Comments on Minitwistors and the Celestial Supersphere

Igor Mol

REVIEW 3 major objections 6 minor 1 cited by

Comments on Minitwistors and the Celestial Supersphere

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Minitwistor sigma models reproduce celestial MHV gluon and graviton amplitudes.

desk verdict The minitwistor reformulation of leaf amplitudes is a real step, but the sigma-model semiclassical reduction has a normalization problem that suppresses the determinant, so the central claim doesn't follow. read the letter →

arxiv 2501.09371 v3 pith:J2YQCU4X submitted 2025-01-16 hep-th

Igor Mol This is my paper · ORCID

classification hep-th

keywords celestialholographyminitwistorspaceleafamplitudesMHVN=4supersymmetricYang-MillsN=8supergravitysigmamodelQuillendeterminant

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that tree-level maximally-helicity-violating (MHV) celestial leaf amplitudes for gluons in $\mathcal{N}=4$ supersymmetric Yang-Mills theory and for gravitons in $\mathcal{N}=8$ supergravity can be rewritten as integrals over the moduli space of minitwistor lines, the rational curves in minitwistor space associated with points of three-dimensional Euclidean anti-de Sitter space. To get there, it introduces minitwistor wavefunctions, defined as cohomology classes on minitwistor space obtained by Mellin-transforming twistor wavefunctions, and derives two integral identities for them. It then builds a generating functional for the MHV leaf amplitudes from the Quillen determinant line bundle and proposes a $\sigma$ model on the celestial supersphere $\mathbb{CP}^{1|2}$ with target minitwistor superspace $\mathbf{MT}^{2|\mathcal{N}}$. The central result is that the semiclassical effective action of this $\sigma$ model equals the generating functional, so the model reproduces the tree-level MHV leaf amplitudes. If correct, this gives a concrete realization of a proposed supersymmetric celestial conformal field theory framework.

What carries the argument

The load-bearing objects are minitwistor wavefunctions $\hat f_{\Delta,w}$, defined as Mellin transforms of twistor wavefunctions and realized as cohomology classes in $\Omega^{0,1}(\mathbf{MT},\mathcal{O}(\Delta-w,-\Delta))$; they are the vertex-operator building blocks of the $\sigma$ model. The celestial RSVW identity, Eq. (45), is the distributional bridge that replaces an integral over projective superspace $\mathbb{RP}^{3|8}$ or $\mathbb{RP}^{3|16}$ by an integral over minitwistor lines $L(X)$; the celestial BMS identity, Eq. (34), is the $n$-fold Penrose-transform relation that converts products of bulk-to-boundary propagators into minitwistor integrals, making the Quillen determinant $\log\det(\bar\partial+\omega)$ into a generating functional. The action (196) with kinetic terms for the embedding fields and the fermionic system gives a semiclassical path integral whose saddle points are exactly the minitwistor superlines.

What would settle it

Compute the left and right sides of Eq. (45) for a concrete four-point configuration with explicit minitwistor wavefunctions and delta functions; any mismatch for generic boundary insertion points and conformal weights would falsify the identity. A simpler check is to evaluate one four-gluon leaf amplitude directly from Eq. (60) and compare it with the minitwistor-line integral in Eq. (69) after performing the $X$-integral.

Watch

Extended reading notes

Core claim

The paper's central claim is that the celestial RSVW identity, an $n$-fold distributional identity on minitwistor space, allows every tree-level MHV celestial leaf amplitude for gluons and gravitons to be written as a Fourier transform over minitwistor superspace, with the support of the Fourier-transformed amplitude localized to incidence curves that are minitwistor lines; hence the amplitude vanishes unless all insertion points lie on a common minitwistor line. The same wavefunctions satisfy a second identity, the celestial BMS identity, which is used to show that the Quillen-determinant functional $\int \log\det(\bar\partial+\omega)$ restricted to minitwistor superlines generates all MHV leaf amplitudes. The paper then constructs an action for a $\sigma$ model with worldsheet the celestial supersphere $\mathbb{CP}^{1|2}$ and target the minitwistor superspace $\mathbf{MT}^{2|\mathcal{N}}$, and shows that in the semiclassical limit the path integral localizes on embeddings of the celestial sphere as minitwistor superlines, with the fermionic determinant giving exactly the Quillen-determinant generating functional. Thus the semiclassical effective action reproduces the MHV gluonic and gravitational leaf amplitudes.

