Twistor Space Research Articles

Twistor and Reflector Spaces for Paraquaternionic Contact Manifolds

We consider certain fiber bundles over paraquaternionic contact manifolds, called twistor and reflector spaces. We show that the twistor space carries an integrable CR structure (Cauchy–Riemann structure) and the reflector space is an integrable para-CR structure, both with neutral signatures.

Carrollian $\mathscr Lw_{1+\infty}$ representation from twistor space

We construct an explicit realization of the action of the \mathscr Lw_{1+∞}ℒw1+∞ loop algebra on fields at null infinity. This action is directly derived by Penrose transform of the geometrical action of \mathscr Lw_{1+∞}ℒw1+∞ symmetries in twistor space, ensuring that it forms a representation of the algebra. Finally, we show that this action coincides with the canonical action of \mathscr Lw_{1+∞}ℒw1+∞ Noether charges on the asymptotic phase space.

On Riemannian 4‐manifolds and their twistor spaces: A moving frame approach

AbstractIn this paper, we study the twistor space of an oriented Riemannian 4‐manifold using the moving frame approach, focusing, in particular, on the Einstein, non‐self‐dual setting. We prove that any general first‐order linear condition on the almost complex structures of forces the underlying manifold to be self‐dual, also recovering most of the known related rigidity results. Thus, we are naturally lead to consider first‐order quadratic conditions, showing that the Atiyah–Hitchin–Singer almost Hermitian twistor space of an Einstein 4‐manifold bears a resemblance, in a suitable sense, to a nearly Kähler manifold.

Integrability in gravity from Chern-Simons theory

This paper presents a new perspective on integrability in theories of gravity. We show how the stationary, axisymmetric sector of General Relativity can be described by the boundary dynamics of a four-dimensional Chern-Simons theory. This approach shows promise for simplifying solution generating methods in both General Relativity and higher-dimensional supergravity theories. The four-dimensional Chern-Simons theory presented generalises those for flat space integrable models by introducing a space-time dependent branch cut in the spectral plane. We also make contact with twistor space approaches to integrability, showing how the branch cut defects of four-dimensional Chern-Simons theory arise from a discrete reduction of six-dimensional Chern-Simons theory.

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 N=4 super Yang-Mills. Beginning with four lines in twistor space, we characterize a positive subspace to which an ℓ-loop geometry is attached. The loop geometry then consists of ℓ lines in twistor space satisfying positivity conditions among themselves and with respect to the base. Consequently, the can be viewed as fibration over a . The fibration naturally dissects the base into chambers, in which the degree-4ℓ loop form is unique and distinct for each chamber. Interestingly, up to three loops, the chambers are simply organized by the six ordering of x1,22x3,42, x1,42x2,32, and x1,32x2,42. We explicitly verify our conjecture by computing the loop-forms in terms of a basis of planar conformal integrals up to ℓ=3, which indeed yield correct loop integrands for the four-point correlator. Published by the American Physical Society 2024

Multivector Fields on Quaternionic Kähler Manifolds

In this paper we define a differential operator as a modified Dirac operator. Using the operator, we introduce a quaternionic k -vector field on a quaternionic Kähler manifold and show that any quaternionic k -vector field corresponds to a holomorphic k -vector field on the twistor space.

The Dirac–Higgs Complex and Categorification of (BBB)-Branes

Abstract Let ${\mathcal{M}}_{\operatorname{Dol}}(X,G)$ denote the hyperkähler moduli space of $G$-Higgs bundles over a smooth projective curve $X$. In the context of four dimensional supersymmetric Yang–Mills theory, Kapustin and Witten introduced the notion of (BBB)-brane: boundary conditions that are compatible with the B-model twist in every complex structure of ${\mathcal{M}}_{\operatorname{Dol}}(X,G)$. The geometry of such branes was initially proposed to be hyperkähler submanifolds that support a hyperholomorphic bundle. Gaiotto has suggested a more general type of (BBB)-brane defined by perfect analytic complexes on the Deligne–Hitchin twistor space $\operatorname{Tw}({\mathcal{M}}_{\operatorname{Dol}}(X,G))$. Following Gaiotto’s suggestion, this paper proposes a framework for the categorification of (BBB)-branes, both on the moduli spaces and on the corresponding derived moduli stacks. We do so by introducing the Deligne stack, a derived analytic stack with corresponding moduli space $\operatorname{Tw}({\mathcal{M}}_{\operatorname{Dol}}(X,G))$, defined as a gluing between two analytic Hodge stacks along the Riemann–Hilbert correspondence. We then construct a class of (BBB)-branes using integral functors that arise from higher non-abelian Hodge theory, before discussing their relation to the Wilson functors from the Dolbeault geometric Langlands correspondence.

