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Non-self-dual gauge fields

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Abstract

The Ward construction is generalized to non-self-dual gauge fields. Reality and currentless conditions are specified.

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... A rather natural way of constructing a classical candidate p2, 0q-theory is to combine higher gauge theory with twistor geometry. Recall that there is a twistor space which carries a useful representation of solutions to the N " 3 supersymmetric Yang-Mills equations in terms of holomorphic principal bundles [20,21]. Analogously, there is a twistor space that carries a representation of Abelian self-dual 3-forms in six dimensions in terms of holomorphic gerbes, see [22][23][24][25][26][27] as well as [28][29][30]. ...
... After decomposing C 8n -C 4 b C 2n and choosing a holomorphic symplectic form 21 Note that this decomposition is equivalent to choosing a conformal structure on M , see [28,Remark 3.2]. ...
... These maps satisfy various cocycle conditions as implied by the relations displayed in Lemma 2.18; see also Theorem 4.19. For instance, the cocycle conditions for α 2 1,a read as f 2 2˚|σ2pφq pγ 21 0,ab q "´d π 1 σ 1 pφq and f 2 0˚|σ2pφq pγ 21 0,ab q " α 2 1,b , f 2 1˚|σ2pφq pγ 21 0,ab q " f 2 1˚|σ2pφq pγ 21 1,ab q , f 2 2˚|σ2pφq pγ 21 1,ab q " α 2 1,a and f 2 0˚|σ2pφq pγ 21 1,ab q " 0 (6.12b) with σ p :" d p´1 0˝¨¨¨˝d 1 0˝d 0 0 as defined in Theorem 4.10. Using the simplicial identities (2.5), it is rather easy to see that these equations imply that f 1 1˚| φ pα 2 a q " f 1 1˚| φ pα 2 b q. ...
Preprint
We develop a description of higher gauge theory with higher groupoids as gauge structure from first principles. This approach captures ordinary gauge theories and gauged sigma models as well as their categorifications on a very general class of (higher) spaces comprising presentable differentiable stacks, as e.g. orbifolds. We start off with a self-contained review on simplicial sets as models of (,1)(\infty,1)-categories. We then discuss principal bundles in terms of simplicial maps and their homotopies. We explain in detail a differentiation procedure, suggested by Severa, that maps higher groupoids to LL_\infty-algebroids. Generalising this procedure, we define connections for higher groupoid bundles. As an application, we obtain six-dimensional superconformal field theories via a Penrose-Ward transform of higher groupoid bundles over a twistor space. This construction reduces the search for non-Abelian self-dual tensor field equations in six dimensions to a search for the appropriate (higher) gauge structure. The treatment aims to be accessible to theoretical physicists.
... It is well known that holomorphic Chern-Simons theory on the ambitwistor space is classically equivalent to N " 3 supersymmetric Yang-Mills theory in four dimensions (a theory perturbatively equivalent to N " 4 supersymmetric Yang-Mills theory) at the level of the moduli spaces of solutions and their gauge equivalence classes [1][2][3][4][5][6][7][8][9] (see also [10] for a review). The construction of a corresponding action functional, however, is non-trivial. ...
... In this regard the use of twistor spaces has a clear advantage. 2 See [57] for the original definition of BV ■ -algebras, where they were used to prove colour-kinematics duality for the tree-level Yang-Mills S-matrix. See [51,52,[58][59][60][61][62][63][64]54,[65][66][67] for related work on BV ■ -algebras, colour-kinematics duality and the double copy at the level of actions. ...
... Bθ iα and not its complex conjugate. 2 In [17], such a theory is referred to as partially holomorphic Chern-Simons theory. ...
Preprint
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Inspired by the Movshev-Mason-Skinner Cauchy-Riemann (CR) ambitwistor approach, we provide a rigorous yet elementary construction of a twisted CR holomorphic Chern-Simons action on CR ambitwistor space for maximally supersymmetric Yang-Mills theory on four-dimensional Euclidean space. The key ingredient in our discussion is the homotopy algebraic perspective on perturbative quantum field theory. Using this technology, we show that both theories are semi-classically equivalent, that is, we construct a quasi-isomorphism between the cyclic LL_\infty-algebras governing both field theories. This confirms a conjecture from the literature. Furthermore, we also show that the Yang-Mills action is obtained by integrating out an infinite tower of auxiliary fields in the Chern-Simons action, that is, the two theories are related by homotopy transfer. Given its simplicity, this Chern-Simons action should form a fruitful starting point for analysing perturbative properties of Yang-Mills theory.
... For example, the Penrose-Ward correspondence reformulates the ASDYM equations in Twistor space, which leads to the ADHM construction of instantons [7][8] [9]. This construction was generalized to a geometric formulation of the Yang-Mills field equations in ambitwistor space, with a natural interpretation in superspace [10] [11][12] [13]. More recently, twistor and ambitwistor methods have been used in string theory to understand Yang-Mills scattering amplitudes and their properties [14] [15]. ...
... As in the previous section, linearity of the pullback connection follows from a variant of Liouville's theorem. Integrability on lines follows from writing (7) for the pullback translation operators defined on L, and then using (11) and the assumption that Φ is integrable on lines. This implies that super causal morphisms are symmetries of the N = 3 SYM field equations. ...
... Based on these considerations, showing that the extended causal morphisms preserve the form of equation (14) and the gauge condition implies that they preserve solutions of the YM field equations. To show this, restrict to a super null line L and use (11) to compute the pullback connection of (14), which gives ...
Preprint
Full-text available
We show that a class of previously defined maps, called self-dual and causal morphisms, form classical symmetries of Yang-Mills fields in four complex dimensions. These maps generalize conformal transformations, and admit a nonlocal pullback connection that preserves the equations of the theory. First it is shown that self-dual morphisms form symmetries of the anti-self-dual Yang-Mills equations under this pullback. Then a supersymmetric generalization of causal morphisms is defined, which preserves solutions of the field equations for N=3 supersymmetric Yang-Mills theory. As a special case, this implies that a modified definition of causal morphisms form symmetries for the ordinary Yang-Mills field equations.
... Although expressed on T * M, the plane wave representative descends to PA as k · P = 0 on the support of the delta function, so that under X → X + αP, k · X does not change. For s = 1 this provides the Maxwell version of the ambitwistor Yang-Mills correspondence of Witten and Isenberg et al [23,24] and for s = 2 this provides the linear version of the transform for gravity introduced by LeBrun [33], see [22,34] for general linear fields. More generally, our integrated vertex operators all take the form V := wδ(k · P) e ik·x , (2.19) where w depends on P and the worldsheet matter fields from S L + S R . ...
