We display the vertex operators for all states in the conformal supergravity sector of the twistor string, as outlined by Berkovits and Witten. These include 'dipole' states, which are pairs of supergravitons that do not diagonalize the translation generators. We use canonical quantization of the open string version of Berkovits, and compute N -point tree level scattering amplitudes for gravitons, gluons and scalars. We reproduce the Berkovits-Witten formula for maximal helicity violating (MHV) amplitudes (which they derived using path integrals), and extend their results to the dipole pairs. We compare these trees with those of Einstein gravity field theory. email: ldolan, jihry@physics.unc.edu IntroductionWe pursue the tree amplitudes for graviton scattering in conformal gravity, described by twistor string theory. The twistor string Conformal gravity field theories [4,5] provided early examples of finite field theories of gravity [6,7]. They are not unitary theories, but have interesting structure and continue to provoke comments about possible uses [8]. Of course the conformal supergraviton states have zero norm, due to the lack of unitarity [9]. Nevertheless, the equivalence of the twistor string with this field theory system can be exploited to derive conformal gravity tree level scattering amplitudes hard to access in the field theory. We compute the gravity trees as a step toward learning how to decouple them in the twistor string. This would result in a perturbative string theory for super Yang-Mills (with no tower of massive states), and the computational advantage one hopes for in a string theory vs. field theory description.Various efforts towards a QCD string are discussed in [10].We work in a spinor helicity basis [11]- [13], and compare the conformal gravity tree amplitudes with those of Einstein gravity [14]- [19]. The conformal gravity trees have fewer poles.We compute the conformal couplings in detail, as they should be important in further study of the loop calculation [20].Computation is done in the Berkovits open string version [2]. We use the twistor string canonical quantization described in [20,21] and follow their notation. In section 2, we give the vertex operators for all states in the conformal supergravity multiplets, as outlined by Berkovits and Witten [3]. These include the dipole states, which form pairs of supergravitons, where one state in each pair does not diagonalize the translation generators, and is not a momentum eigenstate. We show all supergraviton states have zero norm in our basis.In section 3, three-point scattering amplitudes for gluons and gravitons and scalars, with both one and two negative helicities are calculated. We include cases for both members the dipoles, and find a momentum derivative appearing in amplitudes for states that do not diagonalize the translations generators. These amplitudes still have translational invariance.In section 4, we extend our results to N -point tree level amplitudes for these dipole pairs. We reproduce the Berkovits-Witte...

We discuss the integrability structure of deformed, four-dimensional N = 4 super Yang-Mills theories using Yangians. We employ a recent procedure by Beisert and Roiban that generalizes the beta deformation of Lunin and Maldacena to produce N = 1 superconformal gauge theories, which have the superalgebra SU(2, 2|1)×U(1) 2 . The deformed theories, including those with the more general twist, were shown to have retained their integrable structure. Here we examine the Yangian algebra of these deformed theories. In a five field subsector, we compute the two cases of SU(2)×U(1) 3 and SU(2|1)×U(1) 2 as residual symmetries of SU(2, 2|1)×U(1) 2 . We compute a twisted coproduct for these theories, and show that only for the residual symmetry do we retain the standard coproduct. The twisted coproduct thus provides a method for symmetry breaking. However, the full Yangian structure of SU(2|3) is manifest in our subsector, albeit with twisted coproducts, and provides for the integrability of the theory.

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