Twistor theory of the Chen–Teo gravitational instanton**Dedicated to Nick Woodhouse on the occasion of his 75th birthday.

Abstract Toric Ricci--flat metrics in dimension four correspond to certain holomorphic vector bundles over a
twistor space. We construct these bundles explicitly, by exhibiting and characterising their patching matrices, 
for the five--parameter family of Riemannian ALF metrics constructed by Chen and Teo. The Chen--Teo family contains
a two--parameter family of asymptotically flat gravitational instantons. The patching matrices for these 
instantons take a simple rational form.

Cohesive self-dual Yang-Mills modules over four manifolds

This research is motivated by Atiyah, Hitchin, and Singers paper Self-duality in Four-Dimensional Riemannian Geometry , which introduced a relationship between self-dual Yang-Mills fields on smooth manifolds and holomorphic vector bundles on their twistor spaces. Here, self-duality is a specific structure in 4-dimensional manifolds and Yang-Mills fields are gauge fields that satisfy Yang-Mills equations in 4-dimensions and are corresponded to the holomorphic bundles on twistor spaces. In this paper, we extend the relationship from vector bundles to a generalization of the Yang-Mills fields. To achieve this purpose, we apply Atiyah, Hitchin, and Singers theorem to cohesive modules, which was originally introduced by Block in in studying coherent sheaves over complex manifolds and the relations between homomorphic torus and its dual non-commutative torus. We introduce the notion of cohesive self-dual Yang-Mills modules and show that the twistor correspondence actually induces the equivalence between the dg category of cohesive self-dual Yang Mills modules P_(A_SD ) and the dg category of holomorphic cohesive modules P_(A_Hol ) on the twistor spaces.

A hidden 2d CFT for self-dual Yang-Mills on the celestial sphere

Self-dual Yang-Mills theory admits an underlying infinite dimensional symmetry algebra, which has been obtained from mode expansion of Mellin transformed 4d scattering amplitudes and separately, Koszul duality on twistor space. In this paper, we propose to derive an explicit 2d realization of the algebra by performing a particular gauge transformation on the twistor action for self-dual Yang-Mills. The gauge parameter used in the transformation generates pure gauge connections corresponding to large gauge transformations on 4d Minkowski space, which localises part of the twistor action to a ℂℙ1 on ℂ* reduction of twistor space. Under a projection, it can be mapped to the celestial sphere at the light-cone cut of the origin on I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{I} $$\\end{document}. Geometrically, this is the common boundary celestial sphere shared by Euclidean AdS3 or Lorentzian dS3 slices of Minkowski space. We comment on the geometric meaning of the derivation from the perspective of minitwistor spaces of the 3d slices embedded in 4d Minkowski space. Using the action functional of this 2d CFT, we compute its stress-energy tensor and central charge. By a further marginal deformation, we calculate correlation functions of current algebra generators purely from the 2d side which reproduce full 4d tree-level form factors.

Integrable deformations from twistor space

Integrable field theories in two dimensions are known to originate as defect theories of 4d Chern-Simons and as symmetry reductions of the 4d anti-self-dual Yang-Mills equations. Based on ideas of Costello, it has been proposed in work of Bittleston and Skinner that these two approaches can be unified starting from holomorphic Chern-Simons in 6 dimensions. We provide the first complete description of this diamond of integrable theories for a family of deformed sigma models, going beyond the Dirichlet boundary conditions that have been considered thus far. Starting from 6d holomorphic Chern-Simons theory on twistor space with a particular meromorphic 3-form \OmegaΩ, we construct the defect theory to find a novel 4d integrable field theory, whose equations of motion can be recast as the 4d anti-self-dual Yang-Mills equations. Symmetry reducing, we find a multi-parameter 2d integrable model, which specialises to the \lambdaλ-deformation at a certain point in parameter space. The same model is recovered by first symmetry reducing, to give 4d Chern-Simons with generalised boundary conditions, and then constructing the defect theory.