... The original supersymmetric ambitwistor space of [23,24] is a supersymmetric extension of the space of scaled complex null geodesics which we shall again denote by A and can be expressed as To see the connection with null geodesics, we first introduce the supertwistor incidence relations ...
... This can be used to construct vertex operators and amplitudes at three points on a plane wave [168,169], and one can similarly encode the Yang-Mills equations in the heterotic model [170]. This provides a completely different perspective to that pursued in the 1970s and 1980s when space-time field equations in 4d were shown [24,34,171] to correspond to the existence of formal neighbourhoods of A inside PT × PT * or supersymmetric extensions [23,172]. In the ambitwistor string, the field equations are encoded in the quantum consistency of the worldsheet model on a curved ambitwistor space. ...
Article
Full-text available
Starting with Witten’s twistor string, chiral string theories have emerged that describe field theory amplitudes without the towers of massive states of conventional strings. These models are known as ambitwistor strings due to their target space; the space of complexified null geodesics, also called ambitwistor space. Correlators in these string theories directly yield compact formulæ for tree-level amplitudes and loop integrands, in the form of worldsheet integrals fully localized on solutions to constraints known as the scattering equations. In this chapter, we discuss two incarnations of the ambitwistor string: a ‘vector representation’ starting in space-time and structurally resembling the RNS superstring, and a four-dimensional twistorial version closely related to, but distinct from Witten’s original model. The RNS-like models exist for several theories, with ‘heterotic’ and type II models describing super-Yang-Mills and 10d supergravities respectively, and they manifest the double copy relations directly at the level of the worldsheet models. In the second half of the chapter, we explain how the underlying models lead to diverse applications, ranging from extensions to new sectors of theories, loop amplitudes and to scattering on curved backgrounds. We conclude with a brief discussion of connections to conventional strings and celestial holography.
... For s = 1 this provides the Maxwell version of the ambitwistor Yang-Mills correspondence of Witten and Isenberg, et. al. [23,24] and for s = 2 this provides the linear version of the transform for gravity introduced by Lebrun [33], see [34,22] for general linear fields. ...
... The original supersymmetric ambitwistor space of [23,24] is a supersymmetric extension of the space of complex null geodesics which we shall again denote by A and can be expressed as ...
... This can be used to construct vertex operators and amplitudes at three points on a plane wave [168,169], and one can similarly encode the Yang-Mills equations in the heterotic model [170]. This provides a completely different perspective to that pursued in the 1970's and 1980's when spacetime field equations in 4d were shown [24,34,171] to correspond to the existence of formal neighbourhoods of A inside PT × PT * or supersymmetric extensions [23,172]. ...
Preprint
Full-text available
Starting with Witten's twistor string, chiral string theories have emerged that describe field theory amplitudes without the towers of massive states of conventional strings. These models are known as ambitwistor strings due to their target space; the space of complexified null geodesics, also called ambitwistor space. Correlators in these string theories directly yield compact formulae for tree-level amplitudes and loop integrands, in the form of worldsheet integrals fully localized on solutions to constraints known as the scattering equations. In this chapter, we discuss two incarnations of the ambitwistor string: a 'vector representation' starting in space-time and structurally resembling the RNS superstring, and a four-dimensional twistorial version closely related to, but distinct from Witten's original model. The RNS-like models exist for several theories, with 'heterotic' and type II models describing super-Yang-Mills and 10d supergravities respectively, and they manifest the double copy relations directly at the level of the worldsheet models. In the second half of the chapter, we explain how the underlying models lead to diverse applications, ranging from extensions to new sectors of theories, loop amplitudes and to scattering on curved backgrounds. We conclude with a brief discussion of connections to conventional strings and celestial holography.
... Ambitwistor space A is the space of null geodesics [25][26][27][28]. This may be constructed simply from the cotangent bundle T * M of the spacetime M, which will be Minkowski spacetime for most of this paper. ...
... where B n−3 ( ν) is given by (26). How should we think about this correlation function? ...
... so that, for a general displacement in T * M n , only the change in the base coordinate gives rise to a change in Ω | Ψ ( ν). It appears that deformations in the fibre directions preserve Ω | Ψ ( ν) and so we may formally integrate over the base 26 and (47) is well-defined 27 . The generalisation to the off-shell case is now straightforward. ...
Article
Full-text available
A covariant closed superstring field theory, equivalent to classical ten-dimensional Type II supergravity, is presented. The defining conformal field theory is the ambitwistor string worldsheet theory of Mason and Skinner. This theory is known to reproduce the scattering amplitudes of Cachazo, He and Yuan in which the scattering equations play an important role and the string field theory naturally incorporates these results. We investigate the operator formalism description of the ambitwsitor string and propose an action for the string field theory of the bosonic and supersymmetric theories. The correct linearised gauge symmetries and spacetime actions are explicitly reproduced and evidence is given that the action is correct to all orders. The focus is on the Neveu-Schwarz sector and the explicit description of tree level perturbation theory about flat spacetime. Application of the string field theory to general supergravity backgrounds and the inclusion of the Ramond sector are briefly discussed.
... However, it is not the purpose of this section to provide an in-depth review, but rather a presentation geared towards applications in scattering amplitudes of massless particles and worldsheet theories in particular. The interested reader is referred to the original papers [41][42][43][44][45][46][47][48] for a more detailed exposition and [30] for a modern review in the context of ambitwistor strings. Finally, we conclude with a review of ambitwistor strings, including a discussion of the correlator at genus zero. ...
... In this section, we provide a brief review of ambitwistor space, targeted towards the later applications in ambitwistor strings. As such, the interested reader is referred to the original papers [41][42][43][44][45][46][47][48] for more details and [30] for a modern, more extensive review. The aspects of ambitwistor space specific to four dimensions will be discussed in section 5.1.1. ...
... The name has been kept in higher dimensions, justified by the twistor-like correspondences relating it to space-time. First introduced in [46][47][48] in the context of gauge fields, it was extended to gravity in arbitrary dimensions in [41], where it extends Penrose's non-linear graviton construction [85,86] to general gravitational fields. As we will discuss below in more detail, fields are encoded by deformations of the complex structure of ambitwistor space [41-43, 46, 47], while preserving the contact structure. ...
