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Let $X$ be a quasi-projective curve, compactified to $(Y,D)$ with $X=Y-D$. We construct a Deligne-Hitchin twistor space out of moduli spaces of framed $\lambda$-connections of rank $2$ over $Y$ with logarithmic singularities and quasi-parabolic structure along $D$. To do this, one should divide by a Hecke-gauge groupoid. Tame harmonic bundles on $X$ give preferred sections, and the relative tangent bundle along a preferred section has a mixed twistor structure with weights $0,1,2$. The weight $2$ piece corresponds to the deformations of the KMS structure including parabolic weights and the residues of the $\lambda$-connection., Comment: minor changes and a reference

We show that a conformal connection on a closed oriented surface $\Sigma$ of negative Euler characteristic preserves precisely one conformal structure and is furthermore uniquely determined by its unparametrised geodesics. As a corollary it follows that the unparametrised geodesics of a Riemannian metric on $\Sigma$ determine the metric up to constant rescaling. It is also shown that every conformal connection on the $2$-sphere lies in a complex $5$-manifold of conformal connections, all of which share the same unparametrised geodesics., Comment: 16 pages, exposition improved, references added

Many integrable systems can be reformulated as holomorphic vector bundles on twistor space. This is a powerful organizing principle in the theory of integrable systems. One shortcoming is that it is formulated at the level of the equations of motion. From this perspective, it is mysterious that integrable systems have Lagrangians. In this paper, we study a Chern-Simons action on twistor space and use it to derive the Lagrangians of some integrable sigma models. Our focus is on examples that come from dimensionally reduced gravity and supergravity. The dimensional reduction of general relativity to two spacetime dimensions is an integrable coset sigma model coupled to a dilaton and 2d gravity. The dimensional reduction of supergravity to two spacetime dimensions is an integrable coset sigma model coupled to matter fermions, a dilaton, and 2d supergravity. We derive Lax operators and Lagrangians for these 2d integrable systems using the Chern-Simons theory on twistor space. In the supergravity example, we use an extended setup in which twistor Chern-Simons theory is coupled to a pair of matter fermions., 22 pages

The Weyl double copy is a procedure for relating exact solutions in biadjoint scalar, gauge and gravity theories, and relates fields in spacetime directly. Where this procedure comes from, and how general it is, have until recently remained mysterious. In this paper, we show how the current form and scope of the Weyl double copy can be derived from a certain procedure in twistor space. The new formalism shows that the Weyl double copy is more general than previously thought, applying in particular to gravity solutions with arbitrary Petrov types. We comment on how to obtain anti-self-dual as well as self-dual fields, and clarify some conceptual issues in the twistor approach., 38 pages, 1 figure

The worldsheet string theory dual to free 4d N = 4 super Yang-Mills theory was recently proposed in [1]. It is described by a free field sigma model on the twistor space of AdS5 × S5, and is a direct generalisation of the corresponding model for tensionless string theory on AdS3 × S3. As in the case of AdS3, the worldsheet theory contains spectrally flowed representations. We proposed in [1] that in each such sector only a finite set of generalised zero modes (‘wedge modes’) are physical. Here we show that after imposing the appropriate residual gauge conditions, this worldsheet description reproduces precisely the spectrum of the planar gauge theory. Specifically, the states in the sector with w units of spectral flow match with single trace operators built out of w super Yang-Mills fields (‘letters’). The resulting physical picture is a covariant version of the BMN light-cone string, now with a finite number of twistorial string bit constituents of an essentially topological worldsheet., Journal of High Energy Physics, 2021 (11), ISSN:1126-6708, ISSN:1029-8479

We derive an on-shell diagram recursion for tree-level scattering amplitudes in $\mathcal{N}=7$ supergravity. The diagrams are evaluated in terms of Grassmannian integrals and momentum twistors, generalising previous results of Hodges in momentum twistor space to non-MHV amplitudes. In particular, we recast five and six-point NMHV amplitudes in terms of $\mathcal{N}=7$ R-invariants analogous to those of $\mathcal{N}=4$ super-Yang-Mills, which makes cancellation of spurious poles more transparent. Above 5-points, this requires defining momentum twistors with respect to different orderings of the external momenta., Comment: v2. minor changes, published in JHEP

The theory of twistors on foliated manifolds is developed. We construct the twistor space of the normal bundle of a foliation. It is demonstrated that the classical constructions of the twistor theory lead to foliated objects and permit to formulate and prove foliated versions of some well-known results on holomorphic mappings. Since any orbifold can be understood as the leaf space of a suitably defined Riemannian foliation we obtain orbifold versions of the classical results as a simple consequence of the results on foliated mappings.

