arXiv subjects

Chia-Kai Kuo

Publications and source records attributed to Chia-Kai Kuo.

11 recordsLinked to original sources

arXiv2026

Beyond Discontinuities: Cosmological WFCs and the Supersymmetric Orthogonal Grassmannian

We construct an $\mathcal N=2$ supersymmetric Grassmannian representation of tree-level wavefunction coefficients (WFCs) by combining Grassmannian representations of energy discontinuities with an inversion formula. Since the orthogonal Grassmannian captures homogeneous solutions of the spinor conformal Ward identities, while current WFCs satisfy inhomogeneous Ward identities, the full WFC is obtained by reconstructing the energy-dependent prefactors from a basis of discontinuities. We first demonstrate this mechanism at three points, where the triple discontinuity determines the transverse current WFC and admits a supersymmetric uplift. At four points, we invert a spanning set of five current discontinuities and embed the result in momentum superspace using super-orthogonal-Grassmannian invariants generated by $\hatδ(CΩΞ^I)$. This yields the full four-point super WFC in Grassmannian form. We show that the two orthogonal-Grassmannian branches organize distinct supersymmetric invariants and reduce, in the flat-space limit, to different helicity superamplitudes.

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arXiv2026

Notes on off-shell conformal integrals and correlation functions at five points

We study five-point off-shell conformal integrals and the associated half-BPS correlation functions at two loops in the 't Hooft coupling expansion of maximally supersymmetric Yang-Mills theory. We construct a basis of uniform-transcendental (UT) pure integrals spanning six distinct topologies by diagonalizing leading singularities subject to conformal invariance. By fixing conformal frames, this basis can be mapped to known two-loop four-mass integral families. We then compute the integrated results by combining canonical differential equations with integration-by-parts reduction. As an application, we present symbol-level integrated results for the two-loop five-point half-BPS correlators, including both maximal and non-maximal sectors.

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arXiv2026

Leading singularities and chambers of Correlahedron

In this paper, we explore the chamber dissection of the loop-geometry of Correlahedron, which encodes the loop integrand of four-point stress-energy correlators in planar $\mathcal{N}=4$ super Yang-Mills. We demonstrate that at four loops, continuing the pattern of lower loops, the integrand of the four-point correlation function can be written as a sum over products of chamber-forms and local loop integrands. The chambers and their associated forms are identical to those of three loops, indicating that the dissection may be complete to all loop orders. Furthermore, this suggests that the leading singularities at all loops are simply linear combinations of these chamber forms. This is especially intriguing at four loops since it contains elliptic functions. Interestingly, each elliptic function appears in a subset of chambers. Our geometric approach motivates us to ``diagonalize" the representation, where the local integrals only possess a single leading singularity or elliptic cut. In such a representation, all integrands must evaluate to pure functions, including a single pure elliptic integrand. Inspired by this picture, we also present a simplified form of the three-loop correlator in terms of two independent pure functions (weight-$6$ single-valued multiple polylogarithms), which are directly computed from local integrands with unit leading singularities, multiplied by the leading singularities from chamber forms.

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arXiv2025

BDS ansatz in ABJM via scaffolding triangulations

In this work, we analyze the infrared divergence of two-loop amplitudes at arbitrary multiplicity in three-dimensional $\mathcal{N}=6$ Chern-Simons matter theory. We introduce the Bern-Dixon-Smirnov (BDS) integrand, which captures the full infrared structure while remaining free of unphysical cuts. We show that these local integrands, together with their kinematic prefactors, are naturally organized by the scaffolding triangulations of $n=2k$-gon, with distinct triangulations yielding different local representations. Remarkably, this triangulation structure also persists at the level of the integrated functions. This observation provides a graphical proof of both the cancellation of elliptic cuts and the triangulation independence of the integrated result. As a direct consequence, we obtain a simple proof that the integrated BDS integrand coincides with the one-loop maximally-helicity-violating (MHV) amplitude (the BDS ansatz) of $\mathcal{N}=4$ super Yang-Mills theory for all $n=2k$.

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arXiv2024

All-loop geometry for four-point correlation functions

In this letter, we consider a positive geometry conjectured to encode the loop integrand of four-point stress-energy correlators in planar $\mathcal{N}=4$ super Yang-Mills. Beginning with four lines in twistor space, we characterize a positive subspace to which an $\ell$-loop geometry is attached. The loop geometry then consists of $\ell$ lines in twistor space satisfying positivity conditions among themselves and with respect to the base. Consequently, the $\textit{loop geometry}$ can be viewed as fibration over a $\textit{tree geometry}$. The fibration naturally dissects the base into chambers, in which the degree-$4 \ell$ loop form is unique and distinct for each chamber. Interestingly, up to three loops, the chambers are simply organized by the six ordering of $x^2_{1,2}x^2_{3,4}$, $x^2_{1,4}x^2_{2,3}$ and $x^2_{1,3}x^2_{2,4}$. We explicitly verify our conjecture by computing the loop-forms in terms of a basis of planar conformal integrals up to $\ell=3$, which indeed yield correct loop integrands for the four-point correlator.

