arXiv subjects

Paul Heslop

Publications and source records attributed to Paul Heslop.

At least 19 recordsLinked to original sources

arXiv2026

Decompactification and the Flat-Space Limit of AdS$_5\times$ S$^5$ Master Correlators

We study the flat-space limit of AdS$_5$/CFT$_4$ holographic correlators while accounting for the simultaneous decompactification of the internal S$^5$. We show that this limit is naturally formulated in terms of master propagators and correlators, which resum the infinite tower of Kaluza--Klein modes. For boundary correlators, the resulting expressions can be interpreted in terms of amplitudes with kinematics restricted to 5d. We then consider master correlators whose external insertions are kept in the AdS$_5\times$S$^5$ bulk before taking the limit, and argue that they recover flat-space amplitudes with unrestricted kinematics in 10d Minkowski space. Finally, we show that the flat-space limit commutes with the boundary limit if we analytically continue the sphere so that the bulk geometry becomes AdS$_5 \times$dS$_5$, and the resulting formulae suggest an interpretation in terms of amplitudes in flat space with two time directions and with 10d kinematics satisfying two simultaneous null constraints. We discuss the implications of these constructions for flat-space holography and its relation to the Carrollian limit of the dual CFT.

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arXiv2026

Graph Neural Networks for the Graphical Bootstrap

We study a graph classification problem involving over 20 million graphs, arising from high-order perturbative computations of correlators in planar $\mathcal{N}=4$ super-Yang--Mills, a model closely related to the theory of the strong nuclear force. We benchmark graph neural networks, including graph transformers, achieving robust generalization to larger graphs with up to $99.996\%$ ROC AUC. Then, we analyze how the models can be used to gain a computational speedup compared to the traditional graphical bootstrap algorithm, through shrinking the redundant data by up to $85.5\%$ at the level of denominator graphs. Finally, we study the embeddings of the models to investigate their interpretability.

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arXiv2026

Superconformal Weight Shifting Operators

We develop a framework for constructing superconformal blocks for correlators of general supermultiplets in theories with $\mathrm{SU}(m,m|2n)$ symmetry, such as four-dimensional $\mathcal{N}=2$ and $\mathcal{N} = 4$ conformal theories. We use analytic superspace, viewed as the super-Grassmannian $\mathrm{Gr}(m|n,2m|2n)$, which includes 4D Minkowski space ($m=2,n=0$). In this formalism, superblocks for non-half-BPS correlators are analogous to non-supersymmetric conformal blocks for correlators of fields with spin. We construct $\mathrm{SU}(m,m|2n)$-covariant differential operators which generalise the existing conformal weight-shifting operators, and thus allow us to derive all superconformal blocks from the known half-BPS blocks. Our results provide a framework from which to advance the conformal bootstrap in 4D supersymmetric settings, with potential extensions to lower and higher-dimensional SCFTs. The Grassmannian formalism is also seen to offer a natural and often simpler alternative to the embedding space formalism of non-supersymmetric CFTs.

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arXiv2026

A Compact Formula for Conserved Three-Point Tensor Structures in 4D CFT

We derive a compact analytic formula for a complete basis of conformally invariant tensor structures for three-point functions of conserved operators in arbitrary 4D Lorentz representations. The construction follows directly from a novel constraint equivalent to applying conservation conditions at each point: the leading terms in all OPE limits appear as symmetric traceless tensors. We derive this by lifting to a unified $\mathrm{SU}(m,m|2n)$ analytic superspace framework, where the conservation conditions are automatically solved and then reducing back to 4D CFT. The same method is also used for cases involving one non-conserved operator. This formalism further reveals a map of the counting of CFT tensor structures to that of finite-dimensional $\mathrm{SU}(2n)$ representations, solved by Littlewood-Richardson coefficients. All results can be directly re-interpreted as three-point $\mathcal{N}=2$ and $\mathcal{N}=4$ superconformal tensor structures via the unified analytic superspace.