We study a natural functional on the space of holomorphic sections of the Deligne-Hitchin moduli space of a compact Riemann surface, generalizing the energy of equivariant harmonic maps corresponding to twistor lines. We give a link to a natural meromorphic connection on the hyperholomorphic line bundle recently constructed by Hitchin. Moreover, we prove that for a certain class of real holomorphic sections of the Deligne-Hitchin moduli space, the functional is basically given by the Willmore energy of corresponding (equivariant) conformal map to the 3-sphere. As an application we use the functional to distinguish new components of real holomorphic sections of the Deligne-Hitchin moduli space from the space of twistor lines., 33 pages

For smooth families $\mathcal{X} \to S$ of projective algebraic curves and holomorphic line bundles $\mathcal{L, M} \to X$ equipped with flat relative connections, we prove the existence of a canonical and functorial “intersection” connection on the Deligne pairing $\langle \mathcal{L, M} \rangle \to S$. This generalizes the construction of Deligne in the case of Chern connections of hermitian structures on $\mathcal{L}$ and $\mathcal{M}$. A relationship is found with the holomorphic extension of analytic torsion, and in the case of trivial fibrations we show that the Deligne isomorphism is flat with respect to the connections we construct. Finally, we give an application to the construction of a meromorphic connection on the hyperholomorphic line bundle over the twistor space of rank one flat connections on a Riemann surface.

The first part of this thesis is the geometry of non-Abelian Hodge theory, especially the study of geometric properties of moduli spaces.The first main result of this part is the construction of a dynamical system on the moduli space of Higgs bundles, we show that fixed points of this dynamical system are exactly those fixed by the C*-action on the moduli space of Higgs bundles, that is, all C-VHS in the moduli space. At the same time, we study its first variation and asymptotic behaviour.The second main result of this part is the proof of a conjecture (weak form) by Simpson on the stratification of the moduli space of flat bundles, we prove that the oper stratum is the unique closed stratum of minimal dimension by studying the moduli space of holomorphic chains of given type.The third main result of this part is a generalization of Deligne’s construction of Hitchin twistor space in Riemann surface case, we construct holomorphic sections for this new twistor space, namely the de Rham sections. We calculate the normal bundles of these sections, and we found that de Rham sections in the Deligne–Hitchin twistor space also have wight 1 property, so they are ample rational curves. We also show the Torelli-type theorem for this new twistor space.; La premiére partie de cette thèse est la géométrie de la théorie de Hodge non-Abélienne, en particulier l’étude des propriétés géométriques des espaces de modules.Le premier résultat principal de cette partie est la construction d’un système dynamique sur l’espace de modules des fibrés de Higgs, nous montrons que les points fixes de ce système dynamique sont exactement ceux fixés par l’action de C* sur l’espace de modules des fibrés de Higgs, c’est-àdire tous les C-VHS dans l’espace de modules. Dans le même temps, nous étudions sa première variation et son comportement asymptotique.Le deuxième résultat principal de cette partie est la preuve d’une conjecture (forme faible) par Simpson sur la stratification de l’espace de modules des fibrés plats, nous prouvons que la strata d’opérateurs est la strata fermée unique de dimension minimale en étudiant l’espace de modules des chaînes holomorphes de type donné.Le troisième résultat principal de cette partie est une généralisation de la construction par Deligne en l’espace de twistor de Hitchin dans le cas de surface de Riemann, nous construisons des sections holomorphes pour ce nouvel espace de twistor, c’est-à-dire les sections de de Rham. Nous calculons les fibrés normals de ces sections, et nous avons constaté que les sections de de Rham dans l’espace de twistor de Deligne–Hitchin ont également la propriété wight 1, donc ce sont des courbes rationnelles amples. Dans le même temps, nous montrons le théorème de type Torelli pour l’espace de twistor.La deuxième partie de cette thèse est l’étude de certaines spécialisations de la correspondance de Hodge non-Abélienne. Celui-ci comprend principalement deux chapitres, le premier est une preuve fondamentale d’une conjecture liée aux représentations de carquois proposée par Reineke en 2003, nous montrons pour les représentations de carquois de type An , il existe un système de poids tel que les représentations stables par rapport à ce système de poids sont précisément celles indécomposables. Pour la deuxième, nous construisons la correspondance de Kobayashi–Hitchin pour les fibrés de carquois sur les variétés Kähleriennes généralisées.