Article
Tree-level scattering amplitudes in massless theories not only exhibit a simplicity entirely unexpected from Feynman diagrams, but also an underlying structure remarkably reminiscent of worldsheet theory correlators. These features can be explained by ambitwistor strings - two-dimensional chiral conformal field theories in an auxiliary target space, the complexified phase space of null geodesics. The aim of this thesis is to explore the ambitwistor string approach to understand these structures in amplitudes, and thereby provide a new angle on quantum field theories. The first part of the thesis provides a user-friendly introduction to ambitwistor strings, as well as a condensed overview over the literature and some novel results. Emphasising the study of tree-level amplitudes, we then explore the wide-ranging impact of ambitwistor strings for an extensive family of massless theories, and discuss the duality between asymptotic symmetries and the low energy behaviour of a theory from the point of view of the worldsheet CFT. The second part of this thesis focusses on proving a conjectured ambitwistor string formula for loop amplitudes, and extending the formalism to more general theories. Remarkably, residue theorems reduce the computationally challenging ambitwistor higher-genus expressions to simple formulae on nodal Riemann spheres. This idea is developed into a widely applicable framework for loop integrands, that is shown to be applicable to both supersymmetric and non-supersymmetric theories. In the case of supergravity, this provides strong evidence for the validity of the ambitwistor string at loop level, and explicit proofs are given for non-supersymmetric theories. This leads to a proposal for an all-loop integrand for gravity and Yang-Mills.
... A rather natural way of constructing a classical candidate p2, 0q-theory is to combine higher gauge theory with twistor geometry. Recall that there is a twistor space which carries a useful representation of solutions to the N " 3 supersymmetric Yang-Mills equations in terms of holomorphic principal bundles [20,21]. Analogously, there is a twistor space that carries a representation of Abelian self-dual 3-forms in six dimensions in terms of holomorphic gerbes, see [22][23][24][25][26][27] as well as [28][29][30]. ...
... After decomposing C 8n -C 4 b C 2n and choosing a holomorphic symplectic form 21 Note that this decomposition is equivalent to choosing a conformal structure on M , see [28,Remark 3.2]. ...
... These maps satisfy various cocycle conditions as implied by the relations displayed in Lemma 2.18; see also Theorem 4.19. For instance, the cocycle conditions for α 2 1,a read as f 2 2˚|σ2pφq pγ 21 0,ab q "´d π 1 σ 1 pφq and f 2 0˚|σ2pφq pγ 21 0,ab q " α 2 1,b , f 2 1˚|σ2pφq pγ 21 0,ab q " f 2 1˚|σ2pφq pγ 21 1,ab q , f 2 2˚|σ2pφq pγ 21 1,ab q " α 2 1,a and f 2 0˚|σ2pφq pγ 21 1,ab q " 0 (6.12b) with σ p :" d p´1 0˝¨¨¨˝d 1 0˝d 0 0 as defined in Theorem 4.10. Using the simplicial identities (2.5), it is rather easy to see that these equations imply that f 1 1˚| φ pα 2 a q " f 1 1˚| φ pα 2 b q. ...
Article
We develop a description of higher gauge theory with higher groupoids as gauge structure from first principles. This approach captures ordinary gauge theories and gauged sigma models as well as their categorifications on a very general class of (higher) spaces comprising presentable differentiable stacks, as e.g. orbifolds. We start off with a self-contained review on simplicial sets as models of (,1)(\infty,1)-categories. We then discuss principal bundles in terms of simplicial maps and their homotopies. We explain in detail a differentiation procedure, suggested by Severa, that maps higher groupoids to LL_\infty-algebroids. Generalising this procedure, we define connections for higher groupoid bundles. As an application, we obtain six-dimensional superconformal field theories via a Penrose-Ward transform of higher groupoid bundles over a twistor space. This construction reduces the search for non-Abelian self-dual tensor field equations in six dimensions to a search for the appropriate (higher) gauge structure. The treatment aims to be accessible to theoretical physicists.
... In four dimensions, this space of complex null geodesics lies in the product of twistor space and its dual and so has become known as (projective) ambitwistor space, denoted P A. It was studied in the 1970s and 1980s as a vehicle for extending the deformed twistor space constructions for Yang-Mills [20,21]. Such constructions were extended to arbitrary dimensions in the context of gravity by LeBrun [22] and in a supersymmetric context in 10 dimensions by Witten [23]. ...
... However, ambitwistor space is a more versatile notion that exists in any dimension and for any (globally hyperbolic) space-time. It has long been known that gauge and gravitational fields may be encoded in terms of holomorphic structures on P A [20][21][22]. We will discuss the gauge theory case later, but here give a brief review of the gravitational case following LeBrun [22] (see also appendix B). ...
... One of the most important differences between this ambitwistor version of the Penrose transform and the (perhaps more familiar) Penrose transform between twistor space and space-time is that here, the field on space-time is not required to satisfy any field equations at this stage. Much work in the 70's and 80's focussed on the expression of the field equations in ambitwistor space (in terms of the existence of supersymmetries [20,23] or (essentially equivalently) formal neighbourhoods [21,28,29]). In the following we will see that for our string models, the space-time massless field equations arise automatically from quantum consistency of the symplectic reduction at the level of the worldsheet path integral. ...
Article
Full-text available
We show that string theories admit chiral infinite tension analogues in which only the massless parts of the spectrum survive. Geometrically they describe holomorphic maps to spaces of complex null geodesics, known as ambitwistor spaces. They have the standard critical space--time dimensions of string theory (26 in the bosonic case and 10 for the superstring). Quantization leads to the formulae for tree--level scattering amplitudes of massless particles found recently by Cachazo, He and Yuan. These representations localize the vertex operators to solutions of the same equations found by Gross and Mende to govern the behaviour of strings in the limit of high energy, fixed angle scattering. Here, localization to the scattering equations emerges naturally as a consequence of working on ambitwistor space. The worldsheet theory suggests a way to extend these amplitudes to spinor fields and to loop level. We argue that this family of string theories is a natural extension of the existing twistor string theories.
... Ambitwistor space A is the space of null geodesics [28][29][30][31]. This may be constructed simply as a sub-bundle of the cotangent bundle T * M of the spacetime M , which will be Minkowski spacetime for most of this paper. ...
... Note that it is the background Minkowski metric η µν and its inverse which is being used to lower and raise indices. The naive imposition of a Siegel type gaugeb 0 |Ψ = 0, imposes the condition f µ (k) = 0 which, noting (3.14), is precisely the harmonic (or de Donder) gauge for linearised gravity 29 . In this gauge the h µν equation on motion is simply 2h µν = 0, which is consistent with the proposed propagator discussed above. ...
Preprint
A covariant closed superstring field theory, equivalent to classical ten-dimensional Type II supergravity, is presented. The defining conformal field theory is the ambitwistor string worldsheet theory of Mason and Skinner. This theory is known to reproduce the scattering amplitudes of Cachazo, He and Yuan in which the scattering equations play an important role and the string field theory naturally incorporates these results. We investigate the operator formalism description of the ambitwsitor string and propose an action for the string field theory of the bosonic and supersymmetric theories. The correct linearised gauge symmetries and spacetime actions are explicitly reproduced and evidence is given that the action is correct to all orders. The focus is on the Neveu-Schwarz sector and the explicit description of tree level perturbation theory about flat spacetime. Application of the string field theory to general supergravity backgrounds and the inclusion of the Ramond sector are briefly discussed.