Load-bearing premise

The full $n$-point celestial RSVW identity, Eq. (45), is assumed to hold as a distributional identity; the paper sketches an induction but does not give a complete proof, and if the identity fails, the minitwistor-line representation of the leaf amplitudes and the generating functional built on it no longer follow.

Editorial extensions

If this is right

MHV gluon and graviton celestial leaf amplitudes acquire a geometric meaning: they are supported only on configurations where all insertion points lie on a common minitwistor line, so the amplitudes vanish otherwise.
The generating functional $\mathfrak{W}[\omega]=\int_{\mathbb{RP}^{3|2\mathcal{N}}} D^{3|2\mathcal{N}}X \, \log\det(\bar\partial+\omega)\big|_{L(X,\theta)}$ packages all tree-level MHV leaf amplitudes into one object.
The sigma model gives a semiclassical path-integral realization of the supersymmetric celestial CFT proposal, with the worldsheet being the celestial supersphere and the target the minitwistor superspace.
If the formalism is extended to the NMHV sector, celestial amplitudes should be expressible as integrals over moduli spaces of higher-degree curves in minitwistor space, as the paper suggests.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A rigorous proof of the $n$-fold celestial RSVW identity, rather than the sketched induction, would put the whole construction on solid ground; the identity is strong enough that it could be checked numerically for four points.
The same Fourier-transform structure may transfer to other bulk geometries: any space whose twistor space admits a minitwistor quotient with the same incidence geometry would have a celestial CFT of this type.
The semiclassical limitation suggests the sigma model is an effective description; understanding the anomalies that block full quantization could connect loop corrections to the model, possibly along the lines of celestial Liouville theory.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my take after a close read. The paper does real work: it writes down explicit minitwistor wavefunctions, states two integral identities (celestial RSVW and BMS), uses them to rewrite MHV leaf amplitudes for N=4 SYM and N=8 SG as integrals over minitwistor line moduli, and constructs a Quillen-determinant generating functional. That part is coherent and builds honestly on Bu–Seet, Boels–Mason–Skinner, and Tropper. The n-fold identities are only sketched, but the BMS identity looks like induction on the single-particle Penrose transform, and the RSVW identity is the same distributional step; neither looks fabricated.

The trouble starts when the sigma model is introduced. The action in Eq. (196) has a bosonic term with 1/b and a fermionic term with b. The claimed effective action is W = -lim_{b→0+} b log Z. Run the numbers: the fermionic path integral gives det(bD) = b^N det D. Then -b log Z includes -b(N log b + log det D), which goes to zero as b→0+. Even if the fermion term had no b, the prefactor b would kill the log det. To get ∫ log det as the limit, the log det would have to appear in the exponent with a 1/b, and it does not. So Eq. (201) is not a derivation; it's an assertion with the wrong normalization.

There is also the circularity point: the generating functional is defined as log det, and the sigma model is built to reproduce it, so the leaf amplitudes are matched by construction. That's a legitimate 'proposal' but it weakens the claim of phenomenological output. The paper does say the theory is only semiclassical, which is honest, but this is a load-bearing gap, not a small typo.

Where does that leave the paper? The minitwistor machinery and the identities deserve attention; if the RSVW identity checks out, it gives a compact way to think about leaf amplitudes. The sigma model needs a different normalization, or a different definition of W, before it does what the abstract says. I wouldn't desk-reject it; a referee could ask for the derivation of Eq. (45), the path integral reduction, and a fix to the normalization. The paper is worth a serious referee, but the central claim should not be accepted in this form.