The first part of this thesis is the geometry of non-Abelian Hodge theory, especially the study of geometric properties of moduli spaces.The first main result of this part is the construction of a dynamical system on the moduli space of Higgs bundles, we show that fixed points of this dynamical system are exactly those fixed by the C*-action on the moduli space of Higgs bundles, that is, all C-VHS in the moduli space. At the same time, we study its first variation and asymptotic behaviour.The second main result of this part is the proof of a conjecture (weak form) by Simpson on the stratification of the moduli space of flat bundles, we prove that the oper stratum is the unique closed stratum of minimal dimension by studying the moduli space of holomorphic chains of given type.The third main result of this part is a generalization of Deligne’s construction of Hitchin twistor space in Riemann surface case, we construct holomorphic sections for this new twistor space, namely the de Rham sections. We calculate the normal bundles of these sections, and we found that de Rham sections in the Deligne–Hitchin twistor space also have wight 1 property, so they are ample rational curves. We also show the Torelli-type theorem for this new twistor space.; La premiére partie de cette thèse est la géométrie de la théorie de Hodge non-Abélienne, en particulier l’étude des propriétés géométriques des espaces de modules.Le premier résultat principal de cette partie est la construction d’un système dynamique sur l’espace de modules des fibrés de Higgs, nous montrons que les points fixes de ce système dynamique sont exactement ceux fixés par l’action de C* sur l’espace de modules des fibrés de Higgs, c’est-àdire tous les C-VHS dans l’espace de modules. Dans le même temps, nous étudions sa première variation et son comportement asymptotique.Le deuxième résultat principal de cette partie est la preuve d’une conjecture (forme faible) par Simpson sur la stratification de l’espace de modules des fibrés plats, nous prouvons que la strata d’opérateurs est la strata fermée unique de dimension minimale en étudiant l’espace de modules des chaînes holomorphes de type donné.Le troisième résultat principal de cette partie est une généralisation de la construction par Deligne en l’espace de twistor de Hitchin dans le cas de surface de Riemann, nous construisons des sections holomorphes pour ce nouvel espace de twistor, c’est-à-dire les sections de de Rham. Nous calculons les fibrés normals de ces sections, et nous avons constaté que les sections de de Rham dans l’espace de twistor de Deligne–Hitchin ont également la propriété wight 1, donc ce sont des courbes rationnelles amples. Dans le même temps, nous montrons le théorème de type Torelli pour l’espace de twistor.La deuxième partie de cette thèse est l’étude de certaines spécialisations de la correspondance de Hodge non-Abélienne. Celui-ci comprend principalement deux chapitres, le premier est une preuve fondamentale d’une conjecture liée aux représentations de carquois proposée par Reineke en 2003, nous montrons pour les représentations de carquois de type An , il existe un système de poids tel que les représentations stables par rapport à ce système de poids sont précisément celles indécomposables. Pour la deuxième, nous construisons la correspondance de Kobayashi–Hitchin pour les fibrés de carquois sur les variétés Kähleriennes généralisées.