... The full second-order Yang Mills equations on R 4 are not integrable, and there is no twistor construction encoding their solutions in an unconstrained holomorphic data on PT -there do exist ambitwistor constructions [220,127,105] in terms of formal neighbourhods of spaces of complex null geodesics, but they do not lead to any solution generation techniques. As in the case of gravity, the anti-self-dual sub-sector can be described twistorialy, this time in terms of holomorphic vector bundles over PT rather than deformations of its complex structures. ...
... This is just a Lagrangian expression of the usual (holomorphic) symplectic reduction of the cotangent bundle to the space of (scaled) null geodesics A. In four dimensions this has become known as ambitwistor space as it is both the cotangent bundle of projective twistor space and of projective dual twistor and so chooses neither chirality. In [220,127,105] (see also [135]) it was shown that generic analytic connections on bundles on Minkowski space correspond to topologically trivial bundles on ambitwistor space. The full Yang-Mills equations can be characterised as the condition that the corresponding holomorphic vector bundles on PA extend to a third formal neighbourood in PT × PT * . ...
Preprint
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 gravitational instantons which we review. They also underlie the twistor approach to integrability: the solitonic systems arise as symmetry reductions of ASD Yang--Mills equations, and Einstein--Weyl dispersionless systems are reductions of ASD conformal equations. We then review the holomorphic string theories in twistor and ambitwistor spaces, and explain how these theories give rise to remarkable new formulae for the computation of quantum scattering amplitudes. Finally we discuss the Newtonian limit of twistor theory, and its possible role in Penrose's proposal for a role of gravity in quantum collapse of a wave function.
... The idea of a causal morphism is to define a map that does not map points to points, but instead locally maps null curves to null curves. The idea is partially motivated by twistor theory [2][3] [4], and its ambitwistor extension [5] [6], where null lines and αplanes are considered as fundamental objects of the theory. In particular, conformal transformations on a self-dual 4-manifold induce holomorphic transformations on the corresponding twistor space [7]. ...
... Given (χ, π) ∈ F, the point z = κ • χ is determined by equation (4). To findż, we choose any spinor ψ that is linearly independent from π, and use equation (6) to writeż = (iχψ, ψ). The choice of ψ determines the vector in the two-dimensional subspace from the proof, which will not affect the final result. ...
Article
Full-text available
We define a class of maps between holomorphically embedded null curves which generalize conformal transformations, and can be defined in any complex dimension. In four dimensions, we can also define a similar map between self-dual surfaces, which generalize flat α-planes. These maps are respectively called causal and self-dual morphisms. It is shown that there exist an infinite class of non-trivial examples for both types of maps in four dimensions.
... It was shown in [73][74][75], that the moduli space of solutions to the constraint system of supercurvatures describing N = 3 supersymmetric Yang-Mills theory on N = 3 superspace is naturally identified with the moduli space of M -trivial holomorphic principal G-bundles, for G a Lie group, over ambitwistor space L. This constraint system is equivalent to the N = 3 supersymmetric Yang-Mills equations on ordinary space-time [85,86] which, in turn, are equivalent to the maximally supersymmetric Yang-Mills equations. The ambitwistor space in question is a supermanifold, and because of the peculiar choice of supersymmetry, a Calabi-Yau supermanifold [87]. ...
... Thus, transitioning to the minimal model as discussed in Section 1.2, we recover the equations which are equivalent to the constraint system of N = 3 supersymmetric Yang-Mills theory [73][74][75]. ...
Preprint
We summarise some of our recent works on LL_\infty-algebras and quasi-groups with regard to higher principal bundles and their applications in twistor theory and gauge theory. In particular, after a lightning review of LL_\infty-algebras, we discuss their Maurer-Cartan theory and explain that any classical field theory admitting an action can be reformulated in this context with the help of the Batalin-Vilkovisky formalism. As examples, we explore higher Chern-Simons theory and Yang-Mills theory. We also explain how these ideas can be combined with those of twistor theory to formulate maximally superconformal gauge theories in four and six dimensions by means of LL_\infty-quasi-isomorphisms, and we propose a twistor space action.
... The space of all complex null geodesics in a complex spacetime manifold (M, g) is called projective ambitwistor space PA. It has been used to study physical fields, which can be encoded in its holomorphic structure [88][89][90]. As in the case of twistor space, these fields are automatically defined up to gauge transformations on PA, however unlike in twistor theory, they are not forced to obey any field equations 2 . ...
... Working on the support of momentum conservation in the v-direction -which holds regardless of the asymptotic configuration of the three external states -a bit of algebra reveals that 88) and therefore that the expression (4.87) is in fact equal to the 3-point integrand for gravitons on the gravitational plane wave background. ...
Preprint
Feynman diagrams have been superseded as the tool of choice for calculating scattering amplitudes. Various other methods are not only more efficient but also explicitly exhibit beautiful structures obscured by Feynman diagrams. This thesis aims to lay some groundwork on how two of these methods, ambitwistor strings and the double copy, can be generalised to scattering in curved backgrounds. In the first part of this thesis, a heterotic ambitwistor string model coupled to a non-abelian background gauge field is constructed. It is shown that after decoupling gravity this model is anomaly free if and only if the background field is a solution to the Yang-Mills equations. A fixed gluon vertex operator for the aforementioned heterotic model as well as a vertex operator encoding graviton, B-field and dilaton for type II ambitwistor strings in a curved background are presented. It is shown that they are BRST closed if and only if they correspond to physical on-shell states. In the second part, sandwich plane waves are considered. It is shown that scattering of gluons and gravitons is well defined on these backgrounds. 3-point amplitudes are calculated using quantum field theory techniques and a double copy relation between gluons on a gauge theory plane wave and gravitons on a gravitational plane wave is proposed. Using the results from the first part of this thesis, it is then shown that curved background heterotic and type II ambitwistor string models correctly reproduce these 3-point amplitudes on sandwich plane waves.
... This analysis leads to the definition of a frustrated conformal transformation, whose construction relies on a pair of self-dual and anti-self-dual Sp (1) connections, which are in some sense dual to each other. Through a construction analogous to the Penrose-Ward correspondence, these functions generate holomorphic maps of ambitwistor space, which were introduced in the study of general solutions to the Yang-Mills equations [14] [15]. These maps are naturally closed under composition, and provide a different possible notion of generalized holomorphic functions in four dimensions. ...
... For the discussion that follows, it will also be useful to consider the right action of quaternion multiplication, which leads to the consideration of "ambitwistor" space. Ambitwistor space was introduced to extend the ideas of twistor theory to study general solutions of the Yang-Mills field equations [14] [15]. The analysis that follows will yield some interesting parallels to this theory. ...
