We present the inverse Penrose transform (the map from spacetime to twistor space) for self-dual Yang-Mills (SDYM) and its higher-spin extensions on a flat background. The twistor action for the higher-spin extension of SDYM (HS-SDYM) is of BF-type. By considering a deformation away from the self-dual sector of HS-SDYM, we discover a new action that describes a higher-spin extension of Yang-Mills theory (HS-YM). The twistor action for HS-YM is a straightforward generalization of the Yang-Mills one.
Recently, Witten proposed a topological string theory in twistor space that is dual to a weakly coupled gauge theory. In this lectures we will discuss aspects of the twistor string theory. Along the way we will learn new things about Yang-Mills scattering amplitudes. The string theory sheds light on Yang-Mills perturbation theory and leads to new methods for computing Yang-Mills scattering amplitudes.
Four-dimensional super-twistors provide a compact covariant description of on-shell N=4 d=4 super-Yang-Mills. In this paper, ten-dimensional super-twistors are introduced which similarly provide a compact covariant description of on-shell d=10 super-Yang-Mills. The super-twistor variables are Z=(lambda^alpha, mu_alpha, Gamma^m) where lambda^alpha and mu_alpha are constrained bosonic d=10 spinors and Gamma^m is a constrained fermionic d=10 vector. The Penrose map relates the twistor superfield Phi(Z) with the d=10 super-Yang-Mills vertex operator lambda^alpha A_alpha(x,theta) which appears in t…
In this letter, an alternative string theory in twistor space is proposed for describing perturbative N=4 super-Yang-Mills theory. Like the recent proposal of Witten, this string theory uses twistor worldsheet variables and has manifest spacetime superconformal invariance. However, in this proposal, tree-level super-Yang-Mills amplitudes come from open string tree amplitudes as opposed to coming from D-instanton contributions.
We introduce a covariant finite regulator for N = 4 super Yang-Mills theory on S^4. Our formulation is based on holomorphic Chern-Simons theory on twistor space. By switching on a large background flux, the twistor space dissolves into a fuzzy geometry, with a finite number of points. The large N continuum limit of the matrix model naturally approaches ordinary N = 4 SYM. We comment on the relation between our model and the 4D quantum Hall effect.
We explain some details of the construction of composite operators in N=4 SYM that we have elaborated earlier in the context of Lorentz harmonic chiral (LHC) superspace. We give a step-by-step elementary derivation and show that the result coincides with the recent hypothesis put forward in arXiv:1603.04471 within the twistor approach. We provide the appropriate LHC-to-twistors dictionary.
Super twistor space admits a certain (super) complex structure deformation that preserves the Poincare subgroup of the symmetry group PSL(4|4) and depends on 10 parameters. In a previous paper [hep-th/0502076], it was proposed that in twistor string theory this deformation corresponds to augmenting N=4 super Yang-Mills theory by a mass term for the left-chirality spinors. In this paper we analyze this proposal in more detail. We calculate 4-particle scattering amplitudes of fermions, gluons and scalars and show that they are supported on holomorphic curves in the deformed twistor space.
We propose various ways of adding mass terms to three-dimensional twistor string theory. We begin with a review of mini-twistor space--the reduction of D=4 twistor space to D=3. We adapt the two proposals for twistor string theory, Witten's and Berkovits's, to D=3 super Yang-Mills theory. In Berkovits's model, we identify the enhanced R-symmetry. We then construct B-model topological string theories that, we propose, correspond to D=3 Yang-Mills theory with massive spinors and massive and massless scalars in the adjoint representation of the gauge group. We also analyze the counterparts of the…
We consider the twistor space ${\cal P}^6\cong{\mathbb R}^4{\times}{\mathbb C}P^1$ of ${\mathbb R}^4$ with a non-integrable almost complex structure ${\cal J}$ such that the canonical bundle of the almost complex manifold $({\cal P}^6, {\cal J})$ is trivial. It is shown that ${\cal J}$-holomorphic Chern-Simons theory on a real $(6|2)$-dimensional graded extension ${\cal P}^{6|2}$ of the twistor space ${\cal P}^6$ is equivalent to self-dual Yang-Mills theory on Euclidean space ${\mathbb R}^4$ with Lorentz invariant action. It is also shown that adding a local term to a Chern-Simons-type action …
There is evidence that one can compute tree level super Yang-Mills amplitudes using either connected or completely disconnected curves in twistor space. We argue that the two computations are equivalent, if the integration contours are chosen in a specific way, by showing that they can both be reduced to the same integral over a moduli space of singular curves. We also formulate a class of new ``intermediate'' prescriptions to calculate the same amplitudes.