Loops of loops expansion in the amplituhedron

We study a novel geometric expansion for scattering amplitudes in the planar sector of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super Yang-Mills theory, in the context of the Amplituhedron which reproduces the all-loop integrand as a canonical differential form on the positive geometry. In a paper by Arkani-Hamed, Henn and one of the authors, it was shown that this result can be recast in terms of negative geometries with a certain hierarchy of loops (closed cycles) in the space of loop momenta, represented by lines in momentum twistor space. One can then calculate an all-loop order result in the approximation where only tree graphs in the space of all loops are considered. Furthermore, using differential equation methods, it is possible to calculate and resum integrated expressions and obtain strong coupling results. In this paper, we provide a more general framework for the ‘loops of loops’ expansion and outline a powerful method for the determination of differential forms for higher-order geometries. We solve the problem completely for graphs with one internal cycle, but the method can be used more generally for other geometries.

Cartan–Helgason theorem for quaternionic symmetric and twistor spaces

Let (g,k) be a complex quaternionic symmetric pair with k having an ideal sl(2,ℂ), k=sl(2,ℂ)+mc. Consider the representation Sm(ℂ2)=ℂm+1 of k via the projection k→sl(2,ℂ) onto the ideal sl(2,ℂ). We study the finite dimensional irreducible representations V(λ) of g which contain Sm(ℂ2) under k⊆g. We give a characterization of all such representations V(λ) and find the corresponding multiplicity, the dimension of Hom(V(λ)|k,Sm(ℂ2)). We consider also the branching problem of V(λ) under l=u(1)ℂ+mc⊂k and find the multiplicities. Geometrically the Lie subalgebra l⊂k defines a twistor space over the compact symmetric space of the compact real form Gu of Gℂ, Lie(Gℂ)=g, and our results give the decomposition for the L2-spaces of sections of certain vector bundles over the symmetric space and line bundles over the twistor space. This generalizes Cartan–Helgason’s theorem for symmetric spaces (g,k) and Schlichtkrull’s theorem for Hermitian symmetric spaces where one-dimensional representations of k are considered.

Carrollian amplitudes and celestial symmetries

Carrollian holography aims to express gravity in four-dimensional asymptotically flat spacetime in terms of a dual three-dimensional Carrollian CFT living at null infinity. Carrollian amplitudes are massless scattering amplitudes written in terms of asymptotic or null data at I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{I} $$\\end{document}. These position space amplitudes at I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{I} $$\\end{document} are to be re-interpreted as correlation functions in the putative dual Carrollian CFT. We derive basic results concerning tree-level Carrollian amplitudes yielding dynamical constraints on the holographic dual. We obtain surprisingly compact expressions for n-point MHV gluon and graviton amplitudes in position space at I\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{I} $$\\end{document}. We discuss the UV/IR behaviours of Carrollian amplitudes and investigate their collinear limit, which allows us to define a notion of Carrollian OPE. By smearing the OPE along the generators of null infinity, we obtain the action of the celestial symmetries — namely, the S algebra for Yang-Mills theory and Lw1+∞ for gravity — on the Carrollian operators. As a consistency check, we systematically relate our results with celestial amplitudes using the link between the two approaches. Finally, we initiate a direct connection between twistor space and Carrollian amplitudes.

Invariant distributions and the transport twistor space of closed surfaces

AbstractWe study transport equations on the unit tangent bundle of a closed oriented Riemannian surface and their links to the transport twistor space of the surface (a complex surface naturally tailored to the geodesic vector field). We show that fibrewise holomorphic distributions invariant under the geodesic flow — which play an important role in tensor tomography on surfaces — form a unital algebra, that is, multiplication of such distributions is well defined and continuous. We also exhibit a natural bijective correspondence between fibrewise holomorphic invariant distributions and genuine holomorphic functions on twistor space with polynomial blowup on the boundary of the twistor space. Additionally, when the surface is Anosov, we classify holomorphic line bundles over twistor space which are smooth up to the boundary. As a byproduct of our analysis, we obtain a quantitative version of a result of Flaminio [C. R. Acad. Sci. Paris Sér. I Math. 315 (1992) no. 6, 735–738] asserting that invariant distributions of the geodesic flow of a positively curved metric on are determined by their zeroth and first Fourier modes.

