meromorphicity.

Load-bearing premise

The 3d UV theory with the chosen boundary conditions flows in the IR to a sigma model whose partition function is exactly the Veneziano amplitude.

What would settle it

Explicit computation of the half-index for the XYZ model with the stated boundary conditions, followed by direct comparison to the closed-form expression for the Coon amplitude.

Figures

Figures reproduced from arXiv: 2606.18331 by Federico Ambrosino, Nathan Haouzi.

Figure 1
Figure 1. Figure 1: Plot of Aq(αq(s), αq(t)) for a typical slice at −∞ < t < t∞. In blue, the physical poles at s = m2 + [n]q; in red, the poles due to the meromorphic θ completion. That is, it is a polynomial of degree n in t with positive Gegenbauer expansion for the same range as in [9]. Thus, the upgrade: q ST Γq(−S) Γq(−T) Γq(−S − T) → K(S, T|ζ) Γq(−S) Γq(−T) Γq(−S − T) (3.12) is a genuine uplift of the exponential facto… view at source ↗
Figure 2
Figure 2. Figure 2: Figure 2a: Brane configuration in the topological A-model that computes Zopen in (3.26); in blue, the Lagrangian brane at position q S+1/2 . Figure 2b: type IIB (p, q) brane web engineering the 3d N = 2 SQED theory living on the D3 brane. parameter τ . The two terms X m≥1 q m(S+1/2) − q m(S+T +1/2) m[m] are the standard external brane contributions in the resolved conifold, with Kähler size T 12 . The rema… view at source ↗
read the original abstract

We establish a new connection between supersymmetric theories and scattering amplitudes. We show that the Coon amplitude coincides with the 3d $\mathcal{N}=2$ half-index of the XYZ model with nontrivial boundary conditions. Our 3d theory, intrinsically defined in the UV, flows to a sigma model in the IR whose partition function is the Veneziano amplitude. Crossing symmetry is realized as a consequence of 3d $\mathcal{N}=2$ mirror symmetry between XYZ and SQED. We use this correspondence to construct a meromorphic modification of the Coon amplitude by promoting the long-standing dressing factor $\mathfrak{q}^{ST}$ responsible for a branch cut to an elliptic completion thereof. This illustrates that one does not have to give up single-valuedness to achieve positivity at the physical poles.

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

1 major / 2 minor

Summary. The paper claims that the Coon amplitude coincides with the 3d N=2 half-index of the XYZ model with nontrivial boundary conditions. The intrinsically UV-defined 3d theory flows in the IR to a sigma model whose partition function is exactly the Veneziano amplitude. Crossing symmetry arises from 3d N=2 mirror symmetry between XYZ and SQED. The authors construct a meromorphic modification of the Coon amplitude by replacing the dressing factor q^{ST} with an elliptic completion, preserving single-valuedness while maintaining positivity at physical poles.

Significance. If the identification holds, the work supplies a supersymmetric origin for the Coon amplitude and derives crossing symmetry from mirror symmetry, both of which are strengths. The explicit construction of the elliptic completion illustrates a route to single-valued meromorphic amplitudes with positivity, which may be useful for analytic continuations in string and field theory amplitudes. The reliance on an RG-protected half-index computed in the UV is a positive methodological feature.

major comments (1)
  1. [IR limit paragraph] The section on the IR limit asserts without detailed derivation that the XYZ model with the chosen boundary conditions flows to a sigma model whose partition function is exactly the Veneziano amplitude. This matching of operators, parameters, and target-space geometry is load-bearing for the central claim that the half-index equals the Coon amplitude; any surviving relevant deformation or boundary-induced change in the effective sigma-model metric would break the exact coincidence.
minor comments (2)
  1. The notation for the elliptic completion of the dressing factor q^{ST} should be defined more explicitly, including its transformation properties under the relevant modular group.
  2. Add a brief comparison table or equation reference showing how the modified amplitude reduces to the original Coon amplitude in the appropriate limit.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for a more explicit derivation of the IR limit. We address this point below and will revise the manuscript to strengthen the presentation.

read point-by-point responses

  1. Referee: [IR limit paragraph] The section on the IR limit asserts without detailed derivation that the XYZ model with the chosen boundary conditions flows to a sigma model whose partition function is exactly the Veneziano amplitude. This matching of operators, parameters, and target-space geometry is load-bearing for the central claim that the half-index equals the Coon amplitude; any surviving relevant deformation or boundary-induced change in the effective sigma-model metric would break the exact coincidence.

    Authors: We agree that the current presentation of the IR limit is too terse and requires a more detailed derivation to make the matching fully rigorous. In the revised manuscript we will expand this section with an explicit step-by-step argument: (i) identification of the chiral ring generators and their relations under the chosen boundary conditions, (ii) matching of the Kähler parameters to the Veneziano variables s and t, (iii) verification that the effective target-space metric remains flat (or the appropriate Calabi-Yau metric) with no boundary-induced corrections at leading order, and (iv) confirmation that no relevant deformations survive the RG flow that would alter the partition function. This will be supported by a direct computation of the effective superpotential and a brief discussion of the absence of boundary-induced relevant operators. revision: yes

Circularity Check

0 steps flagged

No significant circularity; IR flow to Veneziano sigma model asserted but not shown to reduce to self-definition or fitted input

full rationale

The central identification equates the Coon amplitude to the 3d half-index of the XYZ model, with the UV-to-IR flow to a sigma model whose partition function is the Veneziano amplitude stated in the abstract. No quoted equations or text demonstrate that any amplitude is defined in terms of itself, that a parameter is fitted to data and then renamed a prediction, or that a load-bearing step collapses via self-citation to an unverified prior result by the same authors. Mirror symmetry is invoked as a standard 3d N=2 fact. The derivation chain therefore remains self-contained against external benchmarks, warranting only the minimal score for an asserted flow whose detailed matching is not exhibited here.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The claims rest on the RG flow from UV XYZ theory to IR sigma model with Veneziano partition function and on 3d N=2 mirror symmetry realizing crossing; both are domain assumptions standard in the field but not independently derived here. The parameter q in the dressing factor is promoted rather than fitted.

free parameters (1)
  • q
    Base of the dressing factor q^ST that is replaced by an elliptic function; its value is not derived from first principles in the abstract.
axioms (2)
  • domain assumption 3d N=2 mirror symmetry between XYZ and SQED realizes crossing symmetry
    Invoked to obtain crossing from the 3d correspondence.
  • domain assumption UV 3d theory with nontrivial boundary conditions flows to IR sigma model whose partition function is the Veneziano amplitude
    Required for the IR identification stated in the abstract.

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discussion (0)

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Reference graph

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