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arXiv2024

Integrated correlators in $\mathcal{N}=4$ SYM beyond localisation

We study integrated correlators of four superconformal primaries $\mathcal{O}_{p}$ with arbitrary charges $p$ in $\mathcal{N}{=}4$ super Yang-Mills theory (SYM). The $ \langle \mathcal{O}_{2} \mathcal{O}_{2} \mathcal{O}_{p} \mathcal{O}_{p} \rangle$ integrated correlators can be computed by supersymmetric localisation, whereas correlators with more general charges are currently not accessible from this method and in general contain complicated multi-zeta values. Nevertheless we observe that if one sums over the contributions from all different channels in a given correlator, then all the multi-zeta values (and products of zeta's) cancel leaving only $ζ(2\ell{+}1)$ at $\ell$-loops. We then propose an exact expression of such integrated correlators in the planar limit, valid for arbitrary 't Hooft coupling. The expression matches with the known exact localisation-based results for specific charges, as well as with all existing perturbative and strong-coupling results in the literature for more general charges. As an application, our result is used to determine certain $7$-loop Feynman integral periods and fix previously unknown coefficients in the correlators at strong coupling.

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arXiv2023

The SAGEX Review on Scattering Amplitudes, Chapter 8: Half BPS correlators

Half BPS correlators in N=4 super Yang-Mills theory are key quantities both in the AdS/CFT correspondence as well as in scattering amplitudes research. They are dual at strong coupling to quantum gravity amplitudes. At weak coupling on the other hand they contain all N=4 SYM amplitudes. They have been found to possess a number of hidden symmetries, for example non-trivial permutation symmetry of perturbative four-point integrands and a higher dimensional conformal symmetry. Their study has enjoyed continuous progress since the discovery of AdS/CFT and they are now some of the best understood quantities of any four-dimensional quantum field theory. In this review we outline the current knowledge of half BPS correlators, emphasising these two co-existing relations to scattering amplitudes at strong and weak coupling.

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arXiv2023

The SAGEX Review on Scattering Amplitudes

This is an introduction to, and invitation to read, a series of review articles on scattering amplitudes in gauge theory, gravity, and superstring theory. Our aim is to provide an overview of the field, from basic aspects to a selection of current (2022) research and developments.

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arXiv2022

Internal boundaries of the loop amplituhedron

The strict definition of positive geometry implies that all maximal residues of its canonical form are $\pm 1$. We observe, however, that the loop integrand of the amplitude in planar $\mathcal{N}=4$ super Yang-Mills has maximal residues not equal to $\pm 1$. We find the reason for this is that deep in the boundary structure of the loop amplituhedron there are geometries which contain internal boundaries: codimension one defects separating two regions of opposite orientation. This phenomenon requires a generalisation of the concept of positive geometry and canonical form to include such internal boundaries and also suggests the utility of a further generalisation to `weighted positive geometries'. We re-examine the deepest cut of $\mathcal{N}=4$ amplitudes in light of this and obtain new all order residues.

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arXiv2022

Higher-Dimensional Symmetry of AdS$_2\times$S$^2$ Correlators

It was recently shown that IIB supergravity on AdS$_5\times$S$^5$ enjoys 10d conformal symmetry and that superstring theory on this background can be described using a 10d scalar effective field theory. In this paper we adapt these two complementary approaches to correlators of hypermultiplets in AdS$_2\times$S$^2$. In particular, we show that 4-point correlators of $1/2$-BPS operators in the 1d boundary can be computed using 4d conformal symmetry and a 4d effective action in the bulk. The 4d conformal symmetry is realised by acting with Casimirs of $SU(1,1|2)$, and is generically broken by higher derivative corrections. We point out similar structure underlying $α'$ corrections to IIB supergravity in AdS$_5\times$S$^5$. In particular, while the $α'^3$ corrections can be written in terms of a sixth order Casimir acting on a 10d conformal block, similar structure does not appear in higher-order corrections. We note however that a specific combination of higher derivative corrections can give rise to Witten diagrams with higher dimensional symmetry at the integrand level, with breaking then arising from the measure.