This paper is a supplement to and extension of arXiv:1903.01399. In the internal twistor space of the 4D Vasiliev's higher-spin gravity, we study the star-product eigenfunctions of number operators with generic complex eigenvalues. In particular, we focus on a set of eigenfunctions represented by formulas with generalized Laguerre functions. This set of eigenfunctions can be written as linear combinations of two subsets of eigenfunctions, one of which is closed under the star-multiplication with the creation operator to a generic complex power -- and the other similarly with the annihilation operator. The two subsets intersect when the left and the right eigenvalues differ by an integer. We further investigate how star-multiplications with both the creation and annihilation operators together may change such eigenfunctions and briefly discuss some problems that we are facing in order to use these eigenfunctions as the initial data to construct solutions to Vasiliev's equations., Comment: 16 pages, 3 figures; v4: Section 5 revised and other minor modifications; published on JHEP

The four-point function of length-two half-BPS operators in $\mathcal{N}=4$ SYM receives non-planar corrections starting at four loops. Previous work relied on the analysis of symmetries and logarithmic divergences to fix the integrand up to four constants. In this work, we compute those undetermined coefficients and fix the integrand completely by using the reformulation of $\mathcal{N}=4$ SYM in twistor space. The final integrand can be written as a combination of finite conformal integrals and we have used the method of asymptotic expansions to extract non-planar anomalous dimensions and structure constants for twist-two operators up to spin eight. Some of the results were already know in the literature and we have found agreement with them., 26 pages, 2 figures

Using quaternionic Feix–Kaledin construction, we provide a local classification of quaternion-Kähler metrics with a rotating $$S^1$$S1-symmetry with the fixed point set submanifold S of maximal possible dimension. For any real-analytic Kähler manifold S equipped with a line bundle with a real-analytic unitary connection with curvature proportional to the Kähler form, we explicitly construct a holomorphic contact distribution on the twistor space obtained by the quaternionic Feix–Kaledin construction from these data. Conversely, we show that quaternion-Kähler metrics with a rotating $$S^1$$S1-symmetry induce on the fixed point set of maximal dimension a Kähler metric together with a unitary connection on a holomorphic line bundle with curvature proportional to the Kähler form and the two constructions are inverse to each other. Moreover, we study the case when S is compact, showing that in this case the quaternion-Kähler geometry is determined by the Kähler metric on the fixed point set (of maximal possible dimension) and by the contact line bundle along the corresponding submanifold on the twistor space. Finally, we relate the results to the c-map construction showing that the family of quaternion-Kähler manifolds obtained from a fixed Kähler metric on S by varying the line bundle and the hyperkähler manifold obtained by hyperkähler Feix–Kaledin construction from S are related by hyperkähler/quaternion-Kähler correspondence.

In this paper a bijective correspondence between superminimal surfaces of an oriented Riemannian 4-manifold and particular Lagrangian submanifolds of the twistor space over the 4-manifold is proven. More explicitly, for every superminimal surface a submanifold of the twistor space is constructed which is Lagrangian for all the natural almost Hermitian structures on the twistor space. The twistor fibration restricted to the constructed Lagrangian gives a circle bundle over the superminimal surface. Conversely, if a submanifold of the twistor space is Lagrangian for all the natural almost Hermitian structures, then the Lagrangian projects to a superminimal surface and is contained in the Lagrangian constructed from this surface. In particular this produces many Lagrangian submanifolds of the twistor spaces C P 3 and F 1 , 2 ( C 3 ) with respect to both the Kahler structure as well as the nearly Kahler structure. Moreover, it is shown that these Lagrangian submanifolds are minimal submanifolds.

金沢大学人間社会研究域学校教育系 / Institute of Human and Social science, Teacher Education, We introduce a quaternionic invariant for an inclusive immersion into a quaternionic manifold, which is a quaternionic object corresponding to the Willmore functional. The lower bound of this invariant is given by topological invariant and the equality case can be characterized in terms of the natural twistor lift. When the ambient manifold is the quaternionic projective space and the natural twistor lift is holomorphic, we obtain a relation between the quaternionic invariant and the degree of the image of the natural twistor lift as an algebraic curve. Moreover the first variation formula for the invariant is obtained. As an application of the formula, if the natural twistor lift is a harmonic section, then the surface is a stationary point under any variations such that the induced complex structures do not vary. © 2017, Springer-Verlag Berlin Heidelberg., Embargo Period 12 months