We argue that the string theory dual to a certain sector of the four-dimensional Yang-Mills theory at large-N is the A-model wrapping N-Lagrangian D-branes on the twistor space of the complex two-dimensional projective space, with certain fluxes and Wilson loops, by finding that the target-space beta function of this A-model coincides with the large-N beta function of the Yang-Mills theory
In this PhD thesis we extend the twistor formalism to encompass (partially) off-shell quantities. We describe all gauge-invariant local composite operators in twistor space and show that they immediately generate all tree-level form factors of the MHV type. We use the formalism to compute form factors at NMHV and higher NkMHV level in parallel to how this had been done for amplitudes previously. Finally, we move on to integrability by computing the one-loop dilatation operator in the scalar sector of the theory in twistor space.
We consider the Hermitian-Yang-Mills (HYM) equations for gauge potentials on a complex vector bundle E over an almost complex manifold X^6 which is the twistor space of an oriented Riemannian manifold M^4. Each solution of the HYM equations on such X^6 defines a pseudo-holomorphic structure on the bundle E. It is shown that the pull-back to X^6 of any anti-self-dual gauge field on M^4 is a solution of the HYM equations on X^6. This correspondence allows us to introduce new twistor actions for bosonic and supersymmetric Yang-Mills theories. As examples of X^6 we consider homogeneous nearly Kaeh…
Lagrangian and Hamiltonian dynamics of de Azcarraga-Lukierski N=2 massive superparticle is considered in the framework of twistor-like Lorentz-harmonic approach. The emphasis is on the study of the interaction with external Abelian gauge superfield. The requirement of preservation of all gauge symmetries of the free model including kappa-symmetry yields correct expressions for the superfield strength constraints and determines the form of nonminimal interaction. We also show that for de Azcarraga-Lukierski N=2 massive superparticle the pullback of field strength 2-superform to the superworld l…
Conformal supergravity arises in presently known formulations of twistor-string theory either via closed strings or via gauge-singlet open strings. We explore this sector of twistor-string theory, relating the relevant string modes to the particles and fields of conformal supergravity. We also use the twistor-string theory to compute some tree level scattering amplitudes with supergravitons, and compare to expectations from conformal supergravity. Since the supergravitons interact with the same coupling constant as the Yang-Mills fields, conformal supergravity states will contribute to loop am…
This thesis carries out a detailed investigation of the action for pure Yang- Mills theory which L. Mason formulated in twistor space. The rich structure of twistor space results in greater gauge freedom compared to the theory in ordinary space-time. One particular gauge choice, the CSW gauge, allows simplifications to be made at both the classical and quantum level. The equations of motion have an interesting form in the CSW gauge, which suggests a possible solution procedure. This is explored in three special cases. Explicit solutions are found in each case and connections with earlier work …
These lectures fall into two distinct, although tenouously related, parts. The first part is about fuzzy and noncommutative spaces, and particle mechanics on such spaces, in other words, noncommutative mechanics. The second part is a discussion/review of twistors and how they can be used in the calculation of Yang-Mills amplitudes. The point of connection between these two topics, discussed in the last section, is in the realization of holomorphic maps as the lowest Landau level wave functions, or as wave functions of the Hilbert space used for the fuzzy version of the two-sphere. This article…
Witten has recently proposed a string theory in twistor space whose D-instanton contributions are conjectured to compute N=4 super-Yang-Mills scattering amplitudes. An alternative string theory in twistor space was then proposed whose open string tree amplitudes reproduce the D-instanton computations of maximal degree in Witten's model. In this paper, a cubic open string field theory action is constructed for this alternative string in twistor space, and is shown to be invariant under parity transformations which exchange MHV and googly amplitudes. Since the string field theory action is gauge…