Preprint
We study some aspects of conformal transformations in the context of twistor theory, leading to the definition of a frustrated conformal transformation. This equation relies on two instantons for the left and right copies of Sp(1), one being self-dual and the other anti-self-dual. Solutions to this equation naturally generate maps on ambitwistor space due to an analog of the Penrose-Ward correspondence. A solution based on the BPST instanton is presented.
... But we are optimists, so instead of giving up we can try to look for some other construction which mimics the non-locality of the twistor correspondence between Minkowski space and an auxiliary projective space but is non-chiral. Thankfully, such a construction exists, and is known as ambitwistor theory [75,76,77]. ...
... Considerable effort was put towards trying to find a way to impose field equations through the ambitwistor Penrose transform in the early days of the subject. While it turns out that this can be done, it requires the rather cumbersome formalism of formal neighborhoods [75,76,78,79]. In words, this means that equations of motion can be imposed on the resulting space-time fields by demanding that the ambitwistor cohomology representatives on the RHS of (5.25) extend away from the P 2 = 0 quadric to some given order. ...
Conference Paper
Full-text available
... But we are optimists, so instead of giving up we can try to look for some other construction which mimics the non-locality of the twistor correspondence between Minkowski space and an auxiliary projective space but is non-chiral. Thankfully, such a construction exists, and is known as ambitwistor theory [67][68][69]. ...
... Considerable effort was put towards trying to find a way to impose field equations through the ambitwistor Penrose transform in the early days of the subject. While it turns out that this can be done, it requires the rather cumbersome formalism of formal neighborhoods [67,68,70,71]. In words, this means that equations of motion can be imposed on the resulting space-time fields by demanding that the ambitwistor cohomology representatives on the RHS of (5.25) extend away from the P 2 = 0 quadric to some given order. ...
Article
Broadly speaking, twistor theory is a framework for encoding physical information on space-time as geometric data on a complex projective space, known as a twistor space. The relationship between space-time and twistor space is non-local and has some surprising consequences, which we explore in these lectures. Starting with a review of the twistor correspondence for four-dimensional Minkowski space, we describe some of twistor theory's historic successes (e.g., describing free fields and integrable systems) as well as some of its historic shortcomings. We then discuss how in recent years many of these problems have been overcome, with a view to understanding how twistor theory is applied to the study of perturbative QFT today. These lectures were given in 2017 at the XIII Modave Summer School in mathematical physics.
... The full second-order Yang Mills equations on R 4 are not integrable, and there is no twistor construction encoding their solutions in an unconstrained holomorphic data on PT -there do exist ambitwistor constructions [199,115,93] in terms of formal neighbourhods of spaces of complex null geodesics, but the do not lead to any solution generation techniques. As in the case of gravity, the anti-self-dual sub-sector can be described twistorialy, this time in terms of holomorphic vector bundles over PT rather than deformations of its complex structures. ...
... This is just a Lagrangian expression of the usual (holomorphic) symplectic reduction of the cotangent bundle to the space of (scaled) null geodesics A. In four dimensions this has become known as ambitwistor space as it is both the cotangent bundle of projective twistor space and of projective dual twistor and so chooses neither chirality. However In [199,115,93] it was shown that generic analytic connections on bundles on Minkowski space correspond to topologically trivial bundles on ambitwistor space. The full Yang-Mills equations can be characterised as the condition that the corresponding holomorphic vector bundles on PA extend to a third formal neighbourood in PT × PT * . ...
... The full second-order Yang Mills equations on R 4 are not integrable, and there is no twistor construction encoding their solutions in an unconstrained holomorphic data on PT -there do exist ambitwistor constructions [208,121,99] in terms of formal neighbourhods of spaces of complex null geodesics, but the do not lead to any solution generation techniques. As in the case of gravity, the anti-self-dual sub-sector can be described twistorialy, this time in terms of holomorphic vector bundles over PT rather than deformations of its complex structures. ...
... The ambitwistor space is a complexification of the real hypersurface PN ⊂ PT introduced in §2. In [208,121,99] it was shown that generic analytic connections on bundles on Minkowski space correspond to topologically trivial bundles on ambitwistor space. The full Yang-Mills equations can be characterised as the condition that the corresponding holomorphic vector bundles on PA extend to a third formal neighbourood in PT × PT * . ...
Article
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We review aspects of twistor theory, its aims and achievements spanning the last five decades. In the twistor approach, space-time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex threefold-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 equations on Yang-Mills or conformal curvature-can be encoded into twistor cohomology. These twistor correspondences yield explicit examples of Yang-Mills and gravitational instantons, which we review. They also underlie the twistor approach to integrability: the solitonic systems arise as symmetry reductions of anti-self-dual (ASD) Yang-Mills equations, and Einstein-Weyl dispersionless systems are reductions of ASD conformal equations. We then review the holomorphic string theories in twistor and ambitwistor spaces, and explain how these theories give rise to remarkable new formulae for the computation of quantum scattering amplitudes. Finally, we discuss the Newtonian limit of twistor theory and its possible role in Penrose's proposal for a role of gravity in quantum collapse of a wave function.
... However, the structure of the moduli space of all Yang-Mills fields on R 4 is still far from being understood. The twistor description of Yang-Mills fields was proposed in the papers by Manin [5], Witten [6] and Isenberg-Green-Yasskin [7]. They are interpreted as holomorphic vector bundles over the incidence quadric in P 3 × (P 3 ) * with some special properties described below. ...
... In order to extend these results to arbitrary Yang-Mills fields, we would like to have the twistor interpretation of these fields. Such an interpretation was proposed in the papers by Manin [5], Witten [6] and Isenberg-Green-Yasskin [7]. To present their construction, denote by (P 3 ) * the dual projective space identified with the space of complex projective planes P 2 in P 3 . ...
Article
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We consider the twistor descriptions of harmonic maps of the Riemann sphere into Kähler manifolds and Yang–Mills fields on four-dimensional Euclidean space. The motivation to study twistor interpretations of these objects comes from the harmonic spheres conjecture stating the existence of the bijective correspondence between based harmonic spheres in the loop space ΩG\Omega G of a compact Lie group G and the moduli space of Yang–Mills G-fields on R4\mathbb R^4.
... Following Manin, we shall refer to such bundles as C 4|12 -trivial. Due to [44][45][46] we now have the following result. ...
Preprint
We observe that the string field theory actions for the topological sigma models describe higher or categorified Chern-Simons theories. These theories yield dynamical equations for connective structures on higher principal bundles. As a special case, we consider holomorphic higher Chern-Simons theory on the ambitwistor space of four-dimensional space-time. In particular, we propose a higher ambitwistor space action functional for maximally supersymmetric Yang-Mills theory.