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arXiv2022

Superconformal blocks in diverse dimensions and $BC$ symmetric functions

We uncover a precise relation between superblocks for correlators of superconformal field theories (SCFTs) in various dimensions and symmetric functions related to the $BC$ root system. The theories we consider are defined by two integers $(m,n)$ together with a parameter $θ$ and they include correlators of all half-BPS correlators in 4d theories with ${\cal N}=2n$ supersymmetry, 6d theories with $(n,0)$ supersymmetry and 3d theories with ${\cal N}=4n$ supersymmetry, as well as all scalar correlators in any non SUSY theory in any dimension, and conjecturally various 5d, 2d and 1d superconformal theories. The superblocks are eigenfunctions of the super Casimir of the superconformal group whose action we find to be precisely that of the $BC_{m|n}$ Calogero-Moser-Sutherland (CMS) Hamiltonian. When $m=0$ the blocks are polynomials, and we show how these relate to $BC_n$ Jacobi polynomials. However, differently from $BC_n$ Jacobi polynomials, the $m=0$ blocks possess a crucial stability property that has not been emphasised previously in the literature. This property allows for a novel supersymmetric uplift of the $BC_n$ Jacobi polynomials, which in turn yields the $(m,n;θ)$ superblocks. Superblocks defined in this way are related to Heckman-Opdam hypergeometrics and are non polynomial functions. A fruitful interaction between the mathematics of symmetric functions and SCFT follows, and we give a number of new results on both sides. One such example is a new Cauchy identity which naturally pairs our superconformal blocks with Sergeev-Veselov super Jacobi polynomials and yields the CPW decomposition of any free theory diagram in any dimension.

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arXiv2021

Amplituhedron-like geometries

We consider amplituhedron-like geometries which are defined in a similar way to the intrinsic definition of the amplituhedron but with non-maximal winding number. We propose that for the cases with minimal number of points the canonical form of these geometries corresponds to the product of parity conjugate amplitudes at tree as well as loop level. The product of amplitudes in superspace lifts to a star product in bosonised superspace which we give a precise definition of. We give an alternative definition of amplituhedron-like geometries, analogous to the original amplituhedron definition, and also a characterisation as a sum over pairs of on-shell diagrams that we use to prove the conjecture at tree level. The union of all amplituhedron-like geometries has a very simple definition given by only physical inequalities. Although such a union does not give a positive geometry, a natural extension of the standard definition of canonical form, the globally oriented canonical form, acts on this union and gives the square of the amplitude.

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arXiv2020

Towards the Virasoro-Shapiro Amplitude in AdS5xS5

We propose a systematic procedure for obtaining all single trace 1/2-BPS correlators in N=4 super Yang-Mills corresponding to the four-point tree-level amplitude for type IIB string theory in AdS5xS5. The underlying idea is to compute generalised contact Witten diagrams coming from a 10d effective field theory on AdS5xS5 whose coefficients are fixed by the flat space Virasoro-Shapiro amplitude up to ambiguities related to commutators of the 10d covariant derivatives which require additional information such as localisation. We illustrate this procedure by computing stringy corrections to the supergravity prediction for all single trace 1/2-BPS correlators up to $O(α'^7)$, and spell out a general algorithm for extending this to any order in $α'$.

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arXiv2019

One-loop amplitudes in $AdS_5\times S^5$ supergravity from $\mathcal{N}=4$ SYM at strong coupling

We explore the structure of maximally supersymmetric Yang-Mills correlators in the supergravity regime. We develop an algorithm to construct one-loop supergravity amplitudes of four arbitrary Kaluza-Klein supergravity states, properly dualised into single-particle operators. We illustrate this algorithm by constructing new explicit results for multi-channel correlation functions, and we show that correlators which are degenerate at tree level become distinguishable at one-loop. The algorithm contains a number of subtle features which have not appeared until now. In particular, we address the presence of non-trivial low twist protected operators in the OPE that are crucial for obtaining the correct one-loop results. Finally, we outline how the differential operators $\widehat{\mathcal{D}}_{pqrs}$ and $Δ^{(8)}$, which play a role in the context of the hidden 10d conformal symmetry at tree level, can be used to reorganise our one-loop correlators.