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arXiv2023

The ABJM Amplituhedron

In this paper, we take a major step towards the construction and applications of an all-loop, all-multiplicity amplituhedron for three-dimensional planar $\mathcal{N}=6$ Chern-Simons matter theory, or the $\textit{ABJM amplituhedron}$. We show that by simply changing the overall sign of the positive region of the original amplituhedron for four-dimensional planar $\mathcal{N}=4$ super-Yang-Mills (sYM) and performing a symplectic reduction, only three-dimensional kinematics in the middle sector of even-multiplicity survive. The resulting form of the geometry, combined with its parity images, gives the full loop integrand. This simple modification geometrically enforces the vanishing of odd-multiplicity cuts, and manifests the correct soft cuts as well as two-particle unitarity cuts. Furthermore, the so-called ``bipartite structures" of four-point all-loop negative geometries also directly generalize to all multiplicities. We introduce a novel approach for triangulating loop amplituhedra based on the kinematics of the tree region, resulting in local integrands tailored to ``prescriptive unitarity". This construction sheds fascinating new light on the interplay between loop and tree amplituhedra for both ABJM and $\mathcal{N}=4$ sYM: the loop geometry demands that the tree region must be dissected into $\textit{chambers}$, defined by the simultaneous positivity of maximal cuts. The loop geometry is then the ``fibration" of the tree region. Using the new construction, we give explicit results of one-loop integrands up to ten points and two-loop integrands up to eight points by computing the canonical form of ABJM loop amplituhedron.

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arXiv2023

Emergent unitarity, all-loop cuts and integrations from the ABJM amplituhedron

We elaborate on aspects of a new positive geometry proposed recently, which was conjectured to be the four-point amplituhedron for ABJM theory. We study generalized unitarity cuts from the geometry, and in particular we prove that (1) the four-point integrand satisfies perturbative unitarity (or optical theorem) to all loops, which follows directly from the geometry, and (2) vanishing cuts involving odd-point amplitudes follow from the ``bipartite" nature of the associated ``negative geometries", which justifies their appearance in ABJM theory. We also take a first step in integrating the forms of these negative geometries and obtain an infrared-finite quantity up to two loops, from which we extract the cusp anomalous dimension at leading order.

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arXiv2023

The two-loop eight-point amplitude in ABJM theory

In this paper, we present the two-loop correction to scattering amplitudes in three-dimensional $\mathcal{N}=6$ Chern-Simons matter theory. We use eight-point case as our main example, but the method generalizes to all multiplicities. The integrand is completely fixed by dual conformal symmetry, maximal cuts, constraints from soft-collinear behavior and from vanishing of odd-multiplicity amplitudes. After performing integrations with Higgs regularizations, the integrated results demonstrate that the infrared divergence is again identical to that of ${\cal N}=4$ super Yang-Mills. After subtracting divergences, the finite part is dual conformal invariant, and respects various symmetries; it has uniform transcendentality weight two and exhibits nice analytic structure.

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arXiv2023

All-Loop Four-Point Aharony-Bergman-Jafferis-Maldacena Amplitudes from Dimensional Reduction of the Amplituhedron

We define a new geometry obtained from the all-loop amplituhedron in ${\cal N}=4$ SYM by reducing its four-dimensional external and loop momenta to three dimensions. Focusing on the simplest four-point case, we provide strong evidence that the canonical form of this ``reduced amplituhedron" gives the all-loop integrand of the ABJM four-point amplitude. In addition to various all-loop cuts manifested by the geometry, we present explicitly new results for the integrand up to five loops, which are much simpler than results in ${\cal N}=4$ SYM. One of the reasons for such all-loop simplifications is that only a very small fraction of the so-called negative geometries survive the dimensional reduction, which corresponds to bipartite graphs. Our results suggest an unexpected relation between four-point amplitudes in these two theories.

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arXiv2022

The momentum amplituhedron of SYM and ABJM from twistor-string maps

We study remarkable connections between twistor-string formulas for tree amplitudes in ${\cal N}=4$ SYM and ${\cal N}=6$ ABJM, and the corresponding momentum amplituhedron in the kinematic space of $D=4$ and $D=3$, respectively. Based on the Veronese map to positive Grassmannians, we define a twistor-string map from $G_{+}(2,n)$ to a $(2n{-}4)$-dimensional subspace of the 4d kinematic space where the momentum amplituhedron of SYM lives. We provide strong evidence that the twistor-string map is a diffeomorphism from $G_+(2,n)$ to the interior of momentum amplituhedron; the canonical form of the latter, which is known to give tree amplitudes of SYM, can be obtained as pushforward of that of former. We then move to three dimensions: based on Veronese map to orthogonal positive Grassmannian, we propose a similar twistor-string map from the moduli space ${\cal M}_{0,n}^+$ to a $(n{-}3)$-dimensional subspace of 3d kinematic space. The image gives a new positive geometry which conjecturally serves as the momentum amplituhedron for ABJM; its canonical form gives the tree amplitude with reduced supersymmetries in the theory. We also show how boundaries of compactified ${\cal M}_{0,n}^+$ map to boundaries of momentum amplituhedra for SYM and ABJM corresponding to factorization channels of amplitudes, and in particular for ABJM case the map beautifully excludes all unwanted channels.

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arXiv2018

Dualities for Ising networks

In this note, we study the equivalence between planar Ising networks and cells in the positive orthogonal Grassmannian. We present a microscopic construction based on amalgamation, which establishes the correspondence for any planar Ising network. The equivalence allows us to introduce two recursive methods for computing correlators of Ising networks. The first based on duality moves, which generate networks belonging to the same cell in the Grassmannian. This leads to fractal lattices where the recursion formulas become the exact RG equations of the effective couplings. For the second, we use amalgamation where each iteration doubles the size of the seed lattice. This leads to an efficient way of computing the correlator where the complexity scales logarithmically with respect to the number of spin sites.

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