We describe new ambitwistor string theories that give rise to the recent amplitude formulae for Einstein-Yang-Mills, (Dirac)-Born-Infeld, Galileons and others introduced by Cachazo, He and Yuan. In the case of the Einstein-Yang-Mills amplitudes, an important role is played by a novel worldsheet conformal field theory that provides the appropriate colour factors precisely without the spurious multitrace terms of earlier models that had to be ignored by hand. This is needed to obtain the correct multitrace terms that arise when Yang-Mills is coupled to gravity.

We construct an ambitwistor string that describes Type II supergravity on AdS3×S3 with pure NS flux. The background Einstein equations ensure that the model is anomaly free. The spectrum consists of supergravity fluctuations around this background, with no higher string states. This theory transforms the problem of computing n-point tree-level amplitudes on AdS3 into that of understanding an $$ {\mathfrak{sl}}_2 $$ sl 2 Gaudin integrable system, whose representations are determined by the dual boundary operators and whose spectral parameters correspond to the worldsheet insertion points. The scattering equations take a similar form to flat space, with n(n − 3)/2 parameters τij parametrizing the eigenvalues of the Gaudin model.

We describe new ambitwistor string theories that give rise to the recent amplitude formulae for Einstein-Yang-Mills, (Dirac)-Born-Infeld, Galileons and others introduced by Cachazo, He and Yuan. In the case of the Einstein-Yang-Mills amplitudes, an important role is played by a novel worldsheet conformal field theory that provides the appropriate colour factors precisely without the spurious multitrace terms of earlier models that had to be ignored by hand. This is needed to obtain the correct multitrace terms that arise when Yang-Mills is coupled to gravity.

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 construct an ambitwistor string that describes Type II supergravity on AdS 3 ×S 3 with pure NS flux. The background Einstein equations ensure that the model is anomaly free. The spectrum consists of supergravity fluctuations around this background, with no higher string states. This theory transforms the problem of computing n-point tree-level amplitudes on AdS 3 into that of understanding an sl 2 Gaudin integrable system, whose representations are determined by the dual boundary operators and whose spectral parameters correspond to the worldsheet insertion points. The scattering equations take a similar form to flat space, with n(n − 3)/2 parameters τ ij parametrizing the eigenvalues of the Gaudin model. Contents 1 Introduction 1 2 The classical worldsheet theory 4 2.1 Type II ambitwistor strings 6 3 Quantization 8 3.1 OPEs of the fundamental fields 8 3.2 Einstein equations from OPEs of the currents 10 3.3 The Virasoro algebra 12 4 Vertex Operators 13 4.1 NS vertex operators 14 4.2 Bulk-to-boundary propagators on AdS 3 15 5 Gauge theory on AdS 3 ×S 3 18 6 Worldsheet correlation functions 19 6.1 Correlator of n holomorphic gluons 19 6.1.1 The Gaudin Hamiltonian 20 6.1.2 Completing the evaluation of the worldsheet correlator 23 6.2 General gluon amplitudes 24 6.3 Correlator of n holomorphic gravitons 25 6.4 Bi-adjoint scalar amplitudes 27 7 The scattering equations on AdS 28 7.1 Eigenfunctions of the Gaudin Hamiltonians 30 8 Conclusions 32 A Some properties of group manifolds 34