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arXiv2019

Recursion Relations for Anomalous Dimensions in the 6d $(2,0)$ Theory

We derive recursion relations for the anomalous dimensions of double-trace operators occurring in the conformal block expansion of four-point stress tensor correlators in the 6d $(2,0)$ theory, which encode higher-derivative corrections to supergravity in $AdS_7 \times S^4$ arising from M-theory. As a warm-up, we derive analogous recursion relations for four-point functions of scalar operators in a toy non-supersymmetric 6d conformal field theory.

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arXiv2018

The twistor Wilson loop and the amplituhedron

The amplituhedron provides a beautiful description of perturbative superamplitude integrands in N=4 SYM in terms of purely geometric objects, generalisations of polytopes. On the other hand the Wilson loop in supertwistor space also gives an explicit description of these superamplitudes as a sum of planar Feynman diagrams. Each Feynman diagram can be naturally associated with a geometrical object in the same space as the amplituhedron (although not uniquely). This suggests that these geometric images of the Feynman diagrams give a tessellation of the amplituhedron. This turns out to be the case for NMHV amplitudes. We prove however that beyond NMHV this is not true. Specifically, each Feynman diagram leads to an image with a physical boundary and spurious boundaries. The spurious ones should be "internal", matching with neighbouring diagrams. We however show that there is no choice of geometric image of the Wilson loop Feynman diagrams which yields a geometric object without leaving unmatched spurious boundaries.

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arXiv2018

Multi-Particle Amplitudes from the Four-Point Correlator in Planar N=4 SYM

A non-trivial consequence of the super-correlator/super-amplitude duality is that the integrand of the four-point correlation function of stress-tensor multiplets in planar N=4 super Yang-Mills contains a certain combination of n-point amplitude integrands for any n. This combination is the sum of products of all helicity super-amplitudes with their corresponding helicity conjugates. The four-point correlator itself is described by a single scalar function whose loop level integrands possess a hidden permutation symmetry facilitating its computation up to ten loops. We discover that assuming Yangian symmetry and an appropriate basis of planar dual conformal integrands it is possible to disentangle the contributions from the individual amplitudes from this combination. We test this up to seven points and up to two loops. This suggests that any scattering amplitude for any n, with any helicity structure and at any loop order may be extractable from the four-point correlator.

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arXiv2018

M-theory Beyond The Supergravity Approximation

We analyze the four-point function of stress-tensor multiplets for the 6d quantum field theory with $OSp(8^*|4)$ symmetry which is conjectured to be dual to M-theory on $AdS_7 \times S^4$, and deduce the leading correction to the tree-level supergravity prediction by obtaining a solution of the crossing equations in the large-$N$ limit with the superconformal partial wave expansion truncated to operators with zero spin. This correction corresponds to the M-theoretic analogue of $\mathcal{R}^4$ corrections in string theory. We also find solutions corresponding to higher-spin truncations, but they are subleading compared to the 1-loop supergravity prediction, which has yet to be calculated.

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arXiv2017

The Correlahedron

We introduce a new geometric object, the correlahedron, which we conjecture to be equivalent to stress-energy correlators in planar N=4 super Yang-Mills. Re-expressing the Grassmann dependence of correlation functions of n chiral stress-energy multiplets with Grassmann degree 4k in terms of 4(n+k)-linear bosonic variables, the resulting expressions have an interpretation as volume forms on a Gr(n+k,4+n+k) Grassmannian, analogous to the expressions for planar amplitudes via the amplituhedron. The resulting volume forms are to be naturally associated with the correlahedron geometry. We construct such expressions in this bosonised space both directly, in general, from Feynman diagrams in twistor space, and then more invariantly from specific known correlator expressions in analytic superspace. We give a geometric interpretation of the action of the consecutive lightlike limit and show that under this the correlahedron reduces to the squared amplituhedron both as a geometric object as well as directly on the corresponding volume forms. We give an explicit easily implementable algorithm via cylindrical decompositions for extracting the squared amplituhedron volume form from the squared amplituhedron geometry with explicit examples and discuss the analogous procedure for the correlators.

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