Amplitudes at Strong Coupling as Hyper-Kähler Scalars.

Alday and Maldacena conjectured an equivalence between string amplitudes in AdS_{5}×S^{5} and null polygonal Wilson loops in planar N=4 super-Yang-Mills (SYM) theory. At strong coupling this identifies SYM amplitudes with areas of minimal surfaces in anti-de Sitter space. For minimal surfaces in AdS_{3}, we find that the nontrivial part of these amplitudes, the remainder function, satisfies an integrable system of nonlinear differential equations, and we give its Lax form. The result follows from a new perspective on "Y systems," which defines a new psuedo-hyper-Kähler structure directly on the space of kinematic data, via a natural twistor space defined by the Y-system equations. The remainder function is the (pseudo-)Kähler scalar for this geometry. This connection to pseudo-hyper-Kähler geometry and its twistor theory provides a new ingredient for extending recent conjectures for nonperturbative amplitudes using structures arising at strong coupling.

Calibrated geometry in hyperkähler cones, 3-Sasakian manifolds, and twistor spaces

Abstract We systematically study calibrated geometry in hyperkähler cones $C^{4n+4}$ , their 3-Sasakian links $M^{4n+3}$ , and the corresponding twistor spaces $Z^{4n+2}$ , emphasizing the relationships between submanifold geometries in various spaces. Our analysis highlights the role played by a canonical $\mathrm {Sp}(n)\mathrm {U}(1)$ -structure $\gamma $ on the twistor space Z. We observe that $\mathrm {Re}(e^{- i \theta } \gamma )$ is an $S^1$ -family of semi-calibrations and make a detailed study of their associated calibrated geometries. As an application, we obtain new characterizations of complex Lagrangian and complex isotropic cones in hyperkähler cones, generalizing a result of Ejiri–Tsukada. We also generalize a theorem of Storm on submanifolds of twistor spaces that are Lagrangian with respect to both the Kähler–Einstein and nearly Kähler structures.

Twistor space for local systems on an open curve

Let [Formula: see text] be a smooth quasi-projective curve. We previously constructed a Deligne–Hitchin moduli space with Hecke gauge groupoid for connections of rank [Formula: see text]. We extend this construction to the case of any rank [Formula: see text], although still keeping a genericity hypothesis. The formal neighborhood of a preferred section corresponding to a tame harmonic bundle is governed by a mixed twistor structure.

Actuation selection for multi-loop coupling mechanisms using geometric algebra

Multi-loop coupling mechanisms (MCMs) have been widely used in architectural design antennas and space-deployable. This paper proposes a general method for actuation selection of MCMs using geometric algebra. The constraint space is obtained by collineation and dual operators of twist space. This method determines additional constraints by analyzing the linear correlation between the constrained spaces of the two limbs within the closed-loop and making corrections to the constraint space of the basic limb. Through outer and dual operators, the input screw corresponding to the transmission wrench screw and output twist screw are obtained. The input transmission index (ITI) and output transmission index (OTI) corresponding to this input are calculated, which leads to generation of a local transmission index (LTI). The actuation selection with the best performance can be obtained by comparing and analyzing the motion/force transfer performance of all input combinations. The proposed method has the advantages of simple computation and clear physical significance, and three typical MCMs are applied to validate this method. Finally, an experimental analysis compared the power consumption of a test motor at different actuation positions, further proving the accuracy of the method.

Generalized Deligne–Hitchin twistor spaces: Construction and properties

In this paper, we generalize the construction of Deligne–Hitchin twistor space by gluing two certain Hodge moduli spaces. We investigate some properties of such generalized Deligne–Hitchin twistor space as a complex analytic manifold. More precisely, we show it admits holomorphic sections whose normal bundle contains a semistable subbundle with positive degree and whose energy is semi-negative, and it carries a balanced metric. Moreover, we also study the automorphism groups of the Hodge moduli spaces and the generalized Deligne–Hitchin twistor spaces.