We interpret the A and B model topological strings on CP^{3|4} as equivalent to open N=2 string theory on spacetime with signature (2,2), when covariantized with respect to SO(2,2) and supersymmetrized a la Siegel. We propose that instantons ending on Lagrangian branes wrapping RP^{3|4} deform the self-dual N=4 Yang-Mills sector to ordinary Yang-Mills by generating a `t Hooft like expansion. We conjecture that the A and B versions are S-dual to each other. We also conjecture that mirror symmetry may explain the recent observations of Witten that twistor transformed N=4 Yang-Mills amplitudes li…
Recently N. Berkovits, motivated by the supertwistor description of ${\cal N}=4 D=4$ super Yang-Mills, considered the generalization of the ${\cal N}=1 D=4$ $θ$-twistor construction to D=10 and applied it for a compact covariant description of ${\cal N}=1 D=10$ super Yang-Mills. This supports the relevance of the $θ$-twistor as a supersymmetric twistor alternative to the well-known supertwistor. The minimal breaking of superconformal symmetry is an inherent property of the $θ$-twistor received from its fermionic components, described by a Grassmannian vector instead of a Grassmannian scalar in…
We reformulate twistor-string theory as a heterotic string based on a twisted (0,2) model. The path integral localizes on holomorphic maps, while the (0,2) moduli naturally correspond to the states of N=4 super Yang-Mills and conformal supergravity under the Penrose transform. We show how the standard twistor-string formulae of scattering amplitudes as integrals over the space of curves in supertwistor space may be obtained from our model. The corresponding string field theory gives rise to a twistor action for N=4 conformal supergravity coupled to super Yang-Mills. The model helps to explain …
We present a twistor space that describes super null-lines on six-dimensional N=(1,1) superspace. We then show that there is a one-to-one correspondence between holomorphic vector bundles over this twistor space and solutions to the field equations of N=(1,1) super Yang-Mills theory. Our constructions naturally reduce to those of the twistorial description of maximally supersymmetric Yang-Mills theory in four dimensions.
We show that there are remarkable simplifications when the MHV diagram formalism for N=4 super Yang-Mills is reformulated in momentum twistor space. The vertices are replaced by unity while each propagator becomes a dual superconformal `R-invariant' whose arguments may be read off from the diagram. The momentum twistor MHV rules generate a formula for the full, all-loop planar integrand of the super Yang-Mills S-matrix that is manifestly dual superconformally invariant up to the choice of a reference twistor. We give a general proof of this reformulation and illustrate its use by computing the…
By lifting full Yang-Mills theory to $R_4 \times S_2$, Mason et al. obtained MHV vertices by gauge transformation. Their Lagrangian depended on long intricate twistor manipulations. Spinor $S_2$ harmonics give a one page proof, eliminating all the $CP_3$ apparatus.
The twistor superstring, which describes N=4 super Yang-Mills trees, is taken off-shell (for loops) by generalizing Penrose twistors (which describe on-shell momenta) to Atiyah-Drinfel'd-Hitchin-Manin twistors (which include the usual spacetime coordinates). The resulting string is then shown to be the tensionless limit of a Quantum ChromoDynamics-like superstring.
We study the twistor formulation of the classical N=4 super Yang-Mills theory on the quadric submanifold of CP(3|3) X CP(3|3). We reformulate the twistor equations in six dimension, where the superconformal symmetry is manifest, and find a connection to complexified AdS5.
Different string theories in twistor space have recently been proposed for describing ${\cal N}=4$ super-Yang-Mills. In this paper, my Strings 2003 talk is reviewed in which a string theory in $(x,θ)$ space was constructed for self-dual ${\cal N}=4$ super-Yang-Mills. It is hoped that these results will be useful for understanding the twistor-string proposals and their possible relation with the pure spinor formalism of the $d=10$ superstring.
Four dimensional Yang-Mills theory formulated through an action on twistor space has a larger gauge symmetry than the usual formulation, which in previous work was shown to allow a simple gauge transformation between text-book perturbation theory and the Cachazo-Svrcek-Witten rules. In this paper we study non-supersymmetric twistor Yang-Mills theory at loop level using the background field method. For an appropriate partial quantum field gauge choice it is shown the calculation of the effective action is equivalent to (the twistor lift of) the calculation in ordinary Yang-Mills theory in the C…
We extend the twistor string theory inspired formalism introduced in hep-th/0407214 for calculating loop amplitudes in N=4 super Yang-Mills theory to the case of N=1 (and N=2) super Yang-Mills. Our approach yields a novel representation of the gauge theory amplitudes as dispersion integrals, which are surprisingly simple to evaluate. As an application we calculate one-loop maximally helicity violating (MHV) scattering amplitudes with an arbitrary number of external legs. The result we obtain agrees precisely with the expressions for the N=1 MHV amplitudes derived previously by Bern, Dixon, Dun…
We construct a twistor space action for N=4 super Yang-Mills theory and show that it is equivalent to its four dimensional spacetime counterpart at the level of perturbation theory. We compare our partition function to the original twistor-string proposal, showing that although our theory is closely related to string theory, it is free from conformal supergravity. We also provide twistor actions for gauge theories with N<4 supersymmetry, and show how matter multiplets may be coupled to the gauge sector.
We present a new formula for all single trace tree amplitudes in four dimensional super Yang-Mills coupled to Einstein supergravity. Like the Cachazo-He-Yuan formula, our expression is supported on solutions of the scattering equations, but with momenta written in terms of spinor helicity variables. Supersymmetry and parity are both manifest. In the pure gravity and pure Yang-Mills sectors, it reduces to the known twistor-string formulae. We show that the formula behaves correctly under factorization and sketch how these amplitudes may be obtained from a four-dimensional (ambi)twistor string.
We present a new, explicit formula for all tree-level amplitudes in N=4 super Yang-Mills. The formula is written as a certain contour integral of the connected prescription of Witten's twistor string, expressed in link variables. A very simple deformation of the integrand gives directly the Grassmannian integrand proposed by Arkani-Hamed et al. together with the explicit contour of integration. The integral is derived by iteratively adding particles to the Grassmannian integral, one particle at a time, and makes manifest both parity and soft limits. The formula is shown to be related to those …
We discuss infinite-dimensional hidden symmetry algebras (and hence an infinite number of conserved nonlocal charges) of the N-extented self-dual super Yang-Mills equations for general N\leq4 by using the supertwistor correspondence. Furthermore, by enhancing the supertwistor space, we construct the N-extended self-dual super Yang-Mills hierarchies, which describe infinite sets of graded Abelian symmetries. We also show that the open topological B-model with the enhanced supertwistor space as target manifold will describe the hierarchies. Furthermore, these hierarchies will in turn -- by a sup…
Twistor space constructions and actions are given for full Yang-Mills and conformal gravity using almost complex structures that are not, in general, integrable. These are used as the basis of a derivation of the twistor-string generating functionals for tree level perturbative scattering amplitudes of Yang-Mills and conformal gravity. The derivation follows by expanding and resumming the classical approximation to the path integral obtained from the twistor action. It provides a basis for exploring whether the equivalence can be made to extend beyond tree level and allows one to disentangle c…
We consider the fuzzy 4-sphere $S_N^4$ as a background in the IKKT matrix model and explore the relation between $S_N^4$ and fuzzy twistor space in the semi-classical limit. A novel description for the IKKT-matrix model in terms of spinorial indices is given, which is reminiscent of $\mathcal{N}=4$ super-symmetric Yang-Mills (SYM) in $4d$. On fuzzy twistor space, the interactions of the IKKT model are of gravitational type. The higher-spin (HS) gauge theory emerging in this limit from the IKKT model, denoted as HS-IKKT, on fuzzy twistor space is shown to be a higher-spin extension of $\mathcal…
Institute for Theoretical Physics, Albert Einstein Center for Fundamental Physics, University of Bern（伯尔尼大学理论物理研究所爱因斯坦基础物理中心）；Wilczek Quantum Center, Shanghai Institute for Advanced Studies, University of Science and Technology of China（中国科学技术大学上海高等研究院微尔切克量子中心）
In this thesis, we report on results in non-anticommutative field theory and twistor string theory, trying to be self-contained. We first review the construction of non-anticommutative N=4 super Yang-Mills theory and discuss a Drinfeld-twist which allows to regain a twisted supersymmetry in the non-anticommutative setting. This symmetry then leads to twisted chiral rings and supersymmetric Ward-Takahashi identities, which, when combined with the usual naturalness argument by Seiberg, could yield non-renormalization theorems for non-anticommutative field theories. The major part of this thesis …
Dual superconformal invariance has recently emerged as a hidden symmetry of planar scattering amplitudes in N=4 super Yang-Mills theory. This symmetry can be made manifest by expressing amplitudes in terms of `momentum twistors', as opposed to the usual twistors that make the ordinary superconformal properties manifest. The relation between momentum twistors and on-shell momenta is algebraic, so the translation procedure does not rely on any choice of space-time signature. We show that tree amplitudes and box coefficients are succinctly generated by integration of holomorphic delta-functions i…
We study Yang-Mills connections on holomorphic bundles over complex Kähler manifolds of arbitrary dimension, in the spirit of Hitchin's and Simpson's study of flat connections. The space of non-Hermitian Yang-Mills (NHYM) connections has dimension twice the space of Hermitian Yang-Mills connections, and is locally isomorphic to the complexification of the space of Hermitian Yang-Mills connections (which is, by Uhlenbeck and Yau, the same as the space of stable bundles). Further, we study the NHYM connections over hyperkähler manifolds. We construct direct and inverse twistor transform from NHY…
The simplicity and hidden symmetries of (Super) Yang-Mills and (Super)Gravity scattering amplitudes suggest the existence of a "weak-weak" dual formulation in which these structures are made manifest at the expense of manifest locality. We suggest that this dual description lives in (2,2) signature and is naturally formulated in twistor space. We recast the BCFW recursion relations in an on-shell form that begs to be transformed into twistor space. Our twistor transformation is inspired by Witten's, but differs in treating twistor and dual twistor variables more equally. In these variables the…
We review aspects of twistor theory, its aims and achievements spanning thelast five decades. In the twistor approach, space--time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex three--fold -- the twistor space. After giving an elementary construction of this space we demonstrate how solutions to linear and nonlinear equations of mathematical physics: anti-self-duality (ASD) equations on Yang--Mills, or conformal curvature can be encoded into twistor cohomology. These twistor correspondences yield explicit examples of Yang--Mills, and …
Perturbative scattering amplitudes in Yang-Mills theory have many unexpected properties, such as holomorphy of the maximally helicity violating amplitudes. To interpret these results, we Fourier transform the scattering amplitudes from momentum space to twistor space, and argue that the transformed amplitudes are supported on certain holomorphic curves. This in turn is apparently a consequence of an equivalence between the perturbative expansion of ${\cal N}=4$ super Yang-Mills theory and the $D$-instanton expansion of a certain string theory, namely the topological $B$ model whose target spac…
We consider superstring sigma models that are based on coset superspaces G/H in which H arises as the fixed point set of an order-4 automorphism of G. We show by means of twistor theory that the corresponding first-order system, consisting of the Maurer-Cartan equations and the equations of motion, arises from a dimensional reduction of some generalised self-dual Yang-Mills equations in eight dimensions. Such a relationship might help shed light on the explicit construction of solutions to the superstring equations including their hidden symmetry structures and thus on the properties of their …
The S-matrix for planar N = 4 super Yang-Mills theory can be computed as the correlation function for a holomorphic polygonal Wilson loop in twistor space. In an axial gauge, this leads to the construction of the all-loop integrand via MHV diagrams in twistor space. We show that at MHV, this formulation leads directly to expressions for loop integrands in d log form; i.e., the integrand is a product of exterior derivatives of logarithms of rational functions. For higher MHV degree, it is in d log form multiplied by delta functions. The parameters appearing in the d log form arise geometrically…
A family of new twistor string theories is constructed and shown to be free from world-sheet anomalies. The spectra in space-time are calculated and shown to give Einstein supergravities with second order field equations instead of the higher derivative conformal supergravities that arose from earlier twistor strings. The theories include one with the spectrum of N=8 supergravity, another with the spectrum of N=4 supergravity coupled to N=4 super-Yang-Mills, and a family with $N\ge 0$ supersymmetries with the spectra of self-dual supergravity coupled to self-dual super-Yang-Mills. The non-supe…
The worldsheet string theory dual to free 4d ${\cal N}=4$ super Yang-Mills theory was recently proposed in arXiv:2104.08263. It is described by a free field sigma model on the twistor space of ${\rm AdS}_5\times {\rm S}^5$, and is a direct generalisation of the corresponding model for tensionless string theory on ${\rm AdS}_3\times {\rm S}^3$. As in the case of ${\rm AdS}_3$, the worldsheet theory contains spectrally flowed representations. We proposed in arXiv:2104.08263 that in each such sector only a finite set of generalised zero modes (`wedge modes') are physical. Here we show that after …
Recently, a worldsheet dual to free ${\cal N}=4$ Super Yang-Mills has been proposed in terms of twistor variables for ${\rm AdS}_5$, in parallel to that for the ${\rm AdS}_3$ dual to the free symmetric orbifold CFT. In the latter case, holomorphic covering maps play a central role in determining correlators and are associated to Feynman diagrams. After recasting these maps in terms of the worldsheet twistor variables for ${\rm AdS}_3$, we generalise to ${\rm AdS}_5$. We propose stringy incidence relations and appropriate reality conditions for the twistor covering maps. For some special kinema…
In this talk we review the harmonic space formulation of the twistor transform for the supersymmetric self-dual Yang-Mills equations. The recently established harmonic-twistor correspondence for the N-extended supersymmetric gauge theories is described. It affords an explicit construction of solutions to these equations which displays a remarkable matreoshka-like structure determined by the N=0 core.
Recursion relations are succinctly obtained for $(++... +)$ and $(-++... +)$ amplitudes in the context of the space-cone gauge in QCD. We rely on the helicity symmetry of the problems to dictate our choices of reference twistors and the momentum shifts to complexify the amplitudes. Of great importance is the power of gauge Lorentz invariance, which is enough to determine the soft factors in the latter cases.
Infinite-dimensional algebras of hidden symmetries of the self-dual Yang-Mills equations are considered. A current-type algebra of symmetries and an affine extension of conformal symmetries introduced recently are discussed using the twistor picture. It is shown that the extended conformal symmetries of the self-dual Yang-Mills equations have a simple description in terms of Ward's twistor construction.
MHV diagrams give an efficient Feynman diagram-like formalism for calculating gauge theory scattering amplitudes on momentum space. Although they arise as the Feynman diagrams from an action on twistor space in an axial gauge, the main ingredients were previously expressed only in momentum space and momentum twistor space. Here we show how the formalism can be elegantly derived and expressed entirely in twistor space. This brings out the underlying superconformal invariance of the framework (up to the choice of a reference twistor used to define the axial gauge) and makes the twistor support t…
The geometrical structure and the quantum properties of the recently proposed harmonic space action describing self-dual Yang-Mills (SDYM) theory are analyzed. The geometrical structure that is revealed is closely related to the twistor construction of instanton solutions. The theory gets no quantum corrections and, despite having SDYM as its classical equation of motion, its S matrix is trivial. It is therefore NOT the theory of the N=2 string. We also discuss the 5-dimensional actions that have been proposed for SDYM.