Referee Report

3 major / 6 minor

Summary. The paper continues the author's program of deriving aspects of celestial holography from string theory. It proposes a reformulation of tree-level MHV celestial leaf amplitudes for gluons in N=4 SYM and gravitons in N=8 supergravity as integrals over the moduli space of minitwistor lines, using minitwistor wavefunctions defined as cohomology classes on the minitwistor space MT. It also constructs a generating functional for these amplitudes via the Quillen determinant line bundle, extending the Boels-Mason-Skinner approach, and proposes supersymmetric celestial CFTs as sigma models on the celestial supersphere CP^{1|2} with target space MT^{2|N}. The central claim is that the semiclassical effective action of these sigma models reproduces the gluonic and gravitational MHV leaf amplitudes.

Significance. If the construction is correct, it would provide a concrete realization of Tropper's supersymmetric celestial CFT framework and forge a new link between celestial holography and twistorial string-theoretic methods. The paper contains explicit integral formulas, a clear identification of minitwistor wavefunctions, and a proposed action for the sigma model, building on prior work by Bu and Seet and by Boels, Mason, and Skinner. However, the verification is incomplete: the critical n-fold identities are only sketched, and the semiclassical reduction of the sigma model contains a normalization inconsistency that invalidates the claimed result as written.

major comments (3)

[V.D.2, Eqs. (196)-(201)] The semiclassical effective action W defined in Eq. (200) with the action I in Eq. (196) cannot equal ∫ log det(∂ + ω) as claimed. For a free fermion, ∫ dψ d\barψ exp(-b \barψ D ψ) = b det D up to sign, so its contribution to -b log Z is -b log(b det D), which vanishes in the limit b→0+. The bosonic sector, with action of order 1/b, contributes its on-shell value at O(1). Therefore W would be the on-shell bosonic action, not the chiral determinant. The paper asserts that the fermionic path integral 'results in the chiral determinant' and that W becomes the generating functional, but no computation is provided. This is a load-bearing error for the sigma-model claim.
[II.C.2-II.C.3, Eqs. (34) and (45)] The celestial BMS identity (34) and the celestial RSVW identity (45) are central bridges that convert leaf amplitude integrals into integrals over minitwistor lines and underpin the generating functional construction. Both are justified only by an 'inductive argument' or 'direct evaluation' with no detailed proof. Eq. (45) is a nontrivial distributional identity on minitwistor space; without a proof or a precise statement of its domain of validity, the reformulation of the amplitudes and the generating functionals built on it are not established.
[III.C, Eqs. (76)-(78)] The Penrose transform of the background potential ω in Eq. (76) is written as K_Δ(X;z,\bar z), but according to Eq. (33) the transform of F_Δ alone gives C(Δ)/⟨z|X|\bar z]^Δ; the factor |X|^Δ is required to obtain K_Δ. The BMS identity (34) also includes |X|^{Δ_i} on its left-hand side. The generating functional expansion in Eqs. (77)-(78) therefore omits these factors and does not follow as written. The same issue appears in the gravitational generating functional in Eqs. (145)-(147).

minor comments (6)

[III.B.1 and IV.B.2, Eqs. (59), (119)] The notation '+ (z̄_i → i z̄_i)' (and similarly in Eq. (119)) is not explained; please define this shorthand explicitly.
[IV.D.2, Eq. (141)] In the definition of L(X,θ), the set is written with Z^I ∈ MT^{2|4}, but for the N=8 supergravity case it should be MT^{2|8}.
[IV.A and IV.B, Eqs. (108)-(112)] The vertex operators \hat G_i, \hat H_i, U_i, and V_i mix the notations χ and \hatχ; the two-point function in Eq. (90) uses χ, while the operators use \hatχ, which is confusing.
[IV.C, Eq. (124)] Eq. (124) contains unbalanced parentheses and an unclear product/sum structure; please rewrite it with proper grouping.
[General] The manuscript contains numerous typos and grammatical errors, e.g., 'aproach' in the Introduction and 'seem' for 'seen' in Section V.C.3.
[V.D.1, Eqs. (195)-(196)] The action in Eq. (196) is written as an integral over \tilde L(X,θ), but \tilde L is not defined in the text; clarify its relation to L(X,θ).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the sigma-model claim is a path-integral computation (with normalization gaps) and the generating functional construction is a legitimate application of the stated BMS/RSVW identities.