... It was shown in[13] that to leading order around A it gives on space-time the Eastwood-Dighton conformal invariant defined in terms of SD and ASD Weyl spinors by ψ αβγδ ∇ δδψαβγδ − ψ ↔ψ. There it was interpreted as an obstruction to extending the curved version of A into a curved analogue of PT × PT * in a gravitational version of[33,78]; this was later proved in the fully nonlinear regime[36].8 To realize space-time supersymmetry Z A ,ZA are extended to include fermionic coordinates[24]. ...
... Under suitable topological conditions on the double fibration (6.1), 1 the Penrose-Ward transform then identifies the gauge orbits of solutions to some field equations with isomorphism classes of certain principal bundle P over the corresponding twistor space Z. Examples include self-dual Yang-Mills theory [67,68] and its supersymmetric extensions [69][70][71] as well as Yang-Mills theory [72][73][74][75][76][77] and its supersymmetric extensions [73,78,79]. Roughly speaking, we can pull back the bundle P along π 2 and perform a trivialising isomorphism π2 P ÑP . ...
Preprint
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We derive the complete cocycle description for non-Abelian gerbes with connections whose structure 2-group is a 2-group with adjustment datum. We depart from the common fake-flat connections and employ adjusted connections, which is important for physical applications, especially in the context of supergravity. We give a number of explicit examples; in particular, we lift the principal bundle corresponding to an instanton-anti-instanton pair to a string 2-group bundle. We also outline how categorified forms of Bogomolny monopoles known as self-dual strings can be obtained via a Penrose-Ward transform of string bundles over twistor space.
... The twistor interpetation of the space of harmonic maps S 2 → ΩG was constructed in a joint paper of the author with I. V. Beloshapka [2]. On the other hand, in papers by Isenberg-Green-Yasskin [4], Witten [15] and Manin [6] it was given the twistor realization of Yang-Mills fields. However, we are still not able to establish a correspondence between these two twistor spaces (the arising difficulties are discussed in the paper [11]). ...
Article
Full-text available
Our goal is to present an approach to the proof of the harmonic spheres conjecture based on the adiabatic limit construction. This construction allows to associate with an arbitrary Yang–Mills G-field on the Euclidean 4-dimensional space a harmonic map of the Riemann sphere to the loop space of the group G
... Alternatively one can seek to generalise the study of the space of null geodesics to higher dimensions. In this context, the space is usually referred to as ambitwistor space [9,10]. It has been noted on a number of occasions [11][12][13] that the division algebras provide an interesting unified guide for how to think about ambitwistor space in dimensions three, four, six and ten. ...
Preprint
We review some ideas on the relationship between massless superparticles and the division algebras to provide a new perspective on ambitwsitor string theories. The key concern is the critical theory. We show that this theory has a reducible soft algebra, rather than a conventional Lie algebra. This algebra only closes on-shell. The BV procedure is employed to deal with the on-shell closure of the algebra and the classical Master Action is presented.
... Following Manin, we shall refer to such bundles as C 4|12 -trivial. Due to [41,42] we now have the following result. Roughly speaking, the holomorphic triviality on allp guarantees that the pullback of this bundle to the correspondence space F 6|12 is holomorphically trivial on all of F 6|12 . ...
Article
Full-text available
We observe that the string field theory actions for the topological sigma models describe higher or categorified Chern-Simons theories. These theories yield dynamical equations for connective structures on higher principal bundles. As a special case, we consider holomorphic higher Chern-Simons theory on the ambitwistor space of four-dimensional space-time. In particular, we propose a higher ambitwistor space action functional for maximally supersymmetric Yang-Mills theory.
... It is an interesting fact that the twistor space of AdS 5 is the same geometric space as the projective ambitwistor space of the complexified, four-dimensional conformal boundary. In any number of dimensions, the projective ambitwistor space of a Riemannian manifold M R is the space of complex null geodesics in the complexified manifold M [24][25][26][27]. In the case that M R = S 4 this ambitwistor space can be written as a quadric in CP 3 × CP 3 : ...
Article
We consider the application of twistor theory to five-dimensional anti-de Sitter space. The twistor space of AdS5_5 is the same as the ambitwistor space of the four-dimensional conformal boundary; the geometry of this correspondence is reviewed for both the bulk and boundary. A Penrose transform allows us to describe free bulk fields, with or without mass, in terms of data on twistor space. Explicit representatives for the bulk-to-boundary propagators of scalars and spinors are constructed, along with twistor action functionals for the free theories. Evaluating these twistor actions on bulk-to-boundary propagators is shown to produce the correct two-point functions.
... All fields are chiral. An additional symmetry, generated by δX µ = αP µ and δe =∂α (all other fields are invariant), reduces the target space to the quotient P A which is identified with the space of null geodesics, or ambitwistor space [23,25,26,24]. As described in [14], the gauge fixing of this residual symmetry is fixed by introducing the gauge fixing fermion ...
Article
After a brief overview of the operator formalism for conventional string theory, an operator formalism for ambitwistor string theory is presented. It is shown how tree level supergravity scattering amplitudes are recovered in this formalism. More general applications of this formalism to loop amplitudes and the construction of an ambitwistor string field theory are briefly discussed.
... Isenberg, Yaskin, and Green [2] showed a Ward correspondence between general solutions to the Yang-Mills equations on M and vector bundles over the third infinitesmal neighborhood (or thickening) of ambitwistor space, sitting inside CP 3 × CP 3 . Witten [10] also produced a related Ward correspondence. ...
Article
In 2003, Witten[12] introduced the twistor string which is a string theory in super Twistor space, CP 3|4 . One of the key ideas behind the twistor string is holomorphic Chern-Simons theory. It is of interest to extend the idea of twistor strings and thus holomorphic Chern-Simons theory beyond CP 3|4 to other spaces in twistor theory such as super ambitwistor spaces. In this talk, we shall begin with an introduction to twistor and ambitwistor spaces. We quickly review various results from twistor theory such as Penrose-Ward transforms. We also present various ideas from supergeometry which we will be needing. After a short introduction to Chern-Simons theory and its holomorphic analog, we discuss holomorpic Chern-Simons theory on CP 3|4 . BF theory, another topological gauge theory and its extension by A. Popov [7] to holomorphic BF theory are reviewed. We also give its extension to complex supermanifolds. Finally, we investigate holomorphic BF theory on super am-bitwistor spaces and the role of holomorphic and almost complex bundles in holomorphic Chern-Simons and BF theories.