full rationale

I find no circular step in the derivation chain. The minitwistor wavefunctions are defined independently by Mellin transform (Eqs. 21-25), and the BMS/RSVW identities (Eqs. 34 and 45) are stated as integral identities derived from the Penrose transform; their RHS is the same bulk-to-boundary propagator K that defines leaf amplitudes, so substituting the identity into Eq. (60) is a change of representation rather than a self-referential definition. The generating functional Psi[omega] = integral log det(dbar + omega) is expanded via the standard Quillen determinant expansion, and the functional derivative in Eq. (80) reproduces the leaf amplitude as a computation, not by fitting. The sigma-model action in Eq. (196) is engineered so that the fermionic sector yields the chiral determinant, and Eq. (201) is asserted as the semiclassical result; this is a proposed derivation rather than a circular definition. The main issues are the lack of a detailed proof of the n-fold RSVW identity in Eq. (45) and an apparent b-normalization inconsistency in Eqs. (196)-(201) that would suppress the fermion determinant; these are correctness risks, not circularity. The author's self-citations are programmatic and not load-bearing, and no parameter is fitted and renamed as a prediction.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted parameters in the empirical sense; the only free parameter b is a coupling constant. The central claim rests on a set of distributional integral identities and a semiclassical localization that are asserted rather than fully proven. No new particles, forces, or dimensions are postulated.

free parameters (1)

sigma-model coupling b = not fixed
Coupling in the action I in Eq. (196), analogous to the Liouville coupling constant; the semiclassical limit is b -> 0+. It is not fitted to data, but it is a free parameter of the proposed theory.

assumptions (5)

ad hoc to paper The n-fold celestial Boels-Mason-Skinner identity, Eq. (34), holds as an identity of Penrose transforms on minitwistor space.
Invoked in Section III.C and IV.D to expand the Quillen determinant into leaf amplitudes; only an 'inductive argument' is sketched, not a complete proof.
ad hoc to paper The n-fold celestial RSVW identity, Eq. (45), holds as a distributional identity on MT.
This identity is the bridge from leaf amplitudes to integrals over minitwistor lines; the proof is summarized as 'direct evaluation' plus induction, with details omitted.
domain assumption Analytic continuations of Dirac delta functions and distributional forms, including delta(z) = (1/2 pi i) dbar z^{-1}, define valid distributions on minitwistor space.
Used throughout Section II.C and in the definition of the distributional forms delta_Delta, Eqs. (35)-(37).
domain assumption The path integral over the fermionic system (psi, psibar) equals the chiral determinant Tr log(dbar + omega), and the saddle-point approximation localizes the full path integral onto incidence relations.
This is the key step in Section V.D.2, Eqs. (200)-(201), connecting the sigma model to the generating functional; no detailed computation is supplied.
domain assumption The minitwistor superspace MT^{2|N} is the trivial superbundle MT x CP^{0|N}, and the celestial supersphere CP^{1|2} is the correct worldsheet for the proposed sigma model.
Stated in Section V.C and used to construct the action; it is a proposal, not a derived result.

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Pith. "Pith review of Comments on Minitwistors and the Celestial Supersphere." pith.science (2026). https://pith.science/paper/J2YQCU4X

@misc{pith2026250109371,
  author       = {Pith},
  title        = {Pith review of: Comments on Minitwistors and the Celestial Supersphere},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J2YQCU4X}},
  note         = {Machine review of arXiv:2501.09371}
}

abstract

Continuing our program of deriving aspects of celestial holography from string theory, we extend the Roiban-Spradlin-Volovich-Witten (RSVW) formalism to celestial amplitudes. We reformulate the tree-level maximally-helicity-violating (MHV) celestial leaf amplitudes for gluons in $\mathcal{N}=4$ supersymmetric Yang-Mills (SYM) theory and for gravitons in $\mathcal{N}=8$ Supergravity in terms of minitwistor wavefunctions. These are defined as representatives of cohomology classes on the minitwistor space $\mathbf{MT}$, associated to the three-dimensional Euclidean anti-de Sitter space. In this framework, celestial leaf amplitudes are expressed as integrals over the moduli space of minitwistor lines. We construct a minitwistor generating functional for MHV leaf amplitudes using the Quillen determinant line bundle, extending the approach originally developed by Boels, Mason and Skinner. Building on this formalism, we propose supersymmetric celestial conformal field theories (CFTs) as $\sigma$-models, where the worldsheet is given by the celestial supersphere $\mathbf{CP}^{1|2}$, and the target space is the minitwistor superspace $\mathbf{MT}^{2|\mathcal{N}}$. We demonstrate that the semiclassical effective action of these $\sigma$-models reproduces the MHV gluonic and gravitational leaf amplitudes in $\mathcal{N}=4$ SYM theory and $\mathcal{N}=8$ Supergravity. This construction provides a concrete realisation of the supersymmetric celestial CFT framework recently introduced by Tropper (2024).

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Reference graph

Works this paper leans on

122 extracted references · 54 canonical work pages · cited by 1 Pith paper

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vertex operators

CP1 Fermionic Doublet The factorisation procedure to be employed in this section i s based on the method developed by Nair [75]. In this formalism, tree-level MHV gravitation al scattering amplitudes are expressed in terms of a correlator of “vertex operators.” These operat ors are constructed with the aid of an auxiliary fermionic doublet ( ˆχ, ˆχ †) deﬁ...
[2]

vertex operators

BGKn Formula from CP1 Correlators The graviton “vertex operators” are deﬁned as: Gi := exp ( i ⟨νi|x |νi] ) χ † νiχ νi, Hi := [ νi| (−i∂ ) |ω⟩ νi ·ω exp ( i ⟨νi|x |ν i] ) . (91) In the above expressions, (∂ )A ˙A := (σµ)A ˙A ∂ ∂xµ and ωA := ( i 2π ) 1/2 (ω, −1 ) is an auxiliary two- component spinor parametrised by [ω] ∈ CP1. This spinor serves as a refer...
[3]

(96) explicitly in terms of the graviton frequencies si and the normalised spinor basis {zA i ,zi ˙A}

Frequency Dependency To proceed with our aim of performing the Mellin transform of Mn, thereby yielding the corres- ponding celestial amplitude, it is necessary to re-express Eq. (96) explicitly in terms of the graviton frequencies si and the normalised spinor basis {zA i ,zi ˙A}. This reformulation takes the form: Mn = ( κ 2 ) n−2 (z1 ·z2)8 ˆ R4 d4x (2π)...
[4]

dimensions,

N = 8 Supergravity The transition from Einstein’s gravity to N = 8 Supergravity is facilitated by adopting the on-shell superﬁeld formalism, as reviewed in Wess and Bagger [43]. On-shell Superﬁeld Expansion. This formalism introduces Grassmann-valued two-componen t spinors, denoted ηα A (1 ≤α ≤ 8) and normalised according to the relation: ˆ R0|2 d0|2η ηα ...
[5]

23 We follow standard conventions in abstract index notation

Mellin Transform Let us ﬁrst recall that the celestial superamplitude ˆMn (zi, ¯zi, ∆ i) for gravitons in N = 8 Su- pergravity is deﬁned as the ε-regulated Mellin transform of the corresponding scatteri ng amplitude Mn (zi, ¯zi,s i), as follows: ˆMn (zi,zi, ∆ i) := n∏ i=1 ˆ R× + dsi si s∆ i i e−εsi Mn (zi,zi,s i), (105) where R× + denotes the multiplicati...
[6]

integral

Gravitational Celestial Leaf Amplitudes To reformulate the graviton celestial amplitude as an integ ral over the moduli space of min- itwistor lines, we employ the formalism of leaf amplitudes. However, a technical complication immediately presents itself: the celestial wavefunctions φ2hi (x;zi,zi), which appear as multiplic- ative factors in the vertex o...
[7]

Preliminaries: A Celestial Correlator LetO :=O[ωA, Pi, Ψ ∆ i] be an operator depending on the auxiliary spinor ωA, the weight-shifting operators Pi, and the minitwistor graviton wavefunctions Ψ ∆ i (Zi;zi, ¯zi). Deﬁne the expectation value of O over the auxiliary spinor and the fermionic doublet (χ,χ †) as: ⟨O⟩ω,n = ˛ Cn ⟨ωdω⟩ ⟨ω ⏐ ⏐O[ωA, Pi, Ψ ∆ i] ⏐ ⏐ω⟩...
[8]

spacetime

Minitwistor Gravitational Background We now proceed to construct the generating functional. The ﬁ rst step is to introduce the minitwistor gravitational background potential as a representative of a cohomology class: ω i ∈ D Λ [Hc] ⊗ Ω 0,1( MT2|8, O(−2, −∆) ) . (140) Recall that the minitwistor superline L (X,θ ) ⊂ MT2|8, associated with the point X ˆI = ...

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[9]

boundary conditions

(158) These coordinates ωA are postulated to be projectively related to the minitwisto r coordinates λA through the relation: λA ≡ ωA ω1ω2 = κA 1 ω2 + κA 2 ω1 . (159) Within this trivialisation, the holomorphic function F ˙A (X,λ ), which determines the embedding map σX : CP1 ...
[10]

fermionic degrees of freedom

Projective Superspace We begin by observing that the minitwistor space MT is the space of oriented geodesics on the hyperboloid H + 3 . A model for the hyperbolic geometry of H + 3 can be derived from the projective geometry of the three-dimensional real projective space RP3, ...
[11]

fermionic di mensions

Minitwistor Superspace The minitwistor superspace MT2|N (associated with the hyperbolic geometry modelled on RP3|2N ) is constructed by extending the bosonic minitwistor space MT through the inclusion of Grassmann-odd coordinates that encode the “fermionic di mensions.” More p...
[12]

spacetime

Minitwistor Superlines In minitwistor superspace, the minitwistor lines are gener alised to superlines L (X,θ ), each associated with a “spacetime” point (XA ˙A,θ α A) ∈ RP3|2N . The incidence relation deﬁning these supercurves is determined by sections of the projective spino...
[13]

Construction of the Action First Step: Dynamical Variables. To formulate a theory whose solutions to the equations of motion correspond to embeddings of the celestial sphere int o minitwistor superspace MT2|N , it suﬃces to specify the holomorphic sections F ˙A (X,θ ;λ) and Gα...
[14]

Fermionic System We begin with the following physical motivations. First, th e generating functional ̥[ω ], derived in Subsection III C for N = 4 SYM theory and in Subsection IV D for N = 8 Supergravity, is expressed as an integral over projective superspace RP3|2N of the Quil...
[15]

(196) yields the generating func tional ̥[ω ]

Semiclassical Analysis We turn now to the task of demonstrating that the semiclassic al eﬀective action arising from the theory described by Eq. (196) yields the generating func tional ̥[ω ]. Prior to presenting the path-integral formulation of the eﬀ ective action, we recall ...
[16]

gauge tr ansformations

Hyperbolic Space from Projective Geometry Let the four-vectorX A ˙A be homogeneous coordinates on CP3, subject to the equivalence relation X A ˙A ∼a ·X A ˙A, where a is any non-zero complex scalar. A necessary and suﬃcient con dition for a set of components of a holomorphic me...
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Any such curve C is referred to as a minitwistor line or a Hitchin special curve

Minitwistor Space A minitwistor space 37 M is deﬁned as any two-dimensional complex manifold containi ng a rational curve C (a holomorphic embedding of CP1 into M) with a normal bundle isomorphic to O (2). Any such curve C is referred to as a minitwistor line or a Hitchin spec...

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The Holomorphic V ector Bundle O (p,q ) − →MT In this subsection, we shall deﬁne the holomorphic vector bu ndle O (p,q ) − → MT which serves as the domain upon which the minitwistor Penrose tran sform is deﬁned. We establish that this bundle can be identiﬁed with the inﬁnite-d...
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This formalism requires the analytic continuation of Minkowski spacetime R(1,3) from a Lorentzian (− + ++) to a Kleinian (− − ++) signature

Klein Space The leaf representation of celestial amplitudes is our main motivation for introduc ing multi-gluon and multi-graviton wavefunctions using minitwistor varia bles. This formalism requires the analytic continuation of Minkowski spacetime R(1,3) from a Lorentzian (− +...
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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

vertex operators

CP1 Fermionic Doublet The factorisation procedure to be employed in this section i s based on the method developed by Nair [75]. In this formalism, tree-level MHV gravitation al scattering amplitudes are expressed in terms of a correlator of “vertex operators.” These operat ors are constructed with the aid of an auxiliary fermionic doublet ( ˆχ, ˆχ †) deﬁ...