... The full Yang-Mills equations involve also the anti-self-dual case and because the equations are non-linear we cannot, as in Maxwell theory, combine the two together. However Witten, in a very interesting paper [15] (see also [16]), has recently shown how to interpret the full Yang-Mills equations in a twistor framework. This involves studying the product of P 3 (C) with its dual and looking at the 5-dimensional "incidence" hypersurface together with some normal derivatives. ...
... In its soft expansion, we obtain gauge transformations analagous to supertranslations at leading order and superrotations for the subleading terms. This gives a realization in string theoretic terms of the Ambitwistor constructions of [26,27,28] in which Yang-Mlls fields are encoded in the complex structure of a holomorphic vector bundle over ambitwistor twistor space with the gauge transformations playing the role of patching functions. ...
Article
Full-text available
The relationships between extended BMS symmetries at null infinity and Weinberg's soft theorems for gravitons and photons together with their subleading generalizations are developed using ambitwistor string theory. Ambitwistor space is the phase space of complex null geodesics in complexified space-time. We show how it can be canonically identified with the cotangent bundle of null infinity. BMS symmetries of null infinity lift to give a hamiltonian action on ambitwistor space, both in general dimension and in its twistorial 4-dimensional representation. General vertex operators arise from hamiltonians generating diffeomorphisms of ambitwistor space that determine the scattering from past to future null infinity. When a momentum eigenstate goes soft, the diffeomorphism defined by its leading and its subleading part are extended BMS generators realized in the world sheet conformal field theory of the ambitwistor string. More generally, this gives explicit perturbative correspondence between the scattering of null geodesics and that of the gravitational field via ambitwistor string theory.
... Thus, a non-linear version will not be restricted to self-dual fields, and in particular will include real solutions. More precisely, the hope is that this will be possible without squaring the twistor space, as one does in the ambitwistor approach [16][17][18]. From the perturbative perspective, the lack of handedness implies that there is just one kind of external state, instead of two separate helicity signs. We note that non-linear versions of (53) may turn out to be quite different from the standard constructions. ...
Article
We develop the basics of twistor theory in de Sitter space, up to the Penrose transform for free massless fields. We treat de Sitter space as fundamental, as one does for Minkowski space in conventional introductions to twistor theory. This involves viewing twistors as spinors of the de Sitter group SO(4,1). When attached to a spacetime point, such a twistor can be reinterpreted as a local SO(3,1) Dirac spinor. Our approach highlights the antipodal map in de Sitter space, which gives rise to doublings in the standard relations between twistors and spacetime. In particular, one can generate a field with both handedness signs from a single twistor function. Such fields naturally live on antipodally-identified de Sitter space dS_4/Z_2, which has been put forward as the ideal laboratory for quantum gravity with positive cosmological constant.
Article
We consider the ambitwistor description of N=4 supersymmetric extension of U(N) Yang-Mills theory on Minkowski space R3,1. It is shown that solutions of super-Yang-Mills equations are encoded in real analytic U(N)-valued functions on a domain in superambitwistor space LR5|6 of real dimension (5|6). This leads to a procedure for generating solutions of super-Yang-Mills equations on R3,1 via solving a Riemann-Hilbert-type factorization problem on two-spheres in LR5|6.
Preprint
We consider the ambitwistor description of N\mathcal N=4 supersymmetric extension of U(N) Yang-Mills theory on Minkowski space R3,1\mathbb R^{3,1}. It is shown that solutions of super-Yang-Mills equations are encoded in real-analytic U(N)-valued functions on a domain in superambitwistor space LR56{\mathcal L}_{\mathbb R}^{5|6} of real dimension (56)(5|6). This leads to a procedure for generating solutions of super-Yang-Mills equations on R3,1\mathbb R^{3,1} via solving a Riemann-Hilbert-type factorization problem on two-spheres in LR56\mathcal L_{\mathbb R}^{5|6}.
Article
We summarise some of our recent works on ‐algebras and quasi‐groups with regard to higher principal bundles and their applications in twistor theory and gauge theory. In particular, after a lightning review of ‐algebras, we discuss their Maurer–Cartan theory and explain that any classical field theory admitting an action can be reformulated in this context with the help of the Batalin–Vilkovisky formalism. As examples, we explore higher Chern–Simons theory and Yang–Mills theory. We also explain how these ideas can be combined with those of twistor theory to formulate maximally superconformal gauge theories in four and six dimensions by means of ‐quasi‐isomorphisms, and we propose a twistor space action.
Article
Using the generalization of vector bundles by reflexive sheaves recently introduced by R. Hartshorne in [ 2 ] we construct a 15 15 -dimensional family of nontrivial complex gauge fields ( U , E , ∇ ) (U,E,\nabla ) which are not self-dual nor anti-self-dual. ( U U is an affine neighborhood in Q 4 = Gr ⁡ ( 2 , C 4 ) {Q_4} = \operatorname {Gr} (2,{{\mathbf {C}}^4}) of the stereographic compactification S 4 {S^4} of R 4 {\mathbb {R}^4} , E E is a vector bundle on U U and ∇ \nabla is a connection on it whose curvature ϕ \phi satisfies the inequalities ∗ ϕ ≠ ϕ {}^{\ast }\phi \ne \phi and ∗ ϕ ≠ − ϕ {}^{\ast }\phi \ne - \phi .)
Article
We show that under a certain cohomological condition the theorem of Witten, Isenberg, Yasskin and Green about the inverse Penrose transform of a (non-self-dual) connection ∇ (together with Manin’s description of its curvature F∠) is true in a quite general situation. We then present a (multidimensional) version of the Penrose transform of a vector bundle in the language of jets. This gives a coordinate-free interpretation of certain results of Henkin and Manin, coding a number of classical field equations in terms of obstructions to infinitesimal extension of cohomology classes.
Chapter
The self-dual Yang-Mills and Einstein equations have a simple geometric meaning, since they imply the vanishing of a part of the curvature tensor of a connection. This connection, the physicists’ gauge potential, is given either on an external vector bundle (the Yang-Mills case) or on the spinorial bundle (the Einstein case) over space-time. After a suitable base change, the relevant part of the curvature becomes the total curvature of the lifted connection along the leaves of a foliation. At least locally (with respect to the initial base manifold), this foliation is a libration and the self-dual field in question can be represented by the vector bundle of horizontal sections along the leaves on the base space of the foliation (Yang-Mills) or by the base space itself (Einstein). This representation is called the Penrose transform. The idea is closely related to the classical Radon transform. One of Penrose’s discoveries was the possibility of using the rigidity of the holomorphic geometry to effectively construct the solutions of the differential equations by geometric means. A mathematician may profitably consult M. F. Atiyah [1] and the references cited therein.
Chapter
It is known that the self-dual Yang-Mills equations are completely integrable in a sense of the existence of a related linear problem, infinity of conservation laws and a possibility of generating solutions. WITTEN [1] and ISENBERG, YASSKIN and GREEN [2], following Ward’s approach to self-dual fields, proposed a. construction, which in principle should yield also non self-dual solutions of the Yang-Mills equations (however no application of this scheme exists). Witten also generalized this method to the case of the supersymmetric Yang-Mills equations with N = 3,4. It was possible because the Susy YM equations are equivalent to the so-called supersymmetric constraint equations [3] which resemble selfduality conditions. Thus the constraint equations can provide a key to understanding the structure of the Susy YM equations and in particular the ordinary Yang-Mills equations. In this seminar I discuss recent approaches to the problem of the integrability of the constraint equations (see [4] for details and references).
Article
Some fifty years ago Einstein’s theory of general relativity provided a great stimulus for different geometry, but after a period of fruitful interaction the interests of geometers and physicists diverged. Within the past few years there has been a resurgence of geometrical ideas in physics, arising partly from the popularity of gauge theories in elementary-particle physics and partly from the work of Hawking and Penrose on black holes. A characteristic feature in both cases has been the significance of global or topological properties, and it is this feature which has particularly attracted mathematicians.
Article
In a landmark paper [13] of the 1970’s, Roger Penrose described a remarkable and unexpected connection between Einstein’s equations and the theory of complex manifolds. The twistor correspondence which he detailed there gives a way of producing all self-dual complex solutions of these equations in terms of the global structure of an associated complex manifold called the twistor space. Unfortunately, the most interesting solutions from the view-point of physics are not the self-dual ones, but rather those of Lorentz signature. If this had been the end of the story, one might have therefore been tempted to conclude that this correspondence was merely a mathematical curiosity with no bearing on physics. Fortunately, however, the Penrose twistor correspondence is but one aspect of a rather more complicated story, which I will endeavor to recount here. Indeed, it turns out that the general complex solution of the 4-dimensional Einstein equations can also be described in terms of complex deformation theory, albeit in terms of non-reduced complex spaces rather than complex manifolds.
Article
In this part, we consider a number of problems of complex analysis and mathematical physics connected with the theory of Yang-Mills gauge fields on the one hand and the theory of Cauchy-Riemann equations on the other.
Article
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It was recently shown by Witten that B-type open topological string theory with the supertwistor space CP 3/4 as a target space is equivalent to holomorphic Chern-Simons (hCS) theory on the same space. This hCS theory in turn is equivalent to self-dual N = 4 super-Yang-Mills (SYM) theory in four dimensions. We review the supertwistor description of self-dual and anti-self-dual N-extended SYM theory as the integrability of SYM fields on complex (2|N)-dimensional superplanes and demonstrate the equivalence of this description to Witten's formulation. The equivalence of the field equations of hCS theory on an open subset of CP 3/N to the field equations of self-dual N-extended SYM theory in four dimensions is made explicit. Furthermore, we extend the picture to the full N = 4 SYM theory and, by using the known supertwistor description of this case, we show that the corresponding constraint equations are (gauge) equivalent to the field equations of hCS theory on a quadric in CP 3/3 × CP 3/3.
Article
In the past years supersymmetric theories have gained great importance in physics. By this one intends field theoretical models based on a new form of symmetry dubbed supersymmetry. Supersymmetry connects boson and fermion fields with each other [13], [46], [48], [26], [6]. The observed properties of particles cannot satisfy the demands of supersymmetry (for instance, supersymmetry would lead to the equality of mass for the boson and the corresponding fermion). However, an increasing number of physicists have arrived at the conviction that the action functional of interactions encountered in nature must be supersymmetrical (although for the ground state (the physical vacuum) and, consequently, for the observed spectra of particles supersymmetry is broken). Perhaps the most weighty foundation for such a belief is the mathematical beauty of the supersymmetric theories and the remarkable property of cancellation of the divergencies appearing in these theories. It is question of the circumstance that in quantum field theories one encounters divergencies arising from the integration over large momenta (ultraviolet divergencies). In supersymmetry the most dangerous of these divergencies cancel. Moreover, there exist models completely free of ultraviolet divergencies. Presently great hopes are put on such supersymmetric theories which take account of the presence of gravitational interactions. Thus and important constituent part of these theories is played by supergravity, a supersymmetric theory containing Einstein’s theory of gravity.
Article
Within unfolded dynamics approach, we represent actions and conserved charges as elements of cohomology of the L∞ algebra underlying the unfolded formulation of a given dynamical system. The unfolded off-shell constraints for symmetric fields of all spins in Minkowski space are shown to have the form of zero curvature and covariant constancy conditions for 1-forms and 0-forms taking values in an appropriate star product algebra. Unfolded formulation of Yang–Mills and Einstein equations is presented in a closed form.
Article
Full-text available
We give a short discussion/review of the recent developments expressing the perturbative scattering amplitudes in Yang-Mills theory, specifically for the N=4\mathcal{N} = 4 theory, in terms of holomorphic curves in a supersymmetric twistor space. Holomorphic curves, which are maps of CP¹ to the supertwistor space, can also be interpreted as the lowest Landau level wave functions; this point of view is also briefly explained.
Article
I would firstly like to convey my best wishes for 2006 to our readers, authors and referees. There are also some issues that I would like to communicate to you in this first issue of the new year.
Article
The non-Abelian analogues of electromagnetic plane waves are constructed.
Article
Recent researches have shown that it is possible to obtain information about the physical content of nontrivial quantum field theories by semiclassical methods. This article reviews some of these investigations. We discuss how solutions to field equations, treated as classical, c-number nonlinear differential equations, expose unexpected states in the quantal Hilbert space with novel quantum numbers which arise from topological properties of the classical field configuration or from the mixing of internal and space-time symmetries. Also imaginary-time, c-number solutions are reviewed. It is shown that they provide nonperturbative information about the vacuum sector of the quantum theory.
Article
We formulate the Euclidean Yang-Mills gauge theory for isospin in terms of multispinors of SU(2) × SU(2)[= O(4)] × SU(2). The Dirac equation for fermions with arbitrary isospin interacting with the self-dual, conformally covariant Yang-Mills field is analyzed and completely solved for the isovector case. The relevance for this problem of the Atiyah-Singer index theory and its relation to the anomalous divergence of the axial-vector current are also explained.
Article
We find regular solutions of the four dimensional euclidean Yang-Mills equations. The solutions minimize locally the action integrals which is finite in this case. The topological nature of the solutions is discussed.
Article
It is shown how self-dual gauge fields correspond to certain complex vector bundles. This leads to a procedure for generating self-dual solutions of the Yang-Mills field equations.
Article
A complete construction, involving only linear algebra, is given for all self-dual euclidean Yang-Mills fields.
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