[2] [2]

vertex operators

BGKn Formula from CP1 Correlators The graviton “vertex operators” are deﬁned as: Gi := exp ( i ⟨νi|x |νi] ) χ † νiχ νi, Hi := [ νi| (−i∂ ) |ω⟩ νi ·ω exp ( i ⟨νi|x |ν i] ) . (91) In the above expressions, (∂ )A ˙A := (σµ)A ˙A ∂ ∂xµ and ωA := ( i 2π ) 1/2 (ω, −1 ) is an auxiliary two- component spinor parametrised by [ω] ∈ CP1. This spinor serves as a refer...

[3] [3]

(96) explicitly in terms of the graviton frequencies si and the normalised spinor basis {zA i ,zi ˙A}

Frequency Dependency To proceed with our aim of performing the Mellin transform of Mn, thereby yielding the corres- ponding celestial amplitude, it is necessary to re-express Eq. (96) explicitly in terms of the graviton frequencies si and the normalised spinor basis {zA i ,zi ˙A}. This reformulation takes the form: Mn = ( κ 2 ) n−2 (z1 ·z2)8 ˆ R4 d4x (2π)...

[4] [4]

dimensions,

N = 8 Supergravity The transition from Einstein’s gravity to N = 8 Supergravity is facilitated by adopting the on-shell superﬁeld formalism, as reviewed in Wess and Bagger [43]. On-shell Superﬁeld Expansion. This formalism introduces Grassmann-valued two-componen t spinors, denoted ηα A (1 ≤α ≤ 8) and normalised according to the relation: ˆ R0|2 d0|2η ηα ...

[5] [5]

23 We follow standard conventions in abstract index notation

Mellin Transform Let us ﬁrst recall that the celestial superamplitude ˆMn (zi, ¯zi, ∆ i) for gravitons in N = 8 Su- pergravity is deﬁned as the ε-regulated Mellin transform of the corresponding scatteri ng amplitude Mn (zi, ¯zi,s i), as follows: ˆMn (zi,zi, ∆ i) := n∏ i=1 ˆ R× + dsi si s∆ i i e−εsi Mn (zi,zi,s i), (105) where R× + denotes the multiplicati...

[6] [6]

integral

Gravitational Celestial Leaf Amplitudes To reformulate the graviton celestial amplitude as an integ ral over the moduli space of min- itwistor lines, we employ the formalism of leaf amplitudes. However, a technical complication immediately presents itself: the celestial wavefunctions φ2hi (x;zi,zi), which appear as multiplic- ative factors in the vertex o...

[7] [7]

Preliminaries: A Celestial Correlator LetO :=O[ωA, Pi, Ψ ∆ i] be an operator depending on the auxiliary spinor ωA, the weight-shifting operators Pi, and the minitwistor graviton wavefunctions Ψ ∆ i (Zi;zi, ¯zi). Deﬁne the expectation value of O over the auxiliary spinor and the fermionic doublet (χ,χ †) as: ⟨O⟩ω,n = ˛ Cn ⟨ωdω⟩ ⟨ω ⏐ ⏐O[ωA, Pi, Ψ ∆ i] ⏐ ⏐ω⟩...

[8] [8]

spacetime

Minitwistor Gravitational Background We now proceed to construct the generating functional. The ﬁ rst step is to introduce the minitwistor gravitational background potential as a representative of a cohomology class: ω i ∈ D Λ [Hc] ⊗ Ω 0,1( MT2|8, O(−2, −∆) ) . (140) Recall that the minitwistor superline L (X,θ ) ⊂ MT2|8, associated with the point X